FA-12981 / Numerical aggregation / Open access

Merge central moment 3: Third moment uses reversed mean displacement. · case 01

The reduction disagrees with its explicit aggregation oracle.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Third moment uses reversed mean displacement.

THE FAILURE

Third moment uses reversed mean displacement.

Unsuccessful approach: Taking magnitude also destroys odd-moment orientation.

Case contract

Merge disjoint integer observation blocks into [count, exact mean, unnormalised central moments through order 3]. Empty blocks are identities; moments are Fraction strings, empty summary is zero.

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(blocks):
    n=0
    mean=Fraction(0)
    m2=Fraction(0)
    m3=Fraction(0)
    m4=Fraction(0)
    for xs in blocks:
        if not xs: continue
        b=len(xs)
        u=Fraction(sum(xs),b)
        q2=sum((Fraction(x)-u)**2 for x in xs)
        q3=sum((Fraction(x)-u)**3 for x in xs)
        q4=sum((Fraction(x)-u)**4 for x in xs)
        t=n+b
        d=u-mean
        r4=m4+q4+d**4*n*b*(n*n-n*b+b*b)/t**3+6*d*d*(n*n*q2+b*b*m2)/t**2+4*d*(n*q3-b*m3)/t
        r3=m3+q3-d**3*n*b*(n-b)/t**2-3*d*(n*q2-b*m2)/t
        r2=m2+q2+d*d*n*b/t
        mean=mean+d*b/t
        n=t
        m2,m3,m4=r2,r3,r4
    return [n,str(mean),str(m2),str(m3)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([[8], [1, 3]],)), [3, '4', '26', '36'])
check('regression 2', solve(*([[1, 3], [8]],)), [3, '4', '26', '36'])
check('regression 3', solve(*([[0, 0], [0]],)), [3, '0', '0', '0'])
check('regression 4', solve(*([],)), [0, '0', '0', '0'])
check('regression 5', solve(*([[2, 2, 2], [], [-1, 5]],)), [5, '2', '18', '0'])
check('regression 6', solve(*([[-4], [1, 3], [7, 9]],)), [5, '16/5', '524/5', '-3348/25'])
check("variable repeated symmetric blocks",solve([[0,2]]*N),[2*N,"1",str(2*N),"0"])
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	`[3, '4', '26', '-36']`	`[3, '4', '26', '36']`	Failed
regression 2	`[3, '4', '26', '-36']`	`[3, '4', '26', '36']`	Failed
regression 3	`[3, '0', '0', '0']`	`[3, '0', '0', '0']`	Passed
regression 4	`[0, '0', '0', '0']`	`[0, '0', '0', '0']`	Passed
regression 5	`[5, '2', '18', '0']`	`[5, '2', '18', '0']`	Passed
regression 6	`[5, '16/5', '524/5', '3348/25']`	`[5, '16/5', '524/5', '-3348/25']`	Failed
variable repeated symmetric blocks	`[2, '1', '2', '0']`	`[2, '1', '2', '0']`	Passed

SHA-256 / 531191b857085412354f128f770197a66bd61596bbd07419fd6cf8d5d3a48997

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(blocks):
    n=0
    mean=Fraction(0)
    m2=Fraction(0)
    m3=Fraction(0)
    m4=Fraction(0)
    for xs in blocks:
        if not xs: continue
        b=len(xs)
        u=Fraction(sum(xs),b)
        q2=sum((Fraction(x)-u)**2 for x in xs)
        q3=sum((Fraction(x)-u)**3 for x in xs)
        q4=sum((Fraction(x)-u)**4 for x in xs)
        t=n+b
        d=u-mean
        r4=m4+q4+d**4*n*b*(n*n-n*b+b*b)/t**3+6*d*d*(n*n*q2+b*b*m2)/t**2+4*d*(n*q3-b*m3)/t
        r3=m3+q3+abs(d)**3*n*b*(n-b)/t**2+3*abs(d)*(n*q2-b*m2)/t
        r2=m2+q2+d*d*n*b/t
        mean=mean+d*b/t
        n=t
        m2,m3,m4=r2,r3,r4
    return [n,str(mean),str(m2),str(m3)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('regression 1', solve(*([[8], [1, 3]],)), [3, '4', '26', '36'])
check('regression 2', solve(*([[1, 3], [8]],)), [3, '4', '26', '36'])
check('regression 3', solve(*([[0, 0], [0]],)), [3, '0', '0', '0'])
check('regression 4', solve(*([],)), [0, '0', '0', '0'])
check('regression 5', solve(*([[2, 2, 2], [], [-1, 5]],)), [5, '2', '18', '0'])
check('regression 6', solve(*([[-4], [1, 3], [7, 9]],)), [5, '16/5', '524/5', '-3348/25'])
check("variable repeated symmetric blocks",solve([[0,2]]*N),[2*N,"1",str(2*N),"0"])
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	`[3, '4', '26', '-36']`	`[3, '4', '26', '36']`	Failed
regression 2	`[3, '4', '26', '36']`	`[3, '4', '26', '36']`	Passed
regression 3	`[3, '0', '0', '0']`	`[3, '0', '0', '0']`	Passed
regression 4	`[0, '0', '0', '0']`	`[0, '0', '0', '0']`	Passed
regression 5	`[5, '2', '18', '0']`	`[5, '2', '18', '0']`	Passed
regression 6	`[5, '16/5', '524/5', '-3348/25']`	`[5, '16/5', '524/5', '-3348/25']`	Passed
variable repeated symmetric blocks	`[2, '1', '2', '0']`	`[2, '1', '2', '0']`	Passed

SHA-256 / b5a0f3a5f0220232a6ea2c6cac3f714ede4e8f0ce9fa3b3c524e1c7574f06efa

HELD IN THE MEMBER ARCHIVE

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

This mechanism has 7 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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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 2026-09-29T14:39:02.380590+00:00.

Case digest / 5fcaef7173d115cc28692221d895197af4d466060909ed5295a5d8479f7d895d