FAILURE MAP
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FA-15401 / Numerics / Open access

Rational nullspace basis: free coordinate normalization · case 01

The exact rational nullspace basis result violates the stated contract at free coordinate normalization.

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

ROOT CAUSE

The free coordinate normalization step uses Fraction(-1) instead of Fraction(1).

THE FAILURE

The free coordinate normalization step uses Fraction(-1) instead of Fraction(1).

Unsuccessful approach: The partial repair Fraction(0) still violates the free coordinate normalization 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 = [[([[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, -1]], [[[1, 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]]])], [([[2, -1]], [[[1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 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, 1]], [[[1, 1], [0, 1]]]), ([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[0, -1]], [[[1, 1], [0, 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]]]), ([[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]]])], [([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [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]]]), ([[-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, 0]], [[[0, 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, -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], [0, 1]], [[0, 1], [-1, 1]]][[[1, 1], [0, 1]], [[0, 1], [1, 1]]]Failed
explicit oracle 1[[[3, 1], [1, 2], [15, 2], [-1, 1]]][[[3, 1], [1, 2], [15, 2], [1, 1]]]Failed
explicit oracle 2[[[1, 2], [-1, 1]]][[[1, 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]]]Failed
explicit oracle 5[[[0, 1], [-1, 1]]][[[0, 1], [1, 1]]]Failed
explicit oracle 6[[[-1, 1], [0, 1]]][[[1, 1], [0, 1]]]Failed
explicit oracle 7[[[-1, 2], [-1, 1]]][[[-1, 2], [1, 1]]]Failed

SHA-256 / fa9da2b94e084956aaa464a92a28140af6263777f953096fca05f6d4db986e1a

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(0)
     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 = [[([[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, -1]], [[[1, 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]]])], [([[2, -1]], [[[1, 2], [1, 1]]]), ([[0, 0]], [[[1, 1], [0, 1]], [[0, 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, 1]], [[[1, 1], [0, 1]]]), ([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[0, -1]], [[[1, 1], [0, 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]]]), ([[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]]])], [([[0, 0]], [[[1, 1], [0, 1]], [[0, 1], [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]]]), ([[-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, 0]], [[[0, 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, -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[[[0, 1], [0, 1]], [[0, 1], [0, 1]]][[[1, 1], [0, 1]], [[0, 1], [1, 1]]]Failed
explicit oracle 1[[[3, 1], [1, 2], [15, 2], [0, 1]]][[[3, 1], [1, 2], [15, 2], [1, 1]]]Failed
explicit oracle 2[[[1, 2], [0, 1]]][[[1, 2], [1, 1]]]Failed
explicit oracle 3[[[-1, 1], [0, 1]]][[[-1, 1], [1, 1]]]Failed
explicit oracle 4[[[0, 1], [0, 1]], [[0, 1], [0, 1]]][[[1, 1], [0, 1]], [[0, 1], [1, 1]]]Failed
explicit oracle 5[[[0, 1], [0, 1]]][[[0, 1], [1, 1]]]Failed
explicit oracle 6[[[0, 1], [0, 1]]][[[1, 1], [0, 1]]]Failed
explicit oracle 7[[[-1, 2], [0, 1]]][[[-1, 2], [1, 1]]]Failed

SHA-256 / b960bd7f93de4389ee143942e2b109bd98d42215889e707a9097c52cd6faee94

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.

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

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 2026-09-29T14:39:26.522151+00:00.

Case digest / dd66a26999460d5cb5503fe45b169dce25273620565eb83cd2116b2679aed4b8