FA-13811 / Numerical aggregation / Open access
Huber residual total: The linear tail omits its continuity offset. · case 01
The reduction disagrees with its explicit aggregation oracle.
ROOT CAUSE
The linear tail omits its continuity offset.
THE FAILURE
The linear tail omits its continuity offset.
Unsuccessful approach: Subtracting the full threshold makes the tail discontinuous below the quadratic boundary.
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<=d else d*a
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 | 4 | 7/2 | Failed |
| regression 6 | 17 | 25/2 | Failed |
| regression 7 | 0 | 0 | Passed |
| regression 8 | 7 | 7 | Passed |
| variable Huber tail | 12 | 8 | Failed |
SHA-256 / c59d5b8b3402eecb137f41f8c74e0c8eb17b0612101f838d47189e5426a4c4a7
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 else d*(a-d)
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 | 9 | 13 | Failed |
| regression 3 | 0 | 0 | Passed |
| regression 4 | 4 | 4 | Passed |
| regression 5 | 3 | 7/2 | Failed |
| regression 6 | 8 | 25/2 | Failed |
| regression 7 | 0 | 0 | Passed |
| regression 8 | 7 | 7 | Passed |
| variable Huber tail | 4 | 8 | Failed |
SHA-256 / df006189a39ff9d4243045e5a279f4ab480bf3f620ae2d0f486ca8c4b177e4af
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.
Member access is invitation-based. Sign in with your invited account to inspect the repair.
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 / e36dd985f10c33999c8213f07eefa8f9147f2e34808b2191e5de20694328e1a2