FAILURE MAP
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FA-15771 / Numerics / Open access

Montgomery residue product: wide product retention · case 01

The exact montgomery residue product result violates the stated contract at wide product retention.

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

ROOT CAUSE

The wide product retention step uses (a*b)%R instead of a*b.

THE FAILURE

The wide product retention step uses (a*b)%R instead of a*b.

Unsuccessful approach: The partial repair a+b still violates the wide product retention invariant.

Case contract

Input [a,b,m,R], odd modulus m<R and R power of2, 0<=a,b<m; return a*b*R^-1 modulo m using Montgomery reduction.

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,b,m,R=x
    nprime=(-pow(m,-1,R))%R
    t=(a*b)%R
    u=(t*nprime)%R
    v=(t+u*m)//R
    if v>=m:v-=m
    return v
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([2, 4, 5, 8], 1), ([0, 1, 3, 8], 0), ([0, 0, 3, 8], 0), ([30, 30, 31, 32], 1), ([0, 2, 3, 8], 0), ([1, 0, 3, 8], 0), ([1, 1, 3, 8], 2), ([1, 2, 3, 8], 1)], [([3, 3, 5, 8], 3), ([1, 1, 3, 8], 2), ([0, 0, 3, 8], 0), ([30, 30, 31, 32], 1), ([1, 3, 5, 8], 1), ([1, 4, 5, 8], 3), ([2, 0, 5, 8], 0), ([2, 1, 5, 8], 4)], [([3, 4, 5, 8], 4), ([2, 1, 3, 8], 1), ([0, 0, 3, 8], 0), ([30, 30, 31, 32], 1), ([0, 0, 7, 8], 0), ([0, 1, 7, 8], 0), ([0, 2, 7, 8], 0), ([0, 3, 7, 8], 0)], [([4, 2, 5, 8], 1), ([0, 3, 5, 8], 0), ([0, 0, 3, 8], 0), ([30, 30, 31, 32], 1), ([2, 3, 7, 8], 6), ([2, 4, 7, 8], 1), ([2, 5, 7, 8], 3), ([2, 6, 7, 8], 5)], [([4, 3, 5, 8], 4), ([1, 1, 5, 8], 2), ([0, 0, 3, 8], 0), ([30, 30, 31, 32], 1), ([4, 6, 7, 8], 3), ([5, 0, 7, 8], 0), ([5, 1, 7, 8], 5), ([5, 2, 7, 8], 3)]]
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 001Failed
explicit oracle 100Passed
explicit oracle 200Passed
explicit oracle 341Failed
explicit oracle 400Passed
explicit oracle 500Passed
explicit oracle 622Passed
explicit oracle 711Passed

SHA-256 / 7e305c92d3ebb4ad0dd13ae1e4fd5a229eccbadc8a210bf24cda8e949a7d7635

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,b,m,R=x
    nprime=(-pow(m,-1,R))%R
    t=a+b
    u=(t*nprime)%R
    v=(t+u*m)//R
    if v>=m:v-=m
    return v
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([2, 4, 5, 8], 1), ([0, 1, 3, 8], 0), ([0, 0, 3, 8], 0), ([30, 30, 31, 32], 1), ([0, 2, 3, 8], 0), ([1, 0, 3, 8], 0), ([1, 1, 3, 8], 2), ([1, 2, 3, 8], 1)], [([3, 3, 5, 8], 3), ([1, 1, 3, 8], 2), ([0, 0, 3, 8], 0), ([30, 30, 31, 32], 1), ([1, 3, 5, 8], 1), ([1, 4, 5, 8], 3), ([2, 0, 5, 8], 0), ([2, 1, 5, 8], 4)], [([3, 4, 5, 8], 4), ([2, 1, 3, 8], 1), ([0, 0, 3, 8], 0), ([30, 30, 31, 32], 1), ([0, 0, 7, 8], 0), ([0, 1, 7, 8], 0), ([0, 2, 7, 8], 0), ([0, 3, 7, 8], 0)], [([4, 2, 5, 8], 1), ([0, 3, 5, 8], 0), ([0, 0, 3, 8], 0), ([30, 30, 31, 32], 1), ([2, 3, 7, 8], 6), ([2, 4, 7, 8], 1), ([2, 5, 7, 8], 3), ([2, 6, 7, 8], 5)], [([4, 3, 5, 8], 4), ([1, 1, 5, 8], 2), ([0, 0, 3, 8], 0), ([30, 30, 31, 32], 1), ([4, 6, 7, 8], 3), ([5, 0, 7, 8], 0), ([5, 1, 7, 8], 5), ([5, 2, 7, 8], 3)]]
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 021Failed
explicit oracle 120Failed
explicit oracle 200Passed
explicit oracle 3291Failed
explicit oracle 410Failed
explicit oracle 520Failed
explicit oracle 612Failed
explicit oracle 701Failed

SHA-256 / c44c31cfe18933b7f712fd75013988d36c1b6be21f81bb6c79fafea09645557a

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 2026-09-29T14:39:30.117271+00:00.

Case digest / 1850dfd7f2e22766106a9eef377875f724fbb1585cc30c5b93f131ad8c8fe80e