FA-13721 / Numerical aggregation / Open access
Tie adjusted concordance squared: Squaring the numerator loses negative association direction. · case 01
The reduction disagrees with its explicit aggregation oracle.
ROOT CAUSE
Squaring the numerator loses negative association direction.
VERIFIED REPAIR
Preserve the tie adjusted concordance squared contract at the identified reduction decision.
Unsuccessful approach: Always negative also loses the observed direction.
Case contract
For paired ordinal observations return sign(tau_b)*tau_b**2 as an exact Fraction string. Every unordered observation pair contributes its coordinate comparison; normalize signed concordance squared by products of non-tied x/y pair counts. Undefined if either coordinate has no varying pairs.
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(points):
concordant=discordant=tx=ty=0
for i in range(len(points)):
for j in range(i+1,len(points)):
dx=points[i][0]-points[j][0]
dy=points[i][1]-points[j][1]
if dx: tx+=1
if dy: ty+=1
concordant+=int(dx*dy>0)
discordant+=int(dx*dy<0)
if not tx or not ty: return None
d=concordant-discordant
return str(Fraction(d*d,tx*ty))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([(1, 1), (2, 2), (3, 3)],)), '1')
check('regression 2', solve(*([(1, 3), (2, 2), (3, 1)],)), '-1')
check('regression 3', solve(*([(1, 1), (1, 2), (2, 2), (3, 1)],)), '-1/20')
check('regression 4', solve(*([],)), None)
check('regression 5', solve(*([(1, 1), (1, 2)],)), None)
check('regression 6', solve(*([(1, 1), (2, 2), (2, 2), (4, 3)],)), '1')
check('regression 7', solve(*([(1, 2), (3, 1), (4, 4), (2, 5)],)), '0')
check("variable discordance direction",solve([(0,N),(N,0)]),"-1")
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 | 1 | 1 | Passed |
| regression 2 | 1 | -1 | Failed |
| regression 3 | 1/20 | -1/20 | Failed |
| regression 4 | None | None | Passed |
| regression 5 | None | None | Passed |
| regression 6 | 1 | 1 | Passed |
| regression 7 | 0 | 0 | Passed |
| variable discordance direction | 1 | -1 | Failed |
SHA-256 / 9c00be52cb6952a16db3c709a95f6a11593f8eeb5c504571763abb47f1815abb
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(points):
concordant=discordant=tx=ty=0
for i in range(len(points)):
for j in range(i+1,len(points)):
dx=points[i][0]-points[j][0]
dy=points[i][1]-points[j][1]
if dx: tx+=1
if dy: ty+=1
concordant+=int(dx*dy>0)
discordant+=int(dx*dy<0)
if not tx or not ty: return None
d=concordant-discordant
return str(Fraction(-d*d,tx*ty))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([(1, 1), (2, 2), (3, 3)],)), '1')
check('regression 2', solve(*([(1, 3), (2, 2), (3, 1)],)), '-1')
check('regression 3', solve(*([(1, 1), (1, 2), (2, 2), (3, 1)],)), '-1/20')
check('regression 4', solve(*([],)), None)
check('regression 5', solve(*([(1, 1), (1, 2)],)), None)
check('regression 6', solve(*([(1, 1), (2, 2), (2, 2), (4, 3)],)), '1')
check('regression 7', solve(*([(1, 2), (3, 1), (4, 4), (2, 5)],)), '0')
check("variable discordance direction",solve([(0,N),(N,0)]),"-1")
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 | -1 | 1 | Failed |
| regression 2 | -1 | -1 | Passed |
| regression 3 | -1/20 | -1/20 | Passed |
| regression 4 | None | None | Passed |
| regression 5 | None | None | Passed |
| regression 6 | -1 | 1 | Failed |
| regression 7 | 0 | 0 | Passed |
| variable discordance direction | -1 | -1 | Passed |
SHA-256 / f5b18b4c5d1952075c68d0e6c350b40a6b6cbb3bd120d49b1c5135768ae2bf52
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(points):
concordant=discordant=tx=ty=0
for i in range(len(points)):
for j in range(i+1,len(points)):
dx=points[i][0]-points[j][0]
dy=points[i][1]-points[j][1]
if dx: tx+=1
if dy: ty+=1
concordant+=int(dx*dy>0)
discordant+=int(dx*dy<0)
if not tx or not ty: return None
d=concordant-discordant
return str(Fraction((1 if d>=0 else -1)*d*d,tx*ty))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([(1, 1), (2, 2), (3, 3)],)), '1')
check('regression 2', solve(*([(1, 3), (2, 2), (3, 1)],)), '-1')
check('regression 3', solve(*([(1, 1), (1, 2), (2, 2), (3, 1)],)), '-1/20')
check('regression 4', solve(*([],)), None)
check('regression 5', solve(*([(1, 1), (1, 2)],)), None)
check('regression 6', solve(*([(1, 1), (2, 2), (2, 2), (4, 3)],)), '1')
check('regression 7', solve(*([(1, 2), (3, 1), (4, 4), (2, 5)],)), '0')
check("variable discordance direction",solve([(0,N),(N,0)]),"-1")
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 | 1 | 1 | Passed |
| regression 2 | -1 | -1 | Passed |
| regression 3 | -1/20 | -1/20 | Passed |
| regression 4 | None | None | Passed |
| regression 5 | None | None | Passed |
| regression 6 | 1 | 1 | Passed |
| regression 7 | 0 | 0 | Passed |
| variable discordance direction | -1 | -1 | Passed |
SHA-256 / 4af632310cc2bf180cf06298cff0fd8a24e43365131b2d9f51d804bc0b9d6e7b
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 2026-09-29T14:39:09.948637+00:00.
Case digest / e130f0d3d9a470ce7509b141186ee4fc39b1896983fa02663c13e252348413c1