FA-13831 / Numerical aggregation / Open access
Huber residual total: The quadratic branch threshold is compared to squared residual magnitude. · case 01
The reduction disagrees with its explicit aggregation oracle.
ROOT CAUSE
The quadratic branch threshold is compared to squared residual magnitude.
THE FAILURE
The quadratic branch threshold is compared to squared residual magnitude.
Unsuccessful approach: Squaring the threshold instead moves the quadratic/linear transition.
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,2) if a*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 | 12 | 25/2 | Failed |
| regression 7 | 0 | 0 | Passed |
| regression 8 | 13/2 | 7 | Failed |
| variable Huber tail | 8 | 8 | Passed |
SHA-256 / 6c35c9929e91a0e5a5475b522f7cce82102399d0cfef33b3f8dc747b924ff4f3
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,2) if a<=d*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 | 12 | 10 | Failed |
| regression 2 | 17 | 13 | Failed |
| regression 3 | 0 | 0 | Passed |
| regression 4 | 4 | 4 | Passed |
| regression 5 | 7/2 | 7/2 | Passed |
| regression 6 | 29/2 | 25/2 | Failed |
| regression 7 | 0 | 0 | Passed |
| regression 8 | 7 | 7 | Passed |
| variable Huber tail | 9 | 8 | Failed |
SHA-256 / 389509241390e8441d1f26f85f9872febbcbb7cfc056c1f4a62c06a4eb96a2fa
HELD IN THE MEMBER ARCHIVE
The verified repair and its recorded checks are member-only.
This mechanism has 9 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.
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Sign in to the archive ↗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 2026-09-29T14:39:11.110664+00:00.
Case digest / a5eba82c5595e366045dc8a0648b4a5520eb38c1c637ee07da63f5475245e5b0