FA-58891 / Payroll withholding rules / Open access

Annualized percentage-method withholding: marginal slice accumulation · case 01

Withholding jumps discontinuously when income crosses a bracket, taxing all income at the top reached rate.

Verified by executionVariant 1 · 9 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Each matching bracket overwrites the tax with the flat rate applied to all adjusted income.

VERIFIED REPAIR

Restore the contract rule at the marginal slice accumulation step: use `tax += (top - lo) * rate`.

Unsuccessful approach: The attempt accumulates slices but forgets to subtract the bracket floor, taxing from zero in every bracket.

Case contract

Input {wage, freq, allow, status}. Annualize: wage*periods (weekly 52, biweekly 26, semimonthly 24, monthly 12); subtract 4,300.00 per allowance, floor 0. Marginal brackets over 4,000/15,000/50,000 dollars at 10/12/22% (married thresholds doubled). Annual tax (in cent-percent units) is divided by periods and rounded half-up to cents once.

Why this case matters

Percentage-method withholding depends on marginal bracket slices and on rounding only after de-annualizing.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(x):
    periods = {'weekly': 52, 'biweekly': 26, 'semimonthly': 24, 'monthly': 12}[x['freq']]
    annual = x['wage'] * periods
    adjusted = max(0, annual - x['allow'] * 430000)
    mult = 2 if x['status'] == 'married' else 1
    brackets = [(400000 * mult, 10), (1500000 * mult, 12), (5000000 * mult, 22)]
    tax = 0
    for i, (lo, rate) in enumerate(brackets):
        hi = brackets[i + 1][0] if i + 1 < len(brackets) else None
        if adjusted > lo:
            top = adjusted if hi is None else min(adjusted, hi)
            tax = adjusted * rate
    wh = (tax * 2 + 100 * periods) // (200 * periods)
    return wh
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression (boundary)', {'wage': 57693, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 5577), ('regression (boundary)', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('partial-repair probe', {'wage': 998152, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 168152), ('partial-repair probe', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 22359, 'freq': 'semimonthly', 'allow': 5, 'status': 'married'}, 0), ('normal control', {'wage': 89742, 'freq': 'monthly', 'allow': 1, 'status': 'married'}, 0), ('normal control', {'wage': 17607, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 0), ('normal control', {'wage': 6134, 'freq': 'semimonthly', 'allow': 0, 'status': 'single'}, 0)], [('regression (boundary)', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('regression', {'wage': 998152, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 168152), ('partial-repair probe', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('partial-repair probe', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 52312, 'freq': 'semimonthly', 'allow': 5, 'status': 'single'}, 0), ('normal control', {'wage': 28087, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 0), ('normal control', {'wage': 97334, 'freq': 'monthly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 18119, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 0)], [('regression', {'wage': 998152, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 168152), ('regression', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('partial-repair probe', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('partial-repair probe', {'wage': 124657, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 1966), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 10871, 'freq': 'weekly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 836, 'freq': 'semimonthly', 'allow': 1, 'status': 'single'}, 0), ('normal control', {'wage': 30453, 'freq': 'biweekly', 'allow': 2, 'status': 'married'}, 0), ('normal control', {'wage': 27188, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 0)], [('regression', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('regression', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('partial-repair probe', {'wage': 157996, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 5300), ('partial-repair probe', {'wage': 1223812, 'freq': 'monthly', 'allow': 5, 'status': 'married'}, 134822), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 23859, 'freq': 'semimonthly', 'allow': 2, 'status': 'married'}, 0), ('normal control', {'wage': 50426, 'freq': 'semimonthly', 'allow': 5, 'status': 'married'}, 0), ('normal control', {'wage': 128102, 'freq': 'monthly', 'allow': 5, 'status': 'married'}, 0), ('normal control', {'wage': 6608, 'freq': 'semimonthly', 'allow': 0, 'status': 'married'}, 0)], [('regression', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('regression', {'wage': 124657, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 1966), ('partial-repair probe', {'wage': 143245, 'freq': 'biweekly', 'allow': 5, 'status': 'married'}, 2978), ('partial-repair probe', {'wage': 475908, 'freq': 'monthly', 'allow': 6, 'status': 'single'}, 25476), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 49359, 'freq': 'weekly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 56499, 'freq': 'monthly', 'allow': 4, 'status': 'married'}, 0), ('normal control', {'wage': 6762, 'freq': 'weekly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 30428, 'freq': 'monthly', 'allow': 4, 'status': 'married'}, 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 (boundary) 0	`6923`	`5577`	Failed
regression (boundary) 1	`10708`	`7375`	Failed
partial-repair probe 2	`215652`	`168152`	Failed
partial-repair probe 3	`212825`	`190902`	Failed
boundary control 4	`0`	`0`	Passed
normal control 5	`0`	`0`	Passed
normal control 6	`0`	`0`	Passed
normal control 7	`0`	`0`	Passed
normal control 8	`0`	`0`	Passed

SHA-256 / 5993a5d3a0db5c6378a41d6861d720e5c757784c6692ea8137232e6e553d9a48

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(x):
    periods = {'weekly': 52, 'biweekly': 26, 'semimonthly': 24, 'monthly': 12}[x['freq']]
    annual = x['wage'] * periods
    adjusted = max(0, annual - x['allow'] * 430000)
    mult = 2 if x['status'] == 'married' else 1
    brackets = [(400000 * mult, 10), (1500000 * mult, 12), (5000000 * mult, 22)]
    tax = 0
    for i, (lo, rate) in enumerate(brackets):
        hi = brackets[i + 1][0] if i + 1 < len(brackets) else None
        if adjusted > lo:
            top = adjusted if hi is None else min(adjusted, hi)
            tax += top * rate
    wh = (tax * 2 + 100 * periods) // (200 * periods)
    return wh
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression (boundary)', {'wage': 57693, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 5577), ('regression (boundary)', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('partial-repair probe', {'wage': 998152, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 168152), ('partial-repair probe', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 22359, 'freq': 'semimonthly', 'allow': 5, 'status': 'married'}, 0), ('normal control', {'wage': 89742, 'freq': 'monthly', 'allow': 1, 'status': 'married'}, 0), ('normal control', {'wage': 17607, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 0), ('normal control', {'wage': 6134, 'freq': 'semimonthly', 'allow': 0, 'status': 'single'}, 0)], [('regression (boundary)', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('regression', {'wage': 998152, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 168152), ('partial-repair probe', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('partial-repair probe', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 52312, 'freq': 'semimonthly', 'allow': 5, 'status': 'single'}, 0), ('normal control', {'wage': 28087, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 0), ('normal control', {'wage': 97334, 'freq': 'monthly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 18119, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 0)], [('regression', {'wage': 998152, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 168152), ('regression', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('partial-repair probe', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('partial-repair probe', {'wage': 124657, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 1966), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 10871, 'freq': 'weekly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 836, 'freq': 'semimonthly', 'allow': 1, 'status': 'single'}, 0), ('normal control', {'wage': 30453, 'freq': 'biweekly', 'allow': 2, 'status': 'married'}, 0), ('normal control', {'wage': 27188, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 0)], [('regression', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('regression', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('partial-repair probe', {'wage': 157996, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 5300), ('partial-repair probe', {'wage': 1223812, 'freq': 'monthly', 'allow': 5, 'status': 'married'}, 134822), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 23859, 'freq': 'semimonthly', 'allow': 2, 'status': 'married'}, 0), ('normal control', {'wage': 50426, 'freq': 'semimonthly', 'allow': 5, 'status': 'married'}, 0), ('normal control', {'wage': 128102, 'freq': 'monthly', 'allow': 5, 'status': 'married'}, 0), ('normal control', {'wage': 6608, 'freq': 'semimonthly', 'allow': 0, 'status': 'married'}, 0)], [('regression', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('regression', {'wage': 124657, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 1966), ('partial-repair probe', {'wage': 143245, 'freq': 'biweekly', 'allow': 5, 'status': 'married'}, 2978), ('partial-repair probe', {'wage': 475908, 'freq': 'monthly', 'allow': 6, 'status': 'single'}, 25476), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 49359, 'freq': 'weekly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 56499, 'freq': 'monthly', 'allow': 4, 'status': 'married'}, 0), ('normal control', {'wage': 6762, 'freq': 'weekly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 30428, 'freq': 'monthly', 'allow': 4, 'status': 'married'}, 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 (boundary) 0	`9808`	`5577`	Failed
regression (boundary) 1	`10708`	`7375`	Failed
partial-repair probe 2	`278152`	`168152`	Failed
partial-repair probe 3	`241671`	`190902`	Failed
boundary control 4	`0`	`0`	Passed
normal control 5	`0`	`0`	Passed
normal control 6	`0`	`0`	Passed
normal control 7	`0`	`0`	Passed
normal control 8	`0`	`0`	Passed

SHA-256 / 28792d7665103d6898e1ada5e12855f542bc4c5a1a054cbd40ab71ac181591cf

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(x):
    periods = {'weekly': 52, 'biweekly': 26, 'semimonthly': 24, 'monthly': 12}[x['freq']]
    annual = x['wage'] * periods
    adjusted = max(0, annual - x['allow'] * 430000)
    mult = 2 if x['status'] == 'married' else 1
    brackets = [(400000 * mult, 10), (1500000 * mult, 12), (5000000 * mult, 22)]
    tax = 0
    for i, (lo, rate) in enumerate(brackets):
        hi = brackets[i + 1][0] if i + 1 < len(brackets) else None
        if adjusted > lo:
            top = adjusted if hi is None else min(adjusted, hi)
            tax += (top - lo) * rate
    wh = (tax * 2 + 100 * periods) // (200 * periods)
    return wh
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression (boundary)', {'wage': 57693, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 5577), ('regression (boundary)', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('partial-repair probe', {'wage': 998152, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 168152), ('partial-repair probe', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 22359, 'freq': 'semimonthly', 'allow': 5, 'status': 'married'}, 0), ('normal control', {'wage': 89742, 'freq': 'monthly', 'allow': 1, 'status': 'married'}, 0), ('normal control', {'wage': 17607, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 0), ('normal control', {'wage': 6134, 'freq': 'semimonthly', 'allow': 0, 'status': 'single'}, 0)], [('regression (boundary)', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('regression', {'wage': 998152, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 168152), ('partial-repair probe', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('partial-repair probe', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 52312, 'freq': 'semimonthly', 'allow': 5, 'status': 'single'}, 0), ('normal control', {'wage': 28087, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 0), ('normal control', {'wage': 97334, 'freq': 'monthly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 18119, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 0)], [('regression', {'wage': 998152, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 168152), ('regression', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('partial-repair probe', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('partial-repair probe', {'wage': 124657, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 1966), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 10871, 'freq': 'weekly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 836, 'freq': 'semimonthly', 'allow': 1, 'status': 'single'}, 0), ('normal control', {'wage': 30453, 'freq': 'biweekly', 'allow': 2, 'status': 'married'}, 0), ('normal control', {'wage': 27188, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 0)], [('regression', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('regression', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('partial-repair probe', {'wage': 157996, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 5300), ('partial-repair probe', {'wage': 1223812, 'freq': 'monthly', 'allow': 5, 'status': 'married'}, 134822), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 23859, 'freq': 'semimonthly', 'allow': 2, 'status': 'married'}, 0), ('normal control', {'wage': 50426, 'freq': 'semimonthly', 'allow': 5, 'status': 'married'}, 0), ('normal control', {'wage': 128102, 'freq': 'monthly', 'allow': 5, 'status': 'married'}, 0), ('normal control', {'wage': 6608, 'freq': 'semimonthly', 'allow': 0, 'status': 'married'}, 0)], [('regression', {'wage': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483), ('regression', {'wage': 124657, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 1966), ('partial-repair probe', {'wage': 143245, 'freq': 'biweekly', 'allow': 5, 'status': 'married'}, 2978), ('partial-repair probe', {'wage': 475908, 'freq': 'monthly', 'allow': 6, 'status': 'single'}, 25476), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('normal control', {'wage': 49359, 'freq': 'weekly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 56499, 'freq': 'monthly', 'allow': 4, 'status': 'married'}, 0), ('normal control', {'wage': 6762, 'freq': 'weekly', 'allow': 6, 'status': 'single'}, 0), ('normal control', {'wage': 30428, 'freq': 'monthly', 'allow': 4, 'status': 'married'}, 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 (boundary) 0	`5577`	`5577`	Passed
regression (boundary) 1	`7375`	`7375`	Passed
partial-repair probe 2	`168152`	`168152`	Passed
partial-repair probe 3	`190902`	`190902`	Passed
boundary control 4	`0`	`0`	Passed
normal control 5	`0`	`0`	Passed
normal control 6	`0`	`0`	Passed
normal control 7	`0`	`0`	Passed
normal control 8	`0`	`0`	Passed

SHA-256 / f17255084d5f9895fc44fce505e511c523e1cca9b68a2b5f46489d84a4f916ea

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 2026-09-29T14:46:30.996873+00:00.

Case digest / 10a167db77abbe88e526857e451a41cafd31ac4acddb94393c9da60fcc26873a