FA-13711 / Numerical aggregation / Open access

Tie adjusted concordance squared: Pairs tied in a coordinate are counted as concordant. · case 01

The reduction disagrees with its explicit aggregation oracle.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Pairs tied in a coordinate are counted as concordant.

THE FAILURE

Pairs tied in a coordinate are counted as concordant.

Unsuccessful approach: Equal coordinate differences are not the definition of ordinal concordance.

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((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	`1/5`	`-1/20`	Failed
regression 4	`None`	`None`	Passed
regression 5	`None`	`None`	Passed
regression 6	`36/25`	`1`	Failed
regression 7	`0`	`0`	Passed
variable discordance direction	`-1`	`-1`	Passed

SHA-256 / bc06efc110fdbe776fb8368483d5b5837a5ddddd3af8319a487bf779a72c54a7

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)
            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	`9/25`	`1`	Failed
regression 7	`-1/4`	`0`	Failed
variable discordance direction	`-1`	`-1`	Passed

SHA-256 / b4cda1f6a0830bec4620c108f5f2940517917df2f6e30fe3b36ec83ad7eb9141

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 8 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.

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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 2026-09-29T14:39:10.314103+00:00.

Case digest / dbc7370a4fb5d9f4f61af3d9b98611cfa5dcc50c65cbb1adf017408cc8f116f2