FA-58886 / Payroll withholding rules / Open access

Annualized percentage-method withholding: bracket slice upper bound · case 01

Employees whose income reaches a higher bracket are withheld far too much because lower brackets tax the whole income.

Verified by executionVariant 1 · 10 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Each bracket slice runs to the full adjusted income instead of stopping at the next bracket threshold.

THE FAILURE

Each bracket slice runs to the full adjusted income instead of stopping at the next bracket threshold.

Unsuccessful approach: The attempt clips at one cent below the next threshold, leaving one cent per crossed bracket taxed at the lower rate.

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
            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', {'wage': 497372, 'freq': 'semimonthly', 'allow': 2, 'status': 'single'}, 77789), ('regression (boundary)', {'wage': 57693, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 5577), ('partial-repair probe', {'wage': 460175, 'freq': 'biweekly', 'allow': 4, 'status': 'married'}, 42839), ('partial-repair probe', {'wage': 223432, 'freq': 'monthly', 'allow': 0, 'status': 'single'}, 20979), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('boundary control', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('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', {'wage': 460175, 'freq': 'biweekly', 'allow': 4, 'status': 'married'}, 42839), ('regression', {'wage': 998152, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 168152), ('partial-repair probe', {'wage': 223432, 'freq': 'monthly', 'allow': 0, 'status': 'single'}, 20979), ('partial-repair probe', {'wage': 497372, 'freq': 'semimonthly', 'allow': 2, 'status': 'single'}, 77789), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('boundary control', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('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': 223432, 'freq': 'monthly', 'allow': 0, 'status': 'single'}, 20979), ('regression', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('partial-repair probe', {'wage': 460175, 'freq': 'biweekly', 'allow': 4, 'status': 'married'}, 42839), ('partial-repair probe', {'wage': 497372, 'freq': 'semimonthly', 'allow': 2, 'status': 'single'}, 77789), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('boundary control', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('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': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483)], [('regression', {'wage': 497372, 'freq': 'semimonthly', 'allow': 2, 'status': 'single'}, 77789), ('regression', {'wage': 1223812, 'freq': 'monthly', 'allow': 5, 'status': 'married'}, 134822), ('partial-repair probe', {'wage': 460175, 'freq': 'biweekly', 'allow': 4, 'status': 'married'}, 42839), ('partial-repair probe', {'wage': 223432, 'freq': 'monthly', 'allow': 0, 'status': 'single'}, 20979), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('boundary control', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('normal control', {'wage': 124657, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 1966), ('normal control', {'wage': 27188, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 0), ('normal control', {'wage': 157996, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 5300), ('normal control', {'wage': 143245, 'freq': 'biweekly', 'allow': 5, 'status': 'married'}, 2978)], [('regression', {'wage': 460175, 'freq': 'biweekly', 'allow': 4, 'status': 'married'}, 42839), ('regression', {'wage': 475908, 'freq': 'monthly', 'allow': 6, 'status': 'single'}, 25476), ('partial-repair probe', {'wage': 223432, 'freq': 'monthly', 'allow': 0, 'status': 'single'}, 20979), ('partial-repair probe', {'wage': 497372, 'freq': 'semimonthly', 'allow': 2, 'status': 'single'}, 77789), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('boundary control', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('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)]]
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	`148077`	`77789`	Failed
regression (boundary) 1	`8462`	`5577`	Failed
partial-repair probe 2	`71831`	`42839`	Failed
partial-repair probe 3	`30822`	`20979`	Failed
boundary control 4	`0`	`0`	Passed
boundary control 5	`7375`	`7375`	Passed
normal control 6	`0`	`0`	Passed
normal control 7	`0`	`0`	Passed
normal control 8	`0`	`0`	Passed
normal control 9	`0`	`0`	Passed

SHA-256 / ff344c9ce75e437a141d8a19dcfdf45349bbfb531fd667e039e80145f6111dff

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 - 1)
            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', {'wage': 497372, 'freq': 'semimonthly', 'allow': 2, 'status': 'single'}, 77789), ('regression (boundary)', {'wage': 57693, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 5577), ('partial-repair probe', {'wage': 460175, 'freq': 'biweekly', 'allow': 4, 'status': 'married'}, 42839), ('partial-repair probe', {'wage': 223432, 'freq': 'monthly', 'allow': 0, 'status': 'single'}, 20979), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('boundary control', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('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', {'wage': 460175, 'freq': 'biweekly', 'allow': 4, 'status': 'married'}, 42839), ('regression', {'wage': 998152, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 168152), ('partial-repair probe', {'wage': 223432, 'freq': 'monthly', 'allow': 0, 'status': 'single'}, 20979), ('partial-repair probe', {'wage': 497372, 'freq': 'semimonthly', 'allow': 2, 'status': 'single'}, 77789), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('boundary control', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('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': 223432, 'freq': 'monthly', 'allow': 0, 'status': 'single'}, 20979), ('regression', {'wage': 992194, 'freq': 'weekly', 'allow': 3, 'status': 'married'}, 190902), ('partial-repair probe', {'wage': 460175, 'freq': 'biweekly', 'allow': 4, 'status': 'married'}, 42839), ('partial-repair probe', {'wage': 497372, 'freq': 'semimonthly', 'allow': 2, 'status': 'single'}, 77789), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('boundary control', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('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': 36754, 'freq': 'biweekly', 'allow': 1, 'status': 'single'}, 483)], [('regression', {'wage': 497372, 'freq': 'semimonthly', 'allow': 2, 'status': 'single'}, 77789), ('regression', {'wage': 1223812, 'freq': 'monthly', 'allow': 5, 'status': 'married'}, 134822), ('partial-repair probe', {'wage': 460175, 'freq': 'biweekly', 'allow': 4, 'status': 'married'}, 42839), ('partial-repair probe', {'wage': 223432, 'freq': 'monthly', 'allow': 0, 'status': 'single'}, 20979), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('boundary control', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('normal control', {'wage': 124657, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 1966), ('normal control', {'wage': 27188, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 0), ('normal control', {'wage': 157996, 'freq': 'semimonthly', 'allow': 4, 'status': 'married'}, 5300), ('normal control', {'wage': 143245, 'freq': 'biweekly', 'allow': 5, 'status': 'married'}, 2978)], [('regression', {'wage': 460175, 'freq': 'biweekly', 'allow': 4, 'status': 'married'}, 42839), ('regression', {'wage': 475908, 'freq': 'monthly', 'allow': 6, 'status': 'single'}, 25476), ('partial-repair probe', {'wage': 223432, 'freq': 'monthly', 'allow': 0, 'status': 'single'}, 20979), ('partial-repair probe', {'wage': 497372, 'freq': 'semimonthly', 'allow': 2, 'status': 'single'}, 77789), ('boundary control', {'wage': 0, 'freq': 'weekly', 'allow': 0, 'status': 'single'}, 0), ('boundary control', {'wage': 125000, 'freq': 'semimonthly', 'allow': 1, 'status': 'married'}, 7375), ('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)]]
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	`77788`	`77789`	Failed
regression (boundary) 1	`5577`	`5577`	Passed
partial-repair probe 2	`42838`	`42839`	Failed
partial-repair probe 3	`20978`	`20979`	Failed
boundary control 4	`0`	`0`	Passed
boundary control 5	`7375`	`7375`	Passed
normal control 6	`0`	`0`	Passed
normal control 7	`0`	`0`	Passed
normal control 8	`0`	`0`	Passed
normal control 9	`0`	`0`	Passed

SHA-256 / 4deb4c97149611253692a199964d2c4d3c8fb68cf05d96ca864c8076ee52b411

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 10 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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 2026-09-29T14:46:30.935401+00:00.

Case digest / 3f6c335dedefb5dd2a9fd638c2a0719d150c50edb973e2177c5c7c0731f574d4