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FA-15676 / Numerics / Open access

Binary field trace: trace field addition · case 01

The exact binary field trace result violates the stated contract at trace field addition.

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

ROOT CAUSE

The trace field addition step uses out+v instead of out^v.

THE FAILURE

The trace field addition step uses out+v instead of out^v.

Unsuccessful approach: The partial repair out|v still violates the trace field addition invariant.

Case contract

Input [a,irreducible bitmask]; return absolute field trace a+a^2+...+a^(2^(n-1)), an F2 element represented integer 0 or1.

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,poly=x;n=poly.bit_length()-1
    def mul(a,b):
     z=0
     while b:
      if b&1:z^=a
      b>>=1;a<<=1
      if a&(1<<n):a^=poly
     return z
    v=a;out=0
    for _ in range(n):
     out=out+v
     v=mul(v,v)
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 7], 0), ([0, 7], 0), ([63, 67], 1), ([2, 7], 1), ([3, 7], 1), ([0, 11], 0), ([1, 11], 1), ([2, 11], 0)], [([2, 7], 1), ([2, 11], 0), ([0, 7], 0), ([63, 67], 1), ([5, 19], 0), ([6, 19], 0), ([7, 19], 0), ([8, 19], 1)], [([3, 7], 1), ([5, 11], 1), ([0, 7], 0), ([63, 67], 1), ([6, 25], 0), ([7, 25], 0), ([8, 25], 1), ([9, 25], 1)], [([1, 11], 1), ([1, 19], 0), ([0, 7], 0), ([63, 67], 1), ([7, 31], 0), ([8, 31], 1), ([9, 31], 1), ([10, 31], 0)], [([2, 11], 0), ([4, 19], 0), ([0, 7], 0), ([63, 67], 1), ([8, 37], 1), ([9, 37], 0), ([10, 37], 1), ([11, 37], 0)]]
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 020Failed
explicit oracle 100Passed
explicit oracle 23071Failed
explicit oracle 351Failed
explicit oracle 451Failed
explicit oracle 500Passed
explicit oracle 631Failed
explicit oracle 7120Failed

SHA-256 / 5b20731f41e87fd5d28295be9b1ae1219bb177505d01d0872e7aeeff6d977c23

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,poly=x;n=poly.bit_length()-1
    def mul(a,b):
     z=0
     while b:
      if b&1:z^=a
      b>>=1;a<<=1
      if a&(1<<n):a^=poly
     return z
    v=a;out=0
    for _ in range(n):
     out=out|v
     v=mul(v,v)
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 7], 0), ([0, 7], 0), ([63, 67], 1), ([2, 7], 1), ([3, 7], 1), ([0, 11], 0), ([1, 11], 1), ([2, 11], 0)], [([2, 7], 1), ([2, 11], 0), ([0, 7], 0), ([63, 67], 1), ([5, 19], 0), ([6, 19], 0), ([7, 19], 0), ([8, 19], 1)], [([3, 7], 1), ([5, 11], 1), ([0, 7], 0), ([63, 67], 1), ([6, 25], 0), ([7, 25], 0), ([8, 25], 1), ([9, 25], 1)], [([1, 11], 1), ([1, 19], 0), ([0, 7], 0), ([63, 67], 1), ([7, 31], 0), ([8, 31], 1), ([9, 31], 1), ([10, 31], 0)], [([2, 11], 0), ([4, 19], 0), ([0, 7], 0), ([63, 67], 1), ([8, 37], 1), ([9, 37], 0), ([10, 37], 1), ([11, 37], 0)]]
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 010Failed
explicit oracle 100Passed
explicit oracle 2631Failed
explicit oracle 331Failed
explicit oracle 431Failed
explicit oracle 500Passed
explicit oracle 611Passed
explicit oracle 760Failed

SHA-256 / e525c7dc417e4fd7c16e561ba539a3315aa4c257588eed40fc3244b8b9feb13f

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 2026-09-29T14:39:29.104773+00:00.

Case digest / 354465a4b27e428c5973472747021e918e6033116fa3f852580433c420b5126d