FAILURE MAP
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FA-15791 / Numerics / Open access

Singular hensel root tree: root tree lift depth · case 01

The exact singular hensel root tree result violates the stated contract at root tree lift depth.

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

ROOT CAUSE

The root tree lift depth step uses range(k) instead of range(1,k).

THE FAILURE

The root tree lift depth step uses range(k) instead of range(1,k).

Unsuccessful approach: The partial repair range(1,k-1) still violates the root tree lift depth invariant.

Case contract

Input [polynomial coefficients,p,k], p prime and k>=1; enumerate all roots modulo p^k, including singular roots.

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):
    coeff,p,k=x
    roots=[r for r in range(p) if sum(c*r**i for i,c in enumerate(coeff))%p==0];mod=p
    for _ in range(k):
     nextmod=mod*p
     out=[]
     for r in roots:
      for digit in range(p):
       s=r+digit*mod
       if sum(c*s**i for i,c in enumerate(coeff))%nextmod==0:out.append(s)
     roots=sorted(set(out));mod=nextmod
    return [roots,mod]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[0, 0, 1], 2, 1], [[0], 2]), ([[0, 0, 1], 2, 2], [[0, 2], 4]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 0, 1], 2, 3], [[0, 4], 8]), ([[0, 0, 1], 2, 4], [[0, 4, 8, 12], 16]), ([[0, 0, 1], 3, 1], [[0], 3]), ([[0, 0, 1], 3, 2], [[0, 3, 6], 9]), ([[0, 0, 1], 3, 3], [[0, 9, 18], 27])], [([[0, 0, 1], 2, 2], [[0, 2], 4]), ([[0, 0, 1], 3, 2], [[0, 3, 6], 9]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[-1, 0, 1], 2, 2], [[1, 3], 4]), ([[-1, 0, 1], 2, 3], [[1, 3, 5, 7], 8]), ([[-1, 0, 1], 2, 4], [[1, 7, 9, 15], 16]), ([[-1, 0, 1], 3, 1], [[1, 2], 3])], [([[0, 0, 1], 2, 3], [[0, 4], 8]), ([[0, 0, 1], 5, 2], [[0, 5, 10, 15, 20], 25]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 2, 1], 2, 3], [[0, 2, 4, 6], 8]), ([[0, 2, 1], 2, 4], [[0, 6, 8, 14], 16]), ([[0, 2, 1], 3, 1], [[0, 1], 3]), ([[0, 2, 1], 3, 2], [[0, 7], 9])], [([[0, 0, 1], 2, 4], [[0, 4, 8, 12], 16]), ([[0, 0, 1], 7, 2], [[0, 7, 14, 21, 28, 35, 42], 49]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[-4, 0, 1], 2, 4], [[2, 6, 10, 14], 16]), ([[-4, 0, 1], 3, 1], [[1, 2], 3]), ([[-4, 0, 1], 3, 2], [[2, 7], 9]), ([[-4, 0, 1], 3, 3], [[2, 25], 27])], [([[0, 0, 1], 3, 1], [[0], 3]), ([[-1, 0, 1], 2, 2], [[1, 3], 4]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 0, 0, 1], 3, 1], [[0], 3]), ([[0, 0, 0, 1], 3, 2], [[0, 3, 6], 9]), ([[0, 0, 0, 1], 3, 3], [[0, 3, 6, 9, 12, 15, 18, 21, 24], 27]), ([[0, 0, 0, 1], 3, 4], [[0, 9, 18, 27, 36, 45, 54, 63, 72], 81])]]
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[[0, 2], 4][[0], 2]Failed
explicit oracle 1[[0, 4], 8][[0, 2], 4]Failed
explicit oracle 2[[1, 16805], 16807][[1, 2399], 2401]Failed
explicit oracle 3[[0, 4, 8, 12], 16][[0, 4], 8]Failed
explicit oracle 4[[0, 8, 16, 24], 32][[0, 4, 8, 12], 16]Failed
explicit oracle 5[[0, 3, 6], 9][[0], 3]Failed
explicit oracle 6[[0, 9, 18], 27][[0, 3, 6], 9]Failed
explicit oracle 7[[0, 9, 18, 27, 36, 45, 54, 63, 72], 81][[0, 9, 18], 27]Failed

SHA-256 / d00942094136baa3993bfe0e0e288a0c17b27f30fa692eaf34954e561f52f966

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):
    coeff,p,k=x
    roots=[r for r in range(p) if sum(c*r**i for i,c in enumerate(coeff))%p==0];mod=p
    for _ in range(1,k-1):
     nextmod=mod*p
     out=[]
     for r in roots:
      for digit in range(p):
       s=r+digit*mod
       if sum(c*s**i for i,c in enumerate(coeff))%nextmod==0:out.append(s)
     roots=sorted(set(out));mod=nextmod
    return [roots,mod]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[0, 0, 1], 2, 1], [[0], 2]), ([[0, 0, 1], 2, 2], [[0, 2], 4]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 0, 1], 2, 3], [[0, 4], 8]), ([[0, 0, 1], 2, 4], [[0, 4, 8, 12], 16]), ([[0, 0, 1], 3, 1], [[0], 3]), ([[0, 0, 1], 3, 2], [[0, 3, 6], 9]), ([[0, 0, 1], 3, 3], [[0, 9, 18], 27])], [([[0, 0, 1], 2, 2], [[0, 2], 4]), ([[0, 0, 1], 3, 2], [[0, 3, 6], 9]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[-1, 0, 1], 2, 2], [[1, 3], 4]), ([[-1, 0, 1], 2, 3], [[1, 3, 5, 7], 8]), ([[-1, 0, 1], 2, 4], [[1, 7, 9, 15], 16]), ([[-1, 0, 1], 3, 1], [[1, 2], 3])], [([[0, 0, 1], 2, 3], [[0, 4], 8]), ([[0, 0, 1], 5, 2], [[0, 5, 10, 15, 20], 25]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 2, 1], 2, 3], [[0, 2, 4, 6], 8]), ([[0, 2, 1], 2, 4], [[0, 6, 8, 14], 16]), ([[0, 2, 1], 3, 1], [[0, 1], 3]), ([[0, 2, 1], 3, 2], [[0, 7], 9])], [([[0, 0, 1], 2, 4], [[0, 4, 8, 12], 16]), ([[0, 0, 1], 7, 2], [[0, 7, 14, 21, 28, 35, 42], 49]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[-4, 0, 1], 2, 4], [[2, 6, 10, 14], 16]), ([[-4, 0, 1], 3, 1], [[1, 2], 3]), ([[-4, 0, 1], 3, 2], [[2, 7], 9]), ([[-4, 0, 1], 3, 3], [[2, 25], 27])], [([[0, 0, 1], 3, 1], [[0], 3]), ([[-1, 0, 1], 2, 2], [[1, 3], 4]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 0, 0, 1], 3, 1], [[0], 3]), ([[0, 0, 0, 1], 3, 2], [[0, 3, 6], 9]), ([[0, 0, 0, 1], 3, 3], [[0, 3, 6, 9, 12, 15, 18, 21, 24], 27]), ([[0, 0, 0, 1], 3, 4], [[0, 9, 18, 27, 36, 45, 54, 63, 72], 81])]]
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[[0], 2][[0], 2]Passed
explicit oracle 1[[0], 2][[0, 2], 4]Failed
explicit oracle 2[[1, 341], 343][[1, 2399], 2401]Failed
explicit oracle 3[[0, 2], 4][[0, 4], 8]Failed
explicit oracle 4[[0, 4], 8][[0, 4, 8, 12], 16]Failed
explicit oracle 5[[0], 3][[0], 3]Passed
explicit oracle 6[[0], 3][[0, 3, 6], 9]Failed
explicit oracle 7[[0, 3, 6], 9][[0, 9, 18], 27]Failed

SHA-256 / 80dcb4767d07f73e63bf088993aa1c461d06bcd0cbc54ec50f390ffd9082e0d7

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 2026-09-29T14:39:30.159909+00:00.

Case digest / 26c7492b97c7446c4a624f2c28bdba099207344ed275dc78d23193ebf71a90bb