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FA-15456 / Numerics / Open access

Exact ldl decomposition: off diagonal schur complement · case 01

The exact exact ldl decomposition result violates the stated contract at off diagonal schur complement.

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

ROOT CAUSE

The off diagonal schur complement step uses a[i][j]+sum(L[i][k]*L[j][k]*D[k] for k in range(j)) instead of a[i][j]-sum(L[i][k]*L[j][k]*D[k] for k in range(j)).

VERIFIED REPAIR

Use a[i][j]-sum(L[i][k]*L[j][k]*D[k] for k in range(j)) at the off diagonal schur complement step.

Unsuccessful approach: The partial repair a[i][j]-sum(L[i][k]*L[j][k] for k in range(j)) still violates the off diagonal schur complement invariant.

Case contract

Input symmetric positive definite integer matrix; return unit lower L and diagonal D as rational pairs such that A=L*D*Ltranspose.

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,row)) for row in x];n=len(a);L=[[Fraction(int(i==j)) for j in range(n)] for i in range(n)];D=[Fraction(0)]*n
    for j in range(n):
     D[j]=a[j][j]-sum(L[j][k]**2*D[k] for k in range(j))
     if not D[j]:return None
     for i in range(j+1,n):
      L[i][j]=(a[i][j]+sum(L[i][k]*L[j][k]*D[k] for k in range(j)))/(D[j])
    return [[[[v.numerator,v.denominator] for v in row] for row in L],[[v.numerator,v.denominator] for v in D]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[15, 14, 1], [14, 20, -1], [1, -1, 6]], [[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]])], [([[11, 2, 6], [2, 14, 6], [6, 6, 9]], [[[[1, 1], [0, 1], [0, 1]], [[2, 11], [1, 1], [0, 1]], [[6, 11], [9, 25], [1, 1]]], [[11, 1], [150, 11], [99, 25]]]), ([[7, 2, -10], [2, 5, -4], [-10, -4, 18]], [[[[1, 1], [0, 1], [0, 1]], [[2, 7], [1, 1], [0, 1]], [[-10, 7], [-8, 31], [1, 1]]], [[7, 1], [31, 7], [106, 31]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[20, -4, -6], [-4, 15, 11], [-6, 11, 15]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 5], [1, 1], [0, 1]], [[-3, 10], [49, 71], [1, 1]]], [[20, 1], [71, 5], [457, 71]]]), ([[28, 6, -3], [6, 23, -5], [-3, -5, 14]], [[[[1, 1], [0, 1], [0, 1]], [[3, 14], [1, 1], [0, 1]], [[-3, 28], [-61, 304], [1, 1]]], [[28, 1], [152, 7], [7785, 608]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]])], [([[7, 2, -10], [2, 5, -4], [-10, -4, 18]], [[[[1, 1], [0, 1], [0, 1]], [[2, 7], [1, 1], [0, 1]], [[-10, 7], [-8, 31], [1, 1]]], [[7, 1], [31, 7], [106, 31]]]), ([[12, -6, -4], [-6, 19, -3], [-4, -3, 7]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 2], [1, 1], [0, 1]], [[-1, 3], [-5, 16], [1, 1]]], [[12, 1], [16, 1], [197, 48]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[18, -12, -2], [-12, 14, 2], [-2, 2, 13]], [[[[1, 1], [0, 1], [0, 1]], [[-2, 3], [1, 1], [0, 1]], [[-1, 9], [1, 9], [1, 1]]], [[18, 1], [6, 1], [343, 27]]]), ([[2, -2, 2], [-2, 10, -2], [2, -2, 13]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 1], [1, 1], [0, 1]], [[1, 1], [0, 1], [1, 1]]], [[2, 1], [8, 1], [11, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[11, 6], [6, 5]], [[[[1, 1], [0, 1]], [[6, 11], [1, 1]]], [[11, 1], [19, 11]]]), ([[1, 0], [0, 19]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[1, 1], [19, 1]]]), ([[3, 2], [2, 5]], [[[[1, 1], [0, 1]], [[2, 3], [1, 1]]], [[3, 1], [11, 3]]]), ([[9, 0], [0, 1]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[9, 1], [1, 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]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-1, 104], [1, 1]]], [[15, 1], [104, 15], [617, 104]]][[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]]Failed
explicit oracle 1[[[[1, 1]]], [[2, 1]]][[[[1, 1]]], [[2, 1]]]Passed
explicit oracle 2[[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-25, 8], [1, 1], [0, 1]], [[-3, 20], [-29, 24], [-53, 115], [1, 1]]], [[20, 1], [24, 5], [-345, 8], [8177, 345]]][[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]Failed
explicit oracle 3[[[[1, 1]]], [[5, 1]]][[[[1, 1]]], [[5, 1]]]Passed
explicit oracle 4[[[[1, 1]]], [[5, 1]]][[[[1, 1]]], [[5, 1]]]Passed
explicit oracle 5[[[[1, 1]]], [[2, 1]]][[[[1, 1]]], [[2, 1]]]Passed
explicit oracle 6[[[[1, 1]]], [[10, 1]]][[[[1, 1]]], [[10, 1]]]Passed
explicit oracle 7[[[[1, 1]]], [[2, 1]]][[[[1, 1]]], [[2, 1]]]Passed

SHA-256 / de34c96cefeb70b425a7774d565c8857b1bf701f33e73bf10aa1287af61d42d7

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,row)) for row in x];n=len(a);L=[[Fraction(int(i==j)) for j in range(n)] for i in range(n)];D=[Fraction(0)]*n
    for j in range(n):
     D[j]=a[j][j]-sum(L[j][k]**2*D[k] for k in range(j))
     if not D[j]:return None
     for i in range(j+1,n):
      L[i][j]=(a[i][j]-sum(L[i][k]*L[j][k] for k in range(j)))/(D[j])
    return [[[[v.numerator,v.denominator] for v in row] for row in L],[[v.numerator,v.denominator] for v in D]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[15, 14, 1], [14, 20, -1], [1, -1, 6]], [[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]])], [([[11, 2, 6], [2, 14, 6], [6, 6, 9]], [[[[1, 1], [0, 1], [0, 1]], [[2, 11], [1, 1], [0, 1]], [[6, 11], [9, 25], [1, 1]]], [[11, 1], [150, 11], [99, 25]]]), ([[7, 2, -10], [2, 5, -4], [-10, -4, 18]], [[[[1, 1], [0, 1], [0, 1]], [[2, 7], [1, 1], [0, 1]], [[-10, 7], [-8, 31], [1, 1]]], [[7, 1], [31, 7], [106, 31]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[20, -4, -6], [-4, 15, 11], [-6, 11, 15]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 5], [1, 1], [0, 1]], [[-3, 10], [49, 71], [1, 1]]], [[20, 1], [71, 5], [457, 71]]]), ([[28, 6, -3], [6, 23, -5], [-3, -5, 14]], [[[[1, 1], [0, 1], [0, 1]], [[3, 14], [1, 1], [0, 1]], [[-3, 28], [-61, 304], [1, 1]]], [[28, 1], [152, 7], [7785, 608]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]])], [([[7, 2, -10], [2, 5, -4], [-10, -4, 18]], [[[[1, 1], [0, 1], [0, 1]], [[2, 7], [1, 1], [0, 1]], [[-10, 7], [-8, 31], [1, 1]]], [[7, 1], [31, 7], [106, 31]]]), ([[12, -6, -4], [-6, 19, -3], [-4, -3, 7]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 2], [1, 1], [0, 1]], [[-1, 3], [-5, 16], [1, 1]]], [[12, 1], [16, 1], [197, 48]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[18, -12, -2], [-12, 14, 2], [-2, 2, 13]], [[[[1, 1], [0, 1], [0, 1]], [[-2, 3], [1, 1], [0, 1]], [[-1, 9], [1, 9], [1, 1]]], [[18, 1], [6, 1], [343, 27]]]), ([[2, -2, 2], [-2, 10, -2], [2, -2, 13]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 1], [1, 1], [0, 1]], [[1, 1], [0, 1], [1, 1]]], [[2, 1], [8, 1], [11, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[11, 6], [6, 5]], [[[[1, 1], [0, 1]], [[6, 11], [1, 1]]], [[11, 1], [19, 11]]]), ([[1, 0], [0, 19]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[1, 1], [19, 1]]]), ([[3, 2], [2, 5]], [[[[1, 1], [0, 1]], [[2, 3], [1, 1]]], [[3, 1], [11, 3]]]), ([[9, 0], [0, 1]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[9, 1], [1, 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]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-239, 1560], [1, 1]]], [[15, 1], [104, 15], [2025479, 351000]]][[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]]Failed
explicit oracle 1[[[[1, 1]]], [[2, 1]]][[[[1, 1]]], [[2, 1]]]Passed
explicit oracle 2[[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-29, 16], [1, 1], [0, 1]], [[-3, 20], [-353, 240], [-5555, 46152], [1, 1]]], [[20, 1], [24, 5], [-1923, 160], [50243071681, 4430592000]]][[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]Failed
explicit oracle 3[[[[1, 1]]], [[5, 1]]][[[[1, 1]]], [[5, 1]]]Passed
explicit oracle 4[[[[1, 1]]], [[5, 1]]][[[[1, 1]]], [[5, 1]]]Passed
explicit oracle 5[[[[1, 1]]], [[2, 1]]][[[[1, 1]]], [[2, 1]]]Passed
explicit oracle 6[[[[1, 1]]], [[10, 1]]][[[[1, 1]]], [[10, 1]]]Passed
explicit oracle 7[[[[1, 1]]], [[2, 1]]][[[[1, 1]]], [[2, 1]]]Passed

SHA-256 / 07e33399c736c8798b63ec8fcf5b5b8949b0d7d4d6025e349d46ca0b6f74f6a0

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,row)) for row in x];n=len(a);L=[[Fraction(int(i==j)) for j in range(n)] for i in range(n)];D=[Fraction(0)]*n
    for j in range(n):
     D[j]=a[j][j]-sum(L[j][k]**2*D[k] for k in range(j))
     if not D[j]:return None
     for i in range(j+1,n):
      L[i][j]=(a[i][j]-sum(L[i][k]*L[j][k]*D[k] for k in range(j)))/(D[j])
    return [[[[v.numerator,v.denominator] for v in row] for row in L],[[v.numerator,v.denominator] for v in D]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[15, 14, 1], [14, 20, -1], [1, -1, 6]], [[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]])], [([[11, 2, 6], [2, 14, 6], [6, 6, 9]], [[[[1, 1], [0, 1], [0, 1]], [[2, 11], [1, 1], [0, 1]], [[6, 11], [9, 25], [1, 1]]], [[11, 1], [150, 11], [99, 25]]]), ([[7, 2, -10], [2, 5, -4], [-10, -4, 18]], [[[[1, 1], [0, 1], [0, 1]], [[2, 7], [1, 1], [0, 1]], [[-10, 7], [-8, 31], [1, 1]]], [[7, 1], [31, 7], [106, 31]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[20, -4, -6], [-4, 15, 11], [-6, 11, 15]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 5], [1, 1], [0, 1]], [[-3, 10], [49, 71], [1, 1]]], [[20, 1], [71, 5], [457, 71]]]), ([[28, 6, -3], [6, 23, -5], [-3, -5, 14]], [[[[1, 1], [0, 1], [0, 1]], [[3, 14], [1, 1], [0, 1]], [[-3, 28], [-61, 304], [1, 1]]], [[28, 1], [152, 7], [7785, 608]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]])], [([[7, 2, -10], [2, 5, -4], [-10, -4, 18]], [[[[1, 1], [0, 1], [0, 1]], [[2, 7], [1, 1], [0, 1]], [[-10, 7], [-8, 31], [1, 1]]], [[7, 1], [31, 7], [106, 31]]]), ([[12, -6, -4], [-6, 19, -3], [-4, -3, 7]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 2], [1, 1], [0, 1]], [[-1, 3], [-5, 16], [1, 1]]], [[12, 1], [16, 1], [197, 48]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[18, -12, -2], [-12, 14, 2], [-2, 2, 13]], [[[[1, 1], [0, 1], [0, 1]], [[-2, 3], [1, 1], [0, 1]], [[-1, 9], [1, 9], [1, 1]]], [[18, 1], [6, 1], [343, 27]]]), ([[2, -2, 2], [-2, 10, -2], [2, -2, 13]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 1], [1, 1], [0, 1]], [[1, 1], [0, 1], [1, 1]]], [[2, 1], [8, 1], [11, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[11, 6], [6, 5]], [[[[1, 1], [0, 1]], [[6, 11], [1, 1]]], [[11, 1], [19, 11]]]), ([[1, 0], [0, 19]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[1, 1], [19, 1]]]), ([[3, 2], [2, 5]], [[[[1, 1], [0, 1]], [[2, 3], [1, 1]]], [[3, 1], [11, 3]]]), ([[9, 0], [0, 1]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[9, 1], [1, 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]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]][[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]]Passed
explicit oracle 1[[[[1, 1]]], [[2, 1]]][[[[1, 1]]], [[2, 1]]]Passed
explicit oracle 2[[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]][[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]Passed
explicit oracle 3[[[[1, 1]]], [[5, 1]]][[[[1, 1]]], [[5, 1]]]Passed
explicit oracle 4[[[[1, 1]]], [[5, 1]]][[[[1, 1]]], [[5, 1]]]Passed
explicit oracle 5[[[[1, 1]]], [[2, 1]]][[[[1, 1]]], [[2, 1]]]Passed
explicit oracle 6[[[[1, 1]]], [[10, 1]]][[[[1, 1]]], [[10, 1]]]Passed
explicit oracle 7[[[[1, 1]]], [[2, 1]]][[[[1, 1]]], [[2, 1]]]Passed

SHA-256 / 16ce899fcba43d5617daa0bafd6dcda3671e2c38af5b5aafd19610fdbdab510c

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 2026-09-29T14:39:26.997098+00:00.

Case digest / 6f0cda4ddd08427409ed5827a71882145583a0a86b2448b5ff06504f89673618