FAILURE MAP
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FA-14461 / Numerics / Open access

Continued fraction convergents: lagged state transfer · case 01

The exact continued fraction convergents result violates the stated contract at lagged state transfer.

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

ROOT CAUSE

The lagged state transfer step uses p,p1,q,q1 instead of p1,p,q1,q.

THE FAILURE

The lagged state transfer step uses p,p1,q,q1 instead of p1,p,q1,q.

Unsuccessful approach: The partial repair p,p,q,q still violates the lagged state transfer invariant.

Case contract

Input nonempty coefficients with positive terms after first; return every convergent as [numerator,positive denominator].

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=x
    p0,p1=0,1
    q0,q1=1,0
    out=[]
    for c in a:
     p=c*p1+p0
     q=c*q1+q0
     out.append([p,q])
     p0,p1,q0,q1=p,p1,q,q1
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 2], [[1, 1], [3, 2]]), ([1, 1], [[1, 1], [2, 1]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([2], [[2, 1]]), ([3], [[3, 1]]), ([4], [[4, 1]]), ([1, 3], [[1, 1], [4, 3]])], [([1, 3], [[1, 1], [4, 3]]), ([1, 4], [[1, 1], [5, 4]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([4, 2], [[4, 1], [9, 2]]), ([4, 3], [[4, 1], [13, 3]]), ([4, 4], [[4, 1], [17, 4]]), ([1, 1, 1], [[1, 1], [2, 1], [3, 2]])], [([1, 4], [[1, 1], [5, 4]]), ([2, 3], [[2, 1], [7, 3]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([1, 4, 3], [[1, 1], [5, 4], [16, 13]]), ([1, 4, 4], [[1, 1], [5, 4], [21, 17]]), ([2, 1, 1], [[2, 1], [3, 1], [5, 2]]), ([2, 1, 2], [[2, 1], [3, 1], [8, 3]])], [([2, 2], [[2, 1], [5, 2]]), ([3, 2], [[3, 1], [7, 2]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([2, 4, 4], [[2, 1], [9, 4], [38, 17]]), ([3, 1, 1], [[3, 1], [4, 1], [7, 2]]), ([3, 1, 2], [[3, 1], [4, 1], [11, 3]]), ([3, 1, 3], [[3, 1], [4, 1], [15, 4]])], [([2, 3], [[2, 1], [7, 3]]), ([4, 1], [[4, 1], [5, 1]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([4, 1, 1], [[4, 1], [5, 1], [9, 2]]), ([4, 1, 2], [[4, 1], [5, 1], [14, 3]]), ([4, 1, 3], [[4, 1], [5, 1], [19, 4]]), ([4, 1, 4], [[4, 1], [5, 1], [24, 5]])]]
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[[1, 1], [3, 1]][[1, 1], [3, 2]]Failed
explicit oracle 1[[1, 1], [2, 1]][[1, 1], [2, 1]]Passed
explicit oracle 2[[1, 1]][[1, 1]]Passed
explicit oracle 3[[-2, 1], [1, 1], [5, 1]][[-2, 1], [-5, 3], [-22, 13]]Failed
explicit oracle 4[[2, 1]][[2, 1]]Passed
explicit oracle 5[[3, 1]][[3, 1]]Passed
explicit oracle 6[[4, 1]][[4, 1]]Passed
explicit oracle 7[[1, 1], [4, 1]][[1, 1], [4, 3]]Failed

SHA-256 / 7c572a210cb98dc33a0c78d3aba9ee3d01e1b0f34245438a37b9ea1a7f3a2e36

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=x
    p0,p1=0,1
    q0,q1=1,0
    out=[]
    for c in a:
     p=c*p1+p0
     q=c*q1+q0
     out.append([p,q])
     p0,p1,q0,q1=p,p,q,q
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 2], [[1, 1], [3, 2]]), ([1, 1], [[1, 1], [2, 1]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([2], [[2, 1]]), ([3], [[3, 1]]), ([4], [[4, 1]]), ([1, 3], [[1, 1], [4, 3]])], [([1, 3], [[1, 1], [4, 3]]), ([1, 4], [[1, 1], [5, 4]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([4, 2], [[4, 1], [9, 2]]), ([4, 3], [[4, 1], [13, 3]]), ([4, 4], [[4, 1], [17, 4]]), ([1, 1, 1], [[1, 1], [2, 1], [3, 2]])], [([1, 4], [[1, 1], [5, 4]]), ([2, 3], [[2, 1], [7, 3]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([1, 4, 3], [[1, 1], [5, 4], [16, 13]]), ([1, 4, 4], [[1, 1], [5, 4], [21, 17]]), ([2, 1, 1], [[2, 1], [3, 1], [5, 2]]), ([2, 1, 2], [[2, 1], [3, 1], [8, 3]])], [([2, 2], [[2, 1], [5, 2]]), ([3, 2], [[3, 1], [7, 2]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([2, 4, 4], [[2, 1], [9, 4], [38, 17]]), ([3, 1, 1], [[3, 1], [4, 1], [7, 2]]), ([3, 1, 2], [[3, 1], [4, 1], [11, 3]]), ([3, 1, 3], [[3, 1], [4, 1], [15, 4]])], [([2, 3], [[2, 1], [7, 3]]), ([4, 1], [[4, 1], [5, 1]]), ([1], [[1, 1]]), ([-2, 3, 4], [[-2, 1], [-5, 3], [-22, 13]]), ([4, 1, 1], [[4, 1], [5, 1], [9, 2]]), ([4, 1, 2], [[4, 1], [5, 1], [14, 3]]), ([4, 1, 3], [[4, 1], [5, 1], [19, 4]]), ([4, 1, 4], [[4, 1], [5, 1], [24, 5]])]]
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[[1, 1], [3, 3]][[1, 1], [3, 2]]Failed
explicit oracle 1[[1, 1], [2, 2]][[1, 1], [2, 1]]Failed
explicit oracle 2[[1, 1]][[1, 1]]Passed
explicit oracle 3[[-2, 1], [-8, 4], [-40, 20]][[-2, 1], [-5, 3], [-22, 13]]Failed
explicit oracle 4[[2, 1]][[2, 1]]Passed
explicit oracle 5[[3, 1]][[3, 1]]Passed
explicit oracle 6[[4, 1]][[4, 1]]Passed
explicit oracle 7[[1, 1], [4, 4]][[1, 1], [4, 3]]Failed

SHA-256 / 136a6db4f4a15ac990f673c3e958c31871106e4f268422dd754bc8caf76efc1a

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 2026-09-29T14:39:17.139135+00:00.

Case digest / 33742baae5dd17b9e6639b84dcedb17d6c38766a0e2a2cde7fcc00137ad62408