FA-58881 / Payroll withholding rules / Open access
Additional Medicare withholding threshold: additional rate unit · case 01
Additional Medicare withholding is ten times too large once the threshold is crossed.
ROOT CAUSE
The 0.9% rate is applied as 9% because the denominator drops one power of ten.
VERIFIED REPAIR
Restore the contract rule at the additional rate unit step: use `extra = (extra_wages * 9 + 500) // 1000`.
Unsuccessful approach: The attempt overcorrects to 0.09%, withholding a tenth of the required additional tax.
Case contract
Input [ytd_wages, wages] in cents. Regular Medicare is 1.45% of wages rounded half-up. Additional Medicare is 0.9% of the part of this paycheck that pushes year-to-date wages above 200,000.00, rounded half-up separately. Return [medicare, additional_wages, medicare + additional].
Why this case matters
The employer must begin the extra 0.9% exactly with the paycheck that crosses the threshold, and only on the excess portion.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
ytd, wages = x
base_tax = (wages * 145 + 5000) // 10000
over_before = max(0, ytd - 20000000)
over_after = max(0, ytd + wages - 20000000)
extra_wages = over_after - over_before
extra = (extra_wages * 9 + 50) // 100
return [base_tax, extra_wages, base_tax + extra]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression (boundary)', [20000000, 100000], [1450, 100000, 2350]), ('regression (boundary)', [19950000, 100000], [1450, 50000, 1900]), ('partial-repair probe (boundary)', [25000000, 12345], [179, 12345, 290]), ('partial-repair probe', [20158685, 407195], [5904, 407195, 9569]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [7554383, 209883], [3043, 0, 3043]), ('normal control', [8001356, 2662166], [38601, 0, 38601]), ('normal control', [2439490, 49414], [717, 0, 717]), ('normal control', [19686537, 15872], [230, 0, 230])], [('regression (boundary)', [19950000, 100000], [1450, 50000, 1900]), ('regression (boundary)', [25000000, 12345], [179, 12345, 290]), ('partial-repair probe', [20158685, 407195], [5904, 407195, 9569]), ('partial-repair probe', [20518887, 2208012], [32016, 2208012, 51888]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [3879860, 1040777], [15091, 0, 15091]), ('normal control', [4213397, 2332986], [33828, 0, 33828]), ('normal control', [624582, 1378603], [19990, 0, 19990]), ('normal control', [4705614, 33884], [491, 0, 491])], [('regression (boundary)', [25000000, 12345], [179, 12345, 290]), ('regression', [20158685, 407195], [5904, 407195, 9569]), ('partial-repair probe', [20518887, 2208012], [32016, 2208012, 51888]), ('partial-repair probe', [20000000, 1829965], [26534, 1829965, 43004]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [19201069, 465833], [6755, 0, 6755]), ('normal control', [4401788, 1347616], [19540, 0, 19540]), ('normal control', [703223, 1018779], [14772, 0, 14772]), ('normal control', [916863, 25193], [365, 0, 365])], [('regression', [20158685, 407195], [5904, 407195, 9569]), ('regression', [20518887, 2208012], [32016, 2208012, 51888]), ('partial-repair probe', [19681988, 1173456], [17015, 855444, 24714]), ('partial-repair probe', [19655408, 1735735], [25168, 1391143, 37688]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [10047708, 404743], [5869, 0, 5869]), ('normal control', [10176860, 61801], [896, 0, 896]), ('normal control', [10855940, 2686693], [38957, 0, 38957]), ('normal control', [19631063, 17474], [253, 0, 253])], [('regression', [20518887, 2208012], [32016, 2208012, 51888]), ('regression', [20000000, 1829965], [26534, 1829965, 43004]), ('partial-repair probe', [20723172, 31906], [463, 31906, 750]), ('partial-repair probe', [20767841, 19138], [278, 19138, 450]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [14097993, 22841], [331, 0, 331]), ('normal control', [8593667, 1082072], [15690, 0, 15690]), ('normal control', [3135753, 2538919], [36814, 0, 36814]), ('normal control', [8401894, 797881], [11569, 0, 11569])]]
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 | [1450, 100000, 10450] | [1450, 100000, 2350] | Failed |
| regression (boundary) 1 | [1450, 50000, 5950] | [1450, 50000, 1900] | Failed |
| partial-repair probe (boundary) 2 | [179, 12345, 1290] | [179, 12345, 290] | Failed |
| partial-repair probe 3 | [5904, 407195, 42552] | [5904, 407195, 9569] | Failed |
| boundary control 4 | [0, 0, 0] | [0, 0, 0] | Passed |
| boundary control 5 | [0, 0, 0] | [0, 0, 0] | Passed |
| normal control 6 | [3043, 0, 3043] | [3043, 0, 3043] | Passed |
| normal control 7 | [38601, 0, 38601] | [38601, 0, 38601] | Passed |
| normal control 8 | [717, 0, 717] | [717, 0, 717] | Passed |
| normal control 9 | [230, 0, 230] | [230, 0, 230] | Passed |
SHA-256 / d0ad26700286389f56e109d06fd4ca231e6ecb49eb53572539b336642ffda624
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
ytd, wages = x
base_tax = (wages * 145 + 5000) // 10000
over_before = max(0, ytd - 20000000)
over_after = max(0, ytd + wages - 20000000)
extra_wages = over_after - over_before
extra = (extra_wages * 9 + 5000) // 10000
return [base_tax, extra_wages, base_tax + extra]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression (boundary)', [20000000, 100000], [1450, 100000, 2350]), ('regression (boundary)', [19950000, 100000], [1450, 50000, 1900]), ('partial-repair probe (boundary)', [25000000, 12345], [179, 12345, 290]), ('partial-repair probe', [20158685, 407195], [5904, 407195, 9569]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [7554383, 209883], [3043, 0, 3043]), ('normal control', [8001356, 2662166], [38601, 0, 38601]), ('normal control', [2439490, 49414], [717, 0, 717]), ('normal control', [19686537, 15872], [230, 0, 230])], [('regression (boundary)', [19950000, 100000], [1450, 50000, 1900]), ('regression (boundary)', [25000000, 12345], [179, 12345, 290]), ('partial-repair probe', [20158685, 407195], [5904, 407195, 9569]), ('partial-repair probe', [20518887, 2208012], [32016, 2208012, 51888]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [3879860, 1040777], [15091, 0, 15091]), ('normal control', [4213397, 2332986], [33828, 0, 33828]), ('normal control', [624582, 1378603], [19990, 0, 19990]), ('normal control', [4705614, 33884], [491, 0, 491])], [('regression (boundary)', [25000000, 12345], [179, 12345, 290]), ('regression', [20158685, 407195], [5904, 407195, 9569]), ('partial-repair probe', [20518887, 2208012], [32016, 2208012, 51888]), ('partial-repair probe', [20000000, 1829965], [26534, 1829965, 43004]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [19201069, 465833], [6755, 0, 6755]), ('normal control', [4401788, 1347616], [19540, 0, 19540]), ('normal control', [703223, 1018779], [14772, 0, 14772]), ('normal control', [916863, 25193], [365, 0, 365])], [('regression', [20158685, 407195], [5904, 407195, 9569]), ('regression', [20518887, 2208012], [32016, 2208012, 51888]), ('partial-repair probe', [19681988, 1173456], [17015, 855444, 24714]), ('partial-repair probe', [19655408, 1735735], [25168, 1391143, 37688]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [10047708, 404743], [5869, 0, 5869]), ('normal control', [10176860, 61801], [896, 0, 896]), ('normal control', [10855940, 2686693], [38957, 0, 38957]), ('normal control', [19631063, 17474], [253, 0, 253])], [('regression', [20518887, 2208012], [32016, 2208012, 51888]), ('regression', [20000000, 1829965], [26534, 1829965, 43004]), ('partial-repair probe', [20723172, 31906], [463, 31906, 750]), ('partial-repair probe', [20767841, 19138], [278, 19138, 450]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [14097993, 22841], [331, 0, 331]), ('normal control', [8593667, 1082072], [15690, 0, 15690]), ('normal control', [3135753, 2538919], [36814, 0, 36814]), ('normal control', [8401894, 797881], [11569, 0, 11569])]]
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 | [1450, 100000, 1540] | [1450, 100000, 2350] | Failed |
| regression (boundary) 1 | [1450, 50000, 1495] | [1450, 50000, 1900] | Failed |
| partial-repair probe (boundary) 2 | [179, 12345, 190] | [179, 12345, 290] | Failed |
| partial-repair probe 3 | [5904, 407195, 6270] | [5904, 407195, 9569] | Failed |
| boundary control 4 | [0, 0, 0] | [0, 0, 0] | Passed |
| boundary control 5 | [0, 0, 0] | [0, 0, 0] | Passed |
| normal control 6 | [3043, 0, 3043] | [3043, 0, 3043] | Passed |
| normal control 7 | [38601, 0, 38601] | [38601, 0, 38601] | Passed |
| normal control 8 | [717, 0, 717] | [717, 0, 717] | Passed |
| normal control 9 | [230, 0, 230] | [230, 0, 230] | Passed |
SHA-256 / f6e49ff5993d7b92b36a9d004093a1548ad25166a01114ac4ef8b85203b8f2bb
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
ytd, wages = x
base_tax = (wages * 145 + 5000) // 10000
over_before = max(0, ytd - 20000000)
over_after = max(0, ytd + wages - 20000000)
extra_wages = over_after - over_before
extra = (extra_wages * 9 + 500) // 1000
return [base_tax, extra_wages, base_tax + extra]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression (boundary)', [20000000, 100000], [1450, 100000, 2350]), ('regression (boundary)', [19950000, 100000], [1450, 50000, 1900]), ('partial-repair probe (boundary)', [25000000, 12345], [179, 12345, 290]), ('partial-repair probe', [20158685, 407195], [5904, 407195, 9569]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [7554383, 209883], [3043, 0, 3043]), ('normal control', [8001356, 2662166], [38601, 0, 38601]), ('normal control', [2439490, 49414], [717, 0, 717]), ('normal control', [19686537, 15872], [230, 0, 230])], [('regression (boundary)', [19950000, 100000], [1450, 50000, 1900]), ('regression (boundary)', [25000000, 12345], [179, 12345, 290]), ('partial-repair probe', [20158685, 407195], [5904, 407195, 9569]), ('partial-repair probe', [20518887, 2208012], [32016, 2208012, 51888]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [3879860, 1040777], [15091, 0, 15091]), ('normal control', [4213397, 2332986], [33828, 0, 33828]), ('normal control', [624582, 1378603], [19990, 0, 19990]), ('normal control', [4705614, 33884], [491, 0, 491])], [('regression (boundary)', [25000000, 12345], [179, 12345, 290]), ('regression', [20158685, 407195], [5904, 407195, 9569]), ('partial-repair probe', [20518887, 2208012], [32016, 2208012, 51888]), ('partial-repair probe', [20000000, 1829965], [26534, 1829965, 43004]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [19201069, 465833], [6755, 0, 6755]), ('normal control', [4401788, 1347616], [19540, 0, 19540]), ('normal control', [703223, 1018779], [14772, 0, 14772]), ('normal control', [916863, 25193], [365, 0, 365])], [('regression', [20158685, 407195], [5904, 407195, 9569]), ('regression', [20518887, 2208012], [32016, 2208012, 51888]), ('partial-repair probe', [19681988, 1173456], [17015, 855444, 24714]), ('partial-repair probe', [19655408, 1735735], [25168, 1391143, 37688]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [10047708, 404743], [5869, 0, 5869]), ('normal control', [10176860, 61801], [896, 0, 896]), ('normal control', [10855940, 2686693], [38957, 0, 38957]), ('normal control', [19631063, 17474], [253, 0, 253])], [('regression', [20518887, 2208012], [32016, 2208012, 51888]), ('regression', [20000000, 1829965], [26534, 1829965, 43004]), ('partial-repair probe', [20723172, 31906], [463, 31906, 750]), ('partial-repair probe', [20767841, 19138], [278, 19138, 450]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [19999999, 1], [0, 0, 0]), ('normal control', [14097993, 22841], [331, 0, 331]), ('normal control', [8593667, 1082072], [15690, 0, 15690]), ('normal control', [3135753, 2538919], [36814, 0, 36814]), ('normal control', [8401894, 797881], [11569, 0, 11569])]]
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 | [1450, 100000, 2350] | [1450, 100000, 2350] | Passed |
| regression (boundary) 1 | [1450, 50000, 1900] | [1450, 50000, 1900] | Passed |
| partial-repair probe (boundary) 2 | [179, 12345, 290] | [179, 12345, 290] | Passed |
| partial-repair probe 3 | [5904, 407195, 9569] | [5904, 407195, 9569] | Passed |
| boundary control 4 | [0, 0, 0] | [0, 0, 0] | Passed |
| boundary control 5 | [0, 0, 0] | [0, 0, 0] | Passed |
| normal control 6 | [3043, 0, 3043] | [3043, 0, 3043] | Passed |
| normal control 7 | [38601, 0, 38601] | [38601, 0, 38601] | Passed |
| normal control 8 | [717, 0, 717] | [717, 0, 717] | Passed |
| normal control 9 | [230, 0, 230] | [230, 0, 230] | Passed |
SHA-256 / 27844e4b413540299bd7d31bbb879b49e5722643471e984f0c81048007a16f55
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 2026-09-29T14:46:30.874647+00:00.
Case digest / 67db1c8799a3901766c4ec647550f26c7b906f812db4b407981dfdba458d38ea