FAILURE MAP
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FA-15391 / Numerics / Open access

Rational nullspace basis: free column classification · case 01

The exact rational nullspace basis result violates the stated contract at free column classification.

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

ROOT CAUSE

The free column classification step uses pivots instead of [c for c in range(cols) if c not in pivots].

THE FAILURE

The free column classification step uses pivots instead of [c for c in range(cols) if c not in pivots].

Unsuccessful approach: The partial repair [c for c in range(cols) if c>=len(pivots)] still violates the free column classification 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=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, 1]], [[[1, 1], [0, 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]]]), ([[2, 1]], [[[-1, 2], [1, 1]]])], [([[2, -1]], [[[1, 2], [1, 1]]]), ([[0, 1]], [[[1, 1], [0, 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, -1]], [[[-1, 2], [1, 1]]]), ([[-2, -1]], [[[-1, 2], [1, 1]]]), ([[0, -1]], [[[1, 1], [0, 1]]]), ([[0, -2]], [[[1, 1], [0, 1]]])], [([[1, 1]], [[[-1, 1], [1, 1]]]), ([[0, -1]], [[[1, 1], [0, 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]]]), ([[0, 2]], [[[1, 1], [0, 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]]]), ([[0, 0], [0, -1]], [[[1, 1], [0, 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]]]Failed
explicit oracle 1[[[0, 1], [-1, 1]]][[[1, 1], [0, 1]]]Failed
explicit oracle 2[[[-1, 1], [0, 1], [0, 1], [0, 1]], [[0, 1], [-1, 1], [0, 1], [0, 1]], [[0, 1], [0, 1], [-1, 1], [0, 1]]][[[3, 1], [1, 2], [15, 2], [1, 1]]]Failed
explicit oracle 3[[[-1, 1], [0, 1]]][[[1, 2], [1, 1]]]Failed
explicit oracle 4[[[-1, 1], [0, 1]]][[[-1, 1], [1, 1]]]Failed
explicit oracle 5[][[[1, 1], [0, 1]], [[0, 1], [1, 1]]]Failed
explicit oracle 6[[[-1, 1], [0, 1]]][[[0, 1], [1, 1]]]Failed
explicit oracle 7[[[-1, 1], [0, 1]]][[[-1, 2], [1, 1]]]Failed

SHA-256 / 5e85c6adc15f1979285e2f776517bceab1e92544e9eb333aab18e772cd46f3f2

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

SHA-256 / 1bb5bc7a68beafb33120835078f5d45d499e5da1c3e350f4705ef73057b2a5f8

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.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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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 2026-09-29T14:39:26.299417+00:00.

Case digest / 4bba12d6be18dfb5343f84f6520004474b5ac21d38c25cd34290e5ea423a19d1