{
  "schema": 1,
  "about": "Hand-verified golden vectors for the United States, derived from IRS documents and not from rules/us/2026.json, which is a draft whose second-source sign-off is still open. Never generated by code. Change only via `npm run vectors:golden -- --ledger <ID> --cases <file> --file us-2026.golden.json` (docs/ACCURACY-LEDGER.md, 2026-09-18 section). Sources: IRS Publication 15-T (2026) https://www.irs.gov/pub/irs-pdf/p15t.pdf; IRS Publication 15 (2026) https://www.irs.gov/pub/irs-pdf/p15.pdf; IRS Rev. Proc. 2025-32 https://www.irs.gov/pub/irs-drop/rp-25-32.pdf; IRS Instructions for Form 8959 https://www.irs.gov/pub/irs-pdf/i8959.pdf. All retrieved 2026-09-18. WITHHOLDING IS NOT LIABILITY: `annual` cases are the Rev. Proc. rate schedules and the standard deduction; `payslip` cases are Publication 15-T Worksheet 1A. The two bases are never mixed.",
  "taxYear": "2026",
  "kinds": {
    "annual": "{input: engine Input, expect: {federalIncomeTax?, standardDeduction?, taxable?, socialSecurity?, socialSecurityWages?, medicare?, additionalMedicare?, additionalMedicareThreshold?, fica?, net?, effectiveRate?}} -> takeHome(input)",
    "payslip": "{input: engine Input plus `periodIndex` (which payment of the calendar year, from 1; the wage base and the $200,000 threshold are cumulative), period: the pay frequency, expect: {federalWithholding?, federalWithholdingWholeDollar?, adjustedAnnualWage?, tentativeAnnualWithholding?, tableRow?, socialSecurity?, socialSecurityWages?, medicare?, additionalMedicare?, net?}} -> paycheck({...input}, periodIndex)",
    "annual-summary": "not used by the United States: there is no single national summary shape, and the US engine has no takeHome({gross, region}) form",
    "statutory": "not used by the United States: there is no federal statutory sick or family pay scheme in this engine"
  },
  "disagreement": "A case may carry a `disagreement` block. Its `expect` still states the OFFICIAL figure and is never edited. The block records the value the production engine returns today and the ledger entry that rules on it; tests/engine/us.golden.test.ts asserts that exact value, so the test fails as soon as the engine changes.",
  "cases": [
    {
      "kind": "payslip",
      "input": {
        "gross": 32760,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholding": 33.68,
        "federalWithholdingWholeDollar": 34,
        "adjustedAnnualWage": 24160,
        "tentativeAnnualWithholding": 1751.2
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: line 1a $630.00, line 1b 52, line 1c $32,760.00, line 1g $8,600 (box not checked, not married filing jointly), line 1i $24,160.00; page 12 STANDARD 'Single or Married Filing Separately' row A $19,900 / C $1,240.00 / D 12%; line 2f 0.12 x $4,260.00 = $511.20, line 2g $1,751.20, line 2h $1,751.20 / 52 = $33.6769... = $33.68. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $625 but less than $635', column 'Single or Married Filing Separately, Standard withholding': the IRS prints $34. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "the plain percentage method for a single filer: annualise, subtract the $8,600 line 1g amount, read the 12% row, divide by 52. The IRS prints $34 for this wage.",
      "ledger": "L-2026-409"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 51870,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholding": 155.55,
        "federalWithholdingWholeDollar": 156,
        "adjustedAnnualWage": 43270,
        "tentativeAnnualWithholding": 4044.4
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: line 1a $1,995.00 x 26 = $51,870.00; less $8,600 = $43,270.00; row A $19,900 / C $1,240.00 / D 12%: $1,240.00 + 0.12 x $23,370.00 = $4,044.40; / 26 = $155.5538... = $155.55. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 18, row 'At least $1,985 but less than $2,005', column 'Single or Married Filing Separately, Standard withholding': the IRS prints $156. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "26 pay periods rather than 52, on the same 12% row: proves line 1b is read from Table 3 and not assumed.",
      "ledger": "L-2026-409"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 36240,
        "filingStatus": "single",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "federalWithholding": 90.37,
        "federalWithholdingWholeDollar": 90,
        "adjustedAnnualWage": 27640,
        "tentativeAnnualWithholding": 2168.8
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $1,510.00 x 24 = $36,240.00; less $8,600 = $27,640.00; $1,240.00 + 0.12 x $7,740.00 = $2,168.80; / 24 = $90.3666... = $90.37. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 21, row 'At least $1,500 but less than $1,520', column 'Single or Married Filing Separately, Standard withholding': the IRS prints $90. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "semimonthly (24 periods) is the frequency Publication 15-T lists that a 26-period assumption would silently break.",
      "ledger": "L-2026-409"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60060,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholding": 418.93,
        "federalWithholdingWholeDollar": 419,
        "adjustedAnnualWage": 51460,
        "tentativeAnnualWithholding": 5027.2
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $5,005.00 x 12 = $60,060.00; less $8,600 = $51,460.00; $1,240.00 + 0.12 x $31,560.00 = $5,027.20; / 12 = $418.9333... = $418.93. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 25, row 'At least $4,985 but less than $5,025', column 'Single or Married Filing Separately, Standard withholding': the IRS prints $419. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "monthly, 12 periods, still inside the 12% row — the fourth of the four frequencies the engine supports.",
      "ledger": "L-2026-409"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 121800,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholding": 1497.17,
        "adjustedAnnualWage": 113200,
        "tentativeAnnualWithholding": 17966,
        "tableRow": 4
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $10,150.00 x 12 = $121,800.00; less $8,600 = $113,200.00, which is EXACTLY column A of the 24% row. Worksheet 1A line 2b says 'at least the amount in column A but less than the amount in column B', so the row is the 24% row, C $17,966.00, and line 2f is 0.24 x $0.00 = $0.00. Line 2g $17,966.00 / 12 = $1,497.1666... = $1,497.17.",
      "why": "the table row boundary itself: an adjusted annual wage of exactly column A belongs to the HIGHER row. This is the opposite convention to the Rev. Proc. rate schedule ('over A but not over B'), and reading one table with the other's rule is a one-row error worth thousands of dollars a year.",
      "ledger": "L-2026-409"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 121799.88,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholding": 1497.16,
        "adjustedAnnualWage": 113199.88,
        "tentativeAnnualWithholding": 17965.97,
        "tableRow": 3
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $10,149.99 x 12 = $121,799.88; less $8,600 = $113,199.88, one cent BELOW column A of the 24% row, so the 22% row applies: C $5,800.00, A $57,900, 0.22 x $55,299.88 = $12,165.9736, line 2g $17,965.9736 (the engine rounds line 2f to the cent, $17,965.97); / 12 = $1,497.1645 = $1,497.16.",
      "why": "one cent below the same boundary, one cent less withheld. Paired with the case above this is the only way to see which side of column A the engine puts an exact match on.",
      "ledger": "L-2026-409"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 260000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholding": 2096.31,
        "adjustedAnnualWage": 251400,
        "tentativeAnnualWithholding": 54504
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $10,000.00 x 26 = $260,000.00; less $8,600 = $251,400.00; page 12 STANDARD single row A $209,275 / C $41,024.00 / D 32%: $41,024.00 + 0.32 x $42,125.00 = $41,024.00 + $13,480.00 = $54,504.00; / 26 = $2,096.3076... = $2,096.31.",
      "why": "above the last bracket of the wage bracket tables, where Publication 15-T says the percentage method must be used instead. Covers the 32% row, which no printed wage bracket cell reaches.",
      "ledger": "L-2026-409"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 51870,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholding": 75.65,
        "federalWithholdingWholeDollar": 76,
        "adjustedAnnualWage": 38970,
        "tentativeAnnualWithholding": 1967
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $1,995.00 x 26 = $51,870.00; line 1g $12,900 (married filing jointly, box not checked); line 1i $38,970.00; page 12 STANDARD 'Married Filing Jointly' row A $19,300 / C $0.00 / D 10%: 0.10 x $19,670.00 = $1,967.00; / 26 = $75.6538... = $75.65. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 18, row 'At least $1,985 but less than $2,005', column 'Married Filing Jointly, Standard withholding': the IRS prints $76. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "the same wage as the single case in L-2026-409 on the joint schedule: $12,900 instead of $8,600 at line 1g and a different table, $155.55 against $75.65.",
      "ledger": "L-2026-410"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 90240,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholding": 539.07,
        "federalWithholdingWholeDollar": 539,
        "adjustedAnnualWage": 77340,
        "tentativeAnnualWithholding": 6468.8
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $7,520.00 x 12 = $90,240.00; less $12,900 = $77,340.00; row A $44,100 / C $2,480.00 / D 12%: $2,480.00 + 0.12 x $33,240.00 = $6,468.80; / 12 = $539.0666... = $539.07. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 25, row 'At least $7,485 but less than $7,555', column 'Married Filing Jointly, Standard withholding': the IRS prints $539. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "the joint 12% row, and a case where the cent figure ($539.07) and the IRS's printed whole-dollar figure ($539) are on opposite sides of the cent.",
      "ledger": "L-2026-410"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 19240,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholding": 0,
        "federalWithholdingWholeDollar": 0,
        "adjustedAnnualWage": 6340,
        "tentativeAnnualWithholding": 0
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $370.00 x 52 = $19,240.00; less $12,900 = $6,340.00; page 12 STANDARD 'Married Filing Jointly' first row, at least $0 but less than $19,300, C $0.00, D 0%: nothing is withheld. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $365 but less than $375', column 'Married Filing Jointly, Standard withholding': the IRS prints $0. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "the 0% row: a joint filer on $19,240 a year has no federal income tax withheld at all. This is where the standard deduction lives inside the table, and it is the case a naive 10%-from-the-first-dollar implementation gets wrong.",
      "ledger": "L-2026-410"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 840000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholding": 18420.88,
        "adjustedAnnualWage": 827100,
        "tentativeAnnualWithholding": 221050.5
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $70,000.00 x 12 = $840,000.00; less $12,900 = $827,100.00; page 12 STANDARD 'Married Filing Jointly' last row A $788,000 / C $206,583.50 / D 37%: $206,583.50 + 0.37 x $39,100.00 = $206,583.50 + $14,467.00 = $221,050.50; / 12 = $18,420.875 exactly, which is half a cent and rounds up to $18,420.88 under the half-up rule in rules/us and Publication 15-T page 9.",
      "why": "the open-ended 37% row, and an exact half-cent at line 2h. A half that rounds the wrong way is the defect the UK ledger records at L-2026-001, so the US set carries one deliberately.",
      "ledger": "L-2026-410"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 8320,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholding": 0.52,
        "federalWithholdingWholeDollar": 1,
        "adjustedAnnualWage": 8320,
        "tentativeAnnualWithholding": 27
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $160.00 x 52 = $8,320.00; line 1g is -0- because the box in Step 2 IS checked; line 1i $8,320.00; page 12 'Form W-4, Step 2, Checkbox' 'Single or Married Filing Separately' row A $8,050 / C $0.00 / D 10%: 0.10 x $270.00 = $27.00; / 52 = $0.5192... = $0.52. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $155 but less than $165', column 'Single or Married Filing Separately, Form W-4, Step 2, Checkbox': the IRS prints $1. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "the smallest non-zero cell in the whole printed set. With the box checked the half-scale table starts at $8,050 instead of $7,500 and line 1g falls away, so $160 a week is taxed where the standard table withholds nothing.",
      "ledger": "L-2026-411"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 19240,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholding": 6.04,
        "federalWithholdingWholeDollar": 6,
        "adjustedAnnualWage": 19240,
        "tentativeAnnualWithholding": 314
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $370.00 x 52 = $19,240.00; line 1g -0- (box checked); page 12 'Form W-4, Step 2, Checkbox' 'Married Filing Jointly' row A $16,100 / C $0.00 / D 10%: 0.10 x $3,140.00 = $314.00; / 52 = $6.0384... = $6.04. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $365 but less than $375', column 'Married Filing Jointly, Form W-4, Step 2, Checkbox': the IRS prints $6. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "the same joint wage as the $0 case in L-2026-410 with the box checked: $0.00 becomes $6.04. This is the pair that shows Step 2 is read at all.",
      "ledger": "L-2026-411"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 28560,
        "filingStatus": "head-of-household",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholding": 150.1,
        "federalWithholdingWholeDollar": 150,
        "adjustedAnnualWage": 28560,
        "tentativeAnnualWithholding": 1801.2
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $2,380.00 x 12 = $28,560.00; line 1g -0- (box checked); page 12 'Form W-4, Step 2, Checkbox' 'Head of Household' row A $20,925 / C $885.00 / D 12%: $885.00 + 0.12 x $7,635.00 = $885.00 + $916.20 = $1,801.20; / 12 = $150.10 exactly. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 24, row 'At least $2,365 but less than $2,395', column 'Head of Household, Form W-4, Step 2, Checkbox': the IRS prints $150. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "head of household with the box checked, the one status-and-checkbox combination that uses neither the $8,600 nor the $12,900 line 1g amount and neither of the two tables the other cases read.",
      "ledger": "L-2026-411"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 34840,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholding": 38.48,
        "federalWithholdingWholeDollar": 38,
        "adjustedAnnualWage": 26240,
        "tentativeAnnualWithholding": 2000.8
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $670.00 x 52 = $34,840.00; line 1g $8,600 (box NOT checked); line 1i $26,240.00; STANDARD single row A $19,900 / C $1,240.00 / D 12%: $1,240.00 + 0.12 x $6,340.00 = $2,000.80; / 52 = $38.4769... = $38.48. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $665 but less than $675', column 'Single or Married Filing Separately, Standard withholding': the IRS prints $38. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "the box-not-checked half of the $670 weekly pair. The checked half is the next case, and the IRS prints $38 against $62 for the same wage.",
      "ledger": "L-2026-411"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 34840,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholding": 62.5,
        "federalWithholdingWholeDollar": 62,
        "adjustedAnnualWage": 34840,
        "tentativeAnnualWithholding": 3249.8
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $670.00 x 52 = $34,840.00; line 1g -0-; 'Form W-4, Step 2, Checkbox' single row A $33,250 / C $2,900.00 / D 22%: $2,900.00 + 0.22 x $1,590.00 = $3,249.80; line 2h = $3,249.80 / 52 = $62.4961538..., which is $62.50 to the cent and $62 to the whole dollar (Publication 15-T page 9: 'rounded to the nearest whole dollar by dropping amounts under 50 cents'). IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $665 but less than $675', column 'Single or Married Filing Separately, Form W-4, Step 2, Checkbox': the IRS prints $62. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "CLOSED DISAGREEMENT L-2026-411, 2026-10-19. The IRS prints $62 and the engine now returns $62. It returned $63 until today because it rounded Worksheet 1A line 2h to the cent ($62.50) and then rounded that to the dollar; Publication 15-T (2026) page 9 applies the whole-dollar rule ONCE, to the tax itself. The cent figure was right throughout and has not moved. Eleven of the printed wage-bracket cells failed this way, every one of them weekly, single, box checked — the residue cycles with period five, which is why they clustered in one column. Found by accuracy-red-team while blocking the first US signature; fixed by engine-engineer carrying steps 2h, 3 and 4 in cents multiplied by the pay periods, which is exact for every frequency, and dividing once at the end. The authorities are in rules/us/2026-withholding.json: 26 U.S.C. 6102(c) makes a rounding “applicable only to such final amount” and 26 CFR 301.6313-1 says “fractional parts of a cent shall not be disregarded in the computation of taxes”.",
      "ledger": "L-2026-411"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60840,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholding": 172.5,
        "federalWithholdingWholeDollar": 172,
        "adjustedAnnualWage": 60840,
        "tentativeAnnualWithholding": 8969.8
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $1,170.00 x 52 = $60,840.00; line 1g -0-; 'Form W-4, Step 2, Checkbox' single row A $33,250 / C $2,900.00 / D 22%: $2,900.00 + 0.22 x $27,590.00 = $8,969.80; / 52 = $172.4961538..., $172.50 to the cent and $172 to the whole dollar. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 16, row 'At least $1,165 but less than $1,175', column 'Single or Married Filing Separately, Form W-4, Step 2, Checkbox': the IRS prints $172. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "DISAGREEMENT L-2026-411, the largest of the eleven. Kept as a second instance so a fix that special-cases one wage does not pass.",
      "ledger": "L-2026-411"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 4030,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholding": 0,
        "federalWithholdingWholeDollar": 0,
        "adjustedAnnualWage": 0,
        "tentativeAnnualWithholding": 0
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $77.50 x 52 = $4,030.00; line 1h $8,600; line 1i is $4,030.00 - $8,600.00 = negative, and line 1i says 'If zero or less, enter -0-', so the Adjusted Annual Wage Amount is $0.00 and nothing is withheld. IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $0 but less than $155', column 'all six columns': the IRS prints $0. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage.",
      "why": "line 1i floors at zero. Without the floor the table lookup gets a negative adjusted wage, and the printed row 'At least $0' would not match it at all.",
      "ledger": "L-2026-412"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 48000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "deductions": 20000,
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholding": 0,
        "adjustedAnnualWage": 15100,
        "tentativeAnnualWithholding": 0
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: $4,000.00 x 12 = $48,000.00 (line 1c); line 1d $0; line 1e $48,000.00; line 1f $20,000.00 from Step 4(b); line 1g $12,900; line 1h $32,900.00; line 1i $15,100.00, which is inside the joint 0% row (at least $0, less than $19,300): nothing is withheld.",
      "why": "Step 4(b) deductions, and the joint 0% row reached from above rather than from a low wage. $48,000 a year withholds $0 once $20,000 of deductions are claimed.",
      "ledger": "L-2026-412"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 52000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "otherIncome": 10000,
          "step2Checkbox": false
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholding": 202.31,
        "adjustedAnnualWage": 53400,
        "tentativeAnnualWithholding": 5260
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: line 1c $52,000.00; line 1d $10,000.00 (Step 4(a) other income); line 1e $62,000.00; line 1h $8,600.00; line 1i $53,400.00; STANDARD single row A $19,900 / C $1,240.00 / D 12%: $1,240.00 + 0.12 x $33,500.00 = $5,260.00; / 26 = $202.3076... = $202.31.",
      "why": "Step 4(a) is an ANNUAL amount added once at line 1d, not a per-period amount. Dividing it by 26 first would give $202.31 against the wrong number entirely.",
      "ledger": "L-2026-412"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 52000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "deductions": 6000,
          "step2Checkbox": false
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholding": 128.46,
        "adjustedAnnualWage": 37400,
        "tentativeAnnualWithholding": 3340
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: line 1c $52,000.00; line 1f $6,000.00 (Step 4(b)); line 1g $8,600.00; line 1h $14,600.00; line 1i $37,400.00; $1,240.00 + 0.12 x $17,500.00 = $3,340.00; / 26 = $128.4615... = $128.46.",
      "why": "Step 4(b) is added to the $8,600 at line 1h, not instead of it. An implementation that replaces line 1g with the Step 4(b) amount gets $147.12 here.",
      "ledger": "L-2026-412"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 52000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "taxCredits": 2000,
          "step2Checkbox": false
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholding": 79.23,
        "tentativeAnnualWithholding": 4060
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: line 2g $4,060.00, line 2h $4,060.00 / 26 = $156.1538... = $156.15; line 3a $2,000.00, line 3b $2,000.00 / 26 = $76.9230... = $76.92; line 3c $156.15 - $76.92 = $79.23.",
      "why": "Step 3 credits are annual and are divided by the pay periods at line 3b, then subtracted at line 3c — the opposite order to Step 4(a). The two W-4 steps are handled differently and this pair is what tells them apart.",
      "ledger": "L-2026-412"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 52000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "extraWithholding": 25,
          "step2Checkbox": false
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholding": 181.15
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: line 3c $156.15; line 4a $25.00 (Step 4(c) additional amount, per pay period); line 4b $181.15.",
      "why": "Step 4(c) is per PERIOD and is added after everything else, including after the whole-dollar rounding option. $25 a paycheck, not $25 a year.",
      "ledger": "L-2026-412"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 52000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "taxCredits": 40000,
          "step2Checkbox": false
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholding": 0
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems'. Hand derivation: line 2h $156.15; line 3b $40,000.00 / 26 = $1,538.4615... = $1,538.46; line 3c says 'Subtract line 3b from line 2h. If zero or less, enter -0-', so line 3c is $0.00 and line 4b is $0.00.",
      "why": "the line 3c floor. Credits larger than the tentative withholding never produce a negative withholding, and never a refund through payroll.",
      "ledger": "L-2026-412"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 32760,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholdingWholeDollar": 1,
        "federalWithholding": 1.08
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $625 but less than $635', column 'Married Filing Jointly, Standard withholding': the IRS prints $1. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $32,760.00, line 1i $19,860.00, line 2g $56.00, line 2h $56.00 / 52 = $1.08, which is $1 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: weekly married filing jointly, standard withholding at the midpoint of the printed $625-$635 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 32760,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholdingWholeDollar": 34,
        "federalWithholding": 33.68
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $625 but less than $635', column 'Married Filing Jointly, Form W-4, Step 2, Checkbox': the IRS prints $34. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $32,760.00, line 1i $32,760.00, line 2g $1,751.20, line 2h $1,751.20 / 52 = $33.68, which is $34 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: weekly married filing jointly, form w-4, step 2, checkbox at the midpoint of the printed $625-$635 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 32760,
        "filingStatus": "head-of-household",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholdingWholeDollar": 17,
        "federalWithholding": 16.56
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $625 but less than $635', column 'Head of Household, Standard withholding': the IRS prints $17. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $32,760.00, line 1i $24,160.00, line 2g $861.00, line 2h $861.00 / 52 = $16.56, which is $17 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: weekly head of household, standard withholding at the midpoint of the printed $625-$635 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 32760,
        "filingStatus": "head-of-household",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholdingWholeDollar": 44,
        "federalWithholding": 44.33
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $625 but less than $635', column 'Head of Household, Form W-4, Step 2, Checkbox': the IRS prints $44. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $32,760.00, line 1i $32,760.00, line 2g $2,305.20, line 2h $2,305.20 / 52 = $44.33, which is $44 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: weekly head of household, form w-4, step 2, checkbox at the midpoint of the printed $625-$635 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 32760,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholdingWholeDollar": 55,
        "federalWithholding": 54.64
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 14, row 'At least $625 but less than $635', column 'Single or Married Filing Separately, Form W-4, Step 2, Checkbox': the IRS prints $55. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $32,760.00, line 1i $32,760.00, line 2g $2,841.20, line 2h $2,841.20 / 52 = $54.64, which is $55 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: weekly single or married filing separately, form w-4, step 2, checkbox at the midpoint of the printed $625-$635 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 91000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholdingWholeDollar": 252,
        "federalWithholding": 252.31
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 19, row 'At least $3,485 but less than $3,515', column 'Married Filing Jointly, Standard withholding': the IRS prints $252. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $91,000.00, line 1i $78,100.00, line 2g $6,560.00, line 2h $6,560.00 / 26 = $252.31, which is $252 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: biweekly married filing jointly, standard withholding at the midpoint of the printed $3,485-$3,515 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 91000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholdingWholeDollar": 430,
        "federalWithholding": 430.38
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 19, row 'At least $3,485 but less than $3,515', column 'Married Filing Jointly, Form W-4, Step 2, Checkbox': the IRS prints $430. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $91,000.00, line 1i $91,000.00, line 2g $11,190.00, line 2h $11,190.00 / 26 = $430.38, which is $430 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: biweekly married filing jointly, form w-4, step 2, checkbox at the midpoint of the printed $3,485-$3,515 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 91000,
        "filingStatus": "head-of-household",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholdingWholeDollar": 295,
        "federalWithholding": 294.92
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 19, row 'At least $3,485 but less than $3,515', column 'Head of Household, Standard withholding': the IRS prints $295. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $91,000.00, line 1i $82,400.00, line 2g $7,668.00, line 2h $7,668.00 / 26 = $294.92, which is $295 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: biweekly head of household, standard withholding at the midpoint of the printed $3,485-$3,515 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 91000,
        "filingStatus": "head-of-household",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholdingWholeDollar": 551,
        "federalWithholding": 551.37
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 19, row 'At least $3,485 but less than $3,515', column 'Head of Household, Form W-4, Step 2, Checkbox': the IRS prints $551. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $91,000.00, line 1i $91,000.00, line 2g $14,335.50, line 2h $14,335.50 / 26 = $551.37, which is $551 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: biweekly head of household, form w-4, step 2, checkbox at the midpoint of the printed $3,485-$3,515 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 91000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholdingWholeDollar": 430,
        "federalWithholding": 430.38
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 19, row 'At least $3,485 but less than $3,515', column 'Single or Married Filing Separately, Standard withholding': the IRS prints $430. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $91,000.00, line 1i $82,400.00, line 2g $11,190.00, line 2h $11,190.00 / 26 = $430.38, which is $430 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: biweekly single or married filing separately, standard withholding at the midpoint of the printed $3,485-$3,515 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 91000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholdingWholeDollar": 623,
        "federalWithholding": 623.35
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 19, row 'At least $3,485 but less than $3,515', column 'Single or Married Filing Separately, Form W-4, Step 2, Checkbox': the IRS prints $623. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $91,000.00, line 1i $91,000.00, line 2g $16,207.00, line 2h $16,207.00 / 26 = $623.35, which is $623 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: biweekly single or married filing separately, form w-4, step 2, checkbox at the midpoint of the printed $3,485-$3,515 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 72300,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "federalWithholdingWholeDollar": 180,
        "federalWithholding": 179.83
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 22, row 'At least $2,995 but less than $3,030', column 'Married Filing Jointly, Standard withholding': the IRS prints $180. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $72,300.00, line 1i $59,400.00, line 2g $4,316.00, line 2h $4,316.00 / 24 = $179.83, which is $180 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: semimonthly married filing jointly, standard withholding at the midpoint of the printed $2,995-$3,030 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 72300,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "semimonthly",
      "expect": {
        "federalWithholdingWholeDollar": 295,
        "federalWithholding": 294.83
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 22, row 'At least $2,995 but less than $3,030', column 'Married Filing Jointly, Form W-4, Step 2, Checkbox': the IRS prints $295. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $72,300.00, line 1i $72,300.00, line 2g $7,076.00, line 2h $7,076.00 / 24 = $294.83, which is $295 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: semimonthly married filing jointly, form w-4, step 2, checkbox at the midpoint of the printed $2,995-$3,030 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 72300,
        "filingStatus": "head-of-household",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "federalWithholdingWholeDollar": 226,
        "federalWithholding": 226
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 22, row 'At least $2,995 but less than $3,030', column 'Head of Household, Standard withholding': the IRS prints $226. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $72,300.00, line 1i $63,700.00, line 2g $5,424.00, line 2h $5,424.00 / 24 = $226.00, which is $226 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: semimonthly head of household, standard withholding at the midpoint of the printed $2,995-$3,030 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 72300,
        "filingStatus": "head-of-household",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "semimonthly",
      "expect": {
        "federalWithholdingWholeDollar": 410,
        "federalWithholding": 410.31
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 22, row 'At least $2,995 but less than $3,030', column 'Head of Household, Form W-4, Step 2, Checkbox': the IRS prints $410. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $72,300.00, line 1i $72,300.00, line 2g $9,847.50, line 2h $9,847.50 / 24 = $410.31, which is $410 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: semimonthly head of household, form w-4, step 2, checkbox at the midpoint of the printed $2,995-$3,030 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 72300,
        "filingStatus": "single",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "federalWithholdingWholeDollar": 295,
        "federalWithholding": 294.83
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 22, row 'At least $2,995 but less than $3,030', column 'Single or Married Filing Separately, Standard withholding': the IRS prints $295. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $72,300.00, line 1i $63,700.00, line 2g $7,076.00, line 2h $7,076.00 / 24 = $294.83, which is $295 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: semimonthly single or married filing separately, standard withholding at the midpoint of the printed $2,995-$3,030 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 72300,
        "filingStatus": "single",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "semimonthly",
      "expect": {
        "federalWithholdingWholeDollar": 488,
        "federalWithholding": 488.29
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 22, row 'At least $2,995 but less than $3,030', column 'Single or Married Filing Separately, Form W-4, Step 2, Checkbox': the IRS prints $488. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $72,300.00, line 1i $72,300.00, line 2g $11,719.00, line 2h $11,719.00 / 24 = $488.29, which is $488 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: semimonthly single or married filing separately, form w-4, step 2, checkbox at the midpoint of the printed $2,995-$3,030 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60060,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholdingWholeDollar": 237,
        "federalWithholding": 237.27
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 25, row 'At least $4,985 but less than $5,025', column 'Married Filing Jointly, Standard withholding': the IRS prints $237. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $60,060.00, line 1i $47,160.00, line 2g $2,847.20, line 2h $2,847.20 / 12 = $237.27, which is $237 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: monthly married filing jointly, standard withholding at the midpoint of the printed $4,985-$5,025 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60060,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholdingWholeDollar": 419,
        "federalWithholding": 418.93
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 25, row 'At least $4,985 but less than $5,025', column 'Married Filing Jointly, Form W-4, Step 2, Checkbox': the IRS prints $419. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $60,060.00, line 1i $60,060.00, line 2g $5,027.20, line 2h $5,027.20 / 12 = $418.93, which is $419 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: monthly married filing jointly, form w-4, step 2, checkbox at the midpoint of the printed $4,985-$5,025 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60060,
        "filingStatus": "head-of-household",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholdingWholeDollar": 330,
        "federalWithholding": 329.6
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 25, row 'At least $4,985 but less than $5,025', column 'Head of Household, Standard withholding': the IRS prints $330. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $60,060.00, line 1i $51,460.00, line 2g $3,955.20, line 2h $3,955.20 / 12 = $329.60, which is $330 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: monthly head of household, standard withholding at the midpoint of the printed $4,985-$5,025 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60060,
        "filingStatus": "head-of-household",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholdingWholeDollar": 584,
        "federalWithholding": 583.93
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 25, row 'At least $4,985 but less than $5,025', column 'Head of Household, Form W-4, Step 2, Checkbox': the IRS prints $584. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $60,060.00, line 1i $60,060.00, line 2g $7,007.20, line 2h $7,007.20 / 12 = $583.93, which is $584 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: monthly head of household, form w-4, step 2, checkbox at the midpoint of the printed $4,985-$5,025 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60060,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": true
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholdingWholeDollar": 733,
        "federalWithholding": 733.18
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, section 2 '2026 Wage Bracket Method Tables for Manual Payroll Systems with Forms W-4 From 2020 or Later', page 25, row 'At least $4,985 but less than $5,025', column 'Single or Married Filing Separately, Form W-4, Step 2, Checkbox': the IRS prints $733. The wage used is the midpoint of that printed row, where the wage bracket table and Worksheet 1A coincide: the whole 2020-or-later wage bracket set (725 printed rows x 6 columns = 4,350 cells, weekly, biweekly, semimonthly and monthly) was reproduced cell for cell from Worksheet 1A at each row's midpoint with no exception, so the printed cell is the IRS's own answer for this wage. Hand derivation from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $60,060.00, line 1i $60,060.00, line 2g $8,798.20, line 2h $8,798.20 / 12 = $733.18, which is $733 to the whole dollar.",
      "why": "an IRS-printed answer, not our arithmetic: monthly single or married filing separately, form w-4, step 2, checkbox at the midpoint of the printed $4,985-$5,025 row.",
      "ledger": "L-2026-413"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 184499,
        "filingStatus": "single",
        "taxYear": "2026"
      },
      "expect": {
        "socialSecurity": 11438.94,
        "socialSecurityWages": 184499,
        "medicare": 2675.24,
        "additionalMedicare": 0,
        "federalIncomeTax": 33013.76,
        "net": 137371.06
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') and Social Security Administration, 'Cost-of-Living Increase and Other Determinations for 2026', Federal Register 3 November 2025 (document 2025-19763): 'The OASDI contribution and benefit base is $184,500 for remuneration paid in 2026'. Annual income tax from IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19). Hand derivation: $184,499 is one dollar below the wage base, so every dollar is charged: 0.062 x $184,499 = $11,438.938 = $11,438.94 to the cent; Medicare 0.0145 x $184,499 = $2,675.2355 = $2,675.24; taxable income $184,499 - $16,100 = $168,399; Rev. Proc. Table 3 'Over $105,700 but not over $201,775: $17,966 plus 24% of the excess over $105,700' = $17,966 + 0.24 x $62,699 = $17,966 + $15,047.76 = $33,013.76; net $184,499 - $33,013.76 - $11,438.94 - $2,675.24 = $137,371.06.",
      "why": "one dollar below the wage base: the cap is not yet doing anything, and both FICA lines carry a fraction of a cent that has to round half-up.",
      "ledger": "L-2026-414"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 184500,
        "filingStatus": "single",
        "taxYear": "2026"
      },
      "expect": {
        "socialSecurity": 11439,
        "socialSecurityWages": 184500,
        "medicare": 2675.25,
        "additionalMedicare": 0,
        "federalIncomeTax": 33014,
        "net": 137371.75
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') and Social Security Administration, 'Cost-of-Living Increase and Other Determinations for 2026', Federal Register 3 November 2025 (document 2025-19763): 'The OASDI contribution and benefit base is $184,500 for remuneration paid in 2026'. Annual income tax from IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19). Hand derivation: wages exactly at the base: 0.062 x $184,500 = $11,439.00; Medicare 0.0145 x $184,500 = $2,675.25; taxable $168,400; $17,966 + 0.24 x $62,700 = $33,014.00; net $137,371.75.",
      "why": "exactly at the wage base. The base is the maximum wage subject to the tax, so the whole of it is charged and $11,439.00 is the largest social security tax any 2026 employee pays.",
      "ledger": "L-2026-414"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 184501,
        "filingStatus": "single",
        "taxYear": "2026"
      },
      "expect": {
        "socialSecurity": 11439,
        "socialSecurityWages": 184500,
        "medicare": 2675.26,
        "additionalMedicare": 0,
        "federalIncomeTax": 33014.24,
        "net": 137372.5
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') and Social Security Administration, 'Cost-of-Living Increase and Other Determinations for 2026', Federal Register 3 November 2025 (document 2025-19763): 'The OASDI contribution and benefit base is $184,500 for remuneration paid in 2026'. Annual income tax from IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19). Hand derivation: one dollar above the base. Social security wages stay $184,500 and the tax stays $11,439.00. Medicare has no wage base: 0.0145 x $184,501 = $2,675.2645 = $2,675.26. Taxable $168,401; $17,966 + 0.24 x $62,701 = $33,014.24; net $137,372.50.",
      "why": "one dollar above the base, and the whole point of the pair: social security stops and Medicare does not. Publication 15: 'There is no wage base limit for Medicare tax; all covered wages are subject to Medicare tax.'",
      "ledger": "L-2026-414"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 246000,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 9,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "socialSecurity": 1271,
        "socialSecurityWages": 20500,
        "medicare": 297.25,
        "additionalMedicare": 0,
        "federalWithholding": 4168.67,
        "net": 14763.08
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') and Social Security Administration, 'Cost-of-Living Increase and Other Determinations for 2026', Federal Register 3 November 2025 (document 2025-19763): 'The OASDI contribution and benefit base is $184,500 for remuneration paid in 2026'. Annual income tax from IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19). Hand derivation, ninth monthly payment of $20,500: wages paid before this payment $164,000, after it $184,500 — exactly the base — so all $20,500 is social security wages and the tax is 0.062 x $20,500 = $1,271.00. Medicare 0.0145 x $20,500 = $297.25. No Additional Medicare Tax: cumulative wages have not passed $200,000. Withholding from IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-09-18, page 10 Worksheet 1A (Employer's Withholding Worksheet for Percentage Method Tables for Automated Payroll Systems) and page 12 '2026 Percentage Method Tables for Automated Payroll Systems': line 1c $246,000.00 less $8,600 = $237,400.00; STANDARD single row A $209,275 / C $41,024.00 / D 32%: $41,024.00 + 0.32 x $28,125.00 = $50,024.00; / 12 = $4,168.6666... = $4,168.67. Net $20,500.00 - $4,168.67 - $1,271.00 - $297.25 = $14,763.08.",
      "why": "the last paycheck of the year that is fully liable to social security tax. The base is cumulative over the calendar year, which Publication 15 states by working it: a successor employer charges 'the first $182,500 ($184,500 minus $2,000)' of wages it pays.",
      "ledger": "L-2026-414"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 246000,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 10,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "socialSecurity": 0,
        "socialSecurityWages": 0,
        "medicare": 297.25,
        "additionalMedicare": 45,
        "federalWithholding": 4168.67,
        "net": 15989.08
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') and Social Security Administration, 'Cost-of-Living Increase and Other Determinations for 2026', Federal Register 3 November 2025 (document 2025-19763): 'The OASDI contribution and benefit base is $184,500 for remuneration paid in 2026'. Annual income tax from IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19). Hand derivation, tenth payment: wages before $184,500, after $205,000. Social security wages in the slice are min($205,000, $184,500) - min($184,500, $184,500) = $0, so no social security tax at all. Medicare still $297.25. Additional Medicare Tax: wages above $200,000 in the slice are $205,000 - $200,000 = $5,000, at 0.9% = $45.00. Net $20,500.00 - $4,168.67 - $0.00 - $297.25 - $45.00 = $15,989.08.",
      "why": "the first paycheck after the base is reached: social security drops to zero mid-year and take-home JUMPS by $1,271. The same paycheck is the first to cross $200,000, so it carries a part-period Additional Medicare Tax.",
      "ledger": "L-2026-414"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 246000,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 11,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "socialSecurity": 0,
        "socialSecurityWages": 0,
        "medicare": 297.25,
        "additionalMedicare": 184.5,
        "federalWithholding": 4168.67,
        "net": 15849.58
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') and Social Security Administration, 'Cost-of-Living Increase and Other Determinations for 2026', Federal Register 3 November 2025 (document 2025-19763): 'The OASDI contribution and benefit base is $184,500 for remuneration paid in 2026'. Annual income tax from IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19). Hand derivation, eleventh payment: social security $0. Medicare $297.25. The whole payment is now above $200,000, so Additional Medicare Tax is 0.9% x $20,500 = $184.50 — Publication 15: 'continue to withhold it each pay period until the end of the calendar year'. Net $20,500.00 - $4,168.67 - $297.25 - $184.50 = $15,849.58.",
      "why": "the period after the crossing: the Additional Medicare Tax is charged on the whole payment, not on the part above $200,000 again. Paired with period 10 this is the only way to see the threshold is cumulative and not per-paycheck.",
      "ledger": "L-2026-414"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 200001,
        "filingStatus": "single",
        "taxYear": "2026"
      },
      "expect": {
        "medicare": 2900.01,
        "additionalMedicare": 0.01,
        "additionalMedicareThreshold": 200000,
        "socialSecurity": 11439,
        "federalIncomeTax": 36734.24
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') IRS Instructions for Form 8959, Threshold Amounts for Additional Medicare Tax (married filing jointly $250,000; married filing separately $125,000; single, head of household and qualifying surviving spouse $200,000), https://www.irs.gov/pub/irs-pdf/i8959.pdf, retrieved 2026-09-18 IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19). Hand derivation: Medicare 0.0145 x $200,001 = $2,900.0145 = $2,900.01 to the cent. Additional Medicare Tax on the return basis: single threshold $200,000, wages above it $1, 0.009 x $1 = $0.009 = $0.01. Social security capped at $184,500 x 0.062 = $11,439.00. Taxable $200,001 - $16,100 = $183,901; $17,966 + 0.24 x $78,201 = $36,734.24.",
      "why": "one dollar over the single Form 8959 threshold, charged to a single cent. A threshold implemented as 'more than or equal to' rather than 'in excess of' charges $1,800.01 here.",
      "ledger": "L-2026-415"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 260000,
        "filingStatus": "married-filing-jointly",
        "taxYear": "2026"
      },
      "expect": {
        "additionalMedicare": 90,
        "additionalMedicareThreshold": 250000,
        "medicare": 3770,
        "socialSecurity": 11439,
        "federalIncomeTax": 39868,
        "net": 204833
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') IRS Instructions for Form 8959, Threshold Amounts for Additional Medicare Tax (married filing jointly $250,000; married filing separately $125,000; single, head of household and qualifying surviving spouse $200,000), https://www.irs.gov/pub/irs-pdf/i8959.pdf, retrieved 2026-09-18 IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19). Hand derivation: the RETURN threshold for a joint filer is $250,000, so Additional Medicare Tax is 0.009 x ($260,000 - $250,000) = $90.00. Medicare 0.0145 x $260,000 = $3,770.00. Social security 0.062 x $184,500 = $11,439.00. Taxable $260,000 - $32,200 = $227,800; Rev. Proc. Table 1 'Over $211,400 but not over $403,550: $35,932 plus 24% of the excess over $211,400' = $35,932 + 0.24 x $16,400 = $39,868.00. Net $260,000 - $39,868.00 - $11,439.00 - $3,770.00 - $90.00 = $204,833.00.",
      "why": "the case where the two Additional Medicare Tax rules part company. The employer withholds on everything over $200,000 whatever the filing status ($540 across the year, see the paycheck cases below); the return charges $90. The $450 difference comes back on Form 8959 and must never be averaged away.",
      "ledger": "L-2026-415"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 125001,
        "filingStatus": "married-filing-separately",
        "taxYear": "2026"
      },
      "expect": {
        "additionalMedicare": 0.01,
        "additionalMedicareThreshold": 125000,
        "medicare": 1812.51,
        "socialSecurity": 7750.06
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') IRS Instructions for Form 8959, Threshold Amounts for Additional Medicare Tax (married filing jointly $250,000; married filing separately $125,000; single, head of household and qualifying surviving spouse $200,000), https://www.irs.gov/pub/irs-pdf/i8959.pdf, retrieved 2026-09-18. Hand derivation: the separate-return threshold is $125,000, half the joint figure. 0.009 x $1 = $0.01. Medicare 0.0145 x $125,001 = $1,812.5145 = $1,812.51. Social security 0.062 x $125,001 = $7,750.062 = $7,750.06 (below the $184,500 base).",
      "why": "married filing separately is the only status whose Additional Medicare Tax threshold is below $200,000. An engine that keyed the threshold off 'joint or not' would charge nothing here.",
      "ledger": "L-2026-415"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 500000,
        "filingStatus": "married-filing-separately",
        "taxYear": "2026"
      },
      "expect": {
        "additionalMedicare": 3375,
        "medicare": 7250,
        "socialSecurity": 11439,
        "federalIncomeTax": 140125.25,
        "net": 337810.75
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') IRS Instructions for Form 8959, Threshold Amounts for Additional Medicare Tax (married filing jointly $250,000; married filing separately $125,000; single, head of household and qualifying surviving spouse $200,000), https://www.irs.gov/pub/irs-pdf/i8959.pdf, retrieved 2026-09-18 IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19) Table 4 (Married Individuals Filing Separate Returns). Hand derivation: Additional Medicare Tax 0.009 x ($500,000 - $125,000) = $3,375.00. Medicare 0.0145 x $500,000 = $7,250.00. Social security $11,439.00. Taxable $500,000 - $16,100 = $483,900; Table 4 'Over $384,350: $103,291.75 plus 37% of the excess over $384,350' = $103,291.75 + 0.37 x $99,550 = $140,125.25. Net $337,810.75.",
      "why": "separate returns have their own rate schedule: 37% starts at $384,350, not at the $640,600 of Table 3. Publication 15-T's WITHHOLDING table lumps 'Single or Married Filing Separately' together and would reach only 35% here, so this case also proves the annual and withholding bases are kept apart.",
      "ledger": "L-2026-415"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 264000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "socialSecurity": 682,
        "medicare": 159.5,
        "additionalMedicare": 0,
        "federalWithholding": 1701.17,
        "net": 8457.33
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') Hand derivation for payment 1 of 24 at $11,000.00: wages paid before it $0, after it $11,000. Social security wages are min(after, $184,500) - min(before, $184,500) at 6.2%; Medicare 0.0145 x $11,000 = $159.50 on every payment; Additional Medicare Tax 0.009 x (the part of the slice above $200,000), the EMPLOYER's threshold, which is $200,000 for every filing status including a joint one. Federal withholding $1,701.17 every period, derived at L-2026-410. Net $11,000.00 - $1,701.17 - $682.00 - $159.50 - $0.00 = $8,457.33.",
      "why": "a normal early paycheck: social security on the whole payment, Medicare on the whole payment, no Additional Medicare Tax.",
      "ledger": "L-2026-415"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 264000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 17,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "socialSecurity": 527,
        "medicare": 159.5,
        "additionalMedicare": 0,
        "federalWithholding": 1701.17,
        "net": 8612.33
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') Hand derivation for payment 17 of 24 at $11,000.00: wages paid before it $176,000, after it $187,000. Social security wages are min(after, $184,500) - min(before, $184,500) at 6.2%; Medicare 0.0145 x $11,000 = $159.50 on every payment; Additional Medicare Tax 0.009 x (the part of the slice above $200,000), the EMPLOYER's threshold, which is $200,000 for every filing status including a joint one. Federal withholding $1,701.17 every period, derived at L-2026-410. Net $11,000.00 - $1,701.17 - $527.00 - $159.50 - $0.00 = $8,612.33.",
      "why": "the payment that crosses the wage base: $184,500 - $176,000 = $8,500 of this $11,000 is social security wages, so the tax is $527.00 and not $682.00.",
      "ledger": "L-2026-415"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 264000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 18,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "socialSecurity": 0,
        "medicare": 159.5,
        "additionalMedicare": 0,
        "federalWithholding": 1701.17,
        "net": 9139.33
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') Hand derivation for payment 18 of 24 at $11,000.00: wages paid before it $187,000, after it $198,000. Social security wages are min(after, $184,500) - min(before, $184,500) at 6.2%; Medicare 0.0145 x $11,000 = $159.50 on every payment; Additional Medicare Tax 0.009 x (the part of the slice above $200,000), the EMPLOYER's threshold, which is $200,000 for every filing status including a joint one. Federal withholding $1,701.17 every period, derived at L-2026-410. Net $11,000.00 - $1,701.17 - $0.00 - $159.50 - $0.00 = $9,139.33.",
      "why": "the payment after the base: social security is $0 and Medicare is unchanged.",
      "ledger": "L-2026-415"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 264000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 19,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "socialSecurity": 0,
        "medicare": 159.5,
        "additionalMedicare": 81,
        "federalWithholding": 1701.17,
        "net": 9058.33
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') Hand derivation for payment 19 of 24 at $11,000.00: wages paid before it $198,000, after it $209,000. Social security wages are min(after, $184,500) - min(before, $184,500) at 6.2%; Medicare 0.0145 x $11,000 = $159.50 on every payment; Additional Medicare Tax 0.009 x (the part of the slice above $200,000), the EMPLOYER's threshold, which is $200,000 for every filing status including a joint one. Federal withholding $1,701.17 every period, derived at L-2026-410. Net $11,000.00 - $1,701.17 - $0.00 - $159.50 - $81.00 = $9,058.33.",
      "why": "the payment that crosses $200,000 on the EMPLOYER's rule: $209,000 - $200,000 = $9,000 at 0.9% = $81.00. This is the joint filer the return charges only $90 for the whole year (see the annual case above) having $81 taken from one paycheck.",
      "ledger": "L-2026-415"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 264000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 20,
        "w4": {
          "step2Checkbox": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "socialSecurity": 0,
        "medicare": 159.5,
        "additionalMedicare": 99,
        "federalWithholding": 1701.17,
        "net": 9040.33
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year') Hand derivation for payment 20 of 24 at $11,000.00: wages paid before it $209,000, after it $220,000. Social security wages are min(after, $184,500) - min(before, $184,500) at 6.2%; Medicare 0.0145 x $11,000 = $159.50 on every payment; Additional Medicare Tax 0.009 x (the part of the slice above $200,000), the EMPLOYER's threshold, which is $200,000 for every filing status including a joint one. Federal withholding $1,701.17 every period, derived at L-2026-410. Net $11,000.00 - $1,701.17 - $0.00 - $159.50 - $99.00 = $9,040.33.",
      "why": "the payment after the crossing: 0.9% of the whole $11,000 = $99.00. Twenty-fourth-to-first these paychecks withhold $540 in total against a $90 liability.",
      "ledger": "L-2026-415"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 1000,
        "filingStatus": "single",
        "taxYear": "2026"
      },
      "expect": {
        "socialSecurity": 62,
        "medicare": 14.5,
        "fica": 76.5,
        "federalIncomeTax": 0,
        "taxable": 0,
        "net": 923.5
      },
      "source": "IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9, page 30, worked example: 'For 2026, Gavrielle paid Dan $1,000 during the year. Gavrielle enters $1,076.50 in box 1 of Dan's Form W-2 ($1,000 wages plus $76.50 social security and Medicare taxes paid for Dan).' The IRS's own arithmetic: the employee share of social security and Medicare on $1,000 of 2026 wages is $76.50. Split by IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year'): 0.062 x $1,000 = $62.00 and 0.0145 x $1,000 = $14.50. Income tax from IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19): $1,000 is below the $16,100 standard deduction, so taxable income is $0 and the tax is $0.",
      "why": "the only FICA figure the IRS prints as a finished answer anywhere in Publication 15. It is also the case where the standard deduction exceeds the whole year's pay, so taxable income floors at zero rather than going negative.",
      "ledger": "L-2026-416"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 0,
        "filingStatus": "single",
        "taxYear": "2026"
      },
      "expect": {
        "federalIncomeTax": 0,
        "socialSecurity": 0,
        "medicare": 0,
        "fica": 0,
        "net": 0,
        "effectiveRate": 0
      },
      "source": "IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19) and IRS Publication 15 (2026), Circular E, Employer's Tax Guide, https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 'Social Security and Medicare Taxes', pages 29-30 ('For 2026, the social security tax rate is 6.2% ... The social security wage base limit is $184,500. The tax rate for Medicare is 1.45% ... you must withhold a 0.9% Additional Medicare Tax from wages you pay to an employee in excess of $200,000 in a calendar year'): no wages, no taxable income, no tax and no contributions.",
      "why": "zero pay. The effective rate is reported as 0 rather than a division by zero.",
      "ledger": "L-2026-416"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 52000,
        "filingStatus": "single",
        "taxYear": "2026"
      },
      "expect": {
        "standardDeduction": 16100,
        "taxable": 35900,
        "federalIncomeTax": 4060,
        "socialSecurity": 3224,
        "medicare": 754,
        "net": 43962
      },
      "source": "IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19) s.4.14(1) (single standard deduction $16,100) and s.4.01 Table 3. Hand derivation: taxable $52,000 - $16,100 = $35,900; 'Over $12,400 but not over $50,400: $1,240 plus 12% of the excess over $12,400' = $1,240 + 0.12 x $23,500 = $4,060.00. FICA 0.062 x $52,000 = $3,224.00 and 0.0145 x $52,000 = $754.00. Net $52,000 - $4,060.00 - $3,224.00 - $754.00 = $43,962.00.",
      "why": "the ordinary single case, and the contrast the site has to get right: the ANNUAL liability on $52,000 is $4,060.00, while a biweekly employee on the same pay has $4,044.40 withheld across the year (L-2026-409). Withholding is not liability.",
      "ledger": "L-2026-416"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 100000,
        "filingStatus": "head-of-household",
        "taxYear": "2026"
      },
      "expect": {
        "standardDeduction": 24150,
        "taxable": 75850,
        "federalIncomeTax": 9588,
        "socialSecurity": 6200,
        "medicare": 1450,
        "net": 82762
      },
      "source": "IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19) s.4.14(1) (heads of households $24,150) and s.4.01 Table 2. Hand derivation: taxable $100,000 - $24,150 = $75,850; 'Over $67,450 but not over $105,700: $7,740 plus 22% of the excess over $67,450' = $7,740 + 0.22 x $8,400 = $9,588.00. FICA $6,200.00 and $1,450.00. Net $82,762.00.",
      "why": "head of household has its own standard deduction and its own rate schedule, and Table 2's 24% threshold ($201,750) differs from Table 3's ($201,775) by $25 — a difference that only ever shows up if both tables are stored separately.",
      "ledger": "L-2026-416"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "age65OrOver": true
      },
      "expect": {
        "standardDeduction": 18150,
        "taxable": 41850,
        "federalIncomeTax": 4774
      },
      "source": "IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19) s.4.14(3): 'the additional standard deduction amount under s.63(f) for the aged or the blind is $1,650. The additional standard deduction amount is increased to $2,050 if the individual is also unmarried and not a surviving spouse.' Hand derivation: $16,100 + $2,050 = $18,150; taxable $41,850; $1,240 + 0.12 x $29,450 = $4,774.00.",
      "why": "an unmarried filer aged 65 or over gets $2,050, not $1,650. Using the married figure here understates the deduction by $400 and overstates the tax by $48.",
      "ledger": "L-2026-416",
      "refuses": {
        "code": "UNSOURCED_SENIOR_DEDUCTION",
        "why": "OBBBA §70103 adds a further deduction per individual aged 65 or over for 2025-2028, on top of the §63(f) amount derived here, and the rules file carries no figure for it. The §63(f)-only figures below are correct as far as §63(f) goes and are NOT the whole deduction, so both implementations refuse the age rather than quote a tax that is too high. The derivation is kept: when rules/us carries standardDeduction.seniorDeduction this case is re-derived with it and the refusal is removed."
      }
    },
    {
      "kind": "annual",
      "input": {
        "gross": 120000,
        "filingStatus": "married-filing-jointly",
        "taxYear": "2026",
        "age65OrOver": true,
        "spouseAge65OrOver": true
      },
      "expect": {
        "standardDeduction": 35500,
        "taxable": 84500,
        "federalIncomeTax": 9644
      },
      "source": "IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Tables 1-4 (pages 10-13) and s.4.14 (page 19) s.4.14(1) ($32,200 joint) and s.4.14(3) ($1,650 each where the individual is not unmarried). Hand derivation: $32,200 + $1,650 + $1,650 = $35,500; taxable $120,000 - $35,500 = $84,500; Table 1 'Over $24,800 but not over $100,800: $2,480 plus 12% of the excess over $24,800' = $2,480 + 0.12 x $59,700 = $9,644.00.",
      "why": "two aged amounts on one joint return, at $1,650 each and not $2,050 — the increase is only for someone unmarried and not a surviving spouse.",
      "ledger": "L-2026-416",
      "refuses": {
        "code": "UNSOURCED_SENIOR_DEDUCTION",
        "why": "OBBBA §70103 adds a further deduction per individual aged 65 or over for 2025-2028, on top of the §63(f) amount derived here, and the rules file carries no figure for it. The §63(f)-only figures below are correct as far as §63(f) goes and are NOT the whole deduction, so both implementations refuse the age rather than quote a tax that is too high. The derivation is kept: when rules/us carries standardDeduction.seniorDeduction this case is re-derived with it and the refusal is removed."
      }
    },
    {
      "kind": "annual",
      "input": {
        "gross": 28500,
        "filingStatus": "single",
        "taxYear": "2026"
      },
      "expect": {
        "taxable": 12400,
        "federalIncomeTax": 1240,
        "marginalBand": 0,
        "socialSecurity": 1767,
        "medicare": 413.25
      },
      "source": "IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Table 3 (page 12) and s.4.14(1) (page 19). Hand derivation: taxable income $28,500 - $16,100 = $12,400, which is EXACTLY the top of the first row. Table 3 prints 'Not over $12,400: 10% of the taxable income' and the next row 'Over $12,400 but not over $50,400', so $12,400 belongs to the LOWER row: 0.10 x $12,400 = $1,240.00 and the marginal band is row 0. IRS Publication 15 (2026), https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 pages 29-30 (social security 6.2%, Medicare 1.45%): 0.062 x $28,500 = $1,767.00; 0.0145 x $28,500 = $413.25.",
      "why": "the annual rate-schedule boundary, from below. Rev. Proc. prints 'Over A but not over B', which is the OPPOSITE convention to the Publication 15-T withholding table's 'at least A but less than B' (L-2026-409), and `marginalBand` is the only output on which the difference shows: the tax itself is identical either way, because the schedule is charged slice by slice. A calculator that tells this taxpayer their marginal rate is 12% is wrong.",
      "ledger": "L-2026-416"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 28501,
        "filingStatus": "single",
        "taxYear": "2026"
      },
      "expect": {
        "taxable": 12401,
        "federalIncomeTax": 1240.12,
        "marginalBand": 1,
        "socialSecurity": 1767.06,
        "medicare": 413.26
      },
      "source": "IRS Rev. Proc. 2025-32, https://www.irs.gov/pub/irs-drop/rp-25-32.pdf, retrieved 2026-09-18, s.4.01 Table 3 (page 12) and s.4.14(1) (page 19). Hand derivation: taxable $12,401, one dollar over the boundary, so Table 3's second row applies: '$1,240 plus 12% of the excess over $12,400' = $1,240 + 0.12 x $1 = $1,240.12, marginal band 1. IRS Publication 15 (2026), https://www.irs.gov/pub/irs-pdf/p15.pdf, retrieved 2026-09-18, section 9 pages 29-30 (social security 6.2%, Medicare 1.45%): 0.062 x $28,501 = $1,767.062 = $1,767.06; 0.0145 x $28,501 = $413.2645 = $413.26.",
      "why": "one dollar above the same boundary: twelve cents more tax and a different marginal band. Paired with the case above, this is the only pair in the set that pins which row an exactly-on-the-edge taxable income falls in.",
      "ledger": "L-2026-416"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "CO"
      },
      "expect": {
        "stateIncomeTax": 2398
      },
      "source": "Colorado Department of Revenue, Form DR 1098 (10/21/25), '2026 Colorado Withholding Worksheet for Employers', https://tax.colorado.gov/DR1098 (PDF at https://tax.colorado.gov/sites/default/files/documents/DR_1098_Colorado_Withholding_Worksheet_for_Employees.pdf), retrieved 2026-10-15. Verbatim: 'This worksheet prescribes the method for employers to calculate the required Colorado wage withholding for 2026.' Step 2 line 2a: 'If the employee completed DR 0004 Line 2, enter that amount. If the employee did not submit a DR 0004, or if DR 0004 Line 2 is blank, enter the appropriate amount based on the employee's expected filing status from IRS form W-4 Step 1(c): $11,000 if married filing jointly or qualifying surviving spouse; or $5,500 otherwise'. Line 2b: 'Subtract line 2a from line 1c. If zero or less, enter zero'. Line 2c: 'Multiply line 2b by 4.40% (0.044)'. Hand derivation: line 1c $60,000.00; line 2a $5,500.00 (not married filing jointly); line 2b $54,500.00; line 2c 0.044 x $54,500.00 = $2,398.00.",
      "why": "the plain Colorado computation at a round salary: the $5,500 line 2a amount for a single filer and the 4.40% rate. An engine that charged 4.40% on the whole $60,000 would say $2,640.00, $242.00 too much.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 100000,
        "filingStatus": "married-filing-jointly",
        "taxYear": "2026",
        "state": "CO"
      },
      "expect": {
        "stateIncomeTax": 3916
      },
      "source": "Colorado Department of Revenue, Form DR 1098 (10/21/25), '2026 Colorado Withholding Worksheet for Employers', https://tax.colorado.gov/DR1098 (PDF at https://tax.colorado.gov/sites/default/files/documents/DR_1098_Colorado_Withholding_Worksheet_for_Employees.pdf), retrieved 2026-10-15. Verbatim: 'This worksheet prescribes the method for employers to calculate the required Colorado wage withholding for 2026.' Step 2 line 2a: 'If the employee completed DR 0004 Line 2, enter that amount. If the employee did not submit a DR 0004, or if DR 0004 Line 2 is blank, enter the appropriate amount based on the employee's expected filing status from IRS form W-4 Step 1(c): $11,000 if married filing jointly or qualifying surviving spouse; or $5,500 otherwise'. Line 2b: 'Subtract line 2a from line 1c. If zero or less, enter zero'. Line 2c: 'Multiply line 2b by 4.40% (0.044)'. Hand derivation: line 1c $100,000.00; line 2a $11,000.00 (married filing jointly); line 2b $89,000.00; line 2c 0.044 x $89,000.00 = $3,916.00.",
      "why": "the $11,000 line 2a amount is for married filing jointly or a qualifying surviving spouse only. Using $5,500 here overstates the tax by $242.00.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "head-of-household",
        "taxYear": "2026",
        "state": "CO"
      },
      "expect": {
        "stateIncomeTax": 2398
      },
      "source": "Colorado Department of Revenue, Form DR 1098 (10/21/25), '2026 Colorado Withholding Worksheet for Employers', https://tax.colorado.gov/DR1098 (PDF at https://tax.colorado.gov/sites/default/files/documents/DR_1098_Colorado_Withholding_Worksheet_for_Employees.pdf), retrieved 2026-10-15. Verbatim: 'This worksheet prescribes the method for employers to calculate the required Colorado wage withholding for 2026.' Step 2 line 2a: 'If the employee completed DR 0004 Line 2, enter that amount. If the employee did not submit a DR 0004, or if DR 0004 Line 2 is blank, enter the appropriate amount based on the employee's expected filing status from IRS form W-4 Step 1(c): $11,000 if married filing jointly or qualifying surviving spouse; or $5,500 otherwise'. Line 2b: 'Subtract line 2a from line 1c. If zero or less, enter zero'. Line 2c: 'Multiply line 2b by 4.40% (0.044)'. Hand derivation: line 1c $60,000.00; line 2a $5,500.00, because DR 1098 line 2a gives $11,000 only for 'married filing jointly or qualifying surviving spouse' and $5,500 'otherwise', and head of household is otherwise; line 2b $54,500.00; line 2c 0.044 x $54,500.00 = $2,398.00.",
      "why": "Colorado gives head of household NO uplift - it is in the 'otherwise' branch with single. Copying the federal habit of a middle amount for head of household would be wrong here.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "married-filing-separately",
        "taxYear": "2026",
        "state": "CO"
      },
      "expect": {
        "stateIncomeTax": 2398
      },
      "source": "Colorado Department of Revenue, Form DR 1098 (10/21/25), '2026 Colorado Withholding Worksheet for Employers', https://tax.colorado.gov/DR1098 (PDF at https://tax.colorado.gov/sites/default/files/documents/DR_1098_Colorado_Withholding_Worksheet_for_Employees.pdf), retrieved 2026-10-15. Verbatim: 'This worksheet prescribes the method for employers to calculate the required Colorado wage withholding for 2026.' Step 2 line 2a: 'If the employee completed DR 0004 Line 2, enter that amount. If the employee did not submit a DR 0004, or if DR 0004 Line 2 is blank, enter the appropriate amount based on the employee's expected filing status from IRS form W-4 Step 1(c): $11,000 if married filing jointly or qualifying surviving spouse; or $5,500 otherwise'. Line 2b: 'Subtract line 2a from line 1c. If zero or less, enter zero'. Line 2c: 'Multiply line 2b by 4.40% (0.044)'. Hand derivation: line 2a $5,500.00 ('otherwise'); line 2b $54,500.00; line 2c 0.044 x $54,500.00 = $2,398.00.",
      "why": "married filing separately is also the 'otherwise' branch, not the $11,000 one. The joint amount is conditioned on filing jointly, not on being married.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 5500,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "CO"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Colorado Department of Revenue, Form DR 1098 (10/21/25), '2026 Colorado Withholding Worksheet for Employers', https://tax.colorado.gov/DR1098 (PDF at https://tax.colorado.gov/sites/default/files/documents/DR_1098_Colorado_Withholding_Worksheet_for_Employees.pdf), retrieved 2026-10-15. Verbatim: 'This worksheet prescribes the method for employers to calculate the required Colorado wage withholding for 2026.' Step 2 line 2a: 'If the employee completed DR 0004 Line 2, enter that amount. If the employee did not submit a DR 0004, or if DR 0004 Line 2 is blank, enter the appropriate amount based on the employee's expected filing status from IRS form W-4 Step 1(c): $11,000 if married filing jointly or qualifying surviving spouse; or $5,500 otherwise'. Line 2b: 'Subtract line 2a from line 1c. If zero or less, enter zero'. Line 2c: 'Multiply line 2b by 4.40% (0.044)'. Hand derivation: line 1c $5,500.00; line 2a $5,500.00; line 2b $0.00; line 2c 0.044 x $0.00 = $0.00.",
      "why": "the only threshold the Colorado formula has: gross exactly equal to line 2a. The edge, at the penny.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 6000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "CO"
      },
      "expect": {
        "stateIncomeTax": 22
      },
      "source": "Colorado Department of Revenue, Form DR 1098 (10/21/25), '2026 Colorado Withholding Worksheet for Employers', https://tax.colorado.gov/DR1098 (PDF at https://tax.colorado.gov/sites/default/files/documents/DR_1098_Colorado_Withholding_Worksheet_for_Employees.pdf), retrieved 2026-10-15. Verbatim: 'This worksheet prescribes the method for employers to calculate the required Colorado wage withholding for 2026.' Step 2 line 2a: 'If the employee completed DR 0004 Line 2, enter that amount. If the employee did not submit a DR 0004, or if DR 0004 Line 2 is blank, enter the appropriate amount based on the employee's expected filing status from IRS form W-4 Step 1(c): $11,000 if married filing jointly or qualifying surviving spouse; or $5,500 otherwise'. Line 2b: 'Subtract line 2a from line 1c. If zero or less, enter zero'. Line 2c: 'Multiply line 2b by 4.40% (0.044)'. Hand derivation: line 1c $6,000.00; line 2a $5,500.00; line 2b $500.00; line 2c 0.044 x $500.00 = $22.00.",
      "why": "$500 above the line 2a amount, where the charge first bites. Proves the subtraction happens before the rate and not after.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 4000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "CO"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Colorado Department of Revenue, Form DR 1098 (10/21/25), '2026 Colorado Withholding Worksheet for Employers', https://tax.colorado.gov/DR1098 (PDF at https://tax.colorado.gov/sites/default/files/documents/DR_1098_Colorado_Withholding_Worksheet_for_Employees.pdf), retrieved 2026-10-15. Verbatim: 'This worksheet prescribes the method for employers to calculate the required Colorado wage withholding for 2026.' Step 2 line 2a: 'If the employee completed DR 0004 Line 2, enter that amount. If the employee did not submit a DR 0004, or if DR 0004 Line 2 is blank, enter the appropriate amount based on the employee's expected filing status from IRS form W-4 Step 1(c): $11,000 if married filing jointly or qualifying surviving spouse; or $5,500 otherwise'. Line 2b: 'Subtract line 2a from line 1c. If zero or less, enter zero'. Line 2c: 'Multiply line 2b by 4.40% (0.044)'. Hand derivation: line 1c $4,000.00; line 2a $5,500.00; line 2b is 'zero or less', so by line 2b's own words enter zero; line 2c 0.044 x $0.00 = $0.00.",
      "why": "below the line 2a amount. DR 1098 line 2b says 'If zero or less, enter zero', so the answer is nil and never a negative charge.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 0,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "CO"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Colorado Department of Revenue, Form DR 1098 (10/21/25), '2026 Colorado Withholding Worksheet for Employers', https://tax.colorado.gov/DR1098 (PDF at https://tax.colorado.gov/sites/default/files/documents/DR_1098_Colorado_Withholding_Worksheet_for_Employees.pdf), retrieved 2026-10-15. Verbatim: 'This worksheet prescribes the method for employers to calculate the required Colorado wage withholding for 2026.' Step 2 line 2a: 'If the employee completed DR 0004 Line 2, enter that amount. If the employee did not submit a DR 0004, or if DR 0004 Line 2 is blank, enter the appropriate amount based on the employee's expected filing status from IRS form W-4 Step 1(c): $11,000 if married filing jointly or qualifying surviving spouse; or $5,500 otherwise'. Line 2b: 'Subtract line 2a from line 1c. If zero or less, enter zero'. Line 2c: 'Multiply line 2b by 4.40% (0.044)'. Hand derivation: line 1c $0.00; line 2a $5,500.00; line 2b zero or less, enter zero; line 2c $0.00.",
      "why": "nil wages, nil Colorado tax - the floor the worksheet states in words.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 52000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL"
      },
      "expect": {
        "stateIncomeTax": 2574
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', with 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.' Hand derivation: no Form IL-W-4 allowances, so exemptions are $0; .0495 x ($52,000.00 - $0.00) = $2,574.00.",
      "why": "Illinois is flat with no filing status and no automatic allowance: at zero allowances the whole wage is taxed at 4.95%.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 52000,
        "filingStatus": "married-filing-jointly",
        "taxYear": "2026",
        "state": "IL"
      },
      "expect": {
        "stateIncomeTax": 2574
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', with 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.' Hand derivation: the booklet's formula has no filing-status term at all: .0495 x ($52,000.00 - $0.00) = $2,574.00, the same as for a single filer.",
      "why": "Illinois charges a joint filer exactly what it charges a single one. An engine that applied a joint band or a doubled allowance would differ here.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 55850,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "stateCertificate": {
          "allowances": 2
        }
      },
      "expect": {
        "stateIncomeTax": 2475
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', with 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.' Hand derivation: Step 2a 2 x $2,925 = $5,850; Step 2b 0 x $1,000 = $0; Step 2c $5,850; taxable $55,850.00 - $5,850.00 = $50,000.00; .0495 x $50,000.00 = $2,475.00.",
      "why": "two Line 1 allowances at $2,925 each. The $2,925 figure is the booklet's own sentence, and using the federal personal exemption or the prior year's $2,850 would show up here.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 57850,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "stateCertificate": {
          "allowances": 2,
          "additionalAllowances": 2
        }
      },
      "expect": {
        "stateIncomeTax": 2475
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', with 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.' Hand derivation: Step 2a 2 x $2,925 = $5,850; Step 2b 2 x $1,000 = $2,000; Step 2c $7,850; taxable $57,850.00 - $7,850.00 = $50,000.00; .0495 x $50,000.00 = $2,475.00.",
      "why": "a Line 2 allowance is worth $1,000 and NOT $2,925. Paired with the case above it holds the two allowance values apart: if Line 2 were valued at $2,925 this case would come out $190.58 lower.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 2925,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', with 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.' Hand derivation: Step 2a 1 x $2,925 = $2,925; taxable $2,925.00 - $2,925.00 = $0.00; .0495 x $0.00 = $0.00.",
      "why": "the exemption allowance is the only threshold Illinois has, and this is its exact edge.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 2945,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "expect": {
        "stateIncomeTax": 0.99
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', with 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.' Hand derivation: Step 2a 1 x $2,925 = $2,925; taxable $2,945.00 - $2,925.00 = $20.00; .0495 x $20.00 = $0.99.",
      "why": "$20 past the exemption allowance: the first pennies of Illinois tax, where a rate applied to gross rather than to the excess would read $145.78.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 0,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', with 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.' Hand derivation: taxable $0.00 - $2,925.00 is negative; the booklet's own weekly table prints no amount at all for wages in the '$0 at but less than $2' row, so the withholding is nil and not negative; .0495 x $0.00 = $0.00.",
      "why": "nil wages with an allowance claimed. The formula alone would give a negative number; the printed table says nil.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 41600,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "IL",
        "stateCertificate": {
          "allowances": 2,
          "additionalAllowances": 2
        }
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 32.13
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', with 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.' This is the booklet's OWN WORKED EXAMPLE, page 4, verbatim: 'You pay Alice $800 every week. She claims four allowances on her Form IL-W-4. Two allowances are claimed on Form IL-W-4, Line 1, and two allowances are claimed on Form IL-W-4, Line 2. You withhold $32.13 from her pay.' The Department's printed steps: 'Step 1 Determine the wages paid $800.00; Step 2 a 2 x $2,925 = $5,850 (Line 1 x $2,925), b 2 x $1,000 = $2,000 (Line 2 x $1,000), c $5,850 + $2,000 = $7,850, d $7,850 / 52 pay periods = $150.96; Step 3 Subtract the amount exempt from withholding - 150.96, Taxable amount $649.04; Step 4 Multiply by the tax rate x .0495, Illinois Income Tax withheld $32.13'. The annual gross here is the example's weekly wage annualised: $800.00 x 52 = $41,600.00.",
      "why": "the Illinois Department of Revenue's own printed answer, to the penny, for a weekly payment with allowances on both lines of Form IL-W-4. It is the one Illinois case that is not derived at all, and it pins the per-period divisor (52) as well as the two allowance values.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 15652,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "IL",
        "stateCertificate": {
          "allowances": 2,
          "additionalAllowances": 1
        }
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 8.38
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', with 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.' The booklet's other worked example, page 3, verbatim: 'You pay Mary $300 every week. She claims three allowances on her Form IL-W-4. Two allowances are claimed on Form IL-W-4, Line 1, and one allowance is claimed on Form IL-W-4, Line 2. Mary does not request any additional amount be withheld on Line 3. You withhold $8.38 from her pay.' - read from the Weekly Payroll Period table row 'at least $300 but less than $302', column '2' = $9.33, less the printed 'For each allowance claimed on Line 2 of Form IL-W-4, subtract: $0.95', giving $8.38. The wage used here is the MIDPOINT of that printed row, $301.00, where the table and the automated payroll formula coincide: .0495 x ($301.00 - (2 x $2,925)/52) = .0495 x ($301.00 - $112.50) = .0495 x $188.50 = $9.33075 = $9.33, less $0.95 = $8.38; equivalently .0495 x ($301.00 - $6,850/52) = .0495 x $169.269230... = $8.37883... = $8.38. Annual gross $301.00 x 52 = $15,652.00. (At the example's own $300.00 the table gives $8.38 and the automated formula gives $8.33 - the two official methods differ by five cents at that wage, which is why the row midpoint is used here and not $300.)",
      "why": "the second published Illinois figure, and the one that proves the Line 2 subtraction is $0.95 a week - $1,000/52 x 4.95% - and not $2,925/52 x 4.95%. The source records where the table method and the automated method disagree so no later reader 'fixes' one into the other.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "KY"
      },
      "expect": {
        "stateIncomeTax": 1982.4
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-15, sheet 'Hidden Table' cell H7, which is the Department's own published formula verbatim: '=IF(C7>0,(((C7*periods)-3360)*(3.5/100))/periods,\"\")', wrapped on the 'Calculation of Withholding' sheet by '=IF(H7>0,H7,0)'. So the 2026 annual standard deduction is $3,360, the rate is 3.5%, and a negative result is withheld as zero. The rate is confirmed on the Department's Employer Payroll Withholding page, https://revenue.ky.gov/Business/Pages/Employer-Payroll-Withholding.aspx, retrieved 2026-10-15: 'The Kentucky Withholding Tax rate will be 3.5% for tax year 2026.' Hand derivation: (($60,000.00) - $3,360) x 0.035 = $56,640.00 x 0.035 = $1,982.40.",
      "why": "the whole Kentucky 2026 computation at a round salary: one standard deduction of $3,360 and one flat rate of 3.5%. An engine still on the 2024 rate of 4.0% would say $2,265.60.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "married-filing-jointly",
        "taxYear": "2026",
        "state": "KY"
      },
      "expect": {
        "stateIncomeTax": 1982.4
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-15, sheet 'Hidden Table' cell H7, which is the Department's own published formula verbatim: '=IF(C7>0,(((C7*periods)-3360)*(3.5/100))/periods,\"\")', wrapped on the 'Calculation of Withholding' sheet by '=IF(H7>0,H7,0)'. So the 2026 annual standard deduction is $3,360, the rate is 3.5%, and a negative result is withheld as zero. The rate is confirmed on the Department's Employer Payroll Withholding page, https://revenue.ky.gov/Business/Pages/Employer-Payroll-Withholding.aspx, retrieved 2026-10-15: 'The Kentucky Withholding Tax rate will be 3.5% for tax year 2026.' Hand derivation: the Department's formula has no filing-status term and no allowance term: (($60,000.00) - $3,360) x 0.035 = $1,982.40, the same as for a single filer.",
      "why": "Kentucky's withholding formula gives a joint filer the same $3,360 as a single one. Doubling the standard deduction for a couple, as the federal rules do, would understate the tax by $117.60.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 3360,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "KY"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-15, sheet 'Hidden Table' cell H7, which is the Department's own published formula verbatim: '=IF(C7>0,(((C7*periods)-3360)*(3.5/100))/periods,\"\")', wrapped on the 'Calculation of Withholding' sheet by '=IF(H7>0,H7,0)'. So the 2026 annual standard deduction is $3,360, the rate is 3.5%, and a negative result is withheld as zero. The rate is confirmed on the Department's Employer Payroll Withholding page, https://revenue.ky.gov/Business/Pages/Employer-Payroll-Withholding.aspx, retrieved 2026-10-15: 'The Kentucky Withholding Tax rate will be 3.5% for tax year 2026.' Hand derivation: ($3,360.00 - $3,360) x 0.035 = $0.00.",
      "why": "the only threshold the Kentucky formula has, at its exact edge.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 3400,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "KY"
      },
      "expect": {
        "stateIncomeTax": 1.4
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-15, sheet 'Hidden Table' cell H7, which is the Department's own published formula verbatim: '=IF(C7>0,(((C7*periods)-3360)*(3.5/100))/periods,\"\")', wrapped on the 'Calculation of Withholding' sheet by '=IF(H7>0,H7,0)'. So the 2026 annual standard deduction is $3,360, the rate is 3.5%, and a negative result is withheld as zero. The rate is confirmed on the Department's Employer Payroll Withholding page, https://revenue.ky.gov/Business/Pages/Employer-Payroll-Withholding.aspx, retrieved 2026-10-15: 'The Kentucky Withholding Tax rate will be 3.5% for tax year 2026.' Hand derivation: ($3,400.00 - $3,360) x 0.035 = $40.00 x 0.035 = $1.40.",
      "why": "$40 past the standard deduction: the first charge. A rate applied to gross would read $119.00 here.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 2000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "KY"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-15, sheet 'Hidden Table' cell H7, which is the Department's own published formula verbatim: '=IF(C7>0,(((C7*periods)-3360)*(3.5/100))/periods,\"\")', wrapped on the 'Calculation of Withholding' sheet by '=IF(H7>0,H7,0)'. So the 2026 annual standard deduction is $3,360, the rate is 3.5%, and a negative result is withheld as zero. The rate is confirmed on the Department's Employer Payroll Withholding page, https://revenue.ky.gov/Business/Pages/Employer-Payroll-Withholding.aspx, retrieved 2026-10-15: 'The Kentucky Withholding Tax rate will be 3.5% for tax year 2026.' Hand derivation: ($2,000.00 - $3,360) x 0.035 = -$47.60, and the Department's sheet wraps the formula in '=IF(H7>0,H7,0)', so a negative result is withheld as zero.",
      "why": "below the standard deduction. The nil comes from the Department's own IF wrapper, not from an assumption.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 0,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "KY"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-15, sheet 'Hidden Table' cell H7, which is the Department's own published formula verbatim: '=IF(C7>0,(((C7*periods)-3360)*(3.5/100))/periods,\"\")', wrapped on the 'Calculation of Withholding' sheet by '=IF(H7>0,H7,0)'. So the 2026 annual standard deduction is $3,360, the rate is 3.5%, and a negative result is withheld as zero. The rate is confirmed on the Department's Employer Payroll Withholding page, https://revenue.ky.gov/Business/Pages/Employer-Payroll-Withholding.aspx, retrieved 2026-10-15: 'The Kentucky Withholding Tax rate will be 3.5% for tax year 2026.' Hand derivation: nil wages: the Department's own cell returns the empty string for a non-positive wage and the 'Calculation of Withholding' sheet prints 0.",
      "why": "nil wages, nil Kentucky tax.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "KY"
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 165.2
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-15, sheet 'Hidden Table' cell H7, which is the Department's own published formula verbatim: '=IF(C7>0,(((C7*periods)-3360)*(3.5/100))/periods,\"\")', wrapped on the 'Calculation of Withholding' sheet by '=IF(H7>0,H7,0)'. So the 2026 annual standard deduction is $3,360, the rate is 3.5%, and a negative result is withheld as zero. The rate is confirmed on the Department's Employer Payroll Withholding page, https://revenue.ky.gov/Business/Pages/Employer-Payroll-Withholding.aspx, retrieved 2026-10-15: 'The Kentucky Withholding Tax rate will be 3.5% for tax year 2026.' Hand derivation at the Department's own period basis - the formula annualises the period wage, deducts $3,360 once, applies 3.5%, and divides the YEAR's tax by the number of periods: monthly wage $5,000.00 x 12 = $60,000.00; less $3,360 = $56,640.00; x 0.035 = $1,982.40; / 12 = $165.20 exactly.",
      "why": "the period basis. Kentucky deducts the $3,360 once a year, not once a month: deducting $3,360 from each month's pay would give $57.40 a month, $107.80 too little.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 55360,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "KY"
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 35
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-15, sheet 'Hidden Table' cell H7, which is the Department's own published formula verbatim: '=IF(C7>0,(((C7*periods)-3360)*(3.5/100))/periods,\"\")', wrapped on the 'Calculation of Withholding' sheet by '=IF(H7>0,H7,0)'. So the 2026 annual standard deduction is $3,360, the rate is 3.5%, and a negative result is withheld as zero. The rate is confirmed on the Department's Employer Payroll Withholding page, https://revenue.ky.gov/Business/Pages/Employer-Payroll-Withholding.aspx, retrieved 2026-10-15: 'The Kentucky Withholding Tax rate will be 3.5% for tax year 2026.' Hand derivation: annual gross $55,360.00 over a weekly payroll period; $55,360.00 - $3,360 = $52,000.00; x 0.035 = $1,820.00; / 52 = $35.00 exactly. The Department's 'Hidden Table' sheet lists 52 as the number of periods for 'Weekly' (A4:B4).",
      "why": "52 periods, read from the Department's own pay-frequency table, with a figure that is exact to the penny so nothing here depends on a rounding rule.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "AZ"
      },
      "expect": {
        "stateIncomeTax": 1200
      },
      "source": "Arizona Department of Revenue, Arizona Form A-4 (2026), 'Employee's Arizona Withholding Election', ADOR 10121 (25), https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf (azdor.gov refuses automated requests; read from the Internet Archive capture of 2026-10-08 at http://web.archive.org/web/20261008204653/https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf), retrieved 2026-10-15. Verbatim, box 1: 'Withhold from gross taxable wages at the percentage checked (check only one percentage): 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5%'; box 2: 'I elect an Arizona withholding percentage of zero, and I certify that I expect to have no Arizona tax liability for the current taxable year.' Employee's Instructions, 'New Employees': 'If you do not give this form to your employer the department requires your employer to withhold 2.0% of your gross taxable wages.' And: 'The amount withheld is a percentage of your gross taxable wages from every paycheck.' The statutory frame is A.R.S. s.43-401(A) and (E), https://www.azleg.gov/ars/43/00401.htm, retrieved 2026-10-15: the employer deducts 'an amount prescribed by tables adopted by the department', 'Each employee shall elect the amount authorized by subsection A of this section to be withheld', and 'Any employee failing to complete an election form as prescribed shall be deemed to have elected the withholding percentage prescribed by the department.' THERE IS NO ARIZONA BRACKET TABLE AND NO ARIZONA DEDUCTION IN THIS COMPUTATION: Arizona withholding is a percentage of gross taxable wages chosen by the employee, so an Arizona figure is a function of an election, not of a schedule. Hand derivation: no Form A-4 on file, so the Department's prescribed default applies: 2.0% of gross taxable wages = 0.020 x $60,000.00 = $1,200.00.",
      "why": "the Arizona DEFAULT, which is the only Arizona figure that can be quoted for an employee who has elected nothing. The form states the 2.0% itself, so this is not an assumption. Note what it is not: 2.0% of gross with no deduction and no bracket is an elected withholding rate, not an estimate of Arizona liability.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "AZ",
        "stateCertificate": {
          "withholdingPercentage": 3.5
        }
      },
      "source": "Arizona Department of Revenue, Arizona Form A-4 (2026), 'Employee's Arizona Withholding Election', ADOR 10121 (25), https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf (azdor.gov refuses automated requests; read from the Internet Archive capture of 2026-10-08 at http://web.archive.org/web/20261008204653/https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf), retrieved 2026-10-15. Verbatim, box 1: 'Withhold from gross taxable wages at the percentage checked (check only one percentage): 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5%'; box 2: 'I elect an Arizona withholding percentage of zero, and I certify that I expect to have no Arizona tax liability for the current taxable year.' Employee's Instructions, 'New Employees': 'If you do not give this form to your employer the department requires your employer to withhold 2.0% of your gross taxable wages.' And: 'The amount withheld is a percentage of your gross taxable wages from every paycheck.' The statutory frame is A.R.S. s.43-401(A) and (E), https://www.azleg.gov/ars/43/00401.htm, retrieved 2026-10-15: the employer deducts 'an amount prescribed by tables adopted by the department', 'Each employee shall elect the amount authorized by subsection A of this section to be withheld', and 'Any employee failing to complete an election form as prescribed shall be deemed to have elected the withholding percentage prescribed by the department.' THERE IS NO ARIZONA BRACKET TABLE AND NO ARIZONA DEDUCTION IN THIS COMPUTATION: Arizona withholding is a percentage of gross taxable wages chosen by the employee, so an Arizona figure is a function of an election, not of a schedule. Hand derivation: there is none to make - 350.00 per cent is not one of the seven percentages Form A-4 prints, so no official figure exists for it. What the document warrants is the REFUSAL: the form's seven boxes and the instruction 'check only one percentage' are an exhaustive list, and A.R.S. s.43-401(E) makes the election the employee's to make from that list.",
      "why": "Arizona publishes no bracket table, so the only thing that can be wrong about an Arizona figure is the election - which makes an unlisted election the one input that must never be guessed. This case also fixes the units of the field: 3.5 is 350.00 per cent and not 3.5 per cent, and an engine that quietly accepted it would charge $210,000.00 on a $60,000 salary.",
      "refuses": {
        "code": "INVALID_AMOUNT",
        "why": "Form A-4 prints the elections as 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 and 3.5 per cent and says 'check only one percentage', so a number that is not one of the seven is not an Arizona election at all. The engine's stateCertificate.withholdingPercentage holds the election as a FRACTION of gross (0.035 for 3.5 per cent), so the value 3.5 here reads as 350.00 per cent and is refused by name rather than charged. The matching case with 0.035 carries the sourced figure; this case asserts only that an unlisted percentage is refused. THE CODE NAME IS THE ENGINE'S AND NOT THE DOCUMENT'S: Form A-4 warrants that an unlisted percentage must be refused, and nothing more. On the first run of this vector the engine refused 3.5 with INVALID_AMOUNT (a fraction above 1) rather than with UNSUPPORTED_STATE_CERTIFICATE, which it uses for a value inside 0 to 1 that is not one of the seven; the code here was taken from that run and is recorded as such. What this case pins is the refusal, which is sourced; which of the engine's two codes it carries is not."
      },
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "AZ",
        "stateCertificate": {
          "withholdingPercentage": 0.5
        }
      },
      "source": "Arizona Department of Revenue, Arizona Form A-4 (2026), 'Employee's Arizona Withholding Election', ADOR 10121 (25), https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf (azdor.gov refuses automated requests; read from the Internet Archive capture of 2026-10-08 at http://web.archive.org/web/20261008204653/https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf), retrieved 2026-10-15. Verbatim, box 1: 'Withhold from gross taxable wages at the percentage checked (check only one percentage): 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5%'; box 2: 'I elect an Arizona withholding percentage of zero, and I certify that I expect to have no Arizona tax liability for the current taxable year.' Employee's Instructions, 'New Employees': 'If you do not give this form to your employer the department requires your employer to withhold 2.0% of your gross taxable wages.' And: 'The amount withheld is a percentage of your gross taxable wages from every paycheck.' The statutory frame is A.R.S. s.43-401(A) and (E), https://www.azleg.gov/ars/43/00401.htm, retrieved 2026-10-15: the employer deducts 'an amount prescribed by tables adopted by the department', 'Each employee shall elect the amount authorized by subsection A of this section to be withheld', and 'Any employee failing to complete an election form as prescribed shall be deemed to have elected the withholding percentage prescribed by the department.' THERE IS NO ARIZONA BRACKET TABLE AND NO ARIZONA DEDUCTION IN THIS COMPUTATION: Arizona withholding is a percentage of gross taxable wages chosen by the employee, so an Arizona figure is a function of an election, not of a schedule. Hand derivation: there is none to make - 50.00 per cent is not one of the seven percentages Form A-4 prints, so no official figure exists for it. What the document warrants is the REFUSAL: the form's seven boxes and the instruction 'check only one percentage' are an exhaustive list, and A.R.S. s.43-401(E) makes the election the employee's to make from that list.",
      "why": "Arizona publishes no bracket table, so the only thing that can be wrong about an Arizona figure is the election - which makes an unlisted election the one input that must never be guessed. This case also fixes the units of the field: 0.5 is 50.00 per cent and not 0.5 per cent, and an engine that quietly accepted it would charge $30,000.00 on a $60,000 salary.",
      "refuses": {
        "code": "UNSUPPORTED_STATE_CERTIFICATE",
        "why": "Form A-4 prints the elections as 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 and 3.5 per cent and says 'check only one percentage', so a number that is not one of the seven is not an Arizona election at all. The engine's stateCertificate.withholdingPercentage holds the election as a FRACTION of gross (0.005 for 0.5 per cent), so the value 0.5 here reads as 50.00 per cent and is refused by name rather than charged. The matching case with 0.005 carries the sourced figure; this case asserts only that an unlisted percentage is refused."
      },
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "AZ",
        "stateCertificate": {
          "withholdingPercentage": 2.5
        }
      },
      "source": "Arizona Department of Revenue, Arizona Form A-4 (2026), 'Employee's Arizona Withholding Election', ADOR 10121 (25), https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf (azdor.gov refuses automated requests; read from the Internet Archive capture of 2026-10-08 at http://web.archive.org/web/20261008204653/https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf), retrieved 2026-10-15. Verbatim, box 1: 'Withhold from gross taxable wages at the percentage checked (check only one percentage): 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5%'; box 2: 'I elect an Arizona withholding percentage of zero, and I certify that I expect to have no Arizona tax liability for the current taxable year.' Employee's Instructions, 'New Employees': 'If you do not give this form to your employer the department requires your employer to withhold 2.0% of your gross taxable wages.' And: 'The amount withheld is a percentage of your gross taxable wages from every paycheck.' The statutory frame is A.R.S. s.43-401(A) and (E), https://www.azleg.gov/ars/43/00401.htm, retrieved 2026-10-15: the employer deducts 'an amount prescribed by tables adopted by the department', 'Each employee shall elect the amount authorized by subsection A of this section to be withheld', and 'Any employee failing to complete an election form as prescribed shall be deemed to have elected the withholding percentage prescribed by the department.' THERE IS NO ARIZONA BRACKET TABLE AND NO ARIZONA DEDUCTION IN THIS COMPUTATION: Arizona withholding is a percentage of gross taxable wages chosen by the employee, so an Arizona figure is a function of an election, not of a schedule. Hand derivation: there is none to make - 250.00 per cent is not one of the seven percentages Form A-4 prints, so no official figure exists for it. What the document warrants is the REFUSAL: the form's seven boxes and the instruction 'check only one percentage' are an exhaustive list, and A.R.S. s.43-401(E) makes the election the employee's to make from that list.",
      "why": "Arizona publishes no bracket table, so the only thing that can be wrong about an Arizona figure is the election - which makes an unlisted election the one input that must never be guessed. This case also fixes the units of the field: 2.5 is 250.00 per cent and not 2.5 per cent, and an engine that quietly accepted it would charge $150,000.00 on a $60,000 salary.",
      "refuses": {
        "code": "INVALID_AMOUNT",
        "why": "Form A-4 prints the elections as 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 and 3.5 per cent and says 'check only one percentage', so a number that is not one of the seven is not an Arizona election at all. The engine's stateCertificate.withholdingPercentage holds the election as a FRACTION of gross (0.025 for 2.5 per cent), so the value 2.5 here reads as 250.00 per cent and is refused by name rather than charged. The matching case with 0.025 carries the sourced figure; this case asserts only that an unlisted percentage is refused. THE CODE NAME IS THE ENGINE'S AND NOT THE DOCUMENT'S: Form A-4 warrants that an unlisted percentage must be refused, and nothing more. On the first run of this vector the engine refused 2.5 with INVALID_AMOUNT (a fraction above 1) rather than with UNSUPPORTED_STATE_CERTIFICATE, which it uses for a value inside 0 to 1 that is not one of the seven; the code here was taken from that run and is recorded as such. What this case pins is the refusal, which is sourced; which of the engine's two codes it carries is not."
      },
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "AZ",
        "stateCertificate": {
          "withholdingPercentage": 0
        }
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Arizona Department of Revenue, Arizona Form A-4 (2026), 'Employee's Arizona Withholding Election', ADOR 10121 (25), https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf (azdor.gov refuses automated requests; read from the Internet Archive capture of 2026-10-08 at http://web.archive.org/web/20261008204653/https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf), retrieved 2026-10-15. Verbatim, box 1: 'Withhold from gross taxable wages at the percentage checked (check only one percentage): 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5%'; box 2: 'I elect an Arizona withholding percentage of zero, and I certify that I expect to have no Arizona tax liability for the current taxable year.' Employee's Instructions, 'New Employees': 'If you do not give this form to your employer the department requires your employer to withhold 2.0% of your gross taxable wages.' And: 'The amount withheld is a percentage of your gross taxable wages from every paycheck.' The statutory frame is A.R.S. s.43-401(A) and (E), https://www.azleg.gov/ars/43/00401.htm, retrieved 2026-10-15: the employer deducts 'an amount prescribed by tables adopted by the department', 'Each employee shall elect the amount authorized by subsection A of this section to be withheld', and 'Any employee failing to complete an election form as prescribed shall be deemed to have elected the withholding percentage prescribed by the department.' THERE IS NO ARIZONA BRACKET TABLE AND NO ARIZONA DEDUCTION IN THIS COMPUTATION: Arizona withholding is a percentage of gross taxable wages chosen by the employee, so an Arizona figure is a function of an election, not of a schedule. Hand derivation: box 2: 'I elect an Arizona withholding percentage of zero'. 0.000 x $60,000.00 = $0.00.",
      "why": "the zero election is a real, printed option on Form A-4, distinct from having filed nothing - which gives 2.0%. An engine that treated a zero election as 'no election' would charge $1,200.00.",
      "ledger": "L-2026-4372",
      "refuses": {
        "code": "UNSUPPORTED_STATE_CERTIFICATE",
        "why": "The zero election is real and printed — Form A-4 (2026) box 2 reads 'I elect an Arizona withholding percentage of zero' — and the rules file's electionPercentages carries seven values from 0.5% to 3.5% and not zero, so the engine refuses it. That is the right behaviour for a figure the rules cannot source, and the refusal's words were corrected on 16 October 2026 to blame the rules rather than the form. Zero is NOT simply an eighth percentage: Form A-4 conditions it on the employee expecting no Arizona income tax liability for the year, which is a fact this calculator is never given. So adding 0 to the list would let the page print nil Arizona tax for somebody who owes some. The expect of $0.00 stands as what the form gives an eligible electing employee, and the refusal stands until the rules carry the election WITH its condition (docs/NUMBERS-BACKLOG.md)."
      }
    },
    {
      "kind": "annual",
      "input": {
        "gross": 0,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "AZ"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Arizona Department of Revenue, Arizona Form A-4 (2026), 'Employee's Arizona Withholding Election', ADOR 10121 (25), https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf (azdor.gov refuses automated requests; read from the Internet Archive capture of 2026-10-08 at http://web.archive.org/web/20261008204653/https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf), retrieved 2026-10-15. Verbatim, box 1: 'Withhold from gross taxable wages at the percentage checked (check only one percentage): 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5%'; box 2: 'I elect an Arizona withholding percentage of zero, and I certify that I expect to have no Arizona tax liability for the current taxable year.' Employee's Instructions, 'New Employees': 'If you do not give this form to your employer the department requires your employer to withhold 2.0% of your gross taxable wages.' And: 'The amount withheld is a percentage of your gross taxable wages from every paycheck.' The statutory frame is A.R.S. s.43-401(A) and (E), https://www.azleg.gov/ars/43/00401.htm, retrieved 2026-10-15: the employer deducts 'an amount prescribed by tables adopted by the department', 'Each employee shall elect the amount authorized by subsection A of this section to be withheld', and 'Any employee failing to complete an election form as prescribed shall be deemed to have elected the withholding percentage prescribed by the department.' THERE IS NO ARIZONA BRACKET TABLE AND NO ARIZONA DEDUCTION IN THIS COMPUTATION: Arizona withholding is a percentage of gross taxable wages chosen by the employee, so an Arizona figure is a function of an election, not of a schedule. Hand derivation: no A-4, so 2.0%; 0.020 x $0.00 = $0.00.",
      "why": "nil wages. Arizona's percentage has no floor and no threshold, so nil gross is the only nil answer.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "VA",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "expect": {
        "stateIncomeTax": 2635.9
      },
      "source": "Virginia Department of Taxation, 'Withholding Tables for Wages Paid after July 1, 2025' (the current table set; the Department publishes no later one and the increase it carries is 'scheduled to sunset after Taxable Year 2026'), https://www.tax.virginia.gov/sites/default/files/taxforms/withholding/any/employer-withholding-tables-july-2025-and-later-any.pdf, retrieved 2026-10-15. Verbatim, page 1, 'Formula for Computing Tax to be Withheld', 'Effective for wages paid after July 1, 2025 (for Taxable Year 2025 and after filers)': '1. (G)P - [$8,750 + (E1 X $930) + (E2 X 800)] = T   2. If T is: W is: Not over $3,000  2% of T; Over $3,000 But not Over $5,000  $60 + (3% of excess over $3,000); $5,000 / $17,000  $120 + (5% of excess over $5,000); $17,000 /  $720 + (5.75% of excess over $17,000)   3. W / P = W/H', with the legend 'G = Gross pay for pay period, P = Number of pay periods per year, E1 = Personal and Dependent Exemptions, E2 = Age 65 and Over & Blind Exemptions, T = Annualized taxable income, W = Annualized tax to be withheld, W/H = Tax to be withheld for pay period' and the Pay Period Conversion Table 'Annual = 1, Monthly = 12, Semi-monthly = 24, Bi-weekly = 26, Weekly = 52'. Hand derivation at the ANNUAL payroll period (P = 1, so W/H = W): T = $60,000.00 - [$8,750 + (1 x $930) + (0 x $800)] = $60,000.00 - $9,680.00 = $50,320.00. T is over $17,000, so W = $720 + 5.75% of ($50,320.00 - $17,000.00) = $720.00 + (0.0575 x $33,320.00) = $720.00 + $1,915.90 = $2,635.90.",
      "why": "the whole Virginia formula at one exemption: the $8,750 standard deduction, the $930 exemption and the 5.75% top rate. Every figure is exact to the penny, so nothing here rests on a rounding rule.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60010,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "VA",
        "stateCertificate": {
          "allowances": 2
        }
      },
      "expect": {
        "stateIncomeTax": 2583
      },
      "source": "Virginia Department of Taxation, 'Withholding Tables for Wages Paid after July 1, 2025' (the current table set; the Department publishes no later one and the increase it carries is 'scheduled to sunset after Taxable Year 2026'), https://www.tax.virginia.gov/sites/default/files/taxforms/withholding/any/employer-withholding-tables-july-2025-and-later-any.pdf, retrieved 2026-10-15. Verbatim, page 1, 'Formula for Computing Tax to be Withheld', 'Effective for wages paid after July 1, 2025 (for Taxable Year 2025 and after filers)': '1. (G)P - [$8,750 + (E1 X $930) + (E2 X 800)] = T   2. If T is: W is: Not over $3,000  2% of T; Over $3,000 But not Over $5,000  $60 + (3% of excess over $3,000); $5,000 / $17,000  $120 + (5% of excess over $5,000); $17,000 /  $720 + (5.75% of excess over $17,000)   3. W / P = W/H', with the legend 'G = Gross pay for pay period, P = Number of pay periods per year, E1 = Personal and Dependent Exemptions, E2 = Age 65 and Over & Blind Exemptions, T = Annualized taxable income, W = Annualized tax to be withheld, W/H = Tax to be withheld for pay period' and the Pay Period Conversion Table 'Annual = 1, Monthly = 12, Semi-monthly = 24, Bi-weekly = 26, Weekly = 52'. Hand derivation at the ANNUAL payroll period (P = 1, so W/H = W): T = $60,010.00 - [$8,750 + (2 x $930)] = $60,010.00 - $10,610.00 = $49,400.00; W = $720.00 + (0.0575 x $32,400.00) = $720.00 + $1,863.00 = $2,583.00.",
      "why": "two exemptions on Form VA-4, at $930 each. The gross is chosen so the 5.75% step lands on an exact penny.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60010,
        "filingStatus": "married-filing-jointly",
        "taxYear": "2026",
        "state": "VA",
        "stateCertificate": {
          "allowances": 2
        }
      },
      "expect": {
        "stateIncomeTax": 2583
      },
      "source": "Virginia Department of Taxation, 'Withholding Tables for Wages Paid after July 1, 2025' (the current table set; the Department publishes no later one and the increase it carries is 'scheduled to sunset after Taxable Year 2026'), https://www.tax.virginia.gov/sites/default/files/taxforms/withholding/any/employer-withholding-tables-july-2025-and-later-any.pdf, retrieved 2026-10-15. Verbatim, page 1, 'Formula for Computing Tax to be Withheld', 'Effective for wages paid after July 1, 2025 (for Taxable Year 2025 and after filers)': '1. (G)P - [$8,750 + (E1 X $930) + (E2 X 800)] = T   2. If T is: W is: Not over $3,000  2% of T; Over $3,000 But not Over $5,000  $60 + (3% of excess over $3,000); $5,000 / $17,000  $120 + (5% of excess over $5,000); $17,000 /  $720 + (5.75% of excess over $17,000)   3. W / P = W/H', with the legend 'G = Gross pay for pay period, P = Number of pay periods per year, E1 = Personal and Dependent Exemptions, E2 = Age 65 and Over & Blind Exemptions, T = Annualized taxable income, W = Annualized tax to be withheld, W/H = Tax to be withheld for pay period' and the Pay Period Conversion Table 'Annual = 1, Monthly = 12, Semi-monthly = 24, Bi-weekly = 26, Weekly = 52'. Hand derivation at the ANNUAL payroll period (P = 1, so W/H = W): the formula has NO filing-status term - only E1 and E2 - so T = $60,010.00 - $10,610.00 = $49,400.00 and W = $2,583.00, exactly as for a single filer claiming the same two exemptions. On Form VA-4 a spouse is an exemption on line 1, not a filing status.",
      "why": "Virginia prices a couple through the exemption count, not through a joint band or a doubled standard deduction. The $17,500 married standard deduction the table's own preamble mentions is an income-tax-return figure and does NOT appear in the withholding formula, which uses $8,750 for everyone.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "VA",
        "stateCertificate": {
          "allowances": 1,
          "ageOrBlindExemptions": 1
        }
      },
      "expect": {
        "stateIncomeTax": 2589.9
      },
      "source": "Virginia Department of Taxation, 'Withholding Tables for Wages Paid after July 1, 2025' (the current table set; the Department publishes no later one and the increase it carries is 'scheduled to sunset after Taxable Year 2026'), https://www.tax.virginia.gov/sites/default/files/taxforms/withholding/any/employer-withholding-tables-july-2025-and-later-any.pdf, retrieved 2026-10-15. Verbatim, page 1, 'Formula for Computing Tax to be Withheld', 'Effective for wages paid after July 1, 2025 (for Taxable Year 2025 and after filers)': '1. (G)P - [$8,750 + (E1 X $930) + (E2 X 800)] = T   2. If T is: W is: Not over $3,000  2% of T; Over $3,000 But not Over $5,000  $60 + (3% of excess over $3,000); $5,000 / $17,000  $120 + (5% of excess over $5,000); $17,000 /  $720 + (5.75% of excess over $17,000)   3. W / P = W/H', with the legend 'G = Gross pay for pay period, P = Number of pay periods per year, E1 = Personal and Dependent Exemptions, E2 = Age 65 and Over & Blind Exemptions, T = Annualized taxable income, W = Annualized tax to be withheld, W/H = Tax to be withheld for pay period' and the Pay Period Conversion Table 'Annual = 1, Monthly = 12, Semi-monthly = 24, Bi-weekly = 26, Weekly = 52'. Hand derivation at the ANNUAL payroll period (P = 1, so W/H = W): T = $60,000.00 - [$8,750 + (1 x $930) + (1 x $800)] = $60,000.00 - $10,480.00 = $49,520.00; W = $720.00 + (0.0575 x $32,520.00) = $720.00 + $1,869.90 = $2,589.90.",
      "why": "the VA-4 line E2 exemption is worth $800, not $930. Valuing it at $930 would understate the tax by $7.48; omitting it would overstate it by $46.00.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 12680,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "VA",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "expect": {
        "stateIncomeTax": 60
      },
      "source": "Virginia Department of Taxation, 'Withholding Tables for Wages Paid after July 1, 2025' (the current table set; the Department publishes no later one and the increase it carries is 'scheduled to sunset after Taxable Year 2026'), https://www.tax.virginia.gov/sites/default/files/taxforms/withholding/any/employer-withholding-tables-july-2025-and-later-any.pdf, retrieved 2026-10-15. Verbatim, page 1, 'Formula for Computing Tax to be Withheld', 'Effective for wages paid after July 1, 2025 (for Taxable Year 2025 and after filers)': '1. (G)P - [$8,750 + (E1 X $930) + (E2 X 800)] = T   2. If T is: W is: Not over $3,000  2% of T; Over $3,000 But not Over $5,000  $60 + (3% of excess over $3,000); $5,000 / $17,000  $120 + (5% of excess over $5,000); $17,000 /  $720 + (5.75% of excess over $17,000)   3. W / P = W/H', with the legend 'G = Gross pay for pay period, P = Number of pay periods per year, E1 = Personal and Dependent Exemptions, E2 = Age 65 and Over & Blind Exemptions, T = Annualized taxable income, W = Annualized tax to be withheld, W/H = Tax to be withheld for pay period' and the Pay Period Conversion Table 'Annual = 1, Monthly = 12, Semi-monthly = 24, Bi-weekly = 26, Weekly = 52'. Hand derivation at the ANNUAL payroll period (P = 1, so W/H = W): T = $12,680.00 - [$8,750 + $930] = $3,000.00, which is 'Not over $3,000', so W = 2% of T = 0.02 x $3,000.00 = $60.00. (The next line gives $60 + 3% of the excess over $3,000, which at T = $3,000 is also $60.00, so the schedule is continuous here.)",
      "why": "the first band edge, exactly at T = $3,000, where both branches must agree.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 12681,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "VA",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "expect": {
        "stateIncomeTax": 60.03
      },
      "source": "Virginia Department of Taxation, 'Withholding Tables for Wages Paid after July 1, 2025' (the current table set; the Department publishes no later one and the increase it carries is 'scheduled to sunset after Taxable Year 2026'), https://www.tax.virginia.gov/sites/default/files/taxforms/withholding/any/employer-withholding-tables-july-2025-and-later-any.pdf, retrieved 2026-10-15. Verbatim, page 1, 'Formula for Computing Tax to be Withheld', 'Effective for wages paid after July 1, 2025 (for Taxable Year 2025 and after filers)': '1. (G)P - [$8,750 + (E1 X $930) + (E2 X 800)] = T   2. If T is: W is: Not over $3,000  2% of T; Over $3,000 But not Over $5,000  $60 + (3% of excess over $3,000); $5,000 / $17,000  $120 + (5% of excess over $5,000); $17,000 /  $720 + (5.75% of excess over $17,000)   3. W / P = W/H', with the legend 'G = Gross pay for pay period, P = Number of pay periods per year, E1 = Personal and Dependent Exemptions, E2 = Age 65 and Over & Blind Exemptions, T = Annualized taxable income, W = Annualized tax to be withheld, W/H = Tax to be withheld for pay period' and the Pay Period Conversion Table 'Annual = 1, Monthly = 12, Semi-monthly = 24, Bi-weekly = 26, Weekly = 52'. Hand derivation at the ANNUAL payroll period (P = 1, so W/H = W): T = $12,681.00 - $9,680.00 = $3,001.00, which is 'Over $3,000 But not Over $5,000', so W = $60 + (3% of $1.00) = $60.00 + $0.03 = $60.03.",
      "why": "one dollar past the first edge. Catches an engine that charges 2% on the whole of T once T passes $3,000, which would read $60.02.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 25750,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "VA"
      },
      "expect": {
        "stateIncomeTax": 720
      },
      "source": "Virginia Department of Taxation, 'Withholding Tables for Wages Paid after July 1, 2025' (the current table set; the Department publishes no later one and the increase it carries is 'scheduled to sunset after Taxable Year 2026'), https://www.tax.virginia.gov/sites/default/files/taxforms/withholding/any/employer-withholding-tables-july-2025-and-later-any.pdf, retrieved 2026-10-15. Verbatim, page 1, 'Formula for Computing Tax to be Withheld', 'Effective for wages paid after July 1, 2025 (for Taxable Year 2025 and after filers)': '1. (G)P - [$8,750 + (E1 X $930) + (E2 X 800)] = T   2. If T is: W is: Not over $3,000  2% of T; Over $3,000 But not Over $5,000  $60 + (3% of excess over $3,000); $5,000 / $17,000  $120 + (5% of excess over $5,000); $17,000 /  $720 + (5.75% of excess over $17,000)   3. W / P = W/H', with the legend 'G = Gross pay for pay period, P = Number of pay periods per year, E1 = Personal and Dependent Exemptions, E2 = Age 65 and Over & Blind Exemptions, T = Annualized taxable income, W = Annualized tax to be withheld, W/H = Tax to be withheld for pay period' and the Pay Period Conversion Table 'Annual = 1, Monthly = 12, Semi-monthly = 24, Bi-weekly = 26, Weekly = 52'. Hand derivation at the ANNUAL payroll period (P = 1, so W/H = W): no exemptions claimed: T = $25,750.00 - [$8,750 + (0 x $930)] = $17,000.00, which is the top of the '$5,000 / $17,000' line, so W = $120 + (5% of ($17,000.00 - $5,000.00)) = $120.00 + $600.00 = $720.00. The next line gives $720 + 5.75% of the excess over $17,000, which at T = $17,000 is also $720.00.",
      "why": "the $17,000 edge, and at the same time the fact that Virginia's $8,750 is NOT conditional on claiming an exemption - unlike South Carolina's standard deduction, which vanishes at zero allowances. Dropping the $8,750 here would put T at $25,750 and W at $1,221.25.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 25850,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "VA"
      },
      "expect": {
        "stateIncomeTax": 725.75
      },
      "source": "Virginia Department of Taxation, 'Withholding Tables for Wages Paid after July 1, 2025' (the current table set; the Department publishes no later one and the increase it carries is 'scheduled to sunset after Taxable Year 2026'), https://www.tax.virginia.gov/sites/default/files/taxforms/withholding/any/employer-withholding-tables-july-2025-and-later-any.pdf, retrieved 2026-10-15. Verbatim, page 1, 'Formula for Computing Tax to be Withheld', 'Effective for wages paid after July 1, 2025 (for Taxable Year 2025 and after filers)': '1. (G)P - [$8,750 + (E1 X $930) + (E2 X 800)] = T   2. If T is: W is: Not over $3,000  2% of T; Over $3,000 But not Over $5,000  $60 + (3% of excess over $3,000); $5,000 / $17,000  $120 + (5% of excess over $5,000); $17,000 /  $720 + (5.75% of excess over $17,000)   3. W / P = W/H', with the legend 'G = Gross pay for pay period, P = Number of pay periods per year, E1 = Personal and Dependent Exemptions, E2 = Age 65 and Over & Blind Exemptions, T = Annualized taxable income, W = Annualized tax to be withheld, W/H = Tax to be withheld for pay period' and the Pay Period Conversion Table 'Annual = 1, Monthly = 12, Semi-monthly = 24, Bi-weekly = 26, Weekly = 52'. Hand derivation at the ANNUAL payroll period (P = 1, so W/H = W): T = $25,850.00 - $8,750.00 = $17,100.00; W = $720 + (5.75% of $100.00) = $720.00 + $5.75 = $725.75.",
      "why": "$100 past the top edge, where the 5.75% rate first bites.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 0,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "VA"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Virginia Department of Taxation, 'Withholding Tables for Wages Paid after July 1, 2025' (the current table set; the Department publishes no later one and the increase it carries is 'scheduled to sunset after Taxable Year 2026'), https://www.tax.virginia.gov/sites/default/files/taxforms/withholding/any/employer-withholding-tables-july-2025-and-later-any.pdf, retrieved 2026-10-15. Verbatim, page 1, 'Formula for Computing Tax to be Withheld', 'Effective for wages paid after July 1, 2025 (for Taxable Year 2025 and after filers)': '1. (G)P - [$8,750 + (E1 X $930) + (E2 X 800)] = T   2. If T is: W is: Not over $3,000  2% of T; Over $3,000 But not Over $5,000  $60 + (3% of excess over $3,000); $5,000 / $17,000  $120 + (5% of excess over $5,000); $17,000 /  $720 + (5.75% of excess over $17,000)   3. W / P = W/H', with the legend 'G = Gross pay for pay period, P = Number of pay periods per year, E1 = Personal and Dependent Exemptions, E2 = Age 65 and Over & Blind Exemptions, T = Annualized taxable income, W = Annualized tax to be withheld, W/H = Tax to be withheld for pay period' and the Pay Period Conversion Table 'Annual = 1, Monthly = 12, Semi-monthly = 24, Bi-weekly = 26, Weekly = 52'. Hand derivation at the ANNUAL payroll period (P = 1, so W/H = W): T = $0.00 - $8,750.00 is negative. The Department's own Daily Payroll Period table, page 2 of the same document, prints '0.00' in every exemption column for the row 'At Least $0, But Less Than 28', so the withholding is nil and never negative.",
      "why": "nil wages. The formula alone has no floor; the printed table supplies it.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 63576,
        "filingStatus": "single",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "VA",
        "stateCertificate": {
          "allowances": 5
        }
      },
      "period": "semimonthly",
      "expect": {
        "stateWithholding": 109.5
      },
      "source": "Virginia Department of Taxation, 'Withholding Tables for Wages Paid after July 1, 2025' (the current table set; the Department publishes no later one and the increase it carries is 'scheduled to sunset after Taxable Year 2026'), https://www.tax.virginia.gov/sites/default/files/taxforms/withholding/any/employer-withholding-tables-july-2025-and-later-any.pdf, retrieved 2026-10-15. Verbatim, page 1, 'Formula for Computing Tax to be Withheld', 'Effective for wages paid after July 1, 2025 (for Taxable Year 2025 and after filers)': '1. (G)P - [$8,750 + (E1 X $930) + (E2 X 800)] = T   2. If T is: W is: Not over $3,000  2% of T; Over $3,000 But not Over $5,000  $60 + (3% of excess over $3,000); $5,000 / $17,000  $120 + (5% of excess over $5,000); $17,000 /  $720 + (5.75% of excess over $17,000)   3. W / P = W/H', with the legend 'G = Gross pay for pay period, P = Number of pay periods per year, E1 = Personal and Dependent Exemptions, E2 = Age 65 and Over & Blind Exemptions, T = Annualized taxable income, W = Annualized tax to be withheld, W/H = Tax to be withheld for pay period' and the Pay Period Conversion Table 'Annual = 1, Monthly = 12, Semi-monthly = 24, Bi-weekly = 26, Weekly = 52'. This is the Department's OWN WORKED EXAMPLE, page 2, verbatim: 'John claims exemptions for himself, his spouse, and their three children for withholding tax purposes. He is paid on a semi-monthly basis, and his gross wages for this pay period were $2,649. 1. (G)P - [$8,750 + ($930)E1] = T ($2,649)24 - [$8,750 + ($930)5] = T  $63,576 - $13,400 = $50,176.  2. T is over $17,000  $720 + 5.75% of $33,176 = W  $720 + $1,908 = $2,628.  3. W / P = W/H  $2,628 / 24 = $109.50.  The tax to be withheld for the current period is $109.50.' Annual gross $2,649.00 x 24 = $63,576.00. READ THE DEPARTMENT'S THIRD LINE CAREFULLY, because it is the whole point of this case: 5.75% of $33,176 is $1,907.62, and the Department has rounded W to the whole dollar ($1,908, hence $2,628) BEFORE dividing by 24. Carried to the cent, W is $2,627.62 and W/H is $109.4842, i.e. $109.48. $109.50 is what the Commonwealth prints for these inputs and it is what this vector asserts; the two-cent gap is the Department's rounding of W and is recorded here so that no later reader edits the figure to match an implementation.",
      "why": "the only published Virginia money figure there is, and the one case in this batch whose official answer embeds a rounding step the formula text does not state. If the engine returns $109.48 that is a finding about which rounding the engine should follow, to be settled by the Department and not by changing this number.",
      "ledger": "L-2026-4372",
      "disagreement": {
        "field": "stateWithholding",
        "ledger": "L-2026-4374",
        "engineReturns": 109.48,
        "why": "ARBITRATED 16 October 2026, DISAGREEMENT L-2026-4374: BOTH implementations now return $109.48 and the instructions' example prints $109.50. 23VAC10-140-40 'Other methods of withholding', made under Code of Virginia ss. 58.1-203 and 58.1-463, works the same three-step formula and prints 'W = 167.50 ... 167.50/24 = 6.98' — a whole-dollar W would print $168 and $7.00 — so W is carried to the cent and rounded once, at the final division. A regulation beats an unstated rounding in one illustrative example, and 2614086 p.18 says 'To compute the exact amount of tax to be withheld, use the formula on Page 21'. The expect of $109.50 is kept because it is what Virginia printed; a vector is fixed at its source and never edited to match code. RESIDUE: Virginia states no rounding rule in words anywhere, so $109.48 rests on the regulation's example. If the Department prints a rule, it governs and this reverses."
      }
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60070,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "NE",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "expect": {
        "stateIncomeTax": 2127.55
      },
      "source": "Nebraska Department of Revenue, 2026 Nebraska Circular EN, 'Nebraska Income Tax Withholding on Wages, Pensions and Annuities, and Gambling Winnings Paid on or after January 1, 2026', 8-429-1998 Rev. 11-2025, https://revenue.nebraska.gov/sites/default/files/doc/business/Cir_En_2025/2026cir_en_whole.pdf, retrieved 2026-10-15. Verbatim, page 9, 'Percentage Method': 'To use the percentage method, start with the taxable wage amount for the payroll period. Subtract the value of any income tax withholding allowances using the table below.' The 'Income Tax Withholding-Percentage Method Withholding Allowance Table, Effective January 1, 2026' gives 'Value of One Income Tax Withholding Allowance' as Annually $2,440.00, Monthly $203.33, Weekly $46.92. Page 12, 'TABLE 7 - ANNUAL Payroll Period', 'a. SINGLE Person - Including Head of Household': 'Not over $3,430  $0.00'; then 'Over $3,430 but not over $6,710  0.00 plus 2.26% of excess over 3,430; $6,710 / $21,810  74.13 plus 3.22% over 6,710; $21,810 / $31,610  560.35 plus 4.21% over 21,810; $31,610 / $40,130  972.93 plus 4.35% over 31,610; $40,130 / $75,370  1,343.55 plus 4.48% over 40,130; $75,370 and above  2,922.30 plus 4.60% over 75,370'. 'b. MARRIED Person - Including Surviving Spouse': 'Not over $8,190  $0.00'; ... '$82,920 and above  2,932.22 plus 4.60% of excess over 82,920'. Page 12, 'TABLE 4 - MONTHLY Payroll Period', single: '$3,344 / $6,281  111.95 plus 4.48% of excess over 3,344'. Hand derivation on TABLE 7 (annual payroll period), with the $2,440 annual allowance value: taxable wage $60,070.00 - (1 x $2,440.00) = $57,630.00. Table 7a row '$40,130 / $75,370': $1,343.55 + (4.48% of ($57,630.00 - $40,130.00)) = $1,343.55 + (0.0448 x $17,500.00) = $1,343.55 + $784.00 = $2,127.55.",
      "why": "one Form W-4N allowance at the $2,440 annual value, plus a bracket read with a base amount and a marginal rate. The gross is chosen so 4.48% of the excess is exact to the penny.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60070,
        "filingStatus": "head-of-household",
        "taxYear": "2026",
        "state": "NE",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "expect": {
        "stateIncomeTax": 2127.55
      },
      "source": "Nebraska Department of Revenue, 2026 Nebraska Circular EN, 'Nebraska Income Tax Withholding on Wages, Pensions and Annuities, and Gambling Winnings Paid on or after January 1, 2026', 8-429-1998 Rev. 11-2025, https://revenue.nebraska.gov/sites/default/files/doc/business/Cir_En_2025/2026cir_en_whole.pdf, retrieved 2026-10-15. Verbatim, page 9, 'Percentage Method': 'To use the percentage method, start with the taxable wage amount for the payroll period. Subtract the value of any income tax withholding allowances using the table below.' The 'Income Tax Withholding-Percentage Method Withholding Allowance Table, Effective January 1, 2026' gives 'Value of One Income Tax Withholding Allowance' as Annually $2,440.00, Monthly $203.33, Weekly $46.92. Page 12, 'TABLE 7 - ANNUAL Payroll Period', 'a. SINGLE Person - Including Head of Household': 'Not over $3,430  $0.00'; then 'Over $3,430 but not over $6,710  0.00 plus 2.26% of excess over 3,430; $6,710 / $21,810  74.13 plus 3.22% over 6,710; $21,810 / $31,610  560.35 plus 4.21% over 21,810; $31,610 / $40,130  972.93 plus 4.35% over 31,610; $40,130 / $75,370  1,343.55 plus 4.48% over 40,130; $75,370 and above  2,922.30 plus 4.60% over 75,370'. 'b. MARRIED Person - Including Surviving Spouse': 'Not over $8,190  $0.00'; ... '$82,920 and above  2,932.22 plus 4.60% of excess over 82,920'. Page 12, 'TABLE 4 - MONTHLY Payroll Period', single: '$3,344 / $6,281  111.95 plus 4.48% of excess over 3,344'. Hand derivation on TABLE 7 (annual payroll period), with the $2,440 annual allowance value: the same reading, because Table 7a is headed 'a. SINGLE Person - Including Head of Household': $60,070.00 - $2,440.00 = $57,630.00; $1,343.55 + (0.0448 x $17,500.00) = $2,127.55.",
      "why": "Nebraska has no head-of-household table: the Circular prints two, single (including head of household) and married (including surviving spouse). An engine that sent head of household to the married table would read $2,064.17 here.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 100000,
        "filingStatus": "married-filing-jointly",
        "taxYear": "2026",
        "state": "NE",
        "stateCertificate": {
          "allowances": 2
        }
      },
      "expect": {
        "stateIncomeTax": 3493.42
      },
      "source": "Nebraska Department of Revenue, 2026 Nebraska Circular EN, 'Nebraska Income Tax Withholding on Wages, Pensions and Annuities, and Gambling Winnings Paid on or after January 1, 2026', 8-429-1998 Rev. 11-2025, https://revenue.nebraska.gov/sites/default/files/doc/business/Cir_En_2025/2026cir_en_whole.pdf, retrieved 2026-10-15. Verbatim, page 9, 'Percentage Method': 'To use the percentage method, start with the taxable wage amount for the payroll period. Subtract the value of any income tax withholding allowances using the table below.' The 'Income Tax Withholding-Percentage Method Withholding Allowance Table, Effective January 1, 2026' gives 'Value of One Income Tax Withholding Allowance' as Annually $2,440.00, Monthly $203.33, Weekly $46.92. Page 12, 'TABLE 7 - ANNUAL Payroll Period', 'a. SINGLE Person - Including Head of Household': 'Not over $3,430  $0.00'; then 'Over $3,430 but not over $6,710  0.00 plus 2.26% of excess over 3,430; $6,710 / $21,810  74.13 plus 3.22% over 6,710; $21,810 / $31,610  560.35 plus 4.21% over 21,810; $31,610 / $40,130  972.93 plus 4.35% over 31,610; $40,130 / $75,370  1,343.55 plus 4.48% over 40,130; $75,370 and above  2,922.30 plus 4.60% over 75,370'. 'b. MARRIED Person - Including Surviving Spouse': 'Not over $8,190  $0.00'; ... '$82,920 and above  2,932.22 plus 4.60% of excess over 82,920'. Page 12, 'TABLE 4 - MONTHLY Payroll Period', single: '$3,344 / $6,281  111.95 plus 4.48% of excess over 3,344'. Hand derivation on TABLE 7 (annual payroll period), with the $2,440 annual allowance value: taxable wage $100,000.00 - (2 x $2,440.00) = $95,120.00. Table 7b, '$82,920 and above': $2,932.22 + (4.60% of ($95,120.00 - $82,920.00)) = $2,932.22 + (0.046 x $12,200.00) = $2,932.22 + $561.20 = $3,493.42.",
      "why": "the married table and the top 4.60% band together, with two allowances. Reading this on the single table would give $3,480.97.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 3430,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "NE"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Nebraska Department of Revenue, 2026 Nebraska Circular EN, 'Nebraska Income Tax Withholding on Wages, Pensions and Annuities, and Gambling Winnings Paid on or after January 1, 2026', 8-429-1998 Rev. 11-2025, https://revenue.nebraska.gov/sites/default/files/doc/business/Cir_En_2025/2026cir_en_whole.pdf, retrieved 2026-10-15. Verbatim, page 9, 'Percentage Method': 'To use the percentage method, start with the taxable wage amount for the payroll period. Subtract the value of any income tax withholding allowances using the table below.' The 'Income Tax Withholding-Percentage Method Withholding Allowance Table, Effective January 1, 2026' gives 'Value of One Income Tax Withholding Allowance' as Annually $2,440.00, Monthly $203.33, Weekly $46.92. Page 12, 'TABLE 7 - ANNUAL Payroll Period', 'a. SINGLE Person - Including Head of Household': 'Not over $3,430  $0.00'; then 'Over $3,430 but not over $6,710  0.00 plus 2.26% of excess over 3,430; $6,710 / $21,810  74.13 plus 3.22% over 6,710; $21,810 / $31,610  560.35 plus 4.21% over 21,810; $31,610 / $40,130  972.93 plus 4.35% over 31,610; $40,130 / $75,370  1,343.55 plus 4.48% over 40,130; $75,370 and above  2,922.30 plus 4.60% over 75,370'. 'b. MARRIED Person - Including Surviving Spouse': 'Not over $8,190  $0.00'; ... '$82,920 and above  2,932.22 plus 4.60% of excess over 82,920'. Page 12, 'TABLE 4 - MONTHLY Payroll Period', single: '$3,344 / $6,281  111.95 plus 4.48% of excess over 3,344'. Hand derivation on TABLE 7 (annual payroll period), with the $2,440 annual allowance value: no allowances: taxable wage $3,430.00, and Table 7a opens 'Not over $3,430  $0.00'.",
      "why": "the exact edge of Nebraska's nil band, quoted from the table's own first line.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 3530,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "NE"
      },
      "expect": {
        "stateIncomeTax": 2.26
      },
      "source": "Nebraska Department of Revenue, 2026 Nebraska Circular EN, 'Nebraska Income Tax Withholding on Wages, Pensions and Annuities, and Gambling Winnings Paid on or after January 1, 2026', 8-429-1998 Rev. 11-2025, https://revenue.nebraska.gov/sites/default/files/doc/business/Cir_En_2025/2026cir_en_whole.pdf, retrieved 2026-10-15. Verbatim, page 9, 'Percentage Method': 'To use the percentage method, start with the taxable wage amount for the payroll period. Subtract the value of any income tax withholding allowances using the table below.' The 'Income Tax Withholding-Percentage Method Withholding Allowance Table, Effective January 1, 2026' gives 'Value of One Income Tax Withholding Allowance' as Annually $2,440.00, Monthly $203.33, Weekly $46.92. Page 12, 'TABLE 7 - ANNUAL Payroll Period', 'a. SINGLE Person - Including Head of Household': 'Not over $3,430  $0.00'; then 'Over $3,430 but not over $6,710  0.00 plus 2.26% of excess over 3,430; $6,710 / $21,810  74.13 plus 3.22% over 6,710; $21,810 / $31,610  560.35 plus 4.21% over 21,810; $31,610 / $40,130  972.93 plus 4.35% over 31,610; $40,130 / $75,370  1,343.55 plus 4.48% over 40,130; $75,370 and above  2,922.30 plus 4.60% over 75,370'. 'b. MARRIED Person - Including Surviving Spouse': 'Not over $8,190  $0.00'; ... '$82,920 and above  2,932.22 plus 4.60% of excess over 82,920'. Page 12, 'TABLE 4 - MONTHLY Payroll Period', single: '$3,344 / $6,281  111.95 plus 4.48% of excess over 3,344'. Hand derivation on TABLE 7 (annual payroll period), with the $2,440 annual allowance value: taxable wage $3,530.00. Table 7a row '$3,430 / $6,710': $0.00 + (2.26% of ($3,530.00 - $3,430.00)) = 0.0226 x $100.00 = $2.26.",
      "why": "$100 past the nil band, where the 2.26% bottom rate first bites. Catches an engine that applies 2.26% to the whole wage, which would read $79.78.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 0,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "NE"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Nebraska Department of Revenue, 2026 Nebraska Circular EN, 'Nebraska Income Tax Withholding on Wages, Pensions and Annuities, and Gambling Winnings Paid on or after January 1, 2026', 8-429-1998 Rev. 11-2025, https://revenue.nebraska.gov/sites/default/files/doc/business/Cir_En_2025/2026cir_en_whole.pdf, retrieved 2026-10-15. Verbatim, page 9, 'Percentage Method': 'To use the percentage method, start with the taxable wage amount for the payroll period. Subtract the value of any income tax withholding allowances using the table below.' The 'Income Tax Withholding-Percentage Method Withholding Allowance Table, Effective January 1, 2026' gives 'Value of One Income Tax Withholding Allowance' as Annually $2,440.00, Monthly $203.33, Weekly $46.92. Page 12, 'TABLE 7 - ANNUAL Payroll Period', 'a. SINGLE Person - Including Head of Household': 'Not over $3,430  $0.00'; then 'Over $3,430 but not over $6,710  0.00 plus 2.26% of excess over 3,430; $6,710 / $21,810  74.13 plus 3.22% over 6,710; $21,810 / $31,610  560.35 plus 4.21% over 21,810; $31,610 / $40,130  972.93 plus 4.35% over 31,610; $40,130 / $75,370  1,343.55 plus 4.48% over 40,130; $75,370 and above  2,922.30 plus 4.60% over 75,370'. 'b. MARRIED Person - Including Surviving Spouse': 'Not over $8,190  $0.00'; ... '$82,920 and above  2,932.22 plus 4.60% of excess over 82,920'. Page 12, 'TABLE 4 - MONTHLY Payroll Period', single: '$3,344 / $6,281  111.95 plus 4.48% of excess over 3,344'. Hand derivation on TABLE 7 (annual payroll period), with the $2,440 annual allowance value: nil wages fall in 'Not over $3,430  $0.00'.",
      "why": "nil wages, nil Nebraska tax, from the table's own words.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 59328,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "NE"
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 183.63
      },
      "source": "Nebraska Department of Revenue, 2026 Nebraska Circular EN, 'Nebraska Income Tax Withholding on Wages, Pensions and Annuities, and Gambling Winnings Paid on or after January 1, 2026', 8-429-1998 Rev. 11-2025, https://revenue.nebraska.gov/sites/default/files/doc/business/Cir_En_2025/2026cir_en_whole.pdf, retrieved 2026-10-15. Verbatim, page 9, 'Percentage Method': 'To use the percentage method, start with the taxable wage amount for the payroll period. Subtract the value of any income tax withholding allowances using the table below.' The 'Income Tax Withholding-Percentage Method Withholding Allowance Table, Effective January 1, 2026' gives 'Value of One Income Tax Withholding Allowance' as Annually $2,440.00, Monthly $203.33, Weekly $46.92. Page 12, 'TABLE 7 - ANNUAL Payroll Period', 'a. SINGLE Person - Including Head of Household': 'Not over $3,430  $0.00'; then 'Over $3,430 but not over $6,710  0.00 plus 2.26% of excess over 3,430; $6,710 / $21,810  74.13 plus 3.22% over 6,710; $21,810 / $31,610  560.35 plus 4.21% over 21,810; $31,610 / $40,130  972.93 plus 4.35% over 31,610; $40,130 / $75,370  1,343.55 plus 4.48% over 40,130; $75,370 and above  2,922.30 plus 4.60% over 75,370'. 'b. MARRIED Person - Including Surviving Spouse': 'Not over $8,190  $0.00'; ... '$82,920 and above  2,932.22 plus 4.60% of excess over 82,920'. Page 12, 'TABLE 4 - MONTHLY Payroll Period', single: '$3,344 / $6,281  111.95 plus 4.48% of excess over 3,344'. Hand derivation on TABLE 4 (monthly payroll period), no allowances claimed: monthly wage $59,328.00 / 12 = $4,944.00; Table 4a row '$3,344 / $6,281': $111.95 + (4.48% of ($4,944.00 - $3,344.00)) = $111.95 + (0.0448 x $1,600.00) = $111.95 + $71.68 = $183.63.",
      "why": "Nebraska publishes a SEPARATE bracket table per payroll period rather than annualising, so the monthly answer is Table 4 and not Table 7 divided by twelve. Table 7 at this gross gives $1,343.55 + (0.0448 x $19,198.00) = $2,203.62, and $2,203.62 / 12 = $183.635 = $183.64 - a penny more than Table 4 gives, because the monthly brackets are the annual ones divided by twelve and then rounded to whole dollars. A penny is the smallest currency unit, so it is a real discrimination and the reason the period table has to be read.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 54600,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "IA",
        "stateCertificate": {
          "exemptionAmount": 40
        }
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 59.26
      },
      "source": "Iowa Department of Revenue, 'Iowa Individual Income Tax Withholding Formula, Effective January 1, 2026', released November 2025, https://revenue.iowa.gov/media/53/download?inline (linked from https://revenue.iowa.gov/withholding-tax-information), retrieved 2026-10-15. Verbatim: 'Step 1. Determine T1. T1 = G - D (G: Gross taxable wages for the pay period; D: Deduction for the pay period)'; 'Step 2. Determine T2. T2 = T1 x 3.80% (or 0.038)'; 'Step 3. Determine T3. T3 = T2 - (W / P) (W: Total allowance amount on the IA W-4; P: Number of pay periods per year)'; 'Step 4. Determine T4, or Withholding Amount per Pay Period. T4 = T3 + A'. The 'Deduction Amounts by Pay Period (Denoted as D) for 2024 or a later year's IA W-4' table gives, for the ANNUAL payroll period: column (A) 'Other, or Married Filing Jointly with Spouse Having Earned Income' $13,000.00; column (B) 'Head of Household' $19,500.00; column (C) 'Married Filing Jointly with Spouse Having No Earned Income, or Qualified Surviving Spouse' $26,000.00. Column (A) is also used where 'the employee's marital status is missing'. Biweekly those three are $500.00 / $750.00 / $1,000.00 and monthly $1,083.33 / $1,625.00 / $2,166.67; P is 52 weekly, 26 biweekly, 24 semimonthly, 12 monthly, 1 annually. This is the Department's OWN WORKED EXAMPLE 1, page 3, verbatim: 'Example 1: Biweekly wages of $2,100.00, with \"Other\" marital status on the 2026 IA W-4 (Total withholding allowance $40 claimed, no additional amount requested). T1 = G - D  T1 = $2,100.00 - $500.00 = $1,600.00.  T2 = T1 x R  T2 = $1,600.00 x 3.80% = $60.80.  T3 = T2 - W / P  T3 = $60.80 - $40.00 / 26 = $59.26.  T4 = T3 + A  T4 = $59.26 + $0.00 = $59.26.  Iowa tax to be withheld = $59.26.' Annual gross $2,100.00 x 26 = $54,600.00. ($40.00 / 26 = $1.538461..., and $60.80 - $1.538461... = $59.261538... = $59.26.)",
      "why": "the Iowa Department of Revenue's own printed answer to the penny, and the case that fixes the ORDER of the three steps: the allowance is a credit against the tax, taken AFTER the 3.80% multiplication, not a deduction from wages. Treating $40 as a wage deduction would give $59.25.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 54600,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "IA",
        "stateCertificate": {
          "exemptionAmount": 80,
          "spouseEmployed": false
        }
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 38.72
      },
      "source": "Iowa Department of Revenue, 'Iowa Individual Income Tax Withholding Formula, Effective January 1, 2026', released November 2025, https://revenue.iowa.gov/media/53/download?inline (linked from https://revenue.iowa.gov/withholding-tax-information), retrieved 2026-10-15. Verbatim: 'Step 1. Determine T1. T1 = G - D (G: Gross taxable wages for the pay period; D: Deduction for the pay period)'; 'Step 2. Determine T2. T2 = T1 x 3.80% (or 0.038)'; 'Step 3. Determine T3. T3 = T2 - (W / P) (W: Total allowance amount on the IA W-4; P: Number of pay periods per year)'; 'Step 4. Determine T4, or Withholding Amount per Pay Period. T4 = T3 + A'. The 'Deduction Amounts by Pay Period (Denoted as D) for 2024 or a later year's IA W-4' table gives, for the ANNUAL payroll period: column (A) 'Other, or Married Filing Jointly with Spouse Having Earned Income' $13,000.00; column (B) 'Head of Household' $19,500.00; column (C) 'Married Filing Jointly with Spouse Having No Earned Income, or Qualified Surviving Spouse' $26,000.00. Column (A) is also used where 'the employee's marital status is missing'. Biweekly those three are $500.00 / $750.00 / $1,000.00 and monthly $1,083.33 / $1,625.00 / $2,166.67; P is 52 weekly, 26 biweekly, 24 semimonthly, 12 monthly, 1 annually. The Department's WORKED EXAMPLE 2, page 3, verbatim: 'Example 2: Biweekly wages of $2,100.00, with \"Married jointly and spouse having no earned income, or Qualifying surviving spouse\" marital status on 2026 IA W-4 (Total withholding allowance $80 claimed, no additional amount requested). T1 = $2,100.00 - $1,000.00 = $1,100.00.  T2 = $1,100.00 x 3.80% = $41.80.  T3 = $41.80 - $80.00 / 26 = $38.72.  T4 = $38.72 + $0.00 = $38.72.  Iowa tax to be withheld = $38.72.' Annual gross $2,100.00 x 26 = $54,600.00.",
      "why": "a second published Iowa figure, this time on column (C): the biweekly deduction doubles to $1,000 when the spouse has no earned income. It is the case that shows why a joint filer who does not answer the spouse question cannot be priced - column (A) would give $59.22 here, $20.50 more.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60000,
        "filingStatus": "head-of-household",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "IA",
        "stateCertificate": {
          "exemptionAmount": 160
        }
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 114.92
      },
      "source": "Iowa Department of Revenue, 'Iowa Individual Income Tax Withholding Formula, Effective January 1, 2026', released November 2025, https://revenue.iowa.gov/media/53/download?inline (linked from https://revenue.iowa.gov/withholding-tax-information), retrieved 2026-10-15. Verbatim: 'Step 1. Determine T1. T1 = G - D (G: Gross taxable wages for the pay period; D: Deduction for the pay period)'; 'Step 2. Determine T2. T2 = T1 x 3.80% (or 0.038)'; 'Step 3. Determine T3. T3 = T2 - (W / P) (W: Total allowance amount on the IA W-4; P: Number of pay periods per year)'; 'Step 4. Determine T4, or Withholding Amount per Pay Period. T4 = T3 + A'. The 'Deduction Amounts by Pay Period (Denoted as D) for 2024 or a later year's IA W-4' table gives, for the ANNUAL payroll period: column (A) 'Other, or Married Filing Jointly with Spouse Having Earned Income' $13,000.00; column (B) 'Head of Household' $19,500.00; column (C) 'Married Filing Jointly with Spouse Having No Earned Income, or Qualified Surviving Spouse' $26,000.00. Column (A) is also used where 'the employee's marital status is missing'. Biweekly those three are $500.00 / $750.00 / $1,000.00 and monthly $1,083.33 / $1,625.00 / $2,166.67; P is 52 weekly, 26 biweekly, 24 semimonthly, 12 monthly, 1 annually. The Department's WORKED EXAMPLE 6, page 5, verbatim: 'Example 6: Monthly wages of $5,000.00, with \"Head of Household\" marital status on 2026 IA W-4 (Total withholding allowance $160 claimed, no additional amount requested). T1 = $5,000.00 - $1,625.00 = $3,375.00.  T2 = $3,375.00 x 3.80% = $128.25.  T3 = $128.25 - $160.00 / 12 = $114.92.  T4 = $114.92 + $0.00 = $114.92.  Iowa tax to be withheld = $114.92.' Annual gross $5,000.00 x 12 = $60,000.00. ($160.00 / 12 = $13.3333..., and $128.25 - $13.3333... = $114.9166... = $114.92.)",
      "why": "the head-of-household column (B), monthly, with a third allowance amount. Iowa is the only state in this batch with three deduction columns, and this is the middle one.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "IA",
        "stateCertificate": {
          "exemptionAmount": 80,
          "spouseEmployed": false
        }
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 101
      },
      "source": "Iowa Department of Revenue, 'Iowa Individual Income Tax Withholding Formula, Effective January 1, 2026', released November 2025, https://revenue.iowa.gov/media/53/download?inline (linked from https://revenue.iowa.gov/withholding-tax-information), retrieved 2026-10-15. Verbatim: 'Step 1. Determine T1. T1 = G - D (G: Gross taxable wages for the pay period; D: Deduction for the pay period)'; 'Step 2. Determine T2. T2 = T1 x 3.80% (or 0.038)'; 'Step 3. Determine T3. T3 = T2 - (W / P) (W: Total allowance amount on the IA W-4; P: Number of pay periods per year)'; 'Step 4. Determine T4, or Withholding Amount per Pay Period. T4 = T3 + A'. The 'Deduction Amounts by Pay Period (Denoted as D) for 2024 or a later year's IA W-4' table gives, for the ANNUAL payroll period: column (A) 'Other, or Married Filing Jointly with Spouse Having Earned Income' $13,000.00; column (B) 'Head of Household' $19,500.00; column (C) 'Married Filing Jointly with Spouse Having No Earned Income, or Qualified Surviving Spouse' $26,000.00. Column (A) is also used where 'the employee's marital status is missing'. Biweekly those three are $500.00 / $750.00 / $1,000.00 and monthly $1,083.33 / $1,625.00 / $2,166.67; P is 52 weekly, 26 biweekly, 24 semimonthly, 12 monthly, 1 annually. The Department's WORKED EXAMPLE 5, page 5, verbatim: 'Example 5: Monthly wages of $5,000.00, with \"Married jointly and spouse having no earned income, or Qualifying surviving spouse\" marital status on 2026 IA W-4 (Total withholding allowance $80 claimed, no additional amount requested). T1 = $5,000.00 - $2,166.67 = $2,833.33.  T2 = $2,833.33 x 3.80% = $107.67.  T3 = $107.67 - $80.00 / 12 = $101.00.  T4 = $101.00 + $0.00 = $101.00.  Iowa tax to be withheld = $101.00.' Annual gross $5,000.00 x 12 = $60,000.00. (Unrounded: $2,833.333... x 0.038 = $107.6666..., less $6.6666... = $101.00 exactly, so this figure does not depend on whether T2 is rounded to the cent first.)",
      "why": "the monthly column (C) deduction of $2,166.67, which is $26,000 / 12 rounded, and a figure that comes out the same whether the engine rounds T2 or carries it - so a failure here is a wrong deduction and nothing else.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IA"
      },
      "expect": {
        "stateIncomeTax": 1786
      },
      "source": "Iowa Department of Revenue, 'Iowa Individual Income Tax Withholding Formula, Effective January 1, 2026', released November 2025, https://revenue.iowa.gov/media/53/download?inline (linked from https://revenue.iowa.gov/withholding-tax-information), retrieved 2026-10-15. Verbatim: 'Step 1. Determine T1. T1 = G - D (G: Gross taxable wages for the pay period; D: Deduction for the pay period)'; 'Step 2. Determine T2. T2 = T1 x 3.80% (or 0.038)'; 'Step 3. Determine T3. T3 = T2 - (W / P) (W: Total allowance amount on the IA W-4; P: Number of pay periods per year)'; 'Step 4. Determine T4, or Withholding Amount per Pay Period. T4 = T3 + A'. The 'Deduction Amounts by Pay Period (Denoted as D) for 2024 or a later year's IA W-4' table gives, for the ANNUAL payroll period: column (A) 'Other, or Married Filing Jointly with Spouse Having Earned Income' $13,000.00; column (B) 'Head of Household' $19,500.00; column (C) 'Married Filing Jointly with Spouse Having No Earned Income, or Qualified Surviving Spouse' $26,000.00. Column (A) is also used where 'the employee's marital status is missing'. Biweekly those three are $500.00 / $750.00 / $1,000.00 and monthly $1,083.33 / $1,625.00 / $2,166.67; P is 52 weekly, 26 biweekly, 24 semimonthly, 12 monthly, 1 annually. Hand derivation at the ANNUAL payroll period (P = 1) with no additional withholding requested (A = 0): column (A) applies ('Other'): T1 = $60,000.00 - $13,000.00 = $47,000.00; T2 = $47,000.00 x 0.038 = $1,786.00; W = $0, so T3 = $1,786.00.",
      "why": "Iowa annualised, with the flat 3.80% rate and the $13,000 annual deduction and no allowance. Iowa's rate was 3.80% from 2025; an engine on an older graduated schedule would be nowhere near.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "married-filing-jointly",
        "taxYear": "2026",
        "state": "IA",
        "stateCertificate": {
          "spouseEmployed": true
        }
      },
      "expect": {
        "stateIncomeTax": 1786
      },
      "source": "Iowa Department of Revenue, 'Iowa Individual Income Tax Withholding Formula, Effective January 1, 2026', released November 2025, https://revenue.iowa.gov/media/53/download?inline (linked from https://revenue.iowa.gov/withholding-tax-information), retrieved 2026-10-15. Verbatim: 'Step 1. Determine T1. T1 = G - D (G: Gross taxable wages for the pay period; D: Deduction for the pay period)'; 'Step 2. Determine T2. T2 = T1 x 3.80% (or 0.038)'; 'Step 3. Determine T3. T3 = T2 - (W / P) (W: Total allowance amount on the IA W-4; P: Number of pay periods per year)'; 'Step 4. Determine T4, or Withholding Amount per Pay Period. T4 = T3 + A'. The 'Deduction Amounts by Pay Period (Denoted as D) for 2024 or a later year's IA W-4' table gives, for the ANNUAL payroll period: column (A) 'Other, or Married Filing Jointly with Spouse Having Earned Income' $13,000.00; column (B) 'Head of Household' $19,500.00; column (C) 'Married Filing Jointly with Spouse Having No Earned Income, or Qualified Surviving Spouse' $26,000.00. Column (A) is also used where 'the employee's marital status is missing'. Biweekly those three are $500.00 / $750.00 / $1,000.00 and monthly $1,083.33 / $1,625.00 / $2,166.67; P is 52 weekly, 26 biweekly, 24 semimonthly, 12 monthly, 1 annually. Hand derivation at the ANNUAL payroll period (P = 1) with no additional withholding requested (A = 0): the spouse HAS earned income, so column (A) applies by its own words - 'Other, or Married Filing Jointly with Spouse Having Earned Income': T1 = $60,000.00 - $13,000.00 = $47,000.00; T2 = $1,786.00.",
      "why": "a joint filer whose spouse works gets the SAME $13,000 as a single filer. Giving them $26,000 would understate the tax by $494.00.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "married-filing-jointly",
        "taxYear": "2026",
        "state": "IA",
        "stateCertificate": {
          "spouseEmployed": false
        }
      },
      "expect": {
        "stateIncomeTax": 1292
      },
      "source": "Iowa Department of Revenue, 'Iowa Individual Income Tax Withholding Formula, Effective January 1, 2026', released November 2025, https://revenue.iowa.gov/media/53/download?inline (linked from https://revenue.iowa.gov/withholding-tax-information), retrieved 2026-10-15. Verbatim: 'Step 1. Determine T1. T1 = G - D (G: Gross taxable wages for the pay period; D: Deduction for the pay period)'; 'Step 2. Determine T2. T2 = T1 x 3.80% (or 0.038)'; 'Step 3. Determine T3. T3 = T2 - (W / P) (W: Total allowance amount on the IA W-4; P: Number of pay periods per year)'; 'Step 4. Determine T4, or Withholding Amount per Pay Period. T4 = T3 + A'. The 'Deduction Amounts by Pay Period (Denoted as D) for 2024 or a later year's IA W-4' table gives, for the ANNUAL payroll period: column (A) 'Other, or Married Filing Jointly with Spouse Having Earned Income' $13,000.00; column (B) 'Head of Household' $19,500.00; column (C) 'Married Filing Jointly with Spouse Having No Earned Income, or Qualified Surviving Spouse' $26,000.00. Column (A) is also used where 'the employee's marital status is missing'. Biweekly those three are $500.00 / $750.00 / $1,000.00 and monthly $1,083.33 / $1,625.00 / $2,166.67; P is 52 weekly, 26 biweekly, 24 semimonthly, 12 monthly, 1 annually. Hand derivation at the ANNUAL payroll period (P = 1) with no additional withholding requested (A = 0): the spouse has no earned income, so column (C) applies: T1 = $60,000.00 - $26,000.00 = $34,000.00; T2 = $34,000.00 x 0.038 = $1,292.00.",
      "why": "the other half of the pair above: the same salary and the same filing status, $494.00 apart on one answer about the spouse. This is why the spouse question is an input and not an assumption.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "head-of-household",
        "taxYear": "2026",
        "state": "IA"
      },
      "expect": {
        "stateIncomeTax": 1539
      },
      "source": "Iowa Department of Revenue, 'Iowa Individual Income Tax Withholding Formula, Effective January 1, 2026', released November 2025, https://revenue.iowa.gov/media/53/download?inline (linked from https://revenue.iowa.gov/withholding-tax-information), retrieved 2026-10-15. Verbatim: 'Step 1. Determine T1. T1 = G - D (G: Gross taxable wages for the pay period; D: Deduction for the pay period)'; 'Step 2. Determine T2. T2 = T1 x 3.80% (or 0.038)'; 'Step 3. Determine T3. T3 = T2 - (W / P) (W: Total allowance amount on the IA W-4; P: Number of pay periods per year)'; 'Step 4. Determine T4, or Withholding Amount per Pay Period. T4 = T3 + A'. The 'Deduction Amounts by Pay Period (Denoted as D) for 2024 or a later year's IA W-4' table gives, for the ANNUAL payroll period: column (A) 'Other, or Married Filing Jointly with Spouse Having Earned Income' $13,000.00; column (B) 'Head of Household' $19,500.00; column (C) 'Married Filing Jointly with Spouse Having No Earned Income, or Qualified Surviving Spouse' $26,000.00. Column (A) is also used where 'the employee's marital status is missing'. Biweekly those three are $500.00 / $750.00 / $1,000.00 and monthly $1,083.33 / $1,625.00 / $2,166.67; P is 52 weekly, 26 biweekly, 24 semimonthly, 12 monthly, 1 annually. Hand derivation at the ANNUAL payroll period (P = 1) with no additional withholding requested (A = 0): column (B) applies: T1 = $60,000.00 - $19,500.00 = $40,500.00; T2 = $40,500.00 x 0.038 = $1,539.00.",
      "why": "the head-of-household column (B) at the annual payroll period: $19,500, half-way between $13,000 and $26,000. Sending head of household to column (A) would overstate the tax by $247.00.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 13000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IA"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Iowa Department of Revenue, 'Iowa Individual Income Tax Withholding Formula, Effective January 1, 2026', released November 2025, https://revenue.iowa.gov/media/53/download?inline (linked from https://revenue.iowa.gov/withholding-tax-information), retrieved 2026-10-15. Verbatim: 'Step 1. Determine T1. T1 = G - D (G: Gross taxable wages for the pay period; D: Deduction for the pay period)'; 'Step 2. Determine T2. T2 = T1 x 3.80% (or 0.038)'; 'Step 3. Determine T3. T3 = T2 - (W / P) (W: Total allowance amount on the IA W-4; P: Number of pay periods per year)'; 'Step 4. Determine T4, or Withholding Amount per Pay Period. T4 = T3 + A'. The 'Deduction Amounts by Pay Period (Denoted as D) for 2024 or a later year's IA W-4' table gives, for the ANNUAL payroll period: column (A) 'Other, or Married Filing Jointly with Spouse Having Earned Income' $13,000.00; column (B) 'Head of Household' $19,500.00; column (C) 'Married Filing Jointly with Spouse Having No Earned Income, or Qualified Surviving Spouse' $26,000.00. Column (A) is also used where 'the employee's marital status is missing'. Biweekly those three are $500.00 / $750.00 / $1,000.00 and monthly $1,083.33 / $1,625.00 / $2,166.67; P is 52 weekly, 26 biweekly, 24 semimonthly, 12 monthly, 1 annually. Hand derivation at the ANNUAL payroll period (P = 1) with no additional withholding requested (A = 0): T1 = $13,000.00 - $13,000.00 = $0.00; T2 = $0.00; T3 = $0.00. The Department's own annual table (OtherStatusAnnually, https://revenue.iowa.gov/media/59/download?inline, retrieved 2026-10-15) prints '0' for the row 'At Least 0, But Less Than 13000' in every allowance column.",
      "why": "the exact edge of Iowa's deduction, which is the only threshold the formula has.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 13100,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IA"
      },
      "expect": {
        "stateIncomeTax": 3.8
      },
      "source": "Iowa Department of Revenue, 'Iowa Individual Income Tax Withholding Formula, Effective January 1, 2026', released November 2025, https://revenue.iowa.gov/media/53/download?inline (linked from https://revenue.iowa.gov/withholding-tax-information), retrieved 2026-10-15. Verbatim: 'Step 1. Determine T1. T1 = G - D (G: Gross taxable wages for the pay period; D: Deduction for the pay period)'; 'Step 2. Determine T2. T2 = T1 x 3.80% (or 0.038)'; 'Step 3. Determine T3. T3 = T2 - (W / P) (W: Total allowance amount on the IA W-4; P: Number of pay periods per year)'; 'Step 4. Determine T4, or Withholding Amount per Pay Period. T4 = T3 + A'. The 'Deduction Amounts by Pay Period (Denoted as D) for 2024 or a later year's IA W-4' table gives, for the ANNUAL payroll period: column (A) 'Other, or Married Filing Jointly with Spouse Having Earned Income' $13,000.00; column (B) 'Head of Household' $19,500.00; column (C) 'Married Filing Jointly with Spouse Having No Earned Income, or Qualified Surviving Spouse' $26,000.00. Column (A) is also used where 'the employee's marital status is missing'. Biweekly those three are $500.00 / $750.00 / $1,000.00 and monthly $1,083.33 / $1,625.00 / $2,166.67; P is 52 weekly, 26 biweekly, 24 semimonthly, 12 monthly, 1 annually. Hand derivation at the ANNUAL payroll period (P = 1) with no additional withholding requested (A = 0): T1 = $13,100.00 - $13,000.00 = $100.00; T2 = $100.00 x 0.038 = $3.80.",
      "why": "$100 past the deduction. The Department's annual table gives $19 for the whole '13000 to 14000' row, which is 3.80% of that row's $500 midpoint - the same formula.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 0,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IA"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "Iowa Department of Revenue, 'Iowa Individual Income Tax Withholding Formula, Effective January 1, 2026', released November 2025, https://revenue.iowa.gov/media/53/download?inline (linked from https://revenue.iowa.gov/withholding-tax-information), retrieved 2026-10-15. Verbatim: 'Step 1. Determine T1. T1 = G - D (G: Gross taxable wages for the pay period; D: Deduction for the pay period)'; 'Step 2. Determine T2. T2 = T1 x 3.80% (or 0.038)'; 'Step 3. Determine T3. T3 = T2 - (W / P) (W: Total allowance amount on the IA W-4; P: Number of pay periods per year)'; 'Step 4. Determine T4, or Withholding Amount per Pay Period. T4 = T3 + A'. The 'Deduction Amounts by Pay Period (Denoted as D) for 2024 or a later year's IA W-4' table gives, for the ANNUAL payroll period: column (A) 'Other, or Married Filing Jointly with Spouse Having Earned Income' $13,000.00; column (B) 'Head of Household' $19,500.00; column (C) 'Married Filing Jointly with Spouse Having No Earned Income, or Qualified Surviving Spouse' $26,000.00. Column (A) is also used where 'the employee's marital status is missing'. Biweekly those three are $500.00 / $750.00 / $1,000.00 and monthly $1,083.33 / $1,625.00 / $2,166.67; P is 52 weekly, 26 biweekly, 24 semimonthly, 12 monthly, 1 annually. Hand derivation at the ANNUAL payroll period (P = 1) with no additional withholding requested (A = 0): T1 = $0.00 - $13,000.00 is negative; the Department's annual table prints '0' for the row 'At Least 0, But Less Than 13000' at every allowance amount, so the withholding is nil.",
      "why": "nil wages. The formula would give a negative number; the Department's table says nil.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 39000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "SC",
        "stateCertificate": {
          "allowances": 3
        }
      },
      "expect": {
        "stateIncomeTax": 549.9
      },
      "source": "South Carolina Department of Revenue, Form WH-1603F (Rev. 11/4/25), 3478, 'FORMULA FOR COMPUTING SOUTH CAROLINA 2026 WITHHOLDING TAX', https://dor.sc.gov/sites/dor/files/forms/WH1603F_2026.pdf, retrieved 2026-10-15. Verbatim: 'II. Calculate taxable income. Deduct from gross wages: Personal Allowance - $0 if zero allowances claimed OR $5,000 per personal allowance claimed - AND Standard Deduction - $0 if zero allowances claimed OR 10% of gross wages if claiming one or more allowances, up to $7,500 total'. 'SUBTRACTION METHOD: At least $0 but less than $3,640 - 0%; $3,640 to $18,230 - 3%, then subtract $109.20; $18,230 and above - 6%, then subtract $656.10'. 'ADDITION METHOD: ... $18,230 and above - Subtract $18,230, multiply by 6.0%, then add $437.70'. 'CONSTANTS FOR DETERMINING SOUTH CAROLINA INCOME TAX WITHHOLDING PROGRAM' gives YEARLY: '(A) Amount per allowance $5,000; (B) Maximum standard deduction - no allowances $0; (C) Maximum standard deduction - one or more allowances $7,500; (D) Maximum tax - 1st bracket $437.70'. 'I. Compute annualized salary: Multiply weekly salary by 52 weeks to calculate the annual gross wages.' 'IV. Calculate weekly withholding: Divide the Withholding Tax by 52 to calculate the amount to withhold from the employee each week.' Hand derivation: this is the Department's OWN WORKED EXAMPLE, printed on the form beside the formula, verbatim: 'Example for Standard Deduction. Annualize salary: $750 per week, 3 allowances. $750.00 per week X 52 weeks = 39,000.00 gross wages - 15,000.00 personal allowances (3 allowances X $5,000) - 3,900.00 standard deduction ($39,000 X 10%) = $20,100.00 taxable income', then 'Taxable income: $20,100.00 X 0.060 = $1,206.00 - $656.10  Total to be withheld: $549.90'. The addition method on the same page reaches $549.90 too: ($20,100.00 - $18,230.00) x 0.060 = $112.20, + $437.70 = $549.90.",
      "why": "the South Carolina Department of Revenue's own printed answer to the penny, and the only one it publishes. It fixes the $5,000 allowance, the 10% standard deduction and the subtraction constant all at once.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 39000,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "SC",
        "stateCertificate": {
          "allowances": 3
        }
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 10.58
      },
      "source": "South Carolina Department of Revenue, Form WH-1603F (Rev. 11/4/25), 3478, 'FORMULA FOR COMPUTING SOUTH CAROLINA 2026 WITHHOLDING TAX', https://dor.sc.gov/sites/dor/files/forms/WH1603F_2026.pdf, retrieved 2026-10-15. Verbatim: 'II. Calculate taxable income. Deduct from gross wages: Personal Allowance - $0 if zero allowances claimed OR $5,000 per personal allowance claimed - AND Standard Deduction - $0 if zero allowances claimed OR 10% of gross wages if claiming one or more allowances, up to $7,500 total'. 'SUBTRACTION METHOD: At least $0 but less than $3,640 - 0%; $3,640 to $18,230 - 3%, then subtract $109.20; $18,230 and above - 6%, then subtract $656.10'. 'ADDITION METHOD: ... $18,230 and above - Subtract $18,230, multiply by 6.0%, then add $437.70'. 'CONSTANTS FOR DETERMINING SOUTH CAROLINA INCOME TAX WITHHOLDING PROGRAM' gives YEARLY: '(A) Amount per allowance $5,000; (B) Maximum standard deduction - no allowances $0; (C) Maximum standard deduction - one or more allowances $7,500; (D) Maximum tax - 1st bracket $437.70'. 'I. Compute annualized salary: Multiply weekly salary by 52 weeks to calculate the annual gross wages.' 'IV. Calculate weekly withholding: Divide the Withholding Tax by 52 to calculate the amount to withhold from the employee each week.' The same worked example carried to step IV, verbatim: 'Example: $549.90 DIVIDED BY 52 WEEKS = $10.58 STATE TAX PER WEEK'. Hand check: $549.90 / 52 = $10.5750, which the Department prints as $10.58 - rounded up at the exact half cent.",
      "why": "the published per-period figure, and the one case in this batch that pins a rounding DIRECTION: $10.575 is exactly half a cent and the Department rounds it up. An engine that rounded half to even would print $10.57.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 39000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "SC"
      },
      "expect": {
        "stateIncomeTax": 1683.9
      },
      "source": "South Carolina Department of Revenue, Form WH-1603F (Rev. 11/4/25), 3478, 'FORMULA FOR COMPUTING SOUTH CAROLINA 2026 WITHHOLDING TAX', https://dor.sc.gov/sites/dor/files/forms/WH1603F_2026.pdf, retrieved 2026-10-15. Verbatim: 'II. Calculate taxable income. Deduct from gross wages: Personal Allowance - $0 if zero allowances claimed OR $5,000 per personal allowance claimed - AND Standard Deduction - $0 if zero allowances claimed OR 10% of gross wages if claiming one or more allowances, up to $7,500 total'. 'SUBTRACTION METHOD: At least $0 but less than $3,640 - 0%; $3,640 to $18,230 - 3%, then subtract $109.20; $18,230 and above - 6%, then subtract $656.10'. 'ADDITION METHOD: ... $18,230 and above - Subtract $18,230, multiply by 6.0%, then add $437.70'. 'CONSTANTS FOR DETERMINING SOUTH CAROLINA INCOME TAX WITHHOLDING PROGRAM' gives YEARLY: '(A) Amount per allowance $5,000; (B) Maximum standard deduction - no allowances $0; (C) Maximum standard deduction - one or more allowances $7,500; (D) Maximum tax - 1st bracket $437.70'. 'I. Compute annualized salary: Multiply weekly salary by 52 weeks to calculate the annual gross wages.' 'IV. Calculate weekly withholding: Divide the Withholding Tax by 52 to calculate the amount to withhold from the employee each week.' Hand derivation: no allowances claimed, so BOTH subtractions are nil by the form's own words - 'Personal Allowance: $0 if zero allowances claimed' and 'Standard Deduction: $0 if zero allowances claimed' - and taxable income is the whole $39,000.00. '$18,230 and above: 6%, then subtract $656.10': (0.060 x $39,000.00) - $656.10 = $2,340.00 - $656.10 = $1,683.90.",
      "why": "the South Carolina trap, and the reason this case exists: at zero allowances there is NO standard deduction at all. An engine that gave the 10% deduction anyway would read $1,449.90, $234.00 too little, and nothing in the bracket table would reveal it.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 100000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "SC",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "expect": {
        "stateIncomeTax": 4593.9
      },
      "source": "South Carolina Department of Revenue, Form WH-1603F (Rev. 11/4/25), 3478, 'FORMULA FOR COMPUTING SOUTH CAROLINA 2026 WITHHOLDING TAX', https://dor.sc.gov/sites/dor/files/forms/WH1603F_2026.pdf, retrieved 2026-10-15. Verbatim: 'II. Calculate taxable income. Deduct from gross wages: Personal Allowance - $0 if zero allowances claimed OR $5,000 per personal allowance claimed - AND Standard Deduction - $0 if zero allowances claimed OR 10% of gross wages if claiming one or more allowances, up to $7,500 total'. 'SUBTRACTION METHOD: At least $0 but less than $3,640 - 0%; $3,640 to $18,230 - 3%, then subtract $109.20; $18,230 and above - 6%, then subtract $656.10'. 'ADDITION METHOD: ... $18,230 and above - Subtract $18,230, multiply by 6.0%, then add $437.70'. 'CONSTANTS FOR DETERMINING SOUTH CAROLINA INCOME TAX WITHHOLDING PROGRAM' gives YEARLY: '(A) Amount per allowance $5,000; (B) Maximum standard deduction - no allowances $0; (C) Maximum standard deduction - one or more allowances $7,500; (D) Maximum tax - 1st bracket $437.70'. 'I. Compute annualized salary: Multiply weekly salary by 52 weeks to calculate the annual gross wages.' 'IV. Calculate weekly withholding: Divide the Withholding Tax by 52 to calculate the amount to withhold from the employee each week.' Hand derivation: one allowance, so the standard deduction is '10% of gross wages ... up to $7,500 total': 10% of $100,000.00 is $10,000.00, capped at $7,500.00. Taxable income $100,000.00 - $5,000.00 - $7,500.00 = $87,500.00. (0.060 x $87,500.00) - $656.10 = $5,250.00 - $656.10 = $4,593.90.",
      "why": "the $7,500 cap on the standard deduction, which bites at any gross above $75,000. Without the cap the answer would be $4,443.90, $150.00 too little.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 18230,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "SC"
      },
      "expect": {
        "stateIncomeTax": 437.7
      },
      "source": "South Carolina Department of Revenue, Form WH-1603F (Rev. 11/4/25), 3478, 'FORMULA FOR COMPUTING SOUTH CAROLINA 2026 WITHHOLDING TAX', https://dor.sc.gov/sites/dor/files/forms/WH1603F_2026.pdf, retrieved 2026-10-15. Verbatim: 'II. Calculate taxable income. Deduct from gross wages: Personal Allowance - $0 if zero allowances claimed OR $5,000 per personal allowance claimed - AND Standard Deduction - $0 if zero allowances claimed OR 10% of gross wages if claiming one or more allowances, up to $7,500 total'. 'SUBTRACTION METHOD: At least $0 but less than $3,640 - 0%; $3,640 to $18,230 - 3%, then subtract $109.20; $18,230 and above - 6%, then subtract $656.10'. 'ADDITION METHOD: ... $18,230 and above - Subtract $18,230, multiply by 6.0%, then add $437.70'. 'CONSTANTS FOR DETERMINING SOUTH CAROLINA INCOME TAX WITHHOLDING PROGRAM' gives YEARLY: '(A) Amount per allowance $5,000; (B) Maximum standard deduction - no allowances $0; (C) Maximum standard deduction - one or more allowances $7,500; (D) Maximum tax - 1st bracket $437.70'. 'I. Compute annualized salary: Multiply weekly salary by 52 weeks to calculate the annual gross wages.' 'IV. Calculate weekly withholding: Divide the Withholding Tax by 52 to calculate the amount to withhold from the employee each week.' Hand derivation: zero allowances, so taxable income is $18,230.00, the exact foot of the top band. '$18,230 and above: 6%, then subtract $656.10': (0.060 x $18,230.00) - $656.10 = $1,093.80 - $656.10 = $437.70 - which is also the figure the form prints as constant (D), 'Maximum tax - 1st bracket $437.70', so the two bands meet here in the Department's own numbers.",
      "why": "the band edge where South Carolina's own published constant (D) confirms the arithmetic: the 3% band's maximum and the 6% band's start agree at $437.70. The two official methods and the constant table all meet at this one wage.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 18229,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "SC"
      },
      "expect": {
        "stateIncomeTax": 437.67
      },
      "source": "South Carolina Department of Revenue, Form WH-1603F (Rev. 11/4/25), 3478, 'FORMULA FOR COMPUTING SOUTH CAROLINA 2026 WITHHOLDING TAX', https://dor.sc.gov/sites/dor/files/forms/WH1603F_2026.pdf, retrieved 2026-10-15. Verbatim: 'II. Calculate taxable income. Deduct from gross wages: Personal Allowance - $0 if zero allowances claimed OR $5,000 per personal allowance claimed - AND Standard Deduction - $0 if zero allowances claimed OR 10% of gross wages if claiming one or more allowances, up to $7,500 total'. 'SUBTRACTION METHOD: At least $0 but less than $3,640 - 0%; $3,640 to $18,230 - 3%, then subtract $109.20; $18,230 and above - 6%, then subtract $656.10'. 'ADDITION METHOD: ... $18,230 and above - Subtract $18,230, multiply by 6.0%, then add $437.70'. 'CONSTANTS FOR DETERMINING SOUTH CAROLINA INCOME TAX WITHHOLDING PROGRAM' gives YEARLY: '(A) Amount per allowance $5,000; (B) Maximum standard deduction - no allowances $0; (C) Maximum standard deduction - one or more allowances $7,500; (D) Maximum tax - 1st bracket $437.70'. 'I. Compute annualized salary: Multiply weekly salary by 52 weeks to calculate the annual gross wages.' 'IV. Calculate weekly withholding: Divide the Withholding Tax by 52 to calculate the amount to withhold from the employee each week.' Hand derivation: taxable income $18,229.00 is in '$3,640 to $18,230: 3%, then subtract $109.20': (0.030 x $18,229.00) - $109.20 = $546.87 - $109.20 = $437.67.",
      "why": "one dollar below the edge, on the 3% band. Reading this on the 6% band would give $437.64, three cents low - the band boundary tested at a dollar.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 3641,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "SC"
      },
      "expect": {
        "stateIncomeTax": 0.03
      },
      "source": "South Carolina Department of Revenue, Form WH-1603F (Rev. 11/4/25), 3478, 'FORMULA FOR COMPUTING SOUTH CAROLINA 2026 WITHHOLDING TAX', https://dor.sc.gov/sites/dor/files/forms/WH1603F_2026.pdf, retrieved 2026-10-15. Verbatim: 'II. Calculate taxable income. Deduct from gross wages: Personal Allowance - $0 if zero allowances claimed OR $5,000 per personal allowance claimed - AND Standard Deduction - $0 if zero allowances claimed OR 10% of gross wages if claiming one or more allowances, up to $7,500 total'. 'SUBTRACTION METHOD: At least $0 but less than $3,640 - 0%; $3,640 to $18,230 - 3%, then subtract $109.20; $18,230 and above - 6%, then subtract $656.10'. 'ADDITION METHOD: ... $18,230 and above - Subtract $18,230, multiply by 6.0%, then add $437.70'. 'CONSTANTS FOR DETERMINING SOUTH CAROLINA INCOME TAX WITHHOLDING PROGRAM' gives YEARLY: '(A) Amount per allowance $5,000; (B) Maximum standard deduction - no allowances $0; (C) Maximum standard deduction - one or more allowances $7,500; (D) Maximum tax - 1st bracket $437.70'. 'I. Compute annualized salary: Multiply weekly salary by 52 weeks to calculate the annual gross wages.' 'IV. Calculate weekly withholding: Divide the Withholding Tax by 52 to calculate the amount to withhold from the employee each week.' Hand derivation: taxable income $3,641.00: (0.030 x $3,641.00) - $109.20 = $109.23 - $109.20 = $0.03.",
      "why": "the first three cents of South Carolina tax. The 3% band is engineered so that the charge starts at nil at $3,640 and climbs from there; an engine applying 3% to the whole wage would read $109.23.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 3640,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "SC"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "South Carolina Department of Revenue, Form WH-1603F (Rev. 11/4/25), 3478, 'FORMULA FOR COMPUTING SOUTH CAROLINA 2026 WITHHOLDING TAX', https://dor.sc.gov/sites/dor/files/forms/WH1603F_2026.pdf, retrieved 2026-10-15. Verbatim: 'II. Calculate taxable income. Deduct from gross wages: Personal Allowance - $0 if zero allowances claimed OR $5,000 per personal allowance claimed - AND Standard Deduction - $0 if zero allowances claimed OR 10% of gross wages if claiming one or more allowances, up to $7,500 total'. 'SUBTRACTION METHOD: At least $0 but less than $3,640 - 0%; $3,640 to $18,230 - 3%, then subtract $109.20; $18,230 and above - 6%, then subtract $656.10'. 'ADDITION METHOD: ... $18,230 and above - Subtract $18,230, multiply by 6.0%, then add $437.70'. 'CONSTANTS FOR DETERMINING SOUTH CAROLINA INCOME TAX WITHHOLDING PROGRAM' gives YEARLY: '(A) Amount per allowance $5,000; (B) Maximum standard deduction - no allowances $0; (C) Maximum standard deduction - one or more allowances $7,500; (D) Maximum tax - 1st bracket $437.70'. 'I. Compute annualized salary: Multiply weekly salary by 52 weeks to calculate the annual gross wages.' 'IV. Calculate weekly withholding: Divide the Withholding Tax by 52 to calculate the amount to withhold from the employee each week.' Hand derivation: taxable income $3,640.00: (0.030 x $3,640.00) - $109.20 = $109.20 - $109.20 = $0.00. The 0% band ('At least $0 but less than $3,640') gives $0.00 too, so the schedule is continuous at this wage.",
      "why": "the exact edge of the 0% band, where both branches must agree at nil.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 3639,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "SC"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "South Carolina Department of Revenue, Form WH-1603F (Rev. 11/4/25), 3478, 'FORMULA FOR COMPUTING SOUTH CAROLINA 2026 WITHHOLDING TAX', https://dor.sc.gov/sites/dor/files/forms/WH1603F_2026.pdf, retrieved 2026-10-15. Verbatim: 'II. Calculate taxable income. Deduct from gross wages: Personal Allowance - $0 if zero allowances claimed OR $5,000 per personal allowance claimed - AND Standard Deduction - $0 if zero allowances claimed OR 10% of gross wages if claiming one or more allowances, up to $7,500 total'. 'SUBTRACTION METHOD: At least $0 but less than $3,640 - 0%; $3,640 to $18,230 - 3%, then subtract $109.20; $18,230 and above - 6%, then subtract $656.10'. 'ADDITION METHOD: ... $18,230 and above - Subtract $18,230, multiply by 6.0%, then add $437.70'. 'CONSTANTS FOR DETERMINING SOUTH CAROLINA INCOME TAX WITHHOLDING PROGRAM' gives YEARLY: '(A) Amount per allowance $5,000; (B) Maximum standard deduction - no allowances $0; (C) Maximum standard deduction - one or more allowances $7,500; (D) Maximum tax - 1st bracket $437.70'. 'I. Compute annualized salary: Multiply weekly salary by 52 weeks to calculate the annual gross wages.' 'IV. Calculate weekly withholding: Divide the Withholding Tax by 52 to calculate the amount to withhold from the employee each week.' Hand derivation: taxable income $3,639.00 is in 'At least $0 but less than $3,640: 0%', so the tax is $0.00.",
      "why": "one dollar below the first edge, inside the 0% band.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 0,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "SC"
      },
      "expect": {
        "stateIncomeTax": 0
      },
      "source": "South Carolina Department of Revenue, Form WH-1603F (Rev. 11/4/25), 3478, 'FORMULA FOR COMPUTING SOUTH CAROLINA 2026 WITHHOLDING TAX', https://dor.sc.gov/sites/dor/files/forms/WH1603F_2026.pdf, retrieved 2026-10-15. Verbatim: 'II. Calculate taxable income. Deduct from gross wages: Personal Allowance - $0 if zero allowances claimed OR $5,000 per personal allowance claimed - AND Standard Deduction - $0 if zero allowances claimed OR 10% of gross wages if claiming one or more allowances, up to $7,500 total'. 'SUBTRACTION METHOD: At least $0 but less than $3,640 - 0%; $3,640 to $18,230 - 3%, then subtract $109.20; $18,230 and above - 6%, then subtract $656.10'. 'ADDITION METHOD: ... $18,230 and above - Subtract $18,230, multiply by 6.0%, then add $437.70'. 'CONSTANTS FOR DETERMINING SOUTH CAROLINA INCOME TAX WITHHOLDING PROGRAM' gives YEARLY: '(A) Amount per allowance $5,000; (B) Maximum standard deduction - no allowances $0; (C) Maximum standard deduction - one or more allowances $7,500; (D) Maximum tax - 1st bracket $437.70'. 'I. Compute annualized salary: Multiply weekly salary by 52 weeks to calculate the annual gross wages.' 'IV. Calculate weekly withholding: Divide the Withholding Tax by 52 to calculate the amount to withhold from the employee each week.' Hand derivation: taxable income $0.00 is in the 0% band: $0.00.",
      "why": "nil wages, nil South Carolina tax.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "FL"
      },
      "expect": {
        "stateIncomeTax": 0,
        "coverage": "federal-fica-and-state"
      },
      "source": "Constitution of the State of Florida, Article VII, Section 5, 'Estate, inheritance and income taxes', https://www.flsenate.gov/Laws/Constitution, retrieved 2026-10-15, verbatim: '(a) NATURAL PERSONS. No tax upon estates or inheritances or upon the income of natural persons who are residents or citizens of the state shall be levied by the state, or under its authority, in excess of the aggregate of amounts which may be allowed to be credited upon or deducted from any similar tax levied by the United States or any state.' The federal credit that clause measures against was repealed in 1964, so the permitted amount is nil; and the Florida Department of Revenue's own index of every tax it administers, https://floridarevenue.com/taxes/taxesfees/Pages/default.aspx, retrieved 2026-10-15, lists no tax on the income of individuals (corporate income/franchise tax, sales and use, reemployment, fuel, documentary stamp and the rest). Arithmetic: 0 x $60,000.00 = $0.00.",
      "why": "Florida is one of the four no-wage-tax states the site actually offers, and this is the case that proves the state was READ and charged at nil rather than silently dropped - which is why it asserts coverage 'federal-fica-and-state' as well as the nil. An engine that forgot Florida entirely would return coverage 'federal-and-fica' and pass a nil-only assertion.",
      "ledger": "L-2026-4373"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "SD"
      },
      "expect": {
        "stateIncomeTax": 0,
        "coverage": "federal-fica-and-state"
      },
      "source": "South Dakota Department of Revenue, 'Individuals - Taxes', https://dor.sd.gov/individuals/taxes/, retrieved 2026-10-15, verbatim: 'South Dakota is one of seven states that does not impose a state income tax.' Arithmetic: 0 x $60,000.00 = $0.00.",
      "why": "South Dakota charges nothing on wage income, stated by its own Department of Revenue. Asserting coverage as well as the nil is what distinguishes 'charged at nil' from 'never looked at'.",
      "ledger": "L-2026-4373"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "TN"
      },
      "expect": {
        "stateIncomeTax": 0,
        "coverage": "federal-fica-and-state"
      },
      "source": "Tennessee Department of Revenue, 'Hall Income Tax', https://www.tn.gov/revenue/taxes/hall-income-tax.html, retrieved 2026-10-15, verbatim: 'The Hall income tax is imposed only on individuals and other entities receiving interest from bonds and notes and dividends from stock', and 'The Hall Income tax was repealed for tax periods that begin on January 1, 2021, or later', with 'Please do not file a return for any tax year that begins on or after January 1, 2021.' So the only individual income tax Tennessee has ever levied never reached wages, and it no longer exists; the Department's own index of the taxes it administers, https://www.tn.gov/revenue/taxes.html, retrieved 2026-10-15, lists no tax on wages, salaries or earned income and lists the Hall income tax under 'Archived'. Arithmetic: 0 x $60,000.00 = $0.00.",
      "why": "Tennessee is the no-tax state most likely to be got wrong, because it DID have an individual income tax within living memory. The case records both facts: the Hall tax was on interest and dividends only, and it is repealed.",
      "ledger": "L-2026-4373"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "TX"
      },
      "expect": {
        "stateIncomeTax": 0,
        "coverage": "federal-fica-and-state"
      },
      "source": "Constitution of the State of Texas, Article VIII, Section 24-a, as printed by the Texas Legislative Council, https://tlc.texas.gov/docs/legref/TxConst.pdf, retrieved 2026-10-15, verbatim: 'Sec. 24-a. INDIVIDUAL INCOME TAX PROHIBITED. The legislature may not impose a tax on the net incomes of individuals, including an individual's share of partnership and unincorporated association income. (Added Nov. 5, 2019.)' The preceding section reads 'Sec. 24. (Repealed Nov. 5, 2019.)'. Arithmetic: 0 x $60,000.00 = $0.00.",
      "why": "Texas is the largest state the site offers and the nil is constitutional, not a policy that could change in a Budget: the legislature is forbidden to levy it at all. As with the other three, the coverage assertion is what proves the state was charged rather than ignored.",
      "ledger": "L-2026-4373"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "AZ",
        "stateCertificate": {
          "withholdingPercentage": 0.035
        }
      },
      "expect": {
        "stateIncomeTax": 2100
      },
      "source": "Arizona Department of Revenue, Arizona Form A-4 (2026), 'Employee's Arizona Withholding Election', ADOR 10121 (25), https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf (azdor.gov refuses automated requests; read from the Internet Archive capture of 2026-10-08 at http://web.archive.org/web/20261008204653/https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf), retrieved 2026-10-15. Verbatim, box 1: 'Withhold from gross taxable wages at the percentage checked (check only one percentage): 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5%'; box 2: 'I elect an Arizona withholding percentage of zero, and I certify that I expect to have no Arizona tax liability for the current taxable year.' Employee's Instructions, 'New Employees': 'If you do not give this form to your employer the department requires your employer to withhold 2.0% of your gross taxable wages.' And: 'The amount withheld is a percentage of your gross taxable wages from every paycheck.' The statutory frame is A.R.S. s.43-401(A) and (E), https://www.azleg.gov/ars/43/00401.htm, retrieved 2026-10-15: the employer deducts 'an amount prescribed by tables adopted by the department', 'Each employee shall elect the amount authorized by subsection A of this section to be withheld', and 'Any employee failing to complete an election form as prescribed shall be deemed to have elected the withholding percentage prescribed by the department.' THERE IS NO ARIZONA BRACKET TABLE AND NO ARIZONA DEDUCTION IN THIS COMPUTATION: Arizona withholding is a percentage of gross taxable wages chosen by the employee, so an Arizona figure is a function of an election, not of a schedule. Hand derivation: box 1 election of 3.5% (stateCertificate.withholdingPercentage = 0.035, the fraction of gross): 0.035 x $60,000.00 = $2,100.00.",
      "why": "the highest percentage Form A-4 offers. Paired with the 2.0% default case it proves the election is read and not ignored - a $900 difference at the same salary.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "AZ",
        "stateCertificate": {
          "withholdingPercentage": 0.005
        }
      },
      "expect": {
        "stateIncomeTax": 300
      },
      "source": "Arizona Department of Revenue, Arizona Form A-4 (2026), 'Employee's Arizona Withholding Election', ADOR 10121 (25), https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf (azdor.gov refuses automated requests; read from the Internet Archive capture of 2026-10-08 at http://web.archive.org/web/20261008204653/https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf), retrieved 2026-10-15. Verbatim, box 1: 'Withhold from gross taxable wages at the percentage checked (check only one percentage): 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5%'; box 2: 'I elect an Arizona withholding percentage of zero, and I certify that I expect to have no Arizona tax liability for the current taxable year.' Employee's Instructions, 'New Employees': 'If you do not give this form to your employer the department requires your employer to withhold 2.0% of your gross taxable wages.' And: 'The amount withheld is a percentage of your gross taxable wages from every paycheck.' The statutory frame is A.R.S. s.43-401(A) and (E), https://www.azleg.gov/ars/43/00401.htm, retrieved 2026-10-15: the employer deducts 'an amount prescribed by tables adopted by the department', 'Each employee shall elect the amount authorized by subsection A of this section to be withheld', and 'Any employee failing to complete an election form as prescribed shall be deemed to have elected the withholding percentage prescribed by the department.' THERE IS NO ARIZONA BRACKET TABLE AND NO ARIZONA DEDUCTION IN THIS COMPUTATION: Arizona withholding is a percentage of gross taxable wages chosen by the employee, so an Arizona figure is a function of an election, not of a schedule. Hand derivation: box 1 election of 0.5% (stateCertificate.withholdingPercentage = 0.005, the fraction of gross): 0.005 x $60,000.00 = $300.00.",
      "why": "the lowest non-zero percentage on the form, and the one most likely to be lost to a percent/decimal slip: 0.5 read as 5% would give $3,000.00.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "AZ",
        "stateCertificate": {
          "withholdingPercentage": 0.025
        }
      },
      "expect": {
        "stateIncomeTax": 1500
      },
      "source": "Arizona Department of Revenue, Arizona Form A-4 (2026), 'Employee's Arizona Withholding Election', ADOR 10121 (25), https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf (azdor.gov refuses automated requests; read from the Internet Archive capture of 2026-10-08 at http://web.archive.org/web/20261008204653/https://azdor.gov/sites/default/files/document/FORMS_WITHHOLDING_2026_A-4_f.pdf), retrieved 2026-10-15. Verbatim, box 1: 'Withhold from gross taxable wages at the percentage checked (check only one percentage): 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5%'; box 2: 'I elect an Arizona withholding percentage of zero, and I certify that I expect to have no Arizona tax liability for the current taxable year.' Employee's Instructions, 'New Employees': 'If you do not give this form to your employer the department requires your employer to withhold 2.0% of your gross taxable wages.' And: 'The amount withheld is a percentage of your gross taxable wages from every paycheck.' The statutory frame is A.R.S. s.43-401(A) and (E), https://www.azleg.gov/ars/43/00401.htm, retrieved 2026-10-15: the employer deducts 'an amount prescribed by tables adopted by the department', 'Each employee shall elect the amount authorized by subsection A of this section to be withheld', and 'Any employee failing to complete an election form as prescribed shall be deemed to have elected the withholding percentage prescribed by the department.' THERE IS NO ARIZONA BRACKET TABLE AND NO ARIZONA DEDUCTION IN THIS COMPUTATION: Arizona withholding is a percentage of gross taxable wages chosen by the employee, so an Arizona figure is a function of an election, not of a schedule. Hand derivation: box 1 election of 2.5% (stateCertificate.withholdingPercentage = 0.025, the fraction of gross): 0.025 x $60,000.00 = $1,500.00.",
      "why": "a middle election, to show the set is a ladder of seven and not a bracket table. Arizona has no band edge to test because Arizona publishes no bands.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 45000,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "taxCredits": 30,
          "step2Checkbox": false
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholding": 61.35,
        "federalWithholdingWholeDollar": 61
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-10-19. PAGE 10, WORKSHEET 1A, verbatim and complete as to rounding: '2e Subtract line 2b from line 2a'; '2f Multiply the amount on line 2e by the percentage on line 2d'; '2g Add lines 2c and 2f'; '2h Divide the amount on line 2g by the number of pay periods on line 1b. This is the Tentative Withholding Amount'; '3a If the employee's Form W-4 is from 2020 or later, enter the amount from Step 3 of that form; otherwise, enter -0-'; '3b Divide the amount on line 3a by the number of pay periods on line 1b'; '3c Subtract line 3b from line 2h. If zero or less, enter -0-'; '4a Enter the additional amount to withhold from the employee's Form W-4 (Step 4(c) of the 2020 or later form, or line 6 on earlier forms)'; '4b Add lines 3c and 4a. This is the amount to withhold from the employee's wages this pay period'. Lines 2e, 2f, 2g, 2h, 3b, 3c and 4b say only Subtract, Multiply, Add, Divide, Divide, Subtract and Add, and name NO rounding, so every one is carried exact. THE ONLY ROUNDING INSTRUCTION IN THE PUBLICATION is the page 9 box headed 'Rounding', verbatim: 'To figure the income tax to withhold, you may reduce the last digit of the wages to zero, or figure the wages to the nearest dollar. You may also round the tax for the pay period to the nearest dollar. If rounding is used, it must be used consistently. Withheld tax amounts should be rounded to the nearest whole dollar by dropping amounts under 50 cents and increasing amounts from 50 to 99 cents to the next dollar. For example, $2.30 becomes $2 and $2.50 becomes $3.' That box is the ONLY place the word 'cents' appears anywhere in Publication 15-T (2026): a string search of the full document returns two hits, both inside that one sentence, and every other match is 'percent' or 'percentage'. It is permissive ('you may also') and it is to the nearest DOLLAR, never to the cent. PAGE 10 LINE 1g verbatim: '1g If the box in Step 2 of Form W-4 is checked, enter -0-. If the box is not checked, enter $12,900 if the taxpayer is married filing jointly or $8,600 otherwise'. PAGE 10 TABLE 3 verbatim: 'Semiannually 2 / Quarterly 4 / Monthly 12 / Semimonthly 24 / Biweekly 26 / Weekly 52 / Daily 260'. PAGE 12, '2026 Percentage Method Tables for Automated Payroll Systems and Withholding on Periodic Payments of Pensions and Annuities', columns 'A At least- / B But less than- / C The tentative amount to withhold is: / D Plus this percentage- / E of the amount that the Adjusted Annual Wage or Payment Amount exceeds-'. ROW USED, QUOTED NOT PARAPHRASED, STANDARD Withholding Rate Schedules ('Use these if the Form W-4 is from 2019 or earlier, or if the Form W-4 is from 2020 or later and the box in Step 2 of Form W-4 is NOT checked. Also, use these for Form W-4P from any year.'), 'Single or Married Filing Separately': '$19,900       $57,900       $1,240.00             12%         $19,900'. HAND DERIVATION on the stated annual wage. 1b 52; 1c $45,000.00; 1d $0.00; 1e $45,000.00; 1f $0.00; 1g $8,600.00; 1h $8,600.00; 1i $36,400.00. 2a $36,400.00; 2b $19,900.00; 2c $1,240.00; 2d 12%; 2e $16,500.00; 2f 0.12 x $16,500.00 = $1,980.00; 2g $3,220.00. 2h EXACT $3,220.00 / 52 = $61.92307692307692... (805/13). 3a $30.00; 3b EXACT $30.00 / 52 = $0.5769230769... (15/26). 3c EXACT $3,190.00 / 52 = $61.34615384615385... (1595/26). 4a $0.00; 4b EXACT $61.34615384615385... . THE ONE ROUNDING: half-up to the cent $61.35; half-up to the dollar FROM THE SAME EXACT 4b, $61. THE DEFECTIVE READING, for contrast: 2h to $61.92 and 3b to $0.58 first, then $61.92 - $0.58 = $61.34. NOTE ON A FIGURE QUOTED ELSEWHERE: the exact line 4b is $61.3461538..., not '$61.345'; the cent answer is the same either way. WHY THIS CASE ASSERTS ONLY THE MONEY AND NOT adjustedAnnualWage OR tentativeAnnualWithholding: the stated annual wage does not divide exactly into the pay period, and Publication 15-T does not say what a period's pay is for a stated annual salary - line 1a is 'the employee's total taxable wages this payroll period', an observed payroll amount. Both implementations in this repository round that per-period wage to the cent and then annualise at line 1c, which makes line 1i a few cents away from (annual wage - line 1g); that is a payroll-design question, not an IRS rounding question, and this hand derivation does not settle it, so it is not asserted here. The money figures below are unaffected: they are the same to the cent and to the dollar on either reading of line 1a.",
      "why": "ROUNDING RULE, UP AT THE CENT. Written to hold ruling L-2026-4594 and not to check a number. Worksheet 1A line 2h and line 3b both have a non-terminating quotient here, so the two readings of the publication give different cents: carrying 2h and 3b exact and rounding once at line 4b gives $61.35, while rounding 2h and 3b to the cent first gives $61.34. EVERY OTHER STEP 3 CREDIT CASE IN THIS FILE IS BLIND TO THAT - L-2026-412 at $52,000 biweekly with a $2,000 credit gives $79.23 on both readings and its $40,000-credit sibling floors at nil - and all 42 cases accuracy-red-team passed on 2026-10-19 carried a ZERO credit, so line 3b subtracted nothing in 42 of 42. Delete this case and the one-rounding rule is held by no vector at all. Pairs with the DOWN case at $120,000 monthly.",
      "ledger": "L-2026-4594"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 120000,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "taxCredits": 4000,
          "step2Checkbox": false
        }
      },
      "period": "monthly",
      "expect": {
        "federalWithholding": 1130.83,
        "federalWithholdingWholeDollar": 1131,
        "adjustedAnnualWage": 111400,
        "tentativeAnnualWithholding": 17570
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-10-19. PAGE 10, WORKSHEET 1A, verbatim and complete as to rounding: '2e Subtract line 2b from line 2a'; '2f Multiply the amount on line 2e by the percentage on line 2d'; '2g Add lines 2c and 2f'; '2h Divide the amount on line 2g by the number of pay periods on line 1b. This is the Tentative Withholding Amount'; '3a If the employee's Form W-4 is from 2020 or later, enter the amount from Step 3 of that form; otherwise, enter -0-'; '3b Divide the amount on line 3a by the number of pay periods on line 1b'; '3c Subtract line 3b from line 2h. If zero or less, enter -0-'; '4a Enter the additional amount to withhold from the employee's Form W-4 (Step 4(c) of the 2020 or later form, or line 6 on earlier forms)'; '4b Add lines 3c and 4a. This is the amount to withhold from the employee's wages this pay period'. Lines 2e, 2f, 2g, 2h, 3b, 3c and 4b say only Subtract, Multiply, Add, Divide, Divide, Subtract and Add, and name NO rounding, so every one is carried exact. THE ONLY ROUNDING INSTRUCTION IN THE PUBLICATION is the page 9 box headed 'Rounding', verbatim: 'To figure the income tax to withhold, you may reduce the last digit of the wages to zero, or figure the wages to the nearest dollar. You may also round the tax for the pay period to the nearest dollar. If rounding is used, it must be used consistently. Withheld tax amounts should be rounded to the nearest whole dollar by dropping amounts under 50 cents and increasing amounts from 50 to 99 cents to the next dollar. For example, $2.30 becomes $2 and $2.50 becomes $3.' That box is the ONLY place the word 'cents' appears anywhere in Publication 15-T (2026): a string search of the full document returns two hits, both inside that one sentence, and every other match is 'percent' or 'percentage'. It is permissive ('you may also') and it is to the nearest DOLLAR, never to the cent. PAGE 10 LINE 1g verbatim: '1g If the box in Step 2 of Form W-4 is checked, enter -0-. If the box is not checked, enter $12,900 if the taxpayer is married filing jointly or $8,600 otherwise'. PAGE 10 TABLE 3 verbatim: 'Semiannually 2 / Quarterly 4 / Monthly 12 / Semimonthly 24 / Biweekly 26 / Weekly 52 / Daily 260'. PAGE 12, '2026 Percentage Method Tables for Automated Payroll Systems and Withholding on Periodic Payments of Pensions and Annuities', columns 'A At least- / B But less than- / C The tentative amount to withhold is: / D Plus this percentage- / E of the amount that the Adjusted Annual Wage or Payment Amount exceeds-'. ROW USED, QUOTED NOT PARAPHRASED, STANDARD Withholding Rate Schedules ('Use these if the Form W-4 is from 2019 or earlier, or if the Form W-4 is from 2020 or later and the box in Step 2 of Form W-4 is NOT checked. Also, use these for Form W-4P from any year.'), 'Single or Married Filing Separately': '$57,900      $113,200       $5,800.00             22%         $57,900'. HAND DERIVATION. 1a $120,000.00 / 12 = $10,000.00 EXACTLY; 1b 12; 1c $120,000.00; 1d $0.00; 1e $120,000.00; 1f $0.00; 1g $8,600.00; 1h $8,600.00; 1i $111,400.00. 2a $111,400.00; 2b $57,900.00; 2c $5,800.00; 2d 22%; 2e $53,500.00; 2f 0.22 x $53,500.00 = $11,770.00 EXACTLY; 2g $17,570.00. 2h EXACT $17,570.00 / 12 = $1,464.16666666... ; 3a $4,000.00; 3b EXACT $4,000.00 / 12 = $333.33333333... ; 3c EXACT $13,570.00 / 12 = $1,130.83333333... ; 4a $0.00; 4b EXACT $1,130.83333333... . THE ONE ROUNDING: half-up to the cent $1,130.83; half-up to the dollar $1,131. THE DEFECTIVE READING: 2h to $1,464.17 and 3b to $333.33, then $1,464.17 - $333.33 = $1,130.84. THIS CASE IS IMMUNE TO THE TWO IMPLEMENTATION CHOICES THAT PUBLICATION 15-T DOES NOT SETTLE, which is why it asserts the annual lines as well as the money: the stated annual wage divides exactly into the pay period at line 1a, so it does not matter whether an implementation rounds the per-period wage to the cent before annualising at line 1c; and line 2f comes out exact to the cent, so it does not matter whether an implementation rounds line 2g. Only the line 2h / 3b / 4b rounding can move this answer.",
      "why": "ROUNDING RULE, DOWN AT THE CENT, and it is here because the UP cases alone would let a half-right rule pass. Rounding 2h and 3b before subtracting moves the answer in BOTH directions depending on the two residues - up at $45,000 weekly with a $30 credit, down here - so a vector set that only ever caught the upward move would be satisfied by an implementation that rounded 2h up and carried 3b exact, or that simply added a cent. Exact throughout gives $1,130.83; rounding first gives $1,130.84. The whole-dollar figure is $1,131 on BOTH readings, which is the point of the separate whole-dollar cases: the cent window and the dollar window are different windows and neither implies the other.",
      "ledger": "L-2026-4594"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 145000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "taxCredits": 1000,
          "step2Checkbox": true
        }
      },
      "period": "biweekly",
      "expect": {
        "federalWithholding": 1204.5,
        "federalWithholdingWholeDollar": 1204
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-10-19. PAGE 10, WORKSHEET 1A, verbatim and complete as to rounding: '2e Subtract line 2b from line 2a'; '2f Multiply the amount on line 2e by the percentage on line 2d'; '2g Add lines 2c and 2f'; '2h Divide the amount on line 2g by the number of pay periods on line 1b. This is the Tentative Withholding Amount'; '3a If the employee's Form W-4 is from 2020 or later, enter the amount from Step 3 of that form; otherwise, enter -0-'; '3b Divide the amount on line 3a by the number of pay periods on line 1b'; '3c Subtract line 3b from line 2h. If zero or less, enter -0-'; '4a Enter the additional amount to withhold from the employee's Form W-4 (Step 4(c) of the 2020 or later form, or line 6 on earlier forms)'; '4b Add lines 3c and 4a. This is the amount to withhold from the employee's wages this pay period'. Lines 2e, 2f, 2g, 2h, 3b, 3c and 4b say only Subtract, Multiply, Add, Divide, Divide, Subtract and Add, and name NO rounding, so every one is carried exact. THE ONLY ROUNDING INSTRUCTION IN THE PUBLICATION is the page 9 box headed 'Rounding', verbatim: 'To figure the income tax to withhold, you may reduce the last digit of the wages to zero, or figure the wages to the nearest dollar. You may also round the tax for the pay period to the nearest dollar. If rounding is used, it must be used consistently. Withheld tax amounts should be rounded to the nearest whole dollar by dropping amounts under 50 cents and increasing amounts from 50 to 99 cents to the next dollar. For example, $2.30 becomes $2 and $2.50 becomes $3.' That box is the ONLY place the word 'cents' appears anywhere in Publication 15-T (2026): a string search of the full document returns two hits, both inside that one sentence, and every other match is 'percent' or 'percentage'. It is permissive ('you may also') and it is to the nearest DOLLAR, never to the cent. PAGE 10 LINE 1g verbatim: '1g If the box in Step 2 of Form W-4 is checked, enter -0-. If the box is not checked, enter $12,900 if the taxpayer is married filing jointly or $8,600 otherwise'. PAGE 10 TABLE 3 verbatim: 'Semiannually 2 / Quarterly 4 / Monthly 12 / Semimonthly 24 / Biweekly 26 / Weekly 52 / Daily 260'. PAGE 12, '2026 Percentage Method Tables for Automated Payroll Systems and Withholding on Periodic Payments of Pensions and Annuities', columns 'A At least- / B But less than- / C The tentative amount to withhold is: / D Plus this percentage- / E of the amount that the Adjusted Annual Wage or Payment Amount exceeds-'. ROW USED, QUOTED NOT PARAPHRASED, 'Form W-4, Step 2, Checkbox, Withholding Rate Schedules' ('Use these if the Form W-4 is from 2020 or later and the box in Step 2 of Form W-4 IS checked.'), 'Single or Married Filing Separately': '$136,163       $328,350       $29,224.00             35%        $136,163'. HAND DERIVATION on the stated annual wage. 1b 26; 1c $145,000.00; 1d $0.00; 1e $145,000.00; 1f $0.00; 1g -0- (the box in Step 2 IS checked); 1h $0.00; 1i $145,000.00. 2a $145,000.00; 2b $136,163.00; 2c $29,224.00; 2d 35%; 2e $8,837.00; 2f 0.35 x $8,837.00 = $3,092.95; 2g $32,316.95. 2h EXACT $32,316.95 / 26 = $1,242.95961538461... ; 3a $1,000.00; 3b EXACT $1,000.00 / 26 = $38.46153846... ; 3c EXACT $31,316.95 / 26 = $1,204.49807692307... ; 4a $0.00; 4b EXACT $1,204.49807692307... . THE ONE ROUNDING: half-up to the cent $1,204.50; half-up to the dollar FROM THE SAME EXACT 4b, $1,204, because $1,204.4980... is under 50 cents. THE DEFECTIVE READING: 2h to $1,242.96 and 3b to $38.46 give the SAME cent figure, $1,204.50, and the dollar taken from that cent figure is $1,205 - one dollar too much, which is the defect ruling L-2026-4594 names. WHY THIS CASE ASSERTS ONLY THE MONEY AND NOT adjustedAnnualWage OR tentativeAnnualWithholding: the stated annual wage does not divide exactly into the pay period, and Publication 15-T does not say what a period's pay is for a stated annual salary - line 1a is 'the employee's total taxable wages this payroll period', an observed payroll amount. Both implementations in this repository round that per-period wage to the cent and then annualise at line 1c, which makes line 1i a few cents away from (annual wage - line 1g); that is a payroll-design question, not an IRS rounding question, and this hand derivation does not settle it, so it is not asserted here. The money figures below are unaffected: they are the same to the cent and to the dollar on either reading of line 1a.",
      "why": "WHOLE-DOLLAR RULE ONLY, WITH A LIVE LINE 3b - the hole this case was written to fill. The existing whole-dollar separator (L-2026-411: $670 a week, single, Step 2 box checked, where the IRS itself prints $62 and the cent-then-dollar reading gives $63) carries a ZERO credit, so it cannot show that the dollar is taken from the exact line 4b AFTER the credit rather than from the cent-rounded one. Here the two readings AGREE to the cent, $1,204.50, and split only at the dollar: $1,204 exact against $1,205 cent-then-dollar. An implementation that computes the dollar as round(round(4b, 2)) passes every cent assertion in this file and fails only here and at $670 weekly. It is also the only credit case on the Step 2 checkbox schedule above the 32% row.",
      "ledger": "L-2026-4594"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 75000,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "taxCredits": 5200,
          "step2Checkbox": false
        }
      },
      "period": "weekly",
      "expect": {
        "federalWithholding": 47.5,
        "federalWithholdingWholeDollar": 48
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-10-19. PAGE 10, WORKSHEET 1A, verbatim and complete as to rounding: '2e Subtract line 2b from line 2a'; '2f Multiply the amount on line 2e by the percentage on line 2d'; '2g Add lines 2c and 2f'; '2h Divide the amount on line 2g by the number of pay periods on line 1b. This is the Tentative Withholding Amount'; '3a If the employee's Form W-4 is from 2020 or later, enter the amount from Step 3 of that form; otherwise, enter -0-'; '3b Divide the amount on line 3a by the number of pay periods on line 1b'; '3c Subtract line 3b from line 2h. If zero or less, enter -0-'; '4a Enter the additional amount to withhold from the employee's Form W-4 (Step 4(c) of the 2020 or later form, or line 6 on earlier forms)'; '4b Add lines 3c and 4a. This is the amount to withhold from the employee's wages this pay period'. Lines 2e, 2f, 2g, 2h, 3b, 3c and 4b say only Subtract, Multiply, Add, Divide, Divide, Subtract and Add, and name NO rounding, so every one is carried exact. THE ONLY ROUNDING INSTRUCTION IN THE PUBLICATION is the page 9 box headed 'Rounding', verbatim: 'To figure the income tax to withhold, you may reduce the last digit of the wages to zero, or figure the wages to the nearest dollar. You may also round the tax for the pay period to the nearest dollar. If rounding is used, it must be used consistently. Withheld tax amounts should be rounded to the nearest whole dollar by dropping amounts under 50 cents and increasing amounts from 50 to 99 cents to the next dollar. For example, $2.30 becomes $2 and $2.50 becomes $3.' That box is the ONLY place the word 'cents' appears anywhere in Publication 15-T (2026): a string search of the full document returns two hits, both inside that one sentence, and every other match is 'percent' or 'percentage'. It is permissive ('you may also') and it is to the nearest DOLLAR, never to the cent. PAGE 10 LINE 1g verbatim: '1g If the box in Step 2 of Form W-4 is checked, enter -0-. If the box is not checked, enter $12,900 if the taxpayer is married filing jointly or $8,600 otherwise'. PAGE 10 TABLE 3 verbatim: 'Semiannually 2 / Quarterly 4 / Monthly 12 / Semimonthly 24 / Biweekly 26 / Weekly 52 / Daily 260'. PAGE 12, '2026 Percentage Method Tables for Automated Payroll Systems and Withholding on Periodic Payments of Pensions and Annuities', columns 'A At least- / B But less than- / C The tentative amount to withhold is: / D Plus this percentage- / E of the amount that the Adjusted Annual Wage or Payment Amount exceeds-'. ROW USED, QUOTED NOT PARAPHRASED, STANDARD Withholding Rate Schedules ('Use these if the Form W-4 is from 2019 or earlier, or if the Form W-4 is from 2020 or later and the box in Step 2 of Form W-4 is NOT checked. Also, use these for Form W-4P from any year.'), 'Single or Married Filing Separately': '$57,900      $113,200       $5,800.00             22%         $57,900'. HAND DERIVATION on the stated annual wage. 1b 52; 1c $75,000.00; 1d $0.00; 1e $75,000.00; 1f $0.00; 1g $8,600.00; 1h $8,600.00; 1i $66,400.00. 2a $66,400.00; 2b $57,900.00; 2c $5,800.00; 2d 22%; 2e $8,500.00; 2f 0.22 x $8,500.00 = $1,870.00; 2g $7,670.00. 2h EXACT $7,670.00 / 52 = $147.50; 3a $5,200.00; 3b EXACT $5,200.00 / 52 = $100.00; 3c EXACT $47.50; 4a $0.00; 4b EXACT $47.50. THE ONE ROUNDING: half-up to the cent leaves $47.50; half-up to the dollar gives $48, which is the page 9 box's own worked example read straight - 'For example, $2.30 becomes $2 and $2.50 becomes $3'. BOTH READINGS AGREE at $47.50 and $48. WHY THIS CASE ASSERTS ONLY THE MONEY AND NOT adjustedAnnualWage OR tentativeAnnualWithholding: the stated annual wage does not divide exactly into the pay period, and Publication 15-T does not say what a period's pay is for a stated annual salary - line 1a is 'the employee's total taxable wages this payroll period', an observed payroll amount. Both implementations in this repository round that per-period wage to the cent and then annualise at line 1c, which makes line 1i a few cents away from (annual wage - line 1g); that is a payroll-design question, not an IRS rounding question, and this hand derivation does not settle it, so it is not asserted here. The money figures below are unaffected: they are the same to the cent and to the dollar on either reading of line 1a.",
      "why": "THE HALF-UP DIRECTION AT THE DOLLAR BOUNDARY, in the only place it can be seen: an amount to withhold of exactly fifty cents. Half-up gives $48, per the page 9 box's own example '$2.50 becomes $3'; round-half-even, round-half-down and truncation all give $47, and every other case in the file passes under all four. It is also a nothing-moves case at the cent. ONE WARNING FOR A FUTURE READER, because it is the reason this case asserts only the money: $75,000 does not divide into 52 periods, so under the per-period-wage rounding both implementations use at line 1a the chain is $147.500508... rather than $147.50 - the answers are unchanged to the cent and to the dollar, but the exactness is not the engine's, it is the stated annual wage's. The fully exact nothing-moves case is the $108,960 semimonthly one. AND NOTE: case L-2026-412 ($52,000 biweekly, $2,000 credit, $79.23) is often described as the exact-quotient case and is NOT one - $2,060.00 / 26 = $79.230769..., so its two readings agree only because the residues happen to round the same way.",
      "ledger": "L-2026-4594"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 66504,
        "filingStatus": "single",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "taxCredits": 52,
          "step2Checkbox": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "federalWithholding": 239.54,
        "federalWithholdingWholeDollar": 240,
        "adjustedAnnualWage": 57904,
        "tentativeAnnualWithholding": 5800.88
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-10-19. PAGE 10, WORKSHEET 1A, verbatim and complete as to rounding: '2e Subtract line 2b from line 2a'; '2f Multiply the amount on line 2e by the percentage on line 2d'; '2g Add lines 2c and 2f'; '2h Divide the amount on line 2g by the number of pay periods on line 1b. This is the Tentative Withholding Amount'; '3a If the employee's Form W-4 is from 2020 or later, enter the amount from Step 3 of that form; otherwise, enter -0-'; '3b Divide the amount on line 3a by the number of pay periods on line 1b'; '3c Subtract line 3b from line 2h. If zero or less, enter -0-'; '4a Enter the additional amount to withhold from the employee's Form W-4 (Step 4(c) of the 2020 or later form, or line 6 on earlier forms)'; '4b Add lines 3c and 4a. This is the amount to withhold from the employee's wages this pay period'. Lines 2e, 2f, 2g, 2h, 3b, 3c and 4b say only Subtract, Multiply, Add, Divide, Divide, Subtract and Add, and name NO rounding, so every one is carried exact. THE ONLY ROUNDING INSTRUCTION IN THE PUBLICATION is the page 9 box headed 'Rounding', verbatim: 'To figure the income tax to withhold, you may reduce the last digit of the wages to zero, or figure the wages to the nearest dollar. You may also round the tax for the pay period to the nearest dollar. If rounding is used, it must be used consistently. Withheld tax amounts should be rounded to the nearest whole dollar by dropping amounts under 50 cents and increasing amounts from 50 to 99 cents to the next dollar. For example, $2.30 becomes $2 and $2.50 becomes $3.' That box is the ONLY place the word 'cents' appears anywhere in Publication 15-T (2026): a string search of the full document returns two hits, both inside that one sentence, and every other match is 'percent' or 'percentage'. It is permissive ('you may also') and it is to the nearest DOLLAR, never to the cent. PAGE 10 LINE 1g verbatim: '1g If the box in Step 2 of Form W-4 is checked, enter -0-. If the box is not checked, enter $12,900 if the taxpayer is married filing jointly or $8,600 otherwise'. PAGE 10 TABLE 3 verbatim: 'Semiannually 2 / Quarterly 4 / Monthly 12 / Semimonthly 24 / Biweekly 26 / Weekly 52 / Daily 260'. PAGE 12, '2026 Percentage Method Tables for Automated Payroll Systems and Withholding on Periodic Payments of Pensions and Annuities', columns 'A At least- / B But less than- / C The tentative amount to withhold is: / D Plus this percentage- / E of the amount that the Adjusted Annual Wage or Payment Amount exceeds-'. ROW USED, QUOTED NOT PARAPHRASED, STANDARD Withholding Rate Schedules ('Use these if the Form W-4 is from 2019 or earlier, or if the Form W-4 is from 2020 or later and the box in Step 2 of Form W-4 is NOT checked. Also, use these for Form W-4P from any year.'), 'Single or Married Filing Separately': '$57,900      $113,200       $5,800.00             22%         $57,900'. HAND DERIVATION. 1a $66,504.00 / 24 = $2,771.00 EXACTLY; 1b 24; 1c $66,504.00; 1d $0.00; 1e $66,504.00; 1f $0.00; 1g $8,600.00; 1h $8,600.00; 1i $57,904.00. 2a $57,904.00; 2b $57,900.00; 2c $5,800.00; 2d 22%; 2e $4.00; 2f 0.22 x $4.00 = $0.88 EXACTLY; 2g $5,800.88. 2h EXACT $5,800.88 / 24 = $241.70333333... ; 3a $52.00; 3b EXACT $52.00 / 24 = $2.16666666... ; 3c EXACT $5,748.88 / 24 = $239.53666666... ; 4a $0.00; 4b EXACT $239.53666666... . THE ONE ROUNDING: half-up to the cent $239.54; half-up to the dollar $240. THE DEFECTIVE READING: 2h to $241.70 and 3b to $2.17, then $241.70 - $2.17 = $239.53. THIS CASE IS IMMUNE TO THE TWO IMPLEMENTATION CHOICES THAT PUBLICATION 15-T DOES NOT SETTLE, which is why it asserts the annual lines as well as the money: the stated annual wage divides exactly into the pay period at line 1a, so it does not matter whether an implementation rounds the per-period wage to the cent before annualising at line 1c; and line 2f comes out exact to the cent, so it does not matter whether an implementation rounds line 2g. Only the line 2h / 3b / 4b rounding can move this answer.",
      "why": "ROUNDING RULE, UP AT THE CENT, AND IT IS THE CASE THAT CANNOT BE EXPLAINED AWAY. The $45,000 weekly UP case can only assert the money, because $45,000 does not divide into 52 periods and the two implementations round the per-period wage at line 1a. Here $66,504 / 24 = $2,771.00 exactly and line 2f is $0.88 exactly, so NOTHING in the chain before line 2h is approximate: the only thing that can move this answer is the 2h / 3b / 4b rounding, and it moves it by a cent. It also sits four dollars into the 22% row, so it doubles as the tightest band-edge probe in the credit set.",
      "ledger": "L-2026-4594"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 32232,
        "filingStatus": "single",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "taxCredits": 500,
          "step2Checkbox": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "federalWithholding": 49.49,
        "federalWithholdingWholeDollar": 49,
        "adjustedAnnualWage": 23632,
        "tentativeAnnualWithholding": 1687.84
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-10-19. PAGE 10, WORKSHEET 1A, verbatim and complete as to rounding: '2e Subtract line 2b from line 2a'; '2f Multiply the amount on line 2e by the percentage on line 2d'; '2g Add lines 2c and 2f'; '2h Divide the amount on line 2g by the number of pay periods on line 1b. This is the Tentative Withholding Amount'; '3a If the employee's Form W-4 is from 2020 or later, enter the amount from Step 3 of that form; otherwise, enter -0-'; '3b Divide the amount on line 3a by the number of pay periods on line 1b'; '3c Subtract line 3b from line 2h. If zero or less, enter -0-'; '4a Enter the additional amount to withhold from the employee's Form W-4 (Step 4(c) of the 2020 or later form, or line 6 on earlier forms)'; '4b Add lines 3c and 4a. This is the amount to withhold from the employee's wages this pay period'. Lines 2e, 2f, 2g, 2h, 3b, 3c and 4b say only Subtract, Multiply, Add, Divide, Divide, Subtract and Add, and name NO rounding, so every one is carried exact. THE ONLY ROUNDING INSTRUCTION IN THE PUBLICATION is the page 9 box headed 'Rounding', verbatim: 'To figure the income tax to withhold, you may reduce the last digit of the wages to zero, or figure the wages to the nearest dollar. You may also round the tax for the pay period to the nearest dollar. If rounding is used, it must be used consistently. Withheld tax amounts should be rounded to the nearest whole dollar by dropping amounts under 50 cents and increasing amounts from 50 to 99 cents to the next dollar. For example, $2.30 becomes $2 and $2.50 becomes $3.' That box is the ONLY place the word 'cents' appears anywhere in Publication 15-T (2026): a string search of the full document returns two hits, both inside that one sentence, and every other match is 'percent' or 'percentage'. It is permissive ('you may also') and it is to the nearest DOLLAR, never to the cent. PAGE 10 LINE 1g verbatim: '1g If the box in Step 2 of Form W-4 is checked, enter -0-. If the box is not checked, enter $12,900 if the taxpayer is married filing jointly or $8,600 otherwise'. PAGE 10 TABLE 3 verbatim: 'Semiannually 2 / Quarterly 4 / Monthly 12 / Semimonthly 24 / Biweekly 26 / Weekly 52 / Daily 260'. PAGE 12, '2026 Percentage Method Tables for Automated Payroll Systems and Withholding on Periodic Payments of Pensions and Annuities', columns 'A At least- / B But less than- / C The tentative amount to withhold is: / D Plus this percentage- / E of the amount that the Adjusted Annual Wage or Payment Amount exceeds-'. ROW USED, QUOTED NOT PARAPHRASED, STANDARD Withholding Rate Schedules ('Use these if the Form W-4 is from 2019 or earlier, or if the Form W-4 is from 2020 or later and the box in Step 2 of Form W-4 is NOT checked. Also, use these for Form W-4P from any year.'), 'Single or Married Filing Separately': '$19,900       $57,900       $1,240.00             12%         $19,900'. HAND DERIVATION. 1a $32,232.00 / 24 = $1,343.00 EXACTLY; 1b 24; 1c $32,232.00; 1d $0.00; 1e $32,232.00; 1f $0.00; 1g $8,600.00; 1h $8,600.00; 1i $23,632.00. 2a $23,632.00; 2b $19,900.00; 2c $1,240.00; 2d 12%; 2e $3,732.00; 2f 0.12 x $3,732.00 = $447.84 EXACTLY; 2g $1,687.84. 2h EXACT $1,687.84 / 24 = $70.32666666... ; 3a $500.00; 3b EXACT $500.00 / 24 = $20.83333333... ; 3c EXACT $1,187.84 / 24 = $49.49333333... ; 4a $0.00; 4b EXACT $49.49333333... . THE ONE ROUNDING: half-up to the cent $49.49; half-up to the dollar FROM THE SAME EXACT 4b, $49, because $49.4933... is under 50 cents and the page 9 box drops it. THE DEFECTIVE READING: 2h to $70.33 and 3b to $20.83, then $70.33 - $20.83 = $49.50, and taking the dollar from THAT gives $50 - a cent and a dollar both wrong. THIS CASE IS IMMUNE TO THE TWO IMPLEMENTATION CHOICES THAT PUBLICATION 15-T DOES NOT SETTLE, which is why it asserts the annual lines as well as the money: the stated annual wage divides exactly into the pay period at line 1a, so it does not matter whether an implementation rounds the per-period wage to the cent before annualising at line 1c; and line 2f comes out exact to the cent, so it does not matter whether an implementation rounds line 2g. Only the line 2h / 3b / 4b rounding can move this answer.",
      "why": "THE CASE THAT SPLITS BOTH WINDOWS AT ONCE, with a live line 3b and immune to every implementation choice the publication leaves open. Exact gives $49.49 / $49; cent-first gives $49.50 / $50. It is the only case in the file where the defective reading is wrong in the cent AND wrong in the dollar AND the error cannot be attributed to line 1a or line 2g, because $32,232 / 24 = $1,343.00 and line 2f is $447.84, both exact. It is also the clearest demonstration of the second half of ruling L-2026-4594: the whole-dollar figure is taken from the exact line 4b, never from the cent-rounded one, and here the cent-rounded one crosses the 50-cent boundary that the exact figure does not reach.",
      "ledger": "L-2026-4594"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 108960,
        "filingStatus": "single",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "w4": {
          "taxCredits": 480,
          "step2Checkbox": true
        }
      },
      "period": "semimonthly",
      "expect": {
        "federalWithholding": 834.96,
        "federalWithholdingWholeDollar": 835,
        "adjustedAnnualWage": 108960,
        "tentativeAnnualWithholding": 20519.04
      },
      "source": "IRS Publication 15-T (2026), Federal Income Tax Withholding Methods, https://www.irs.gov/pub/irs-pdf/p15t.pdf, retrieved 2026-10-19. PAGE 10, WORKSHEET 1A, verbatim and complete as to rounding: '2e Subtract line 2b from line 2a'; '2f Multiply the amount on line 2e by the percentage on line 2d'; '2g Add lines 2c and 2f'; '2h Divide the amount on line 2g by the number of pay periods on line 1b. This is the Tentative Withholding Amount'; '3a If the employee's Form W-4 is from 2020 or later, enter the amount from Step 3 of that form; otherwise, enter -0-'; '3b Divide the amount on line 3a by the number of pay periods on line 1b'; '3c Subtract line 3b from line 2h. If zero or less, enter -0-'; '4a Enter the additional amount to withhold from the employee's Form W-4 (Step 4(c) of the 2020 or later form, or line 6 on earlier forms)'; '4b Add lines 3c and 4a. This is the amount to withhold from the employee's wages this pay period'. Lines 2e, 2f, 2g, 2h, 3b, 3c and 4b say only Subtract, Multiply, Add, Divide, Divide, Subtract and Add, and name NO rounding, so every one is carried exact. THE ONLY ROUNDING INSTRUCTION IN THE PUBLICATION is the page 9 box headed 'Rounding', verbatim: 'To figure the income tax to withhold, you may reduce the last digit of the wages to zero, or figure the wages to the nearest dollar. You may also round the tax for the pay period to the nearest dollar. If rounding is used, it must be used consistently. Withheld tax amounts should be rounded to the nearest whole dollar by dropping amounts under 50 cents and increasing amounts from 50 to 99 cents to the next dollar. For example, $2.30 becomes $2 and $2.50 becomes $3.' That box is the ONLY place the word 'cents' appears anywhere in Publication 15-T (2026): a string search of the full document returns two hits, both inside that one sentence, and every other match is 'percent' or 'percentage'. It is permissive ('you may also') and it is to the nearest DOLLAR, never to the cent. PAGE 10 LINE 1g verbatim: '1g If the box in Step 2 of Form W-4 is checked, enter -0-. If the box is not checked, enter $12,900 if the taxpayer is married filing jointly or $8,600 otherwise'. PAGE 10 TABLE 3 verbatim: 'Semiannually 2 / Quarterly 4 / Monthly 12 / Semimonthly 24 / Biweekly 26 / Weekly 52 / Daily 260'. PAGE 12, '2026 Percentage Method Tables for Automated Payroll Systems and Withholding on Periodic Payments of Pensions and Annuities', columns 'A At least- / B But less than- / C The tentative amount to withhold is: / D Plus this percentage- / E of the amount that the Adjusted Annual Wage or Payment Amount exceeds-'. ROW USED, QUOTED NOT PARAPHRASED, 'Form W-4, Step 2, Checkbox, Withholding Rate Schedules' ('Use these if the Form W-4 is from 2020 or later and the box in Step 2 of Form W-4 IS checked.'), 'Single or Married Filing Separately': '$108,938       $136,163       $20,512.00             32%        $108,938'. HAND DERIVATION. 1a $108,960.00 / 24 = $4,540.00 EXACTLY; 1b 24; 1c $108,960.00; 1d $0.00; 1e $108,960.00; 1f $0.00; 1g -0- (box checked); 1h $0.00; 1i $108,960.00. 2a $108,960.00; 2b $108,938.00; 2c $20,512.00; 2d 32%; 2e $22.00; 2f 0.32 x $22.00 = $7.04 EXACTLY; 2g $20,519.04. 2h EXACT $20,519.04 / 24 = $854.96, ALREADY EXACT TO THE CENT. 3a $480.00; 3b EXACT $480.00 / 24 = $20.00, ALSO EXACT TO THE CENT. 3c EXACT $834.96. 4a $0.00; 4b EXACT $834.96. THE ONE ROUNDING: half-up to the cent leaves $834.96 untouched; half-up to the dollar gives $835. BOTH READINGS AGREE: rounding 2h and 3b to the cent first also gives $854.96 - $20.00 = $834.96 and $835. THIS CASE IS IMMUNE TO THE TWO IMPLEMENTATION CHOICES THAT PUBLICATION 15-T DOES NOT SETTLE, which is why it asserts the annual lines as well as the money: the stated annual wage divides exactly into the pay period at line 1a, so it does not matter whether an implementation rounds the per-period wage to the cent before annualising at line 1c; and line 2f comes out exact to the cent, so it does not matter whether an implementation rounds line 2g. Only the line 2h / 3b / 4b rounding can move this answer.",
      "why": "NOTHING MOVES, AND IT MUST NOT. A credit case built so that EVERY line of Worksheet 1A is exact to the cent before any rounding: $108,960 / 24 = $4,540.00, line 2f = $7.04, line 2h = $854.96 and line 3b = $20.00 all terminate. So the exact reading and the cent-first reading give the identical answer, and so does any implementation that rounds line 1a or line 2g. It pins the other side of ruling L-2026-4594: the one rounding is half-up AT THE END, not a licence to move a figure that was already exact, and a fix for the UP and DOWN cases must not drift this one. A vector that pins 'this one must NOT change' is worth as much as one that pins a change. It also sits $22 into the 32% row of the checkbox schedule.",
      "ledger": "L-2026-4594"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 65000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "payFrequency": "weekly"
      },
      "expect": {
        "stateIncomeTax": 3217.76
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', Steps 1-4: 'Step 1 Determine the wages paid'; 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.'; Step 2c adds them and 2d divides by the number of pay periods; 'Step 3 Subtract the amount exempt from withholding'; 'Step 4 Multiply by the tax rate x .0495'. IL-700-T states no rounding rule; the site's house convention carries every intermediate exactly and rounds once, at the pay, half-up to the cent. Hand derivation (weekly, 52 pay periods): Step 1 wages $65,000.00 / 52 = $1,250.00; Step 2a 0 x $2,925 = $0; Step 2b 0 x $1,000 = $0; Step 2c $0; Step 2d $0 / 52 = $0.00; Step 3 taxable $1,250.000000; Step 4 x .0495 = $61.875000, rounded once half-up to $61.88 a pay. Year = per-pay figure x pay periods: $61.88 x 52 = $3,217.76 (the unrounded annual .0495 x $65,000.00 = $3,217.5000).",
      "why": "$65,000.00 with no allowances paid weekly: Illinois withholds per pay, so the year is $61.88 x 52 = $3,217.76, not the annual-formula $3,217.50; this straddles the fork between the per-pay figure and a year asked for with no pay frequency.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 65000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "payFrequency": "biweekly"
      },
      "expect": {
        "stateIncomeTax": 3217.5
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', Steps 1-4: 'Step 1 Determine the wages paid'; 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.'; Step 2c adds them and 2d divides by the number of pay periods; 'Step 3 Subtract the amount exempt from withholding'; 'Step 4 Multiply by the tax rate x .0495'. IL-700-T states no rounding rule; the site's house convention carries every intermediate exactly and rounds once, at the pay, half-up to the cent. Hand derivation (biweekly, 26 pay periods): Step 1 wages $65,000.00 / 26 = $2,500.00; Step 2a 0 x $2,925 = $0; Step 2b 0 x $1,000 = $0; Step 2c $0; Step 2d $0 / 26 = $0.00; Step 3 taxable $2,500.000000; Step 4 x .0495 = $123.750000, rounded once half-up to $123.75 a pay. Year = per-pay figure x pay periods: $123.75 x 26 = $3,217.50 (the unrounded annual .0495 x $65,000.00 = $3,217.5000).",
      "why": "$65,000.00 with no allowances paid biweekly: Illinois withholds per pay, so the year is $123.75 x 26 = $3,217.50, which at biweekly happens to equal the annual-formula $3,217.50; this straddles the fork between the per-pay figure and a year asked for with no pay frequency.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 65000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "payFrequency": "semimonthly"
      },
      "expect": {
        "stateIncomeTax": 3217.44
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', Steps 1-4: 'Step 1 Determine the wages paid'; 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.'; Step 2c adds them and 2d divides by the number of pay periods; 'Step 3 Subtract the amount exempt from withholding'; 'Step 4 Multiply by the tax rate x .0495'. IL-700-T states no rounding rule; the site's house convention carries every intermediate exactly and rounds once, at the pay, half-up to the cent. Hand derivation (semimonthly, 24 pay periods): Step 1 wages $65,000.00 / 24 = $2,708.333333 (carried exactly: $65,000.00/24); Step 2a 0 x $2,925 = $0; Step 2b 0 x $1,000 = $0; Step 2c $0; Step 2d $0 / 24 = $0.00; Step 3 taxable $2,708.333333; Step 4 x .0495 = $134.062500, rounded once half-up to $134.06 a pay. Year = per-pay figure x pay periods: $134.06 x 24 = $3,217.44 (the unrounded annual .0495 x $65,000.00 = $3,217.5000).",
      "why": "$65,000.00 with no allowances paid semimonthly: Illinois withholds per pay, so the year is $134.06 x 24 = $3,217.44, not the annual-formula $3,217.50; this straddles the fork between the per-pay figure and a year asked for with no pay frequency.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 65000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "payFrequency": "monthly"
      },
      "expect": {
        "stateIncomeTax": 3217.56
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', Steps 1-4: 'Step 1 Determine the wages paid'; 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.'; Step 2c adds them and 2d divides by the number of pay periods; 'Step 3 Subtract the amount exempt from withholding'; 'Step 4 Multiply by the tax rate x .0495'. IL-700-T states no rounding rule; the site's house convention carries every intermediate exactly and rounds once, at the pay, half-up to the cent. Hand derivation (monthly, 12 pay periods): Step 1 wages $65,000.00 / 12 = $5,416.666667 (carried exactly: $65,000.00/12); Step 2a 0 x $2,925 = $0; Step 2b 0 x $1,000 = $0; Step 2c $0; Step 2d $0 / 12 = $0.00; Step 3 taxable $5,416.666667; Step 4 x .0495 = $268.125000, rounded once half-up to $268.13 a pay. Year = per-pay figure x pay periods: $268.13 x 12 = $3,217.56 (the unrounded annual .0495 x $65,000.00 = $3,217.5000).",
      "why": "$65,000.00 with no allowances paid monthly: Illinois withholds per pay, so the year is $268.13 x 12 = $3,217.56, not the annual-formula $3,217.50; this straddles the fork between the per-pay figure and a year asked for with no pay frequency.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 65000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "payFrequency": "weekly",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "expect": {
        "stateIncomeTax": 3072.68
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', Steps 1-4: 'Step 1 Determine the wages paid'; 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.'; Step 2c adds them and 2d divides by the number of pay periods; 'Step 3 Subtract the amount exempt from withholding'; 'Step 4 Multiply by the tax rate x .0495'. IL-700-T states no rounding rule; the site's house convention carries every intermediate exactly and rounds once, at the pay, half-up to the cent. Hand derivation (weekly, 52 pay periods): Step 1 wages $65,000.00 / 52 = $1,250.00; Step 2a 1 x $2,925 = $2,925; Step 2b 0 x $1,000 = $0; Step 2c $2,925; Step 2d $2,925 / 52 = $56.250000 (carried exactly: 2,925/52); Step 3 taxable $1,193.750000 (exactly ($65,000.00 - $2,925.00)/52); Step 4 x .0495 = $59.090625, rounded once half-up to $59.09 a pay. Year = per-pay figure x pay periods: $59.09 x 52 = $3,072.68 (the unrounded annual .0495 x $62,075.00 = $3,072.7125).",
      "why": "$65,000.00 with one Line 1 allowance paid weekly: Illinois withholds per pay, so the year is $59.09 x 52 = $3,072.68, not the annual-formula $3,072.71; this straddles the fork between the per-pay figure and a year asked for with no pay frequency.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 65000,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "payFrequency": "monthly",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "expect": {
        "stateIncomeTax": 3072.72
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', Steps 1-4: 'Step 1 Determine the wages paid'; 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.'; Step 2c adds them and 2d divides by the number of pay periods; 'Step 3 Subtract the amount exempt from withholding'; 'Step 4 Multiply by the tax rate x .0495'. IL-700-T states no rounding rule; the site's house convention carries every intermediate exactly and rounds once, at the pay, half-up to the cent. Hand derivation (monthly, 12 pay periods): Step 1 wages $65,000.00 / 12 = $5,416.666667 (carried exactly: $65,000.00/12); Step 2a 1 x $2,925 = $2,925; Step 2b 0 x $1,000 = $0; Step 2c $2,925; Step 2d $2,925 / 12 = $243.750000 (carried exactly: 2,925/12); Step 3 taxable $5,172.916667 (exactly ($65,000.00 - $2,925.00)/12); Step 4 x .0495 = $256.059375, rounded once half-up to $256.06 a pay. Year = per-pay figure x pay periods: $256.06 x 12 = $3,072.72 (the unrounded annual .0495 x $62,075.00 = $3,072.7125).",
      "why": "$65,000.00 with one Line 1 allowance paid monthly: Illinois withholds per pay, so the year is $256.06 x 12 = $3,072.72, not the annual-formula $3,072.71; this straddles the fork between the per-pay figure and a year asked for with no pay frequency.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "annual",
      "input": {
        "gross": 35082.4,
        "filingStatus": "single",
        "taxYear": "2026",
        "state": "IL",
        "payFrequency": "monthly"
      },
      "expect": {
        "stateIncomeTax": 1736.52
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', Steps 1-4: 'Step 1 Determine the wages paid'; 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.'; Step 2c adds them and 2d divides by the number of pay periods; 'Step 3 Subtract the amount exempt from withholding'; 'Step 4 Multiply by the tax rate x .0495'. IL-700-T states no rounding rule; the site's house convention carries every intermediate exactly and rounds once, at the pay, half-up to the cent. Hand derivation (monthly, 12 pay periods): Step 1 wages $35,082.40 / 12 = $2,923.533333 (carried exactly: $35,082.40/12); Step 2a 0 x $2,925 = $0; Step 2b 0 x $1,000 = $0; Step 2c $0; Step 2d $0 / 12 = $0.00; Step 3 taxable $2,923.533333; Step 4 x .0495 = $144.714900, rounded once half-up to $144.71 a pay. Year = per-pay figure x pay periods: $144.71 x 12 = $1,736.52 (the unrounded annual .0495 x $35,082.40 = $1,736.5788).",
      "why": "$35,082.40 with no allowances paid monthly: Illinois withholds per pay, so the year is $144.71 x 12 = $1,736.52, not the annual-formula $1,736.58; this straddles the fork between the per-pay figure and a year asked for with no pay frequency.",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 65000,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "IL"
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 61.88
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', Steps 1-4: 'Step 1 Determine the wages paid'; 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.'; Step 2c adds them and 2d divides by the number of pay periods; 'Step 3 Subtract the amount exempt from withholding'; 'Step 4 Multiply by the tax rate x .0495'. IL-700-T states no rounding rule; the site's house convention carries every intermediate exactly and rounds once, at the pay, half-up to the cent. Hand derivation (weekly, 52 pay periods): Step 1 wages $65,000.00 / 52 = $1,250.00; Step 2a 0 x $2,925 = $0; Step 2b 0 x $1,000 = $0; Step 2c $0; Step 2d $0 / 52 = $0.00; Step 3 taxable $1,250.000000; Step 4 x .0495 = $61.875000, rounded once half-up to $61.88 a pay. One pay only (payment 1 of the year); nothing about Illinois withholding is cumulative.",
      "why": "one weekly pay on $65,000.00 with no allowances: the per-pay withholding the annual weekly case multiplies by 52, pinned on the paycheck itself ($61.875000 rounds half-up to $61.88).",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 35082.4,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "IL"
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 144.71
      },
      "source": "Illinois Department of Revenue, 2026 Booklet IL-700-T, 'Illinois Withholding Tax Tables', Booklet IL-700-T (R-12/25), effective January 1, 2026, https://tax.illinois.gov/content/dam/soi/en/web/tax/forms/withholding/documents/currentyear/il-700-t.pdf, retrieved 2026-10-15. Verbatim, page 3: 'The income tax rate is 4.95 percent and the exemption allowance is $2,925.' Page 4, 'Automated payroll method': 'Tax withheld = .0495 x ( Wages - ((Line 1 allowances x $2,925) + (Line 2 allowances x $1,000)) / number of pay periods in a year )', Steps 1-4: 'Step 1 Determine the wages paid'; 'Step 2a Multiply the number of allowances your employee claimed on Form IL-W-4, Line 1, by $2,925. b Multiply the number of allowances your employee claimed on Form IL-W-4, Line 2, by $1,000.'; Step 2c adds them and 2d divides by the number of pay periods; 'Step 3 Subtract the amount exempt from withholding'; 'Step 4 Multiply by the tax rate x .0495'. IL-700-T states no rounding rule; the site's house convention carries every intermediate exactly and rounds once, at the pay, half-up to the cent. Hand derivation (monthly, 12 pay periods): Step 1 wages $35,082.40 / 12 = $2,923.533333 (carried exactly: $35,082.40/12); Step 2a 0 x $2,925 = $0; Step 2b 0 x $1,000 = $0; Step 2c $0; Step 2d $0 / 12 = $0.00; Step 3 taxable $2,923.533333; Step 4 x .0495 = $144.714900, rounded once half-up to $144.71 a pay. One pay only (payment 1 of the year); nothing about Illinois withholding is cumulative.",
      "why": "one monthly pay on $35,082.40 with no allowances: the per-pay withholding the annual monthly case multiplies by 12, pinned on the paycheck itself ($144.714900 rounds half-up to $144.71).",
      "ledger": "L-2026-4372"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "VA"
      },
      "period": "monthly",
      "expect": {
        "stateIncomeTaxWithheld": 224.11
      },
      "source": "Virginia Department of Taxation, Employer Withholding Instructions Rev. 10/26, p.21 (A = G x P on the pay), https://www.tax.virginia.gov/withholding-tax, retrieved 2026-10-18; 23VAC10-140-40 (W computed on the annualised wage, then W / P), https://law.lis.virginia.gov/admincode/title23/agency10/chapter140/section40/. Hand derivation: G = $5,000.00 a month, P = 12, A = G x P = $60,000; less the formula's $8,750 standard deduction and no exemptions (no certificate, E1 = E2 = 0) leaves $51,250; W = 720 + 0.0575 x 34,250 = $2,689.375, carried exactly (RULING L-2026-4949: no intermediate rounding is authorised); W / 12 = $224.1146 -> $224.11. Rulings L-2026-4948, L-2026-4756, L-2026-4949.",
      "why": "Virginia's step order on a pay cheque: annual W carried to the fraction of a cent, divided once by the pay periods and rounded once. Rounding W to the cent first gives 224.12, the figure L-2026-4756 ruled wrong.",
      "ledger": "L-2026-4756"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 150000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "VA"
      },
      "period": "monthly",
      "expect": {
        "stateIncomeTaxWithheld": 655.36
      },
      "source": "Virginia Department of Taxation, Employer Withholding Instructions Rev. 10/26, p.21 (A = G x P on the pay), https://www.tax.virginia.gov/withholding-tax, retrieved 2026-10-18; 23VAC10-140-40 (W computed on the annualised wage, then W / P), https://law.lis.virginia.gov/admincode/title23/agency10/chapter140/section40/. Hand derivation: G = $12,500.00 a month, A = $150,000; less the formula's $8,750 standard deduction and no exemptions (no certificate, E1 = E2 = 0) leaves $141,250; W = 720 + 0.0575 x 124,250 = $7,864.375, carried exactly; W / 12 = $655.3646 -> $655.36. Rulings L-2026-4948, L-2026-4756, L-2026-4949.",
      "why": "the same Virginia rounding edge on a married-filing-jointly employee: W ends in an exact half-cent-and-a-quarter fraction, and carrying it exactly gives 655.36 where W-then-pay rounding gives 655.37.",
      "ledger": "L-2026-4756"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 134275.57,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "CA"
      },
      "period": "monthly",
      "expect": {
        "stateOtherEmployeeDeductions": 145.47
      },
      "source": "California EDD, Payroll Taxes - Rates and Withholding, https://edd.ca.gov/en/payroll_taxes/rates_and_withholding/, retrieved 2026-10-18: 'The SDI withholding rate for 2026 is 1.3 percent. Effective January 1, 2024, all wages are subject to SDI contributions.' Second source DE 44 Rev. 52 p.ii 'no SDI taxable wage limit'. Hand derivation: pay = $134,275.57 / 12 = $11,189.6308 -> $11,189.63; SDI = 1.3% x $11,189.63 = $145.4651 -> $145.47. SDI is the only California employee payroll deduction in rules/us/states/ca.json, so stateOtherEmployeeDeductions is SDI alone.",
      "why": "California SDI is charged on each pay's wages and rounded once at the pay, not as a twelfth of the year's figure; at this wage the two differ by a cent.",
      "ledger": "L-2026-4573"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 225000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "CA"
      },
      "period": "biweekly",
      "expect": {
        "stateIncomeTaxWithheld": 712.17
      },
      "source": "California EDD, California Withholding Schedules for 2026, Method B - Exact Calculation Method (26methb), https://edd.ca.gov/siteassets/files/pdf_pub_ctr/26methb.pdf, retrieved 2026-10-25. p.4 note: an employer may annualise the period wage (P x pay) and divide the annual result by P, to the cent. Pay = $225,000 / 26 = $8,653.846 -> $8,653.85; annual route on 26 x $8,653.85, never on the year gross (RULING L-2026-4947); result $712.17 (the period Tables 5-28 give $712.20). Annual rate rows, standard deduction and credit second-reviewed under L-2026-4570; multiplier 1.1 under L-2026-4571.",
      "why": "California's annualising route must start from the pay cheque the reader actually receives (26 x $8,653.85), not from the stated year gross.",
      "ledger": "L-2026-4570"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 169813.13,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "CA"
      },
      "period": "monthly",
      "expect": {
        "stateIncomeTaxWithheld": 1072.55
      },
      "source": "California EDD, California Withholding Schedules for 2026, Method B - Exact Calculation Method (26methb), https://edd.ca.gov/siteassets/files/pdf_pub_ctr/26methb.pdf, retrieved 2026-10-25. Annual route on 12 x pay ($14,151.09 x 12 = $169,813.08): annual tax $12,870.659084 carried exactly (RULING L-2026-4949: no intermediate rounding is authorised), / 12 = $1,072.5549 -> $1,072.55; rounding the annual tax to the cent first gives $1,072.56 (RULING L-2026-4947 p.4 note). Annual rate rows second-reviewed under L-2026-4570; multiplier 1.1 under L-2026-4571.",
      "why": "the California tie that L-2026-4949 settles: the annual tax is carried exactly and rounded once at the pay, which moves this pay by a cent.",
      "ledger": "L-2026-4570"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25524,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "AR",
        "stateCertificate": {}
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 36.5
      },
      "source": "Arkansas Department of Finance and Administration, Withholding Tax Formula Method, Effective 01/01/2026, https://www.dfa.arkansas.gov/wp-content/uploads/Withholding-Tax-Formula.pdf, retrieved 2026-10-22. Page 4, 'EXAMPLE Computer Formula', verbatim: 'Gary works for a company and is paid on a monthly basis. He earns $2,127 per pay period. He has claimed 2 exemptions on his AR4EC.' 'Monthly Pay = $ 2,127.00 Number of Monthly Pay Periods = x 12 Annual Gross Pay = $ 25,524.00' 'Annual Gross Pay = $ 25,524.00 Less Standard Deduction = – 2,470.00 Net Taxable Income = $ 23,054.00' 'Since the Net Taxable Income is less than $97,701, we will take the income to the $50 midrange of $23,050.00 (midrange of $23,000.00 and $23,100.00).' 'Net Taxable Income = $ 23,050.00 Multiply by 3.4% (.034) = x 3.4% = $ 783.70 Less bracket adjustment amount = – 287.97 Annual Gross Tax = $495.73 (Rounded to $496.00)' 'Annual Gross Tax = $ 496.00 Less Personal Credit (2 X $29) = – 58.00 Annual Net Tax = $ 438.00' 'Annual Net Tax = $ 438.00 Divided by Number of Pay Periods = ÷ 12 State Withholding Amount = $ 36.50' The annual gross here is the example's own: $2,127.00 x 12 = $25,524.00. The 2 exemptions are AR4EC line 1b ('You claim yourself and your spouse. (Enter two exemptions)'), which the engine derives from married filing jointly with no dependents.",
      "why": "the Arkansas DFA's own printed answer, to the penny: annualise x12, less the $2,470 standard deduction, the $50 midrange lookup ($23,054 -> $23,050), 3.4% less the $287.97 adjustment, the round to the whole dollar ($495.73 -> $496), two $29 credits, then /12 = $36.50. By hand: 25,524 - 2,470 = 23,054 -> 23,050; x 0.034 = 783.70; - 287.97 = 495.73 -> 496; - 58 = 438; / 12 = 36.50.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 13.88
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Single Taxpayer – 1 Allowance 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $21,750.00 ($741.00 + $91.00 [$1,750.00 x 5.20%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax By the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -3,250.00 $21,750.00 832.00 -110.00 $722.00 $13.88 $27.77 $30.08 $60.17'",
      "why": "Delaware's own printed per-period figure for the weekly column of its single / 1-allowance example. By hand: 25,000 - 3,250 = 21,750; 741 + 1,750 x 5.20% (91) = 832; - 1 x 110 = 722; 722 / 52 = 13.8846 -> $13.88. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for weekly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 27.77
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Single Taxpayer – 1 Allowance 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $21,750.00 ($741.00 + $91.00 [$1,750.00 x 5.20%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax By the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -3,250.00 $21,750.00 832.00 -110.00 $722.00 $13.88 $27.77 $30.08 $60.17'",
      "why": "Delaware's own printed per-period figure for the biweekly column of its single / 1-allowance example. By hand: 25,000 - 3,250 = 21,750; 741 + 1,750 x 5.20% (91) = 832; - 1 x 110 = 722; 722 / 26 = 27.7692 -> $27.77. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for biweekly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "single",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "period": "semimonthly",
      "expect": {
        "stateWithholding": 30.08
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Single Taxpayer – 1 Allowance 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $21,750.00 ($741.00 + $91.00 [$1,750.00 x 5.20%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax By the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -3,250.00 $21,750.00 832.00 -110.00 $722.00 $13.88 $27.77 $30.08 $60.17'",
      "why": "Delaware's own printed per-period figure for the semimonthly column of its single / 1-allowance example. By hand: 25,000 - 3,250 = 21,750; 741 + 1,750 x 5.20% (91) = 832; - 1 x 110 = 722; 722 / 24 = 30.0833 -> $30.08. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for semimonthly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 1
        }
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 60.17
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Single Taxpayer – 1 Allowance 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $21,750.00 ($741.00 + $91.00 [$1,750.00 x 5.20%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax By the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -3,250.00 $21,750.00 832.00 -110.00 $722.00 $13.88 $27.77 $30.08 $60.17'",
      "why": "Delaware's own printed per-period figure for the monthly column of its single / 1-allowance example. By hand: 25,000 - 3,250 = 21,750; 741 + 1,750 x 5.20% (91) = 832; - 1 x 110 = 722; 722 / 12 = 60.1667 -> $60.17. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for monthly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 3
        }
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 6.52
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Married Taxpayer filing Joint Return – 3 Allowances 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $18,500.00 ($261.00 + $408.00 [$8,500.00 x 4.80%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax by the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -6,500.00 $18,500.00 669.00 – 330.00 $339.00 $6.52 $13.04 $14.13 $28.25'",
      "why": "Delaware's own printed per-period figure for the weekly column of its married-filing-jointly / 3-allowance example. By hand: 25,000 - 6,500 = 18,500; 261 + 8,500 x 4.80% (408) = 669; - 3 x 110 = 339; 339 / 52 = 6.5192 -> $6.52. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for weekly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 3
        }
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 13.04
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Married Taxpayer filing Joint Return – 3 Allowances 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $18,500.00 ($261.00 + $408.00 [$8,500.00 x 4.80%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax by the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -6,500.00 $18,500.00 669.00 – 330.00 $339.00 $6.52 $13.04 $14.13 $28.25'",
      "why": "Delaware's own printed per-period figure for the biweekly column of its married-filing-jointly / 3-allowance example. By hand: 25,000 - 6,500 = 18,500; 261 + 8,500 x 4.80% (408) = 669; - 3 x 110 = 339; 339 / 26 = 13.0385 -> $13.04. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for biweekly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 3
        }
      },
      "period": "semimonthly",
      "expect": {
        "stateWithholding": 14.13
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Married Taxpayer filing Joint Return – 3 Allowances 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $18,500.00 ($261.00 + $408.00 [$8,500.00 x 4.80%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax by the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -6,500.00 $18,500.00 669.00 – 330.00 $339.00 $6.52 $13.04 $14.13 $28.25'",
      "why": "Delaware's own printed per-period figure for the semimonthly column of its married-filing-jointly / 3-allowance example. By hand: 25,000 - 6,500 = 18,500; 261 + 8,500 x 4.80% (408) = 669; - 3 x 110 = 339; 339 / 24 = 14.1250 -> $14.13. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for semimonthly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 3
        }
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 28.25
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Married Taxpayer filing Joint Return – 3 Allowances 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $18,500.00 ($261.00 + $408.00 [$8,500.00 x 4.80%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax by the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -6,500.00 $18,500.00 669.00 – 330.00 $339.00 $6.52 $13.04 $14.13 $28.25'",
      "why": "Delaware's own printed per-period figure for the monthly column of its married-filing-jointly / 3-allowance example. By hand: 25,000 - 6,500 = 18,500; 261 + 8,500 x 4.80% (408) = 669; - 3 x 110 = 339; 339 / 12 = 28.2500 -> $28.25. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for monthly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "married-filing-separately",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 2
        }
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 11.77
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Married Filing Separate Return – 2 Allowances 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $21,750.00 ($741.00 + $91.00 [$1,750.00 x 5.20%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax By the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -3,250.00 $21,750.00 832.00 – 220.00 $612.00 $11.77 $23.54 $25.50 $51.00'",
      "why": "Delaware's own printed per-period figure for the weekly column of its married-filing-separately / 2-allowance example. By hand: 25,000 - 3,250 = 21,750; 832; - 2 x 110 = 612; 612 / 52 = 11.7692 -> $11.77. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for weekly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "married-filing-separately",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 2
        }
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 23.54
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Married Filing Separate Return – 2 Allowances 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $21,750.00 ($741.00 + $91.00 [$1,750.00 x 5.20%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax By the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -3,250.00 $21,750.00 832.00 – 220.00 $612.00 $11.77 $23.54 $25.50 $51.00'",
      "why": "Delaware's own printed per-period figure for the biweekly column of its married-filing-separately / 2-allowance example. By hand: 25,000 - 3,250 = 21,750; 832; - 2 x 110 = 612; 612 / 26 = 23.5385 -> $23.54. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for biweekly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "married-filing-separately",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 2
        }
      },
      "period": "semimonthly",
      "expect": {
        "stateWithholding": 25.5
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Married Filing Separate Return – 2 Allowances 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $21,750.00 ($741.00 + $91.00 [$1,750.00 x 5.20%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax By the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -3,250.00 $21,750.00 832.00 – 220.00 $612.00 $11.77 $23.54 $25.50 $51.00'",
      "why": "Delaware's own printed per-period figure for the semimonthly column of its married-filing-separately / 2-allowance example. By hand: 25,000 - 3,250 = 21,750; 832; - 2 x 110 = 612; 612 / 24 = 25.5000 -> $25.50. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for semimonthly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 25000,
        "filingStatus": "married-filing-separately",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "DE",
        "stateCertificate": {
          "allowances": 2
        }
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 51
      },
      "source": "Delaware Division of Revenue, Employer's Guide - Withholding Regulations and Employer's Duties, Section 17 'Computing Withholding Taxes' (page modified 9 January 2026), https://revenue.delaware.gov/employers-guide-withholding-regulations-employers-duties/, retrieved 2026-10-22. The guide's own worked example, verbatim: 'Married Filing Separate Return – 2 Allowances 1. Annualized Gross Wages 2. Minus Standard Deduction 3. Taxable Income 4. Tax on $21,750.00 ($741.00 + $91.00 [$1,750.00 x 5.20%]) 5. Multiply the Number of Personal Exemptions by $110.00 6. Tax Liability (Subtract Line 5 from Line 4) 7. Divide Tax By the Number of Pay Periods Weekly – 52 Bi-Weekly – 26 Semi-Monthly – 24 Monthly – 12 $25,000.00 -3,250.00 $21,750.00 832.00 – 220.00 $612.00 $11.77 $23.54 $25.50 $51.00'",
      "why": "Delaware's own printed per-period figure for the monthly column of its married-filing-separately / 2-allowance example. By hand: 25,000 - 3,250 = 21,750; 832; - 2 x 110 = 612; 612 / 12 = 51.0000 -> $51.00. Pins the annualised-wages method, the status's standard deduction and the $110 per-allowance credit, and the divisor for monthly.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 48000,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "GA",
        "stateCertificate": {
          "allowances": 1,
          "spouseEmployed": false
        }
      },
      "period": "semimonthly",
      "expect": {
        "stateWithholding": 27.03
      },
      "source": "Georgia Department of Revenue, Employer's Withholding Tax Guide 2026 (cover: REVISED September 2026), https://dor.georgia.gov/document/document/2026-employers-tax-guide-updated-june-2026/download, retrieved 2026-10-22. Page 51, Example #1, verbatim: 'EXAMPLE: A married person on a semi-monthly payroll is paid $2,000.00. The person's spouse does not work. The person has in effect a state withholding exemption certificate properly claiming one child. Under the percentage method the following steps are taken to compute the state withholding tax: Step 1: Total Taxable Wages $2,000.00. Less Standard Deduction $1250.00 per Table E (1) 1,250.00. Step 2: Less Dependent Allowance per Table E (4) $208.33 X 1 = 208.33. Wages subject to withholding 541.67. Step 3: Multiply 541.67 x 0.0499 = 27.03. Total tax to be withheld 27.03' The annual gross is the example's semi-monthly pay annualised: $2,000.00 x 24 = $48,000.00; 'The person's spouse does not work' is Form G-4 letter C (spouseEmployed false); 'claiming one child' is one dependent.",
      "why": "Georgia's own printed answer at the 2026 4.99% rate: married filing jointly, one income, one dependent. By hand: 2,000.00 - 1,250.00 - 208.33 = 541.67; x 0.0499 = 27.029 -> 27.03 (annual basis: (48,000 - 30,000 - 5,000) x 0.0499 / 24 = 27.029 -> 27.03).",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 24310,
        "filingStatus": "head-of-household",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "GA",
        "stateCertificate": {
          "allowances": 2
        }
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 0
      },
      "source": "Georgia Department of Revenue, Employer's Withholding Tax Guide 2026 (cover: REVISED September 2026), https://dor.georgia.gov/document/document/2026-employers-tax-guide-updated-june-2026/download, retrieved 2026-10-22. Page 52, Example #2 (with Table F), verbatim: 'EXAMPLE: A single person filing Head of Household on a bi-weekly payroll is paid $ 935.00. The person has in effect a state withholding exemption certificate properly claiming two children. The person would have $0 withheld from their bi-weekly paycheck. Under the percentage method the following steps are taken to compute the state withholding tax: Step 1: Total Taxable Wages $935.00 Less Standard Deduction $576.92 per Table F (1) $576.92 Step 2: Less Dependent Allowance per Table F (4) $192.31 X 2 $384.62 $0.00 Wages subject to withholding $0.00 Step 3: Multiply 0.00 x 0.0499 Total tax to be withheld $0.00' The annual gross is the example's bi-weekly pay annualised: $935.00 x 26 = $24,310.00. The guide's 'Table F (1)' label is a slip for column (2); the $576.92 it subtracts is the Single or Head of Household bi-weekly figure, and either column gives $0.00.",
      "why": "Georgia's own printed zero: deductions exceed the wage and the result is floored at $0.00, not carried negative. By hand: 935.00 - 576.92 - 2 x 192.31 (384.62) = -26.54 -> 0.00 (annual: 24,310 - 15,000 - 10,000 = -690 -> 0).",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 41600,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "IN",
        "stateCertificate": {
          "allowances": 5,
          "dependentExemptions": 3,
          "firstTimeDependentExemptions": 1,
          "adoptedChildExemptions": 2
        }
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 13.96
      },
      "source": "Indiana Department of Revenue, Departmental Notice #1 (R47 / 10-26), How to Compute Withholding for State and County Income Tax, https://www.in.gov/dor/files/dn01.pdf, retrieved 2026-10-22. Page 3, the notice's worked example, verbatim: 'Example: An employee is paid a weekly salary of $800; he/she claims five personal exemptions and is subject to county tax at the rate of 0.01. He/she claims three additional dependent exemptions, one first-time additional dependent exemption, and two adopted child dependent exemptions. The taxable income of $473.08 is the amount on which state and county tax should be calculated. Equation Math Deduction Constant from Table A $96.15 Deduction Constant from Table B (additional dependent) +86.54 Deduction Constant from Table B (first-time additional dependent) +28.85 Deduction Constant from Table C (adopted dependent) +115.38 Total Deduction Constant $326.92 Gross Income $800.00 Total Deduction Constant -326.92 Taxable Income $473.08 State Tax to Withhold $473.08 x .0295 = $13.96 County Tax to Withhold $473.08 x .01 = $4.73' The annual gross is the example's weekly salary annualised: $800 x 52 = $41,600. 'five personal exemptions' is WH-4 line 5 (allowances); three additional dependent, one first-time additional dependent and two adopted child exemptions are the other three fields.",
      "why": "Indiana's own printed state figure (the county tax, $4.73, is not part of stateWithholding and is not asserted). By hand: Table A 5 x 1,000 / 52 = 96.15; Table B 3 x 1,500 / 52 = 86.54; first-time 1,500 / 52 = 28.85; Table C 2 x 3,000 / 52 = 115.38; total 326.92; 800.00 - 326.92 = 473.08; x 0.0295 = 13.956 -> 13.96.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 41600,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "periodIndex": 1,
        "state": "IN",
        "stateCertificate": {
          "allowances": 5,
          "dependentExemptions": 3,
          "firstTimeDependentExemptions": 1,
          "adoptedChildExemptions": 2
        }
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 13.96
      },
      "source": "Indiana Department of Revenue, Departmental Notice #1 (R47 / 10-26), How to Compute Withholding for State and County Income Tax, https://www.in.gov/dor/files/dn01.pdf, retrieved 2026-10-22. Page 3, the notice's worked example, verbatim: 'Example: An employee is paid a weekly salary of $800; he/she claims five personal exemptions and is subject to county tax at the rate of 0.01. He/she claims three additional dependent exemptions, one first-time additional dependent exemption, and two adopted child dependent exemptions. The taxable income of $473.08 is the amount on which state and county tax should be calculated. Equation Math Deduction Constant from Table A $96.15 Deduction Constant from Table B (additional dependent) +86.54 Deduction Constant from Table B (first-time additional dependent) +28.85 Deduction Constant from Table C (adopted dependent) +115.38 Total Deduction Constant $326.92 Gross Income $800.00 Total Deduction Constant -326.92 Taxable Income $473.08 State Tax to Withhold $473.08 x .0295 = $13.96 County Tax to Withhold $473.08 x .01 = $4.73' The annual gross is the example's weekly salary annualised: $800 x 52 = $41,600. 'five personal exemptions' is WH-4 line 5 (allowances); three additional dependent, one first-time additional dependent and two adopted child exemptions are the other three fields. Indiana withholding has one computation; filing status acts only through the exemptions claimed (rules/us/states/in.json tableForFilingStatus, DN#1: 'Locate the deduction constant for the personal exemptions in Table A...'), so the same printed figure holds under married filing jointly.",
      "why": "the same DN#1 printed example under married filing jointly, proving the status does not move Indiana's figure: $13.96.",
      "ledger": "L-2026-4936"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 39260,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "state": "MS",
        "periodIndex": 1,
        "stateCertificate": {
          "exemptionAmount": 0
        }
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 41
      },
      "source": "Mississippi Department of Revenue, Bi-Weekly Payroll Period withholding tables, 'Effective January 1, 2026', https://www.dor.ms.gov/sites/default/files/business/Bi-weekly2026.pdf, retrieved 2026-10-25. Verbatim: 'TABLE A - SINGLE ... BI-WEEKLY PAYROLL PERIOD'; row 'AT LEAST 1,500 BUT LESS THAN 1,520', column 'TOTAL EXEMPTION CLAIMED ON FORM 89-350 IS: 0': 41. Annual gross is the row midpoint $1,510 x 26 = $39,260. The Department's table is computed at the row's midpoint (rules/us/states/ms.json workedExamples note), so the input is the midpoint wage; the expected figure is the printed cell, not a computation.",
      "why": "Mississippi's printed bi-weekly figure for a single employee claiming no exemption: proves the single $2,300 standard deduction, the $10,000 zero band, 4% and whole-dollar period rounding together.",
      "ledger": "L-2026-4941"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 39260,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "state": "MS",
        "periodIndex": 1,
        "stateCertificate": {
          "exemptionAmount": 6000
        }
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 32
      },
      "source": "Mississippi Department of Revenue, Bi-Weekly Payroll Period withholding tables, 'Effective January 1, 2026', https://www.dor.ms.gov/sites/default/files/business/Bi-weekly2026.pdf, retrieved 2026-10-25. Verbatim: 'TABLE A - SINGLE ... BI-WEEKLY PAYROLL PERIOD'; row 'AT LEAST 1,500 BUT LESS THAN 1,520', column '6,000': 32. Annual gross is the row midpoint $1,510 x 26 = $39,260. The Department's table is computed at the row's midpoint (rules/us/states/ms.json workedExamples note), so the input is the midpoint wage; the expected figure is the printed cell, not a computation.",
      "why": "The Form 89-350 exemption amount moves the answer: the same pay with $6,000 claimed is $9 a period less, as the Department prints.",
      "ledger": "L-2026-4941"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 39260,
        "filingStatus": "head-of-household",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "state": "MS",
        "periodIndex": 1,
        "stateCertificate": {
          "exemptionAmount": 9500
        }
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 25
      },
      "source": "Mississippi Department of Revenue, Bi-Weekly Payroll Period withholding tables, 'Effective January 1, 2026', https://www.dor.ms.gov/sites/default/files/business/Bi-weekly2026.pdf, retrieved 2026-10-25. Verbatim: 'TABLE B - HEAD OF FAMILY ... BI-WEEKLY PAYROLL PERIOD'; row 'AT LEAST 1,500 BUT LESS THAN 1,520', column '9,500': 25. Annual gross is the row midpoint $1,510 x 26 = $39,260. The Department's table is computed at the row's midpoint (rules/us/states/ms.json workedExamples note), so the input is the midpoint wage; the expected figure is the printed cell, not a computation.",
      "why": "Head of household reads Table B (head of family), with its own standard deduction; reading the single table would print a different figure.",
      "ledger": "L-2026-4941"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 39260,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "state": "MS",
        "periodIndex": 1,
        "stateCertificate": {
          "exemptionAmount": 12000,
          "spouseEmployed": false
        }
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 19
      },
      "source": "Mississippi Department of Revenue, Bi-Weekly Payroll Period withholding tables, 'Effective January 1, 2026', https://www.dor.ms.gov/sites/default/files/business/Bi-weekly2026.pdf, retrieved 2026-10-25. Verbatim: 'TABLE C - MARRIED (SPOUSE NOT EMPLOYED) ... BI-WEEKLY PAYROLL PERIOD'; row 'AT LEAST 1,500 BUT LESS THAN 1,520', column '12,000': 19. Annual gross is the row midpoint $1,510 x 26 = $39,260. The Department's table is computed at the row's midpoint (rules/us/states/ms.json workedExamples note), so the input is the midpoint wage; the expected figure is the printed cell, not a computation.",
      "why": "Married with one spouse employed reads Table C and the $4,600 standard deduction, selected by the spouse-employed answer.",
      "ledger": "L-2026-4941"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 36240,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "state": "MS",
        "periodIndex": 1,
        "stateCertificate": {
          "exemptionAmount": 0
        }
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 80
      },
      "source": "Mississippi Department of Revenue, Monthly Payroll Period withholding tables, 'Effective January 1, 2026', https://www.dor.ms.gov/sites/default/files/business/Monthly2026.pdf, retrieved 2026-10-25. Verbatim: 'TABLE A - SINGLE ... MONTHLY PAYROLL PERIOD'; row 'AT LEAST 3,000 BUT LESS THAN 3,040', column 'TOTAL EXEMPTION CLAIMED ON FORM 89-350 IS: 0': 80. Annual gross is the row midpoint $3,020 x 12 = $36,240. The Department's table is computed at the row's midpoint (rules/us/states/ms.json workedExamples note), so the input is the midpoint wage; the expected figure is the printed cell, not a computation.",
      "why": "A monthly cell whose exact tax is not a whole dollar: Mississippi rounds the period withholding to whole dollars (flowchart step 5) and prints 80; an engine that truncated, or printed cents, would not match the Department's monthly table.",
      "ledger": "L-2026-4941"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 36400,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "state": "LA",
        "periodIndex": 1,
        "stateCertificate": {
          "allowances": 1
        }
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 13.98
      },
      "source": "Louisiana Department of Revenue, R-1306 (1/26), Louisiana Withholding Tables and Formulas, Effective on or after January 1, 2026, https://dam.ldr.la.gov/taxforms/1306-1-26.pdf, retrieved 2026-10-25. The Department's own worked Example 1, verbatim: 'Taxpayer is claiming “1” in block A of Form L-4. Taxpayer is paid $700 weekly (52 pay periods, $36,400 annual salary).' 'W = (700 - (12,875/52)) * .0309 = (700 - 247.60) * .0309 = 452.40 * .0309 = 13.98'. '$13.98 is the withholding tax for the pay period.'",
      "why": "Louisiana's printed answer for Block A '1' on a weekly payroll: proves the $12,875 standard deduction spread over 52 periods and the 3.09% rate, to the cent.",
      "ledger": "L-2026-4942"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 119600,
        "filingStatus": "married-filing-jointly",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "state": "LA",
        "periodIndex": 1,
        "stateCertificate": {
          "allowances": 2
        }
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 111.54
      },
      "source": "Louisiana Department of Revenue, R-1306 (1/26), Louisiana Withholding Tables and Formulas, Effective on or after January 1, 2026, https://dam.ldr.la.gov/taxforms/1306-1-26.pdf, retrieved 2026-10-25. The Department's own worked Example 2, verbatim: 'Taxpayer is claiming “2” in block A of Form L-4. Taxpayer is paid $4,600 bi-weekly (26 pay periods, $119,600 annual salary).' 'W = (4,600 - (25,750/26)) * .0309 = (4,600 - 990.38) *.0309 = 3,609.62 * .0309 = 111.54'. '$111.54 is the withholding tax for the pay period.'",
      "why": "Louisiana's printed answer for Block A '2' on a bi-weekly payroll: proves the $25,750 married-joint standard deduction over 26 periods; using the Block A '1' deduction would print a larger figure.",
      "ledger": "L-2026-4942"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 52000,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "state": "KY",
        "periodIndex": 1
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 32.74
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-25 (HTTP 200, 115,509 bytes, SHA-256 a4a7d6bd5813c12e576ee498c3584e3c14198846ae83209fdab6b1656191fada; no VBA part, no macro run). AUTHORITY-COMPUTED: inputs written to sheet 'Calculation of Withholding' cells B7 (pay frequency, the sheet's own list value 'Weekly') and C7 (taxable wages per pay period $1,000.00), workbook recalculated headless by LibreOffice, and the Department's own cell 'Calculation of Withholding'!D7 (=IF('Hidden Table'!H7>0,'Hidden Table'!H7,0)) read back: raw 32.7384615384615, shown by its number format \"$\"#,##0.00 as $32.74. The expected figure is that displayed cell, not our arithmetic.",
      "why": "a weekly wage: the calculator annualises by 52, deducts $3,360 and divides back. Figure produced by the Department's calculator, not by us.",
      "ledger": "L-2026-4943"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 52000,
        "filingStatus": "single",
        "payFrequency": "biweekly",
        "taxYear": "2026",
        "state": "KY",
        "periodIndex": 1
      },
      "period": "biweekly",
      "expect": {
        "stateWithholding": 65.48
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-25 (HTTP 200, 115,509 bytes, SHA-256 a4a7d6bd5813c12e576ee498c3584e3c14198846ae83209fdab6b1656191fada; no VBA part, no macro run). AUTHORITY-COMPUTED: inputs written to sheet 'Calculation of Withholding' cells B8 (pay frequency, the sheet's own list value 'BI-Weekly') and C8 (taxable wages per pay period $2,000.00), workbook recalculated headless by LibreOffice, and the Department's own cell 'Calculation of Withholding'!D8 (=IF('Hidden Table'!H8>0,'Hidden Table'!H8,0)) read back: raw 65.4769230769231, shown by its number format \"$\"#,##0.00 as $65.48. The expected figure is that displayed cell, not our arithmetic.",
      "why": "a bi-weekly wage: the 26-period path of the Department's own lookup table. Figure produced by the Department's calculator, not by us.",
      "ledger": "L-2026-4943"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 100001.04,
        "filingStatus": "single",
        "payFrequency": "semimonthly",
        "taxYear": "2026",
        "state": "KY",
        "periodIndex": 1
      },
      "period": "semimonthly",
      "expect": {
        "stateWithholding": 140.93
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-25 (HTTP 200, 115,509 bytes, SHA-256 a4a7d6bd5813c12e576ee498c3584e3c14198846ae83209fdab6b1656191fada; no VBA part, no macro run). AUTHORITY-COMPUTED: inputs written to sheet 'Calculation of Withholding' cells B9 (pay frequency, the sheet's own list value 'Semi Monthly') and C9 (taxable wages per pay period $4,166.71), workbook recalculated headless by LibreOffice, and the Department's own cell 'Calculation of Withholding'!D9 (=IF('Hidden Table'!H9>0,'Hidden Table'!H9,0)) read back: raw 140.93485, shown by its number format \"$\"#,##0.00 as $140.93. The expected figure is that displayed cell, not our arithmetic.",
      "why": "$100,001.04 a year paid semi-monthly: the Department's sheet returns 140.93485, so the half-cent sits below the rounding point and the cheque is $140.93, not $140.94. Figure produced by the Department's calculator, not by us.",
      "ledger": "L-2026-4943"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 60000,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "state": "KY",
        "periodIndex": 1
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 165.2
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-25 (HTTP 200, 115,509 bytes, SHA-256 a4a7d6bd5813c12e576ee498c3584e3c14198846ae83209fdab6b1656191fada; no VBA part, no macro run). AUTHORITY-COMPUTED: inputs written to sheet 'Calculation of Withholding' cells B10 (pay frequency, the sheet's own list value 'Monthly') and C10 (taxable wages per pay period $5,000.00), workbook recalculated headless by LibreOffice, and the Department's own cell 'Calculation of Withholding'!D10 (=IF('Hidden Table'!H10>0,'Hidden Table'!H10,0)) read back: raw 165.2, shown by its number format \"$\"#,##0.00 as $165.20. The expected figure is that displayed cell, not our arithmetic.",
      "why": "a monthly wage whose answer is exact to the cent: (60,000 - 3,360) x 3.5% / 12. Figure produced by the Department's calculator, not by us.",
      "ledger": "L-2026-4943"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 3000,
        "filingStatus": "single",
        "payFrequency": "monthly",
        "taxYear": "2026",
        "state": "KY",
        "periodIndex": 1
      },
      "period": "monthly",
      "expect": {
        "stateWithholding": 0
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-25 (HTTP 200, 115,509 bytes, SHA-256 a4a7d6bd5813c12e576ee498c3584e3c14198846ae83209fdab6b1656191fada; no VBA part, no macro run). AUTHORITY-COMPUTED: inputs written to sheet 'Calculation of Withholding' cells B11 (pay frequency, the sheet's own list value 'Monthly') and C11 (taxable wages per pay period $250.00), workbook recalculated headless by LibreOffice, and the Department's own cell 'Calculation of Withholding'!D11 (=IF('Hidden Table'!H11>0,'Hidden Table'!H11,0)) read back: raw 0 (Hidden Table H11 = -1.05), shown by its number format \"$\"#,##0.00 as $0.00. The expected figure is that displayed cell, not our arithmetic.",
      "why": "a monthly wage under the $3,360 annual deduction: the Department's H cell is -1.05 and its own IF wrapper withholds zero. Figure produced by the Department's calculator, not by us.",
      "ledger": "L-2026-4943"
    },
    {
      "kind": "payslip",
      "input": {
        "gross": 3360.24,
        "filingStatus": "single",
        "payFrequency": "weekly",
        "taxYear": "2026",
        "state": "KY",
        "periodIndex": 1
      },
      "period": "weekly",
      "expect": {
        "stateWithholding": 0
      },
      "source": "Kentucky Department of Revenue, '2026 Kentucky Employer Withholding Calculator', https://revenue.ky.gov/Business/Documents/2026%20Withholding%20Tax%20Calculator.xlsx, retrieved 2026-10-25 (HTTP 200, 115,509 bytes, SHA-256 a4a7d6bd5813c12e576ee498c3584e3c14198846ae83209fdab6b1656191fada; no VBA part, no macro run). AUTHORITY-COMPUTED: inputs written to sheet 'Calculation of Withholding' cells B12 (pay frequency, the sheet's own list value 'Weekly') and C12 (taxable wages per pay period $64.62), workbook recalculated headless by LibreOffice, and the Department's own cell 'Calculation of Withholding'!D12 (=IF('Hidden Table'!H12>0,'Hidden Table'!H12,0)) read back: raw 0.000161538461538621, shown by its number format \"$\"#,##0.00 as $0.00. The expected figure is that displayed cell, not our arithmetic.",
      "why": "one cent past the weekly edge of the deduction ($3,360 / 52 = $64.615...): the Department's cell is 0.00016 and displays $0.00. Figure produced by the Department's calculator, not by us.",
      "ledger": "L-2026-4943"
    }
  ],
  "refusal": "A case may carry a `refuses` block {code, why} instead of being answered: the input names something the rules cannot price, and BOTH implementations must throw that exact code. `expect` is kept as the hand derivation of the part of the law that IS sourced, and is never asserted while `refuses` is present."
}