FA-58916 / Payroll withholding rules / Open access
W-4 step adjustment withholding: step 2 deduction scope · case 01
Multiple-job employees who also itemize are over-withheld because half their itemized deductions vanish.
ROOT CAUSE
The step 2 halving is applied to the itemized step 4(b) deductions as well as the standard allowance.
VERIFIED REPAIR
Restore the contract rule at the step 2 deduction scope step: use `ded = (base_std // 2 if x['step2'] else base_std) + x['deductions']`.
Unsuccessful approach: The attempt stops halving anything, ignoring the step 2 checkbox for the standard allowance entirely.
Case contract
Input {wage, periods, status single|mfj|hoh, step2, credits, other_income, deductions, extra}; annual amounts in cents except wage and extra (per period). Annual = wage*periods + other_income. Deduction = base standard (single 8,600, mfj 12,900, hoh 8,600 dollars; halved when step2) + itemized deductions (never halved). Thresholds 7,000/30,000/90,000 dollars at 10/12/22% scaled by status unit (single 2, mfj 4, hoh 3)/2, or /4 when step2. Tentative per-period = annual tax/periods half-up; credits/periods half-up is subtracted, floored at 0, then extra is added.
Why this case matters
Modern W-4 withholding combines annual adjustments with per-period ones, and the order of flooring and adding extra withholding matters.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
p = x['periods']
annual = x['wage'] * p + x['other_income']
base_std = {'single': 860000, 'mfj': 1290000, 'hoh': 860000}[x['status']]
ded = (base_std + x['deductions']) // 2 if x['step2'] else base_std + x['deductions']
taxable = max(0, annual - ded)
unit = {'single': 2, 'mfj': 4, 'hoh': 3}[x['status']]
div = 4 if x['step2'] else 2
bounds = [700000 * unit // div, 3000000 * unit // div, 9000000 * unit // div]
rates = [10, 12, 22]
tax = 0
for i, lo in enumerate(bounds):
hi = bounds[i + 1] if i + 1 < len(bounds) else taxable
if taxable > lo:
tax += (min(taxable, hi) - lo) * rates[i]
tentative = (tax * 2 + 100 * p) // (200 * p)
credit_pp = (x['credits'] * 2 + p) // (2 * p)
wh = max(0, tentative - credit_pp) + x['extra']
return [tentative, wh]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression', {'wage': 69636, 'periods': 24, 'status': 'hoh', 'step2': True, 'credits': 877725, 'other_income': 1721162, 'deductions': 1014955, 'extra': 0}, [5927, 0]), ('regression', {'wage': 285367, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 1182238, 'extra': 2500}, [41104, 43604]), ('partial-repair probe', {'wage': 492438, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 400000, 'other_income': 131536, 'deductions': 0, 'extra': 0}, [59948, 26615]), ('partial-repair probe', {'wage': 170108, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 167672, 'deductions': 0, 'extra': 0}, [12373, 0]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 4942, 'periods': 12, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 1831505, 'deductions': 0, 'extra': 8860}, [2757, 8860]), ('normal control', {'wage': 111690, 'periods': 26, 'status': 'mfj', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [823, 823]), ('normal control', {'wage': 67447, 'periods': 24, 'status': 'mfj', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [0, 2500]), ('normal control', {'wage': 1041418, 'periods': 24, 'status': 'hoh', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [156854, 159354])], [('regression', {'wage': 285367, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 1182238, 'extra': 2500}, [41104, 43604]), ('regression', {'wage': 178354, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 112718, 'extra': 0}, [12905, 12905]), ('partial-repair probe', {'wage': 170108, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 167672, 'deductions': 0, 'extra': 0}, [12373, 0]), ('partial-repair probe', {'wage': 124748, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [7302, 7302]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 245742, 'periods': 24, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [19772, 22272]), ('normal control', {'wage': 196398, 'periods': 12, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [6640, 6640]), ('normal control', {'wage': 64669, 'periods': 24, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [0, 2500]), ('normal control', {'wage': 232685, 'periods': 52, 'status': 'mfj', 'step2': False, 'credits': 200000, 'other_income': 0, 'deductions': 479252, 'extra': 0}, [18839, 14993])], [('regression', {'wage': 178354, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 112718, 'extra': 0}, [12905, 12905]), ('regression', {'wage': 205442, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 1801669, 'deductions': 1137902, 'extra': 0}, [21574, 4907]), ('partial-repair probe', {'wage': 124748, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [7302, 7302]), ('partial-repair probe', {'wage': 90292, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [7968, 10468]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 79014, 'periods': 24, 'status': 'mfj', 'step2': False, 'credits': 400000, 'other_income': 1864011, 'deductions': 0, 'extra': 0}, [4460, 0]), ('normal control', {'wage': 322003, 'periods': 12, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [19207, 21707]), ('normal control', {'wage': 93316, 'periods': 12, 'status': 'mfj', 'step2': False, 'credits': 200000, 'other_income': 1111591, 'deductions': 0, 'extra': 1563}, [0, 1563]), ('normal control', {'wage': 37313, 'periods': 52, 'status': 'mfj', 'step2': False, 'credits': 117737, 'other_income': 1320234, 'deductions': 0, 'extra': 5664}, [1097, 5664])], [('regression', {'wage': 205442, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 1801669, 'deductions': 1137902, 'extra': 0}, [21574, 4907]), ('regression', {'wage': 117375, 'periods': 24, 'status': 'hoh', 'step2': True, 'credits': 546474, 'other_income': 0, 'deductions': 599256, 'extra': 0}, [5261, 0]), ('partial-repair probe', {'wage': 142479, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 551126, 'other_income': 0, 'deductions': 0, 'extra': 3806}, [14670, 7877]), ('partial-repair probe', {'wage': 606158, 'periods': 26, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 1548001, 'deductions': 0, 'extra': 0}, [113103, 113103]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 179669, 'periods': 52, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [17076, 17076]), ('normal control', {'wage': 244499, 'periods': 26, 'status': 'hoh', 'step2': False, 'credits': 305778, 'other_income': 170107, 'deductions': 0, 'extra': 0}, [18656, 6895]), ('normal control', {'wage': 250357, 'periods': 52, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [31632, 34132]), ('normal control', {'wage': 251742, 'periods': 52, 'status': 'hoh', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 392790, 'extra': 0}, [23568, 23568])], [('regression', {'wage': 117375, 'periods': 24, 'status': 'hoh', 'step2': True, 'credits': 546474, 'other_income': 0, 'deductions': 599256, 'extra': 0}, [5261, 0]), ('regression', {'wage': 121406, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 1123754, 'deductions': 405924, 'extra': 0}, [12030, 0]), ('partial-repair probe', {'wage': 228867, 'periods': 52, 'status': 'single', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [38628, 38628]), ('partial-repair probe', {'wage': 176, 'periods': 12, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 1390591, 'deductions': 0, 'extra': 0}, [398, 398]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 290758, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 1100573, 'deductions': 944056, 'extra': 0}, [26644, 26644]), ('normal control', {'wage': 76761, 'periods': 52, 'status': 'hoh', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [4003, 2500]), ('normal control', {'wage': 451331, 'periods': 52, 'status': 'mfj', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [54220, 54220]), ('normal control', {'wage': 37554, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 695042, 'extra': 0}, [0, 0])]]
for i, (label, args, expected) in enumerate(fixtures[N-1]):
check("%s %d" % (label, i), solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression 0 | [8212, 0] | [5927, 0] | Failed |
| regression 1 | [43605, 46105] | [41104, 43604] | Failed |
| partial-repair probe 2 | [59948, 26615] | [59948, 26615] | Passed |
| partial-repair probe 3 | [12373, 0] | [12373, 0] | Passed |
| boundary control 4 | [0, 5000] | [0, 5000] | Passed |
| normal control 5 | [2757, 8860] | [2757, 8860] | Passed |
| normal control 6 | [823, 823] | [823, 823] | Passed |
| normal control 7 | [0, 2500] | [0, 2500] | Passed |
| normal control 8 | [156854, 159354] | [156854, 159354] | Passed |
SHA-256 / 7d35163c1e6ff016ae2f5c251b52f2e776f1d503860a534b2956e1721062fe4d
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
p = x['periods']
annual = x['wage'] * p + x['other_income']
base_std = {'single': 860000, 'mfj': 1290000, 'hoh': 860000}[x['status']]
ded = base_std + x['deductions']
taxable = max(0, annual - ded)
unit = {'single': 2, 'mfj': 4, 'hoh': 3}[x['status']]
div = 4 if x['step2'] else 2
bounds = [700000 * unit // div, 3000000 * unit // div, 9000000 * unit // div]
rates = [10, 12, 22]
tax = 0
for i, lo in enumerate(bounds):
hi = bounds[i + 1] if i + 1 < len(bounds) else taxable
if taxable > lo:
tax += (min(taxable, hi) - lo) * rates[i]
tentative = (tax * 2 + 100 * p) // (200 * p)
credit_pp = (x['credits'] * 2 + p) // (2 * p)
wh = max(0, tentative - credit_pp) + x['extra']
return [tentative, wh]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression', {'wage': 69636, 'periods': 24, 'status': 'hoh', 'step2': True, 'credits': 877725, 'other_income': 1721162, 'deductions': 1014955, 'extra': 0}, [5927, 0]), ('regression', {'wage': 285367, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 1182238, 'extra': 2500}, [41104, 43604]), ('partial-repair probe', {'wage': 492438, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 400000, 'other_income': 131536, 'deductions': 0, 'extra': 0}, [59948, 26615]), ('partial-repair probe', {'wage': 170108, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 167672, 'deductions': 0, 'extra': 0}, [12373, 0]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 4942, 'periods': 12, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 1831505, 'deductions': 0, 'extra': 8860}, [2757, 8860]), ('normal control', {'wage': 111690, 'periods': 26, 'status': 'mfj', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [823, 823]), ('normal control', {'wage': 67447, 'periods': 24, 'status': 'mfj', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [0, 2500]), ('normal control', {'wage': 1041418, 'periods': 24, 'status': 'hoh', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [156854, 159354])], [('regression', {'wage': 285367, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 1182238, 'extra': 2500}, [41104, 43604]), ('regression', {'wage': 178354, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 112718, 'extra': 0}, [12905, 12905]), ('partial-repair probe', {'wage': 170108, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 167672, 'deductions': 0, 'extra': 0}, [12373, 0]), ('partial-repair probe', {'wage': 124748, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [7302, 7302]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 245742, 'periods': 24, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [19772, 22272]), ('normal control', {'wage': 196398, 'periods': 12, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [6640, 6640]), ('normal control', {'wage': 64669, 'periods': 24, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [0, 2500]), ('normal control', {'wage': 232685, 'periods': 52, 'status': 'mfj', 'step2': False, 'credits': 200000, 'other_income': 0, 'deductions': 479252, 'extra': 0}, [18839, 14993])], [('regression', {'wage': 178354, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 112718, 'extra': 0}, [12905, 12905]), ('regression', {'wage': 205442, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 1801669, 'deductions': 1137902, 'extra': 0}, [21574, 4907]), ('partial-repair probe', {'wage': 124748, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [7302, 7302]), ('partial-repair probe', {'wage': 90292, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [7968, 10468]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 79014, 'periods': 24, 'status': 'mfj', 'step2': False, 'credits': 400000, 'other_income': 1864011, 'deductions': 0, 'extra': 0}, [4460, 0]), ('normal control', {'wage': 322003, 'periods': 12, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [19207, 21707]), ('normal control', {'wage': 93316, 'periods': 12, 'status': 'mfj', 'step2': False, 'credits': 200000, 'other_income': 1111591, 'deductions': 0, 'extra': 1563}, [0, 1563]), ('normal control', {'wage': 37313, 'periods': 52, 'status': 'mfj', 'step2': False, 'credits': 117737, 'other_income': 1320234, 'deductions': 0, 'extra': 5664}, [1097, 5664])], [('regression', {'wage': 205442, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 1801669, 'deductions': 1137902, 'extra': 0}, [21574, 4907]), ('regression', {'wage': 117375, 'periods': 24, 'status': 'hoh', 'step2': True, 'credits': 546474, 'other_income': 0, 'deductions': 599256, 'extra': 0}, [5261, 0]), ('partial-repair probe', {'wage': 142479, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 551126, 'other_income': 0, 'deductions': 0, 'extra': 3806}, [14670, 7877]), ('partial-repair probe', {'wage': 606158, 'periods': 26, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 1548001, 'deductions': 0, 'extra': 0}, [113103, 113103]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 179669, 'periods': 52, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [17076, 17076]), ('normal control', {'wage': 244499, 'periods': 26, 'status': 'hoh', 'step2': False, 'credits': 305778, 'other_income': 170107, 'deductions': 0, 'extra': 0}, [18656, 6895]), ('normal control', {'wage': 250357, 'periods': 52, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [31632, 34132]), ('normal control', {'wage': 251742, 'periods': 52, 'status': 'hoh', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 392790, 'extra': 0}, [23568, 23568])], [('regression', {'wage': 117375, 'periods': 24, 'status': 'hoh', 'step2': True, 'credits': 546474, 'other_income': 0, 'deductions': 599256, 'extra': 0}, [5261, 0]), ('regression', {'wage': 121406, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 1123754, 'deductions': 405924, 'extra': 0}, [12030, 0]), ('partial-repair probe', {'wage': 228867, 'periods': 52, 'status': 'single', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [38628, 38628]), ('partial-repair probe', {'wage': 176, 'periods': 12, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 1390591, 'deductions': 0, 'extra': 0}, [398, 398]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 290758, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 1100573, 'deductions': 944056, 'extra': 0}, [26644, 26644]), ('normal control', {'wage': 76761, 'periods': 52, 'status': 'hoh', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [4003, 2500]), ('normal control', {'wage': 451331, 'periods': 52, 'status': 'mfj', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [54220, 54220]), ('normal control', {'wage': 37554, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 695042, 'extra': 0}, [0, 0])]]
for i, (label, args, expected) in enumerate(fixtures[N-1]):
check("%s %d" % (label, i), solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression 0 | [4135, 0] | [5927, 0] | Failed |
| regression 1 | [39285, 41785] | [41104, 43604] | Failed |
| partial-repair probe 2 | [52065, 18732] | [59948, 26615] | Failed |
| partial-repair probe 3 | [8325, 0] | [12373, 0] | Failed |
| boundary control 4 | [0, 5000] | [0, 5000] | Passed |
| normal control 5 | [2757, 8860] | [2757, 8860] | Passed |
| normal control 6 | [823, 823] | [823, 823] | Passed |
| normal control 7 | [0, 2500] | [0, 2500] | Passed |
| normal control 8 | [156854, 159354] | [156854, 159354] | Passed |
SHA-256 / 04bf65b965d8d028ffd541d9560215a6edeec82aeb26bde48ea771a3c0f0e275
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
p = x['periods']
annual = x['wage'] * p + x['other_income']
base_std = {'single': 860000, 'mfj': 1290000, 'hoh': 860000}[x['status']]
ded = (base_std // 2 if x['step2'] else base_std) + x['deductions']
taxable = max(0, annual - ded)
unit = {'single': 2, 'mfj': 4, 'hoh': 3}[x['status']]
div = 4 if x['step2'] else 2
bounds = [700000 * unit // div, 3000000 * unit // div, 9000000 * unit // div]
rates = [10, 12, 22]
tax = 0
for i, lo in enumerate(bounds):
hi = bounds[i + 1] if i + 1 < len(bounds) else taxable
if taxable > lo:
tax += (min(taxable, hi) - lo) * rates[i]
tentative = (tax * 2 + 100 * p) // (200 * p)
credit_pp = (x['credits'] * 2 + p) // (2 * p)
wh = max(0, tentative - credit_pp) + x['extra']
return [tentative, wh]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression', {'wage': 69636, 'periods': 24, 'status': 'hoh', 'step2': True, 'credits': 877725, 'other_income': 1721162, 'deductions': 1014955, 'extra': 0}, [5927, 0]), ('regression', {'wage': 285367, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 1182238, 'extra': 2500}, [41104, 43604]), ('partial-repair probe', {'wage': 492438, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 400000, 'other_income': 131536, 'deductions': 0, 'extra': 0}, [59948, 26615]), ('partial-repair probe', {'wage': 170108, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 167672, 'deductions': 0, 'extra': 0}, [12373, 0]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 4942, 'periods': 12, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 1831505, 'deductions': 0, 'extra': 8860}, [2757, 8860]), ('normal control', {'wage': 111690, 'periods': 26, 'status': 'mfj', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [823, 823]), ('normal control', {'wage': 67447, 'periods': 24, 'status': 'mfj', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [0, 2500]), ('normal control', {'wage': 1041418, 'periods': 24, 'status': 'hoh', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [156854, 159354])], [('regression', {'wage': 285367, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 1182238, 'extra': 2500}, [41104, 43604]), ('regression', {'wage': 178354, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 112718, 'extra': 0}, [12905, 12905]), ('partial-repair probe', {'wage': 170108, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 167672, 'deductions': 0, 'extra': 0}, [12373, 0]), ('partial-repair probe', {'wage': 124748, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [7302, 7302]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 245742, 'periods': 24, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [19772, 22272]), ('normal control', {'wage': 196398, 'periods': 12, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [6640, 6640]), ('normal control', {'wage': 64669, 'periods': 24, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [0, 2500]), ('normal control', {'wage': 232685, 'periods': 52, 'status': 'mfj', 'step2': False, 'credits': 200000, 'other_income': 0, 'deductions': 479252, 'extra': 0}, [18839, 14993])], [('regression', {'wage': 178354, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 112718, 'extra': 0}, [12905, 12905]), ('regression', {'wage': 205442, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 1801669, 'deductions': 1137902, 'extra': 0}, [21574, 4907]), ('partial-repair probe', {'wage': 124748, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [7302, 7302]), ('partial-repair probe', {'wage': 90292, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [7968, 10468]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 79014, 'periods': 24, 'status': 'mfj', 'step2': False, 'credits': 400000, 'other_income': 1864011, 'deductions': 0, 'extra': 0}, [4460, 0]), ('normal control', {'wage': 322003, 'periods': 12, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [19207, 21707]), ('normal control', {'wage': 93316, 'periods': 12, 'status': 'mfj', 'step2': False, 'credits': 200000, 'other_income': 1111591, 'deductions': 0, 'extra': 1563}, [0, 1563]), ('normal control', {'wage': 37313, 'periods': 52, 'status': 'mfj', 'step2': False, 'credits': 117737, 'other_income': 1320234, 'deductions': 0, 'extra': 5664}, [1097, 5664])], [('regression', {'wage': 205442, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 1801669, 'deductions': 1137902, 'extra': 0}, [21574, 4907]), ('regression', {'wage': 117375, 'periods': 24, 'status': 'hoh', 'step2': True, 'credits': 546474, 'other_income': 0, 'deductions': 599256, 'extra': 0}, [5261, 0]), ('partial-repair probe', {'wage': 142479, 'periods': 52, 'status': 'hoh', 'step2': True, 'credits': 551126, 'other_income': 0, 'deductions': 0, 'extra': 3806}, [14670, 7877]), ('partial-repair probe', {'wage': 606158, 'periods': 26, 'status': 'hoh', 'step2': True, 'credits': 0, 'other_income': 1548001, 'deductions': 0, 'extra': 0}, [113103, 113103]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 179669, 'periods': 52, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [17076, 17076]), ('normal control', {'wage': 244499, 'periods': 26, 'status': 'hoh', 'step2': False, 'credits': 305778, 'other_income': 170107, 'deductions': 0, 'extra': 0}, [18656, 6895]), ('normal control', {'wage': 250357, 'periods': 52, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [31632, 34132]), ('normal control', {'wage': 251742, 'periods': 52, 'status': 'hoh', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 392790, 'extra': 0}, [23568, 23568])], [('regression', {'wage': 117375, 'periods': 24, 'status': 'hoh', 'step2': True, 'credits': 546474, 'other_income': 0, 'deductions': 599256, 'extra': 0}, [5261, 0]), ('regression', {'wage': 121406, 'periods': 12, 'status': 'single', 'step2': True, 'credits': 200000, 'other_income': 1123754, 'deductions': 405924, 'extra': 0}, [12030, 0]), ('partial-repair probe', {'wage': 228867, 'periods': 52, 'status': 'single', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [38628, 38628]), ('partial-repair probe', {'wage': 176, 'periods': 12, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 1390591, 'deductions': 0, 'extra': 0}, [398, 398]), ('boundary control', {'wage': 0, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 5000}, [0, 5000]), ('normal control', {'wage': 290758, 'periods': 26, 'status': 'single', 'step2': False, 'credits': 0, 'other_income': 1100573, 'deductions': 944056, 'extra': 0}, [26644, 26644]), ('normal control', {'wage': 76761, 'periods': 52, 'status': 'hoh', 'step2': False, 'credits': 400000, 'other_income': 0, 'deductions': 0, 'extra': 2500}, [4003, 2500]), ('normal control', {'wage': 451331, 'periods': 52, 'status': 'mfj', 'step2': False, 'credits': 0, 'other_income': 0, 'deductions': 0, 'extra': 0}, [54220, 54220]), ('normal control', {'wage': 37554, 'periods': 26, 'status': 'mfj', 'step2': True, 'credits': 0, 'other_income': 0, 'deductions': 695042, 'extra': 0}, [0, 0])]]
for i, (label, args, expected) in enumerate(fixtures[N-1]):
check("%s %d" % (label, i), solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression 0 | [5927, 0] | [5927, 0] | Passed |
| regression 1 | [41104, 43604] | [41104, 43604] | Passed |
| partial-repair probe 2 | [59948, 26615] | [59948, 26615] | Passed |
| partial-repair probe 3 | [12373, 0] | [12373, 0] | Passed |
| boundary control 4 | [0, 5000] | [0, 5000] | Passed |
| normal control 5 | [2757, 8860] | [2757, 8860] | Passed |
| normal control 6 | [823, 823] | [823, 823] | Passed |
| normal control 7 | [0, 2500] | [0, 2500] | Passed |
| normal control 8 | [156854, 159354] | [156854, 159354] | Passed |
SHA-256 / c2a049a6e0a8d94eb0ef2ea39ee3eb56e4471d5f51b9da23c637f7e434f6e9fe
Verification & scope
A deterministic teaching model of a stipulated payroll rule with toy thresholds and rates. It makes no claim of conformance to any tax authority, statute or jurisdiction and is not payroll software. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.
Observations recorded using Python 3.12.14 at 2026-09-29T14:46:31.330646+00:00.
Case digest / 83c2b640a7d7d74b7d6b1eae66cf154ff2a0e1426d1ca89f12e69c7bb1a9fc4f