FA-13816 / Numerical aggregation / Open access
Huber residual total: The quadratic region omits its one-half coefficient. · case 01
The reduction disagrees with its explicit aggregation oracle.
ROOT CAUSE
The quadratic region omits its one-half coefficient.
VERIFIED REPAIR
Preserve the huber residual total contract at the identified reduction decision.
Unsuccessful approach: Halving again gives the wrong quadratic curvature.
Case contract
For integer residuals and nonnegative integer threshold delta, sum the convex Huber loss: e**2/2 inside threshold and delta*(abs(e)-delta/2) outside. Return exact Fraction string; this is a bounded loss reduction, not an optimizer.
Why this case matters
Exact bounded examples isolate a reduction defect without floating-point or external-service effects.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
from collections import Counter, defaultdict
import math
import itertools
N = 1
observations = []
def solve(errors, delta):
if not errors: return "0"
d=Fraction(delta)
total=Fraction(0)
for e in errors:
a=abs(e)
total+=Fraction(e*e) if a<=d else d*(a-d/2)
return str(total)
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([2, 2, 4], 2)), '10')
check('regression 2', solve(*([0, 1, -1, 4, -4], 2)), '13')
check('regression 3', solve(*([], 2)), '0')
check('regression 4', solve(*([2, -2], 2)), '4')
check('regression 5', solve(*([1, -1, 3], 1)), '7/2')
check('regression 6', solve(*([-5, 0, 2], 3)), '25/2')
check('regression 7', solve(*([7, -2, 1], 0)), '0')
check('regression 8', solve(*([1, 2, 3], 4)), '7')
check("variable Huber tail",solve([N+2,-N-2],2),str(4*N+4))
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 1 | 14 | 10 | Failed |
| regression 2 | 14 | 13 | Failed |
| regression 3 | 0 | 0 | Passed |
| regression 4 | 8 | 4 | Failed |
| regression 5 | 9/2 | 7/2 | Failed |
| regression 6 | 29/2 | 25/2 | Failed |
| regression 7 | 0 | 0 | Passed |
| regression 8 | 14 | 7 | Failed |
| variable Huber tail | 8 | 8 | Passed |
SHA-256 / bbec1138d60229d5ba73f8598e4d7d2e6f4b778e6b7cca080d8976a81b422996
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
from collections import Counter, defaultdict
import math
import itertools
N = 1
observations = []
def solve(errors, delta):
if not errors: return "0"
d=Fraction(delta)
total=Fraction(0)
for e in errors:
a=abs(e)
total+=Fraction(e*e,4) if a<=d else d*(a-d/2)
return str(total)
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([2, 2, 4], 2)), '10')
check('regression 2', solve(*([0, 1, -1, 4, -4], 2)), '13')
check('regression 3', solve(*([], 2)), '0')
check('regression 4', solve(*([2, -2], 2)), '4')
check('regression 5', solve(*([1, -1, 3], 1)), '7/2')
check('regression 6', solve(*([-5, 0, 2], 3)), '25/2')
check('regression 7', solve(*([7, -2, 1], 0)), '0')
check('regression 8', solve(*([1, 2, 3], 4)), '7')
check("variable Huber tail",solve([N+2,-N-2],2),str(4*N+4))
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 1 | 8 | 10 | Failed |
| regression 2 | 25/2 | 13 | Failed |
| regression 3 | 0 | 0 | Passed |
| regression 4 | 2 | 4 | Failed |
| regression 5 | 3 | 7/2 | Failed |
| regression 6 | 23/2 | 25/2 | Failed |
| regression 7 | 0 | 0 | Passed |
| regression 8 | 7/2 | 7 | Failed |
| variable Huber tail | 8 | 8 | Passed |
SHA-256 / 5859598278e3270f8f087230481205de66f4e301a4ba989438d253d297951d1b
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
from collections import Counter, defaultdict
import math
import itertools
N = 1
observations = []
def solve(errors, delta):
if not errors: return "0"
d=Fraction(delta)
total=Fraction(0)
for e in errors:
a=abs(e)
total+=Fraction(e*e,2) if a<=d else d*(a-d/2)
return str(total)
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([2, 2, 4], 2)), '10')
check('regression 2', solve(*([0, 1, -1, 4, -4], 2)), '13')
check('regression 3', solve(*([], 2)), '0')
check('regression 4', solve(*([2, -2], 2)), '4')
check('regression 5', solve(*([1, -1, 3], 1)), '7/2')
check('regression 6', solve(*([-5, 0, 2], 3)), '25/2')
check('regression 7', solve(*([7, -2, 1], 0)), '0')
check('regression 8', solve(*([1, 2, 3], 4)), '7')
check("variable Huber tail",solve([N+2,-N-2],2),str(4*N+4))
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 1 | 10 | 10 | Passed |
| regression 2 | 13 | 13 | Passed |
| regression 3 | 0 | 0 | Passed |
| regression 4 | 4 | 4 | Passed |
| regression 5 | 7/2 | 7/2 | Passed |
| regression 6 | 25/2 | 25/2 | Passed |
| regression 7 | 0 | 0 | Passed |
| regression 8 | 7 | 7 | Passed |
| variable Huber tail | 8 | 8 | Passed |
SHA-256 / 78a8c2ac27d7f0b31f615e117198e611fa6bf184213587c0dee4eeb9201c804f
Verification & scope
Small offline integer/rational inputs only; no performance, statistical inference, or production-library conformance claim. 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 2026-09-29T14:39:10.877492+00:00.
Case digest / 169f6b07e54d8b6c9e5b1cad09e54b36b86dd4928493e3cdb4fc5d57d38576bf