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FA-15421 / Numerics / Open access

Exact tridiagonal solve: initial rhs normalization · case 01

The exact exact tridiagonal solve result violates the stated contract at initial rhs normalization.

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

ROOT CAUSE

The initial rhs normalization step uses r[0] instead of r[0]/den.

THE FAILURE

The initial rhs normalization step uses r[0] instead of r[0]/den.

Unsuccessful approach: The partial repair r[0]*den still violates the initial rhs normalization invariant.

Case contract

Input [lower,diag,upper,rhs] integer tridiagonal system with nonzero Thomas pivots; return rational solution [p,q] per coordinate.

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):
    low,d,up,b=x;n=len(d)
    c=list(map(Fraction,up));r=list(map(Fraction,b));den=Fraction(d[0])
    if not den:return None
    if c:c[0]=c[0]/den
    r[0]=r[0]
    for i in range(1,n):
     den=d[i]-low[i-1]*c[i-1]
     if not den:return None
     if i<n-1:c[i]/=den
     r[i]=(r[i]-low[i-1]*r[i-1])/den
    for i in range(n-2,-1,-1):r[i]=r[i]-c[i]*r[i+1]
    return [[v.numerator,v.denominator] for v in r]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[-1], [6, 6], [2], [-3, -5]], [[-4, 19], [-33, 38]]), ([[-2, -2, -1, -2, -2], [6, 6, 6, 6, 6, 6], [0, 0, 1, -2, -1], [3, -2, 5, -3, -2, -4]], [[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]), ([[2], [6, 6], [2], [2, 2]], [[1, 4], [1, 4]]), ([[2], [6, 6], [-1], [-5, 3]], [[-27, 38], [14, 19]]), ([[1], [6, 6], [-1], [-5, -5]], [[-35, 37], [-25, 37]]), ([[1], [6, 6], [0], [-2, 4]], [[-1, 3], [13, 18]]), ([[1], [6, 6], [2], [-1, 4]], [[-7, 17], [25, 34]]), ([[-1], [6, 6], [-1], [4, 5]], [[29, 35], [34, 35]])], [([[2], [6, 6], [2], [2, 2]], [[1, 4], [1, 4]]), ([[1], [6, 6], [-1], [-5, -5]], [[-35, 37], [-25, 37]]), ([[-1], [6, 6], [2], [-3, -5]], [[-4, 19], [-33, 38]]), ([[-2, -2, -1, -2, -2], [6, 6, 6, 6, 6, 6], [0, 0, 1, -2, -1], [3, -2, 5, -3, -2, -4]], [[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]), ([[2], [6, 6], [2], [2, 3]], [[3, 16], [7, 16]]), ([[-2], [6, 6], [2], [1, 1]], [[1, 10], [1, 5]]), ([[0], [6, 6], [1], [0, -1]], [[1, 36], [-1, 6]]), ([[-1], [6, 6], [1], [5, 1]], [[29, 37], [11, 37]])], [([[2], [6, 6], [-1], [-5, 3]], [[-27, 38], [14, 19]]), ([[-1], [6, 6], [-1], [4, 5]], [[29, 35], [34, 35]]), ([[-1], [6, 6], [2], [-3, -5]], [[-4, 19], [-33, 38]]), ([[-2, -2, -1, -2, -2], [6, 6, 6, 6, 6, 6], [0, 0, 1, -2, -1], [3, -2, 5, -3, -2, -4]], [[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]), ([[2], [6, 6], [0], [0, 4]], [[0, 1], [2, 3]]), ([[-2], [6, 6], [0], [3, -4]], [[1, 2], [-1, 2]]), ([[0], [6, 6], [0], [1, -2]], [[1, 6], [-1, 3]]), ([[-2], [6, 6], [1], [5, 2]], [[14, 19], [11, 19]])], [([[1], [6, 6], [-1], [-5, -5]], [[-35, 37], [-25, 37]]), ([[-2], [6, 6], [0], [5, 5]], [[5, 6], [10, 9]]), ([[-1], [6, 6], [2], [-3, -5]], [[-4, 19], [-33, 38]]), ([[-2, -2, -1, -2, -2], [6, 6, 6, 6, 6, 6], [0, 0, 1, -2, -1], [3, -2, 5, -3, -2, -4]], [[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]), ([[0, -1], [6, 6, 6], [2, 1], [5, -2, 4]], [[217, 222], [-16, 37], [22, 37]]), ([[-1, -1], [6, 6, 6], [0, -1], [0, -3, -1]], [[0, 1], [-19, 35], [-9, 35]]), ([[-2, 0], [6, 6, 6], [0, -1], [-3, -5, 3]], [[-1, 2], [-11, 12], [1, 2]]), ([[1, 0], [6, 6, 6], [-1, 0], [-3, -3, 5]], [[-21, 37], [-15, 37], [5, 6]])], [([[1], [6, 6], [0], [-2, 4]], [[-1, 3], [13, 18]]), ([[-1], [6, 6], [2], [5, 1]], [[14, 19], [11, 38]]), ([[-1], [6, 6], [2], [-3, -5]], [[-4, 19], [-33, 38]]), ([[-2, -2, -1, -2, -2], [6, 6, 6, 6, 6, 6], [0, 0, 1, -2, -1], [3, -2, 5, -3, -2, -4]], [[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]), ([[-1, 2], [6, 6, 6], [-2, 1], [5, 3, -4]], [[107, 96], [27, 32], [-91, 96]]), ([[-2, 2], [6, 6, 6], [-2, 2], [2, -2, 0]], [[5, 21], [-2, 7], [2, 21]]), ([[-2, 1], [6, 6, 6], [0, 2], [4, 4, 2]], [[2, 3], [14, 17], [10, 51]]), ([[1, -1], [6, 6, 6], [-2, 0], [-5, -4, 0]], [[-1, 1], [-1, 2], [-1, 12]])]]
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[[-49, 19], [-24, 19]][[-4, 19], [-33, 38]]Failed
explicit oracle 1[[3, 1], [2, 3], [637, 557], [-883, 1671], [-366, 557], [-1480, 1671]][[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]Failed
explicit oracle 2[[17, 8], [-3, 8]][[1, 4], [1, 4]]Failed
explicit oracle 3[[-177, 38], [39, 19]][[-27, 38], [14, 19]]Failed
explicit oracle 4[[-5, 1], [0, 1]][[-35, 37], [-25, 37]]Failed
explicit oracle 5[[-2, 1], [1, 1]][[-1, 3], [13, 18]]Failed
explicit oracle 6[[-22, 17], [15, 17]][[-7, 17], [25, 34]]Failed
explicit oracle 7[[149, 35], [54, 35]][[29, 35], [34, 35]]Failed

SHA-256 / 06e5dcf3fd8cdfeb833faa61ed382ce49205ce059f6f1a55080982dc07789184

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):
    low,d,up,b=x;n=len(d)
    c=list(map(Fraction,up));r=list(map(Fraction,b));den=Fraction(d[0])
    if not den:return None
    if c:c[0]=c[0]/den
    r[0]=r[0]*den
    for i in range(1,n):
     den=d[i]-low[i-1]*c[i-1]
     if not den:return None
     if i<n-1:c[i]/=den
     r[i]=(r[i]-low[i-1]*r[i-1])/den
    for i in range(n-2,-1,-1):r[i]=r[i]-c[i]*r[i+1]
    return [[v.numerator,v.denominator] for v in r]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[-1], [6, 6], [2], [-3, -5]], [[-4, 19], [-33, 38]]), ([[-2, -2, -1, -2, -2], [6, 6, 6, 6, 6, 6], [0, 0, 1, -2, -1], [3, -2, 5, -3, -2, -4]], [[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]), ([[2], [6, 6], [2], [2, 2]], [[1, 4], [1, 4]]), ([[2], [6, 6], [-1], [-5, 3]], [[-27, 38], [14, 19]]), ([[1], [6, 6], [-1], [-5, -5]], [[-35, 37], [-25, 37]]), ([[1], [6, 6], [0], [-2, 4]], [[-1, 3], [13, 18]]), ([[1], [6, 6], [2], [-1, 4]], [[-7, 17], [25, 34]]), ([[-1], [6, 6], [-1], [4, 5]], [[29, 35], [34, 35]])], [([[2], [6, 6], [2], [2, 2]], [[1, 4], [1, 4]]), ([[1], [6, 6], [-1], [-5, -5]], [[-35, 37], [-25, 37]]), ([[-1], [6, 6], [2], [-3, -5]], [[-4, 19], [-33, 38]]), ([[-2, -2, -1, -2, -2], [6, 6, 6, 6, 6, 6], [0, 0, 1, -2, -1], [3, -2, 5, -3, -2, -4]], [[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]), ([[2], [6, 6], [2], [2, 3]], [[3, 16], [7, 16]]), ([[-2], [6, 6], [2], [1, 1]], [[1, 10], [1, 5]]), ([[0], [6, 6], [1], [0, -1]], [[1, 36], [-1, 6]]), ([[-1], [6, 6], [1], [5, 1]], [[29, 37], [11, 37]])], [([[2], [6, 6], [-1], [-5, 3]], [[-27, 38], [14, 19]]), ([[-1], [6, 6], [-1], [4, 5]], [[29, 35], [34, 35]]), ([[-1], [6, 6], [2], [-3, -5]], [[-4, 19], [-33, 38]]), ([[-2, -2, -1, -2, -2], [6, 6, 6, 6, 6, 6], [0, 0, 1, -2, -1], [3, -2, 5, -3, -2, -4]], [[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]), ([[2], [6, 6], [0], [0, 4]], [[0, 1], [2, 3]]), ([[-2], [6, 6], [0], [3, -4]], [[1, 2], [-1, 2]]), ([[0], [6, 6], [0], [1, -2]], [[1, 6], [-1, 3]]), ([[-2], [6, 6], [1], [5, 2]], [[14, 19], [11, 19]])], [([[1], [6, 6], [-1], [-5, -5]], [[-35, 37], [-25, 37]]), ([[-2], [6, 6], [0], [5, 5]], [[5, 6], [10, 9]]), ([[-1], [6, 6], [2], [-3, -5]], [[-4, 19], [-33, 38]]), ([[-2, -2, -1, -2, -2], [6, 6, 6, 6, 6, 6], [0, 0, 1, -2, -1], [3, -2, 5, -3, -2, -4]], [[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]), ([[0, -1], [6, 6, 6], [2, 1], [5, -2, 4]], [[217, 222], [-16, 37], [22, 37]]), ([[-1, -1], [6, 6, 6], [0, -1], [0, -3, -1]], [[0, 1], [-19, 35], [-9, 35]]), ([[-2, 0], [6, 6, 6], [0, -1], [-3, -5, 3]], [[-1, 2], [-11, 12], [1, 2]]), ([[1, 0], [6, 6, 6], [-1, 0], [-3, -3, 5]], [[-21, 37], [-15, 37], [5, 6]])], [([[1], [6, 6], [0], [-2, 4]], [[-1, 3], [13, 18]]), ([[-1], [6, 6], [2], [5, 1]], [[14, 19], [11, 38]]), ([[-1], [6, 6], [2], [-3, -5]], [[-4, 19], [-33, 38]]), ([[-2, -2, -1, -2, -2], [6, 6, 6, 6, 6, 6], [0, 0, 1, -2, -1], [3, -2, 5, -3, -2, -4]], [[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]), ([[-1, 2], [6, 6, 6], [-2, 1], [5, 3, -4]], [[107, 96], [27, 32], [-91, 96]]), ([[-2, 2], [6, 6, 6], [-2, 2], [2, -2, 0]], [[5, 21], [-2, 7], [2, 21]]), ([[-2, 1], [6, 6, 6], [0, 2], [4, 4, 2]], [[2, 3], [14, 17], [10, 51]]), ([[1, -1], [6, 6, 6], [-2, 0], [-5, -4, 0]], [[-1, 1], [-1, 2], [-1, 12]])]]
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[[-319, 19], [-69, 19]][[-4, 19], [-33, 38]]Failed
explicit oracle 1[[18, 1], [17, 3], [1537, 557], [-373, 1671], [-306, 557], [-1420, 1671]][[1, 2], [-1, 6], [487, 557], [-968, 1671], [-376, 557], [-1490, 1671]]Failed
explicit oracle 2[[107, 8], [-33, 8]][[1, 4], [1, 4]]Failed
explicit oracle 3[[-1077, 38], [189, 19]][[-27, 38], [14, 19]]Failed
explicit oracle 4[[-1085, 37], [150, 37]][[-35, 37], [-25, 37]]Failed
explicit oracle 5[[-12, 1], [8, 3]][[-1, 3], [13, 18]]Failed
explicit oracle 6[[-112, 17], [30, 17]][[-7, 17], [25, 34]]Failed
explicit oracle 7[[869, 35], [174, 35]][[29, 35], [34, 35]]Failed

SHA-256 / f1a233e45b42cb88f6a5428903e7e9ed649b9a85b159260216e8bbe6dd4d4af9

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 2026-09-29T14:39:26.666118+00:00.

Case digest / e5e10c945d2d5b10481552caf6212d399d1200a864bcbd684c52ab92acd2e896