FA-15806 / Numerics / Open access

Singular hensel root tree: root residue digit placement · case 01

The exact singular hensel root tree result violates the stated contract at root residue digit placement.

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

ROOT CAUSE

The root residue digit placement step uses r+digit instead of r+digit*mod.

VERIFIED REPAIR

Use r+digit*mod at the root residue digit placement step.

Unsuccessful approach: The partial repair r*p+digit still violates the root residue digit placement 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(1,k):
     nextmod=mod*p
     out=[]
     for r in roots:
      for digit in range(p):
       s=r+digit
       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, 2], [[0, 2], 4]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-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, 3], [[0, 4], 8]), ([[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, 4], [[0, 4, 8, 12], 16]), ([[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], 3, 2], [[0, 3, 6], 9]), ([[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, 3], [[0, 9, 18], 27]), ([[-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 fixture	Actual	Expected	Outcome
explicit oracle 0	`[[0], 4]`	`[[0, 2], 4]`	Failed
explicit oracle 1	`[[0], 2]`	`[[0], 2]`	Passed
explicit oracle 2	`[[1], 2401]`	`[[1, 2399], 2401]`	Failed
explicit oracle 3	`[[0], 8]`	`[[0, 4], 8]`	Failed
explicit oracle 4	`[[0], 16]`	`[[0, 4, 8, 12], 16]`	Failed
explicit oracle 5	`[[0], 3]`	`[[0], 3]`	Passed
explicit oracle 6	`[[0], 9]`	`[[0, 3, 6], 9]`	Failed
explicit oracle 7	`[[0], 27]`	`[[0, 9, 18], 27]`	Failed

SHA-256 / e47af78945d6d1f6da8aa8764e6c4e3cfdcebc322202a657ef1559efe2b424c5

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):
     nextmod=mod*p
     out=[]
     for r in roots:
      for digit in range(p):
       s=r*p+digit
       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, 2], [[0, 2], 4]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-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, 3], [[0, 4], 8]), ([[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, 4], [[0, 4, 8, 12], 16]), ([[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], 3, 2], [[0, 3, 6], 9]), ([[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, 3], [[0, 9, 18], 27]), ([[-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 fixture	Actual	Expected	Outcome
explicit oracle 0	`[[0], 4]`	`[[0, 2], 4]`	Failed
explicit oracle 1	`[[0], 2]`	`[[0], 2]`	Passed
explicit oracle 2	`[[], 2401]`	`[[1, 2399], 2401]`	Failed
explicit oracle 3	`[[0], 8]`	`[[0, 4], 8]`	Failed
explicit oracle 4	`[[0], 16]`	`[[0, 4, 8, 12], 16]`	Failed
explicit oracle 5	`[[0], 3]`	`[[0], 3]`	Passed
explicit oracle 6	`[[0], 9]`	`[[0, 3, 6], 9]`	Failed
explicit oracle 7	`[[0], 27]`	`[[0, 9, 18], 27]`	Failed

SHA-256 / ef8dd417b08e67bdc6bc843a66e1c776c2fb508ffc107e36450f114eacd2f7b2

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):
    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):
     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, 2], [[0, 2], 4]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-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, 3], [[0, 4], 8]), ([[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, 4], [[0, 4, 8, 12], 16]), ([[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], 3, 2], [[0, 3, 6], 9]), ([[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, 3], [[0, 9, 18], 27]), ([[-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 fixture	Actual	Expected	Outcome
explicit oracle 0	`[[0, 2], 4]`	`[[0, 2], 4]`	Passed
explicit oracle 1	`[[0], 2]`	`[[0], 2]`	Passed
explicit oracle 2	`[[1, 2399], 2401]`	`[[1, 2399], 2401]`	Passed
explicit oracle 3	`[[0, 4], 8]`	`[[0, 4], 8]`	Passed
explicit oracle 4	`[[0, 4, 8, 12], 16]`	`[[0, 4, 8, 12], 16]`	Passed
explicit oracle 5	`[[0], 3]`	`[[0], 3]`	Passed
explicit oracle 6	`[[0, 3, 6], 9]`	`[[0, 3, 6], 9]`	Passed
explicit oracle 7	`[[0, 9, 18], 27]`	`[[0, 9, 18], 27]`	Passed

SHA-256 / 4ca5f0bcb1063eb1391b9ea0f6c8c7f331ad0b5229073ba2a6c94c200f222ba5

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 2026-09-29T14:39:30.426570+00:00.

Case digest / 5628e711e1435e7270ce039a23301826fc82c4ef2491ed2f52736d7e41dbac24