FAILURE MAP
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FA-15411 / Numerics / Open access

Rational nullspace basis: pivot equation rearrangement · case 01

The exact rational nullspace basis result violates the stated contract at pivot equation rearrangement.

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

ROOT CAUSE

The pivot equation rearrangement step uses a[i][c] instead of -a[i][c].

VERIFIED REPAIR

Use -a[i][c] at the pivot equation rearrangement step.

Unsuccessful approach: The partial repair -a[i][p] still violates the pivot equation rearrangement invariant.

Case contract

Input nonempty rectangular integer matrix; return canonical nullspace basis, one vector per free column increasing, with that free coordinate one and other free coordinates zero.

Why this case matters

Exact discrete arithmetic with observable algorithmic state; no floating point approximation is used.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    a=[list(map(Fraction,r)) for r in x];rows=len(a);cols=len(a[0]);r=0;pivots=[]
    for c in range(cols):
     if r==rows:break
     pivot=next((i for i in range(r,rows) if a[i][c]!=0),None)
     if pivot is None:continue
     a[r],a[pivot]=a[pivot],a[r];d=a[r][c];a[r]=[v/d for v in a[r]]
     for i in range(rows):
      if i!=r:
       v=a[i][c];a[i]=[a[i][j]-v*a[r][j] for j in range(cols)]
     pivots.append(c);r+=1
    free=[c for c in range(cols) if c not in pivots]
    out=[]
    for c in free:
     v=[Fraction(0)]*cols
     v[c]=Fraction(1)
     for i,p in enumerate(pivots):v[p]=a[i][c]
     out.append([[z.numerator,z.denominator] for z in v])
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[2, -1]], [[[1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[1, 1]], [[[-1, 1], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[-2, 0]], [[[0, 1], [1, 1]]]), ([[0, 1]], [[[1, 1], [0, 1]]]), ([[2, 1]], [[[-1, 2], [1, 1]]])], [([[1, 1]], [[[-1, 1], [1, 1]]]), ([[2, 1]], [[[-1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[0, 1]], [[[1, 1], [0, 1]]]), ([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[0, -1]], [[[1, 1], [0, 1]]])], [([[2, 1]], [[[-1, 2], [1, 1]]]), ([[2, -2]], [[[1, 1], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[2, 2]], [[[-1, 1], [1, 1]]]), ([[2, -1]], [[[1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[1, -1]], [[[1, 1], [1, 1]]])], [([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[-2, 1]], [[[1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[-1, 1], [-1, 0]], []), ([[0, 0], [-1, 2]], [[[2, 1], [1, 1]]]), ([[-2, 1], [1, -2]], []), ([[-1, 0], [0, 0]], [[[0, 1], [1, 1]]])], [([[2, -2]], [[[1, 1], [1, 1]]]), ([[1, 2]], [[[-2, 1], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[0, -2], [2, 0]], []), ([[-1, -2], [0, 1]], []), ([[-2, 0], [-2, 0]], [[[0, 1], [1, 1]]]), ([[0, 0], [0, -1]], [[[1, 1], [0, 1]]])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
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 fixtureActualExpectedOutcome
explicit oracle 0[[[-1, 2], [1, 1]]][[[1, 2], [1, 1]]]Failed
explicit oracle 1[[[1, 1], [0, 1]], [[0, 1], [1, 1]]][[[1, 1], [0, 1]], [[0, 1], [1, 1]]]Passed
explicit oracle 2[[[-3, 1], [-1, 2], [-15, 2], [1, 1]]][[[3, 1], [1, 2], [15, 2], [1, 1]]]Failed
explicit oracle 3[[[1, 1], [1, 1]]][[[-1, 1], [1, 1]]]Failed
explicit oracle 4[[[1, 1], [0, 1]], [[0, 1], [1, 1]]][[[1, 1], [0, 1]], [[0, 1], [1, 1]]]Passed
explicit oracle 5[[[0, 1], [1, 1]]][[[0, 1], [1, 1]]]Passed
explicit oracle 6[[[1, 1], [0, 1]]][[[1, 1], [0, 1]]]Passed
explicit oracle 7[[[1, 2], [1, 1]]][[[-1, 2], [1, 1]]]Failed

SHA-256 / 5c4d10e8e77deae6d1586f70ebc6585a798f12a06cc5ee6322e89b2218bac027

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    a=[list(map(Fraction,r)) for r in x];rows=len(a);cols=len(a[0]);r=0;pivots=[]
    for c in range(cols):
     if r==rows:break
     pivot=next((i for i in range(r,rows) if a[i][c]!=0),None)
     if pivot is None:continue
     a[r],a[pivot]=a[pivot],a[r];d=a[r][c];a[r]=[v/d for v in a[r]]
     for i in range(rows):
      if i!=r:
       v=a[i][c];a[i]=[a[i][j]-v*a[r][j] for j in range(cols)]
     pivots.append(c);r+=1
    free=[c for c in range(cols) if c not in pivots]
    out=[]
    for c in free:
     v=[Fraction(0)]*cols
     v[c]=Fraction(1)
     for i,p in enumerate(pivots):v[p]=-a[i][p]
     out.append([[z.numerator,z.denominator] for z in v])
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[2, -1]], [[[1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[1, 1]], [[[-1, 1], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[-2, 0]], [[[0, 1], [1, 1]]]), ([[0, 1]], [[[1, 1], [0, 1]]]), ([[2, 1]], [[[-1, 2], [1, 1]]])], [([[1, 1]], [[[-1, 1], [1, 1]]]), ([[2, 1]], [[[-1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[0, 1]], [[[1, 1], [0, 1]]]), ([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[0, -1]], [[[1, 1], [0, 1]]])], [([[2, 1]], [[[-1, 2], [1, 1]]]), ([[2, -2]], [[[1, 1], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[2, 2]], [[[-1, 1], [1, 1]]]), ([[2, -1]], [[[1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[1, -1]], [[[1, 1], [1, 1]]])], [([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[-2, 1]], [[[1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[-1, 1], [-1, 0]], []), ([[0, 0], [-1, 2]], [[[2, 1], [1, 1]]]), ([[-2, 1], [1, -2]], []), ([[-1, 0], [0, 0]], [[[0, 1], [1, 1]]])], [([[2, -2]], [[[1, 1], [1, 1]]]), ([[1, 2]], [[[-2, 1], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[0, -2], [2, 0]], []), ([[-1, -2], [0, 1]], []), ([[-2, 0], [-2, 0]], [[[0, 1], [1, 1]]]), ([[0, 0], [0, -1]], [[[1, 1], [0, 1]]])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
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 fixtureActualExpectedOutcome
explicit oracle 0[[[-1, 1], [1, 1]]][[[1, 2], [1, 1]]]Failed
explicit oracle 1[[[1, 1], [0, 1]], [[0, 1], [1, 1]]][[[1, 1], [0, 1]], [[0, 1], [1, 1]]]Passed
explicit oracle 2[[[-1, 1], [-1, 1], [-1, 1], [1, 1]]][[[3, 1], [1, 2], [15, 2], [1, 1]]]Failed
explicit oracle 3[[[-1, 1], [1, 1]]][[[-1, 1], [1, 1]]]Passed
explicit oracle 4[[[1, 1], [0, 1]], [[0, 1], [1, 1]]][[[1, 1], [0, 1]], [[0, 1], [1, 1]]]Passed
explicit oracle 5[[[-1, 1], [1, 1]]][[[0, 1], [1, 1]]]Failed
explicit oracle 6[[[1, 1], [-1, 1]]][[[1, 1], [0, 1]]]Failed
explicit oracle 7[[[-1, 1], [1, 1]]][[[-1, 2], [1, 1]]]Failed

SHA-256 / 37a9d917431544cbbd0f820ec90b84a5c1221fc0087ca23bd9ee8e7b7fbed911

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    a=[list(map(Fraction,r)) for r in x];rows=len(a);cols=len(a[0]);r=0;pivots=[]
    for c in range(cols):
     if r==rows:break
     pivot=next((i for i in range(r,rows) if a[i][c]!=0),None)
     if pivot is None:continue
     a[r],a[pivot]=a[pivot],a[r];d=a[r][c];a[r]=[v/d for v in a[r]]
     for i in range(rows):
      if i!=r:
       v=a[i][c];a[i]=[a[i][j]-v*a[r][j] for j in range(cols)]
     pivots.append(c);r+=1
    free=[c for c in range(cols) if c not in pivots]
    out=[]
    for c in free:
     v=[Fraction(0)]*cols
     v[c]=Fraction(1)
     for i,p in enumerate(pivots):v[p]=-a[i][c]
     out.append([[z.numerator,z.denominator] for z in v])
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[2, -1]], [[[1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[1, 1]], [[[-1, 1], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[-2, 0]], [[[0, 1], [1, 1]]]), ([[0, 1]], [[[1, 1], [0, 1]]]), ([[2, 1]], [[[-1, 2], [1, 1]]])], [([[1, 1]], [[[-1, 1], [1, 1]]]), ([[2, 1]], [[[-1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[0, 1]], [[[1, 1], [0, 1]]]), ([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[0, -1]], [[[1, 1], [0, 1]]])], [([[2, 1]], [[[-1, 2], [1, 1]]]), ([[2, -2]], [[[1, 1], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[2, 2]], [[[-1, 1], [1, 1]]]), ([[2, -1]], [[[1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[1, -1]], [[[1, 1], [1, 1]]])], [([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[-2, 1]], [[[1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[-1, 1], [-1, 0]], []), ([[0, 0], [-1, 2]], [[[2, 1], [1, 1]]]), ([[-2, 1], [1, -2]], []), ([[-1, 0], [0, 0]], [[[0, 1], [1, 1]]])], [([[2, -2]], [[[1, 1], [1, 1]]]), ([[1, 2]], [[[-2, 1], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [1, 1]]]), ([[0, 2, 0, -1], [2, -1, -1, 2], [1, -2, 0, -2]], [[[3, 1], [1, 2], [15, 2], [1, 1]]]), ([[0, -2], [2, 0]], []), ([[-1, -2], [0, 1]], []), ([[-2, 0], [-2, 0]], [[[0, 1], [1, 1]]]), ([[0, 0], [0, -1]], [[[1, 1], [0, 1]]])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
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 fixtureActualExpectedOutcome
explicit oracle 0[[[1, 2], [1, 1]]][[[1, 2], [1, 1]]]Passed
explicit oracle 1[[[1, 1], [0, 1]], [[0, 1], [1, 1]]][[[1, 1], [0, 1]], [[0, 1], [1, 1]]]Passed
explicit oracle 2[[[3, 1], [1, 2], [15, 2], [1, 1]]][[[3, 1], [1, 2], [15, 2], [1, 1]]]Passed
explicit oracle 3[[[-1, 1], [1, 1]]][[[-1, 1], [1, 1]]]Passed
explicit oracle 4[[[1, 1], [0, 1]], [[0, 1], [1, 1]]][[[1, 1], [0, 1]], [[0, 1], [1, 1]]]Passed
explicit oracle 5[[[0, 1], [1, 1]]][[[0, 1], [1, 1]]]Passed
explicit oracle 6[[[1, 1], [0, 1]]][[[1, 1], [0, 1]]]Passed
explicit oracle 7[[[-1, 2], [1, 1]]][[[-1, 2], [1, 1]]]Passed

SHA-256 / ef2ebb4772c6a7d72002d87c3f0790c0296e8ac72a535096b09ee28e72ff034a

Verification & scope

A deterministic bounded teaching model. Inputs are restricted to the explicit contract; this is not a production algebra library. 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 2026-09-29T14:39:26.666392+00:00.

Case digest / 7a7fb5b282d704aaf51165464d1f7e26a085c59f6bb26ec764d702a920599f01