FAILURE MAP
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FA-15301 / Numerics / Open access

Josephus elimination order: elimination cardinality · case 01

The exact josephus elimination order result violates the stated contract at elimination cardinality.

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

ROOT CAUSE

The elimination cardinality step uses range(n-1) instead of range(n).

VERIFIED REPAIR

Use range(n) at the elimination cardinality step.

Unsuccessful approach: The partial repair range(1,n) still violates the elimination cardinality invariant.

Case contract

Input [n,k,start], n>0 k>0 0<=start<n; return removal order counting current position as one.

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):
    n,k,start=x
    ring=list(range(n));index=start;out=[]
    for _ in range(n-1):
     if not ring:break
     index=(index+k-1)%len(ring)
     out.append(ring.pop(index))
     index=index
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([1, 2, 0], [0]), ([1, 3, 0], [0]), ([1, 4, 0], [0]), ([1, 5, 0], [0]), ([1, 6, 0], [0]), ([1, 7, 0], [0])], [([1, 2, 0], [0]), ([1, 4, 0], [0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([2, 5, 1], [1, 0]), ([2, 6, 0], [1, 0]), ([2, 6, 1], [0, 1]), ([2, 7, 0], [0, 1])], [([1, 3, 0], [0]), ([1, 7, 0], [0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([3, 4, 1], [1, 0, 2]), ([3, 4, 2], [2, 1, 0]), ([3, 5, 0], [1, 2, 0]), ([3, 5, 1], [2, 0, 1])], [([1, 4, 0], [0]), ([2, 1, 1], [1, 0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([4, 1, 3], [3, 0, 1, 2]), ([4, 2, 0], [1, 3, 2, 0]), ([4, 2, 1], [2, 0, 3, 1]), ([4, 2, 2], [3, 1, 0, 2])], [([1, 5, 0], [0]), ([2, 3, 0], [0, 1]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([4, 6, 0], [1, 0, 3, 2]), ([4, 6, 1], [2, 1, 0, 3]), ([4, 6, 2], [3, 2, 1, 0]), ([4, 6, 3], [0, 3, 2, 1])]]
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]Failed
explicit oracle 1[6, 3, 1, 0, 2, 5, 9, 8, 4, 10][6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]Failed
explicit oracle 2[][0]Failed
explicit oracle 3[][0]Failed
explicit oracle 4[][0]Failed
explicit oracle 5[][0]Failed
explicit oracle 6[][0]Failed
explicit oracle 7[][0]Failed

SHA-256 / 39b0a7ee8d3c9d1c0ba32e9c2e5e154ec1b2ff0bd4a83412be2767eeabb678c8

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):
    n,k,start=x
    ring=list(range(n));index=start;out=[]
    for _ in range(1,n):
     if not ring:break
     index=(index+k-1)%len(ring)
     out.append(ring.pop(index))
     index=index
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([1, 2, 0], [0]), ([1, 3, 0], [0]), ([1, 4, 0], [0]), ([1, 5, 0], [0]), ([1, 6, 0], [0]), ([1, 7, 0], [0])], [([1, 2, 0], [0]), ([1, 4, 0], [0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([2, 5, 1], [1, 0]), ([2, 6, 0], [1, 0]), ([2, 6, 1], [0, 1]), ([2, 7, 0], [0, 1])], [([1, 3, 0], [0]), ([1, 7, 0], [0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([3, 4, 1], [1, 0, 2]), ([3, 4, 2], [2, 1, 0]), ([3, 5, 0], [1, 2, 0]), ([3, 5, 1], [2, 0, 1])], [([1, 4, 0], [0]), ([2, 1, 1], [1, 0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([4, 1, 3], [3, 0, 1, 2]), ([4, 2, 0], [1, 3, 2, 0]), ([4, 2, 1], [2, 0, 3, 1]), ([4, 2, 2], [3, 1, 0, 2])], [([1, 5, 0], [0]), ([2, 3, 0], [0, 1]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([4, 6, 0], [1, 0, 3, 2]), ([4, 6, 1], [2, 1, 0, 3]), ([4, 6, 2], [3, 2, 1, 0]), ([4, 6, 3], [0, 3, 2, 1])]]
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]Failed
explicit oracle 1[6, 3, 1, 0, 2, 5, 9, 8, 4, 10][6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]Failed
explicit oracle 2[][0]Failed
explicit oracle 3[][0]Failed
explicit oracle 4[][0]Failed
explicit oracle 5[][0]Failed
explicit oracle 6[][0]Failed
explicit oracle 7[][0]Failed

SHA-256 / 557f8237758abf7e381ac15452d41876a4dfde1cc34b72bfd9b4dc8c6d8d8731

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):
    n,k,start=x
    ring=list(range(n));index=start;out=[]
    for _ in range(n):
     if not ring:break
     index=(index+k-1)%len(ring)
     out.append(ring.pop(index))
     index=index
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([1, 2, 0], [0]), ([1, 3, 0], [0]), ([1, 4, 0], [0]), ([1, 5, 0], [0]), ([1, 6, 0], [0]), ([1, 7, 0], [0])], [([1, 2, 0], [0]), ([1, 4, 0], [0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([2, 5, 1], [1, 0]), ([2, 6, 0], [1, 0]), ([2, 6, 1], [0, 1]), ([2, 7, 0], [0, 1])], [([1, 3, 0], [0]), ([1, 7, 0], [0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([3, 4, 1], [1, 0, 2]), ([3, 4, 2], [2, 1, 0]), ([3, 5, 0], [1, 2, 0]), ([3, 5, 1], [2, 0, 1])], [([1, 4, 0], [0]), ([2, 1, 1], [1, 0]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([4, 1, 3], [3, 0, 1, 2]), ([4, 2, 0], [1, 3, 2, 0]), ([4, 2, 1], [2, 0, 3, 1]), ([4, 2, 2], [3, 1, 0, 2])], [([1, 5, 0], [0]), ([2, 3, 0], [0, 1]), ([1, 1, 0], [0]), ([11, 8, 10], [6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]), ([4, 6, 0], [1, 0, 3, 2]), ([4, 6, 1], [2, 1, 0, 3]), ([4, 6, 2], [3, 2, 1, 0]), ([4, 6, 3], [0, 3, 2, 1])]]
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][0]Passed
explicit oracle 1[6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7][6, 3, 1, 0, 2, 5, 9, 8, 4, 10, 7]Passed
explicit oracle 2[0][0]Passed
explicit oracle 3[0][0]Passed
explicit oracle 4[0][0]Passed
explicit oracle 5[0][0]Passed
explicit oracle 6[0][0]Passed
explicit oracle 7[0][0]Passed

SHA-256 / 2168619310369d56d70f1b1ae3ff0050e8a603642664bb27a2d4d65bd45c5679

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 2026-09-29T14:39:25.331340+00:00.

Case digest / 6724505714c6a1dd14f2318f86e1485059fae8747094c46c2b058c7a12af440e