FA-15231 / Numerics / Open access

Multiset permutation unrank: skipped block subtraction · case 01

The exact multiset permutation unrank result violates the stated contract at skipped block subtraction.

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

ROOT CAUSE

The skipped block subtraction step uses r-1 instead of r-block.

VERIFIED REPAIR

Use r-block at the skipped block subtraction step.

Unsuccessful approach: The partial repair r%block still violates the skipped block subtraction invariant.

Case contract

Input [sorted multiset of small integer symbols, rank], valid zero-based lexicographic rank among distinct permutations.

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):
    items,r=x
    from collections import Counter
    counts=Counter(items);out=[]
    def ways(c):
     v=math.factorial(sum(c.values()))
     for n in c.values():v//=math.factorial(n)
     return v
    for _ in range(len(items)):
     for symbol in sorted(counts):
      if counts[symbol]==0:continue
      counts[symbol]-=1
      block=ways(counts)
      if r<block:
       out.append(symbol)
       break
      r=r-1
      counts[symbol]=counts[symbol]+1
     else:return None
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[0, 0, 1], 2], [1, 0, 0]), ([[0, 1, 1, 2], 6], [1, 1, 2, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1], 1], [0, 1, 0]), ([[0, 1, 1, 2], 0], [0, 1, 1, 2]), ([[0, 1, 1, 2], 1], [0, 1, 2, 1]), ([[0, 1, 1, 2], 2], [0, 2, 1, 1])], [([[0, 1, 1, 2], 2], [0, 2, 1, 1]), ([[0, 1, 1, 2], 9], [2, 0, 1, 1]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1, 1], 2], [0, 1, 1, 0]), ([[0, 0, 1, 1], 3], [1, 0, 0, 1]), ([[0, 0, 1, 1], 4], [1, 0, 1, 0]), ([[0, 0, 1, 1], 5], [1, 1, 0, 0])], [([[0, 1, 1, 2], 3], [1, 0, 1, 2]), ([[0, 0, 1, 1], 2], [0, 1, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 0, 1, 2], 13], [1, 0, 0, 2, 0]), ([[0, 0, 0, 1, 2], 14], [1, 0, 2, 0, 0]), ([[0, 0, 0, 1, 2], 15], [1, 2, 0, 0, 0]), ([[0, 0, 0, 1, 2], 16], [2, 0, 0, 0, 1])], [([[0, 1, 1, 2], 4], [1, 0, 2, 1]), ([[0, 1, 2, 3], 4], [0, 3, 1, 2]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 1, 2, 3], 10], [1, 3, 0, 2]), ([[0, 1, 2, 3], 11], [1, 3, 2, 0]), ([[0, 1, 2, 3], 12], [2, 0, 1, 3]), ([[0, 1, 2, 3], 13], [2, 0, 3, 1])], [([[0, 1, 1, 2], 5], [1, 1, 0, 2]), ([[0, 1, 2, 3], 11], [1, 3, 2, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[1, 1, 2, 2, 3], 3], [1, 2, 1, 2, 3]), ([[1, 1, 2, 2, 3], 4], [1, 2, 1, 3, 2]), ([[1, 1, 2, 2, 3], 5], [1, 2, 2, 1, 3]), ([[1, 1, 2, 2, 3], 6], [1, 2, 2, 3, 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 fixture	Actual	Expected	Outcome
explicit oracle 0	`None`	`[1, 0, 0]`	Failed
explicit oracle 1	`None`	`[1, 1, 2, 0]`	Failed
explicit oracle 2	`[0, 0, 1]`	`[0, 0, 1]`	Passed
explicit oracle 3	`None`	`[3, 2, 2, 1, 1]`	Failed
explicit oracle 4	`[0, 1, 0]`	`[0, 1, 0]`	Passed
explicit oracle 5	`[0, 1, 1, 2]`	`[0, 1, 1, 2]`	Passed
explicit oracle 6	`[0, 1, 2, 1]`	`[0, 1, 2, 1]`	Passed
explicit oracle 7	`None`	`[0, 2, 1, 1]`	Failed

SHA-256 / 64658672f73e57d446b6d201d1d320a5898f1266c9e957a67431cb74ec943630

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):
    items,r=x
    from collections import Counter
    counts=Counter(items);out=[]
    def ways(c):
     v=math.factorial(sum(c.values()))
     for n in c.values():v//=math.factorial(n)
     return v
    for _ in range(len(items)):
     for symbol in sorted(counts):
      if counts[symbol]==0:continue
      counts[symbol]-=1
      block=ways(counts)
      if r<block:
       out.append(symbol)
       break
      r=r%block
      counts[symbol]=counts[symbol]+1
     else:return None
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[0, 0, 1], 2], [1, 0, 0]), ([[0, 1, 1, 2], 6], [1, 1, 2, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1], 1], [0, 1, 0]), ([[0, 1, 1, 2], 0], [0, 1, 1, 2]), ([[0, 1, 1, 2], 1], [0, 1, 2, 1]), ([[0, 1, 1, 2], 2], [0, 2, 1, 1])], [([[0, 1, 1, 2], 2], [0, 2, 1, 1]), ([[0, 1, 1, 2], 9], [2, 0, 1, 1]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1, 1], 2], [0, 1, 1, 0]), ([[0, 0, 1, 1], 3], [1, 0, 0, 1]), ([[0, 0, 1, 1], 4], [1, 0, 1, 0]), ([[0, 0, 1, 1], 5], [1, 1, 0, 0])], [([[0, 1, 1, 2], 3], [1, 0, 1, 2]), ([[0, 0, 1, 1], 2], [0, 1, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 0, 1, 2], 13], [1, 0, 0, 2, 0]), ([[0, 0, 0, 1, 2], 14], [1, 0, 2, 0, 0]), ([[0, 0, 0, 1, 2], 15], [1, 2, 0, 0, 0]), ([[0, 0, 0, 1, 2], 16], [2, 0, 0, 0, 1])], [([[0, 1, 1, 2], 4], [1, 0, 2, 1]), ([[0, 1, 2, 3], 4], [0, 3, 1, 2]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 1, 2, 3], 10], [1, 3, 0, 2]), ([[0, 1, 2, 3], 11], [1, 3, 2, 0]), ([[0, 1, 2, 3], 12], [2, 0, 1, 3]), ([[0, 1, 2, 3], 13], [2, 0, 3, 1])], [([[0, 1, 1, 2], 5], [1, 1, 0, 2]), ([[0, 1, 2, 3], 11], [1, 3, 2, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[1, 1, 2, 2, 3], 3], [1, 2, 1, 2, 3]), ([[1, 1, 2, 2, 3], 4], [1, 2, 1, 3, 2]), ([[1, 1, 2, 2, 3], 5], [1, 2, 2, 1, 3]), ([[1, 1, 2, 2, 3], 6], [1, 2, 2, 3, 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 fixture	Actual	Expected	Outcome
explicit oracle 0	`[1, 0, 0]`	`[1, 0, 0]`	Passed
explicit oracle 1	`[1, 0, 1, 2]`	`[1, 1, 2, 0]`	Failed
explicit oracle 2	`[0, 0, 1]`	`[0, 0, 1]`	Passed
explicit oracle 3	`[2, 1, 2, 3, 1]`	`[3, 2, 2, 1, 1]`	Failed
explicit oracle 4	`[0, 1, 0]`	`[0, 1, 0]`	Passed
explicit oracle 5	`[0, 1, 1, 2]`	`[0, 1, 1, 2]`	Passed
explicit oracle 6	`[0, 1, 2, 1]`	`[0, 1, 2, 1]`	Passed
explicit oracle 7	`[0, 2, 1, 1]`	`[0, 2, 1, 1]`	Passed

SHA-256 / d16a6788b37d576b15a568df20c3efc60fe9087bde213e613b22aaf3b3bb3774

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):
    items,r=x
    from collections import Counter
    counts=Counter(items);out=[]
    def ways(c):
     v=math.factorial(sum(c.values()))
     for n in c.values():v//=math.factorial(n)
     return v
    for _ in range(len(items)):
     for symbol in sorted(counts):
      if counts[symbol]==0:continue
      counts[symbol]-=1
      block=ways(counts)
      if r<block:
       out.append(symbol)
       break
      r=r-block
      counts[symbol]=counts[symbol]+1
     else:return None
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([[0, 0, 1], 2], [1, 0, 0]), ([[0, 1, 1, 2], 6], [1, 1, 2, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1], 1], [0, 1, 0]), ([[0, 1, 1, 2], 0], [0, 1, 1, 2]), ([[0, 1, 1, 2], 1], [0, 1, 2, 1]), ([[0, 1, 1, 2], 2], [0, 2, 1, 1])], [([[0, 1, 1, 2], 2], [0, 2, 1, 1]), ([[0, 1, 1, 2], 9], [2, 0, 1, 1]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 1, 1], 2], [0, 1, 1, 0]), ([[0, 0, 1, 1], 3], [1, 0, 0, 1]), ([[0, 0, 1, 1], 4], [1, 0, 1, 0]), ([[0, 0, 1, 1], 5], [1, 1, 0, 0])], [([[0, 1, 1, 2], 3], [1, 0, 1, 2]), ([[0, 0, 1, 1], 2], [0, 1, 1, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 0, 0, 1, 2], 13], [1, 0, 0, 2, 0]), ([[0, 0, 0, 1, 2], 14], [1, 0, 2, 0, 0]), ([[0, 0, 0, 1, 2], 15], [1, 2, 0, 0, 0]), ([[0, 0, 0, 1, 2], 16], [2, 0, 0, 0, 1])], [([[0, 1, 1, 2], 4], [1, 0, 2, 1]), ([[0, 1, 2, 3], 4], [0, 3, 1, 2]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[0, 1, 2, 3], 10], [1, 3, 0, 2]), ([[0, 1, 2, 3], 11], [1, 3, 2, 0]), ([[0, 1, 2, 3], 12], [2, 0, 1, 3]), ([[0, 1, 2, 3], 13], [2, 0, 3, 1])], [([[0, 1, 1, 2], 5], [1, 1, 0, 2]), ([[0, 1, 2, 3], 11], [1, 3, 2, 0]), ([[0, 0, 1], 0], [0, 0, 1]), ([[1, 1, 2, 2, 3], 29], [3, 2, 2, 1, 1]), ([[1, 1, 2, 2, 3], 3], [1, 2, 1, 2, 3]), ([[1, 1, 2, 2, 3], 4], [1, 2, 1, 3, 2]), ([[1, 1, 2, 2, 3], 5], [1, 2, 2, 1, 3]), ([[1, 1, 2, 2, 3], 6], [1, 2, 2, 3, 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 fixture	Actual	Expected	Outcome
explicit oracle 0	`[1, 0, 0]`	`[1, 0, 0]`	Passed
explicit oracle 1	`[1, 1, 2, 0]`	`[1, 1, 2, 0]`	Passed
explicit oracle 2	`[0, 0, 1]`	`[0, 0, 1]`	Passed
explicit oracle 3	`[3, 2, 2, 1, 1]`	`[3, 2, 2, 1, 1]`	Passed
explicit oracle 4	`[0, 1, 0]`	`[0, 1, 0]`	Passed
explicit oracle 5	`[0, 1, 1, 2]`	`[0, 1, 1, 2]`	Passed
explicit oracle 6	`[0, 1, 2, 1]`	`[0, 1, 2, 1]`	Passed
explicit oracle 7	`[0, 2, 1, 1]`	`[0, 2, 1, 1]`	Passed

SHA-256 / cd09f042063b288263c0d2e79fad528616bf3f56db505d8601c4b4795fa3f18a

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 2026-09-29T14:39:24.653701+00:00.

Case digest / accb1fb70e5a04f79fe693b0a56b6908e7e56a0f72ace0392b39c25fd5afd5fb