FA-58926 / Payroll withholding rules / Open access
Supplemental flat-rate withholding tiers: mandatory tier split · case 01
A bonus that crosses the 1M mark is counted twice, once in each tier, over-withholding heavily.
ROOT CAUSE
The whole crossing bonus is placed in the 37% tier while the 22% slice is also kept.
VERIFIED REPAIR
Restore the contract rule at the mandatory tier split step: use `high = bonus - low`.
Unsuccessful approach: The attempt measures the excess from cumulative totals, which is right on the crossing payment but over-counts once year-to-date is already above the limit.
Case contract
Input [ytd_supplemental, bonus] in cents. Supplemental wages up to a cumulative 1,000,000.00 are withheld at 22%; the part of cumulative supplemental wages above that is withheld at 37%. Withholding = (low*22 + high*37) rounded half-up to the cent once. Return [low, high, withholding].
Why this case matters
The mandatory 37% tier depends on cumulative supplemental wages and must split the crossing payment.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
ytd, bonus = x
limit = 100000000
below_room = max(0, limit - ytd)
low = min(bonus, below_room)
high = bonus if ytd + bonus > limit else 0
wh = (low * 22 + high * 37 + 50) // 100
return [low, high, wh]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression (boundary)', [99999950, 101], [50, 51, 30]), ('regression', [99561826, 3108388], [438174, 2670214, 1084377]), ('partial-repair probe (boundary)', [150000000, 33], [0, 33, 12]), ('partial-repair probe', [100849097, 11351881], [0, 11351881, 4200196]), ('boundary control', [100000000, 100], [0, 100, 37]), ('boundary control', [0, 0], [0, 0, 0]), ('normal control', [98022676, 574037], [574037, 0, 126288]), ('normal control', [3276820, 34567110], [34567110, 0, 7604764]), ('normal control', [2185641, 20897159], [20897159, 0, 4597375]), ('normal control', [100000000, 6674805], [0, 6674805, 2469678])], [('regression', [99561826, 3108388], [438174, 2670214, 1084377]), ('regression', [98276481, 26306241], [1723519, 24582722, 9474781]), ('partial-repair probe', [107387654, 40335762], [0, 40335762, 14924232]), ('partial-repair probe', [105717987, 5743298], [0, 5743298, 2125020]), ('boundary control', [99000000, 1000000], [1000000, 0, 220000]), ('boundary control', [100000000, 100], [0, 100, 37]), ('normal control', [49161621, 69310], [69310, 0, 15248]), ('normal control', [66640759, 96623], [96623, 0, 21257]), ('normal control', [100000000, 52968], [0, 52968, 19598]), ('normal control', [100000000, 2071189], [0, 2071189, 766340])], [('regression', [98276481, 26306241], [1723519, 24582722, 9474781]), ('regression', [95783477, 7789713], [4216523, 3573190, 2249715]), ('partial-repair probe', [106100649, 5242825], [0, 5242825, 1939845]), ('partial-repair probe', [108022995, 3827223], [0, 3827223, 1416073]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [99000000, 1000000], [1000000, 0, 220000]), ('normal control', [2545996, 41743464], [41743464, 0, 9183562]), ('normal control', [100000000, 21687395], [0, 21687395, 8024336]), ('normal control', [97846124, 8522], [8522, 0, 1875]), ('normal control', [3045185, 81939], [81939, 0, 18027])], [('regression', [95783477, 7789713], [4216523, 3573190, 2249715]), ('regression', [80430645, 24514483], [19569355, 4945128, 6134955]), ('partial-repair probe', [107848302, 42942181], [0, 42942181, 15888607]), ('partial-repair probe', [100442126, 20427451], [0, 20427451, 7558157]), ('boundary control', [100000000, 100], [0, 100, 37]), ('boundary control', [0, 0], [0, 0, 0]), ('normal control', [374880, 70400], [70400, 0, 15488]), ('normal control', [688889, 1036750], [1036750, 0, 228085]), ('normal control', [63007595, 8068739], [8068739, 0, 1775123]), ('normal control', [64670743, 3021249], [3021249, 0, 664675])], [('regression', [80430645, 24514483], [19569355, 4945128, 6134955]), ('regression', [98797663, 40240022], [1202337, 39037685, 14708458]), ('partial-repair probe', [105107162, 46320149], [0, 46320149, 17138455]), ('partial-repair probe', [107181794, 22743504], [0, 22743504, 8415096]), ('boundary control', [99000000, 1000000], [1000000, 0, 220000]), ('boundary control', [100000000, 100], [0, 100, 37]), ('normal control', [97882960, 95576], [95576, 0, 21027]), ('normal control', [100000000, 75200], [0, 75200, 27824]), ('normal control', [100000000, 1622529], [0, 1622529, 600336]), ('normal control', [63981087, 6343691], [6343691, 0, 1395612])]]
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 | [50, 101, 48] | [50, 51, 30] | Failed |
| regression 1 | [438174, 3108388, 1246502] | [438174, 2670214, 1084377] | Failed |
| partial-repair probe (boundary) 2 | [0, 33, 12] | [0, 33, 12] | Passed |
| partial-repair probe 3 | [0, 11351881, 4200196] | [0, 11351881, 4200196] | Passed |
| boundary control 4 | [0, 100, 37] | [0, 100, 37] | Passed |
| boundary control 5 | [0, 0, 0] | [0, 0, 0] | Passed |
| normal control 6 | [574037, 0, 126288] | [574037, 0, 126288] | Passed |
| normal control 7 | [34567110, 0, 7604764] | [34567110, 0, 7604764] | Passed |
| normal control 8 | [20897159, 0, 4597375] | [20897159, 0, 4597375] | Passed |
| normal control 9 | [0, 6674805, 2469678] | [0, 6674805, 2469678] | Passed |
SHA-256 / 1d928f0c8c45dd3237c4a26c074d0b8dac52adeb160053f4e7adc135d821eeec
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
ytd, bonus = x
limit = 100000000
below_room = max(0, limit - ytd)
low = min(bonus, below_room)
high = max(0, ytd + bonus - limit)
wh = (low * 22 + high * 37 + 50) // 100
return [low, high, wh]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression (boundary)', [99999950, 101], [50, 51, 30]), ('regression', [99561826, 3108388], [438174, 2670214, 1084377]), ('partial-repair probe (boundary)', [150000000, 33], [0, 33, 12]), ('partial-repair probe', [100849097, 11351881], [0, 11351881, 4200196]), ('boundary control', [100000000, 100], [0, 100, 37]), ('boundary control', [0, 0], [0, 0, 0]), ('normal control', [98022676, 574037], [574037, 0, 126288]), ('normal control', [3276820, 34567110], [34567110, 0, 7604764]), ('normal control', [2185641, 20897159], [20897159, 0, 4597375]), ('normal control', [100000000, 6674805], [0, 6674805, 2469678])], [('regression', [99561826, 3108388], [438174, 2670214, 1084377]), ('regression', [98276481, 26306241], [1723519, 24582722, 9474781]), ('partial-repair probe', [107387654, 40335762], [0, 40335762, 14924232]), ('partial-repair probe', [105717987, 5743298], [0, 5743298, 2125020]), ('boundary control', [99000000, 1000000], [1000000, 0, 220000]), ('boundary control', [100000000, 100], [0, 100, 37]), ('normal control', [49161621, 69310], [69310, 0, 15248]), ('normal control', [66640759, 96623], [96623, 0, 21257]), ('normal control', [100000000, 52968], [0, 52968, 19598]), ('normal control', [100000000, 2071189], [0, 2071189, 766340])], [('regression', [98276481, 26306241], [1723519, 24582722, 9474781]), ('regression', [95783477, 7789713], [4216523, 3573190, 2249715]), ('partial-repair probe', [106100649, 5242825], [0, 5242825, 1939845]), ('partial-repair probe', [108022995, 3827223], [0, 3827223, 1416073]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [99000000, 1000000], [1000000, 0, 220000]), ('normal control', [2545996, 41743464], [41743464, 0, 9183562]), ('normal control', [100000000, 21687395], [0, 21687395, 8024336]), ('normal control', [97846124, 8522], [8522, 0, 1875]), ('normal control', [3045185, 81939], [81939, 0, 18027])], [('regression', [95783477, 7789713], [4216523, 3573190, 2249715]), ('regression', [80430645, 24514483], [19569355, 4945128, 6134955]), ('partial-repair probe', [107848302, 42942181], [0, 42942181, 15888607]), ('partial-repair probe', [100442126, 20427451], [0, 20427451, 7558157]), ('boundary control', [100000000, 100], [0, 100, 37]), ('boundary control', [0, 0], [0, 0, 0]), ('normal control', [374880, 70400], [70400, 0, 15488]), ('normal control', [688889, 1036750], [1036750, 0, 228085]), ('normal control', [63007595, 8068739], [8068739, 0, 1775123]), ('normal control', [64670743, 3021249], [3021249, 0, 664675])], [('regression', [80430645, 24514483], [19569355, 4945128, 6134955]), ('regression', [98797663, 40240022], [1202337, 39037685, 14708458]), ('partial-repair probe', [105107162, 46320149], [0, 46320149, 17138455]), ('partial-repair probe', [107181794, 22743504], [0, 22743504, 8415096]), ('boundary control', [99000000, 1000000], [1000000, 0, 220000]), ('boundary control', [100000000, 100], [0, 100, 37]), ('normal control', [97882960, 95576], [95576, 0, 21027]), ('normal control', [100000000, 75200], [0, 75200, 27824]), ('normal control', [100000000, 1622529], [0, 1622529, 600336]), ('normal control', [63981087, 6343691], [6343691, 0, 1395612])]]
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 | [50, 51, 30] | [50, 51, 30] | Passed |
| regression 1 | [438174, 2670214, 1084377] | [438174, 2670214, 1084377] | Passed |
| partial-repair probe (boundary) 2 | [0, 50000033, 18500012] | [0, 33, 12] | Failed |
| partial-repair probe 3 | [0, 12200978, 4514362] | [0, 11351881, 4200196] | Failed |
| boundary control 4 | [0, 100, 37] | [0, 100, 37] | Passed |
| boundary control 5 | [0, 0, 0] | [0, 0, 0] | Passed |
| normal control 6 | [574037, 0, 126288] | [574037, 0, 126288] | Passed |
| normal control 7 | [34567110, 0, 7604764] | [34567110, 0, 7604764] | Passed |
| normal control 8 | [20897159, 0, 4597375] | [20897159, 0, 4597375] | Passed |
| normal control 9 | [0, 6674805, 2469678] | [0, 6674805, 2469678] | Passed |
SHA-256 / 1933953bd395bb3ed34defcb3cce72cad2fd5f40ac49c9ffb9c4fcc1557865a7
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
ytd, bonus = x
limit = 100000000
below_room = max(0, limit - ytd)
low = min(bonus, below_room)
high = bonus - low
wh = (low * 22 + high * 37 + 50) // 100
return [low, high, wh]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression (boundary)', [99999950, 101], [50, 51, 30]), ('regression', [99561826, 3108388], [438174, 2670214, 1084377]), ('partial-repair probe (boundary)', [150000000, 33], [0, 33, 12]), ('partial-repair probe', [100849097, 11351881], [0, 11351881, 4200196]), ('boundary control', [100000000, 100], [0, 100, 37]), ('boundary control', [0, 0], [0, 0, 0]), ('normal control', [98022676, 574037], [574037, 0, 126288]), ('normal control', [3276820, 34567110], [34567110, 0, 7604764]), ('normal control', [2185641, 20897159], [20897159, 0, 4597375]), ('normal control', [100000000, 6674805], [0, 6674805, 2469678])], [('regression', [99561826, 3108388], [438174, 2670214, 1084377]), ('regression', [98276481, 26306241], [1723519, 24582722, 9474781]), ('partial-repair probe', [107387654, 40335762], [0, 40335762, 14924232]), ('partial-repair probe', [105717987, 5743298], [0, 5743298, 2125020]), ('boundary control', [99000000, 1000000], [1000000, 0, 220000]), ('boundary control', [100000000, 100], [0, 100, 37]), ('normal control', [49161621, 69310], [69310, 0, 15248]), ('normal control', [66640759, 96623], [96623, 0, 21257]), ('normal control', [100000000, 52968], [0, 52968, 19598]), ('normal control', [100000000, 2071189], [0, 2071189, 766340])], [('regression', [98276481, 26306241], [1723519, 24582722, 9474781]), ('regression', [95783477, 7789713], [4216523, 3573190, 2249715]), ('partial-repair probe', [106100649, 5242825], [0, 5242825, 1939845]), ('partial-repair probe', [108022995, 3827223], [0, 3827223, 1416073]), ('boundary control', [0, 0], [0, 0, 0]), ('boundary control', [99000000, 1000000], [1000000, 0, 220000]), ('normal control', [2545996, 41743464], [41743464, 0, 9183562]), ('normal control', [100000000, 21687395], [0, 21687395, 8024336]), ('normal control', [97846124, 8522], [8522, 0, 1875]), ('normal control', [3045185, 81939], [81939, 0, 18027])], [('regression', [95783477, 7789713], [4216523, 3573190, 2249715]), ('regression', [80430645, 24514483], [19569355, 4945128, 6134955]), ('partial-repair probe', [107848302, 42942181], [0, 42942181, 15888607]), ('partial-repair probe', [100442126, 20427451], [0, 20427451, 7558157]), ('boundary control', [100000000, 100], [0, 100, 37]), ('boundary control', [0, 0], [0, 0, 0]), ('normal control', [374880, 70400], [70400, 0, 15488]), ('normal control', [688889, 1036750], [1036750, 0, 228085]), ('normal control', [63007595, 8068739], [8068739, 0, 1775123]), ('normal control', [64670743, 3021249], [3021249, 0, 664675])], [('regression', [80430645, 24514483], [19569355, 4945128, 6134955]), ('regression', [98797663, 40240022], [1202337, 39037685, 14708458]), ('partial-repair probe', [105107162, 46320149], [0, 46320149, 17138455]), ('partial-repair probe', [107181794, 22743504], [0, 22743504, 8415096]), ('boundary control', [99000000, 1000000], [1000000, 0, 220000]), ('boundary control', [100000000, 100], [0, 100, 37]), ('normal control', [97882960, 95576], [95576, 0, 21027]), ('normal control', [100000000, 75200], [0, 75200, 27824]), ('normal control', [100000000, 1622529], [0, 1622529, 600336]), ('normal control', [63981087, 6343691], [6343691, 0, 1395612])]]
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 | [50, 51, 30] | [50, 51, 30] | Passed |
| regression 1 | [438174, 2670214, 1084377] | [438174, 2670214, 1084377] | Passed |
| partial-repair probe (boundary) 2 | [0, 33, 12] | [0, 33, 12] | Passed |
| partial-repair probe 3 | [0, 11351881, 4200196] | [0, 11351881, 4200196] | Passed |
| boundary control 4 | [0, 100, 37] | [0, 100, 37] | Passed |
| boundary control 5 | [0, 0, 0] | [0, 0, 0] | Passed |
| normal control 6 | [574037, 0, 126288] | [574037, 0, 126288] | Passed |
| normal control 7 | [34567110, 0, 7604764] | [34567110, 0, 7604764] | Passed |
| normal control 8 | [20897159, 0, 4597375] | [20897159, 0, 4597375] | Passed |
| normal control 9 | [0, 6674805, 2469678] | [0, 6674805, 2469678] | Passed |
SHA-256 / 3a6c6a1c4f14cbedcdf50fc10519bff88df9c2ffdb5501f87a6b0acfbafcf813
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 2026-09-29T14:46:31.421993+00:00.
Case digest / d61198484b98bf50a8f21c374d3c269f6c4253348a37679d265dddaac7f22d3e