FAILURE MAP
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FA-88406 / Mesh topology invariants / Open access

Boundary loops are ordered by first vertex only · case 01

The outer boundary is no longer listed first when a hole has a smaller vertex id.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The sort key ignores loop length.

THE FAILURE

The sort key ignores loop length.

Unsuccessful approach: Sorting by ascending length lists the smallest hole first.

Case contract

Input: consistently oriented polygon faces. A half-edge (a,b) is boundary when (b,a) is not a half-edge. If any vertex starts two boundary half-edges return "ambiguous". Otherwise walk loops along the half-edge direction, each starting at its smallest vertex, and return them sorted by descending length then first vertex.

Why this case matters

Mesh processing pipelines (remeshing, export, simulation, printing) trust these topological counts and adjacency answers; a wrong invariant silently accepts broken meshes or rejects valid ones.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    faces=x
    he=[]
    for f in faces:
        n=len(f)
        for i in range(n):
            he.append((f[i],f[(i+1)%n]))
    hs=set(he)
    nxt={}
    for a,b in he:
        if (b,a) not in hs:
            if a in nxt: return 'ambiguous'
            nxt[a]=b
    loops=[]
    seen=set()
    for s in sorted(nxt):
        if s in seen: continue
        loop=[]
        v=s
        while v not in seen:
            seen.add(v)
            loop.append(v)
            v=nxt[v]
        loops.append(loop)
    return sorted(loops,key=lambda l:l[0])
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['disk', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], [[1, 2, 3, 4, 5]]], ['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], [[0, 1, 2, 3, 4]]], ['annulus', [[[0, 3, 4, 1], [1, 4, 5, 2], [2, 5, 3, 0]]], [[0, 2, 1], [3, 4, 5]]], ['quad grid', [[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]]], [[0, 1, 2, 5, 8, 7, 6, 3]]], ['square with hole', [[[0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]]], [[0, 1, 2, 3], [4, 7, 6, 5]]], ['closed tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], []], ['disk with small far island', [[[5, 6, 7], [5, 7, 8], [5, 8, 9], [5, 9, 10], [5, 10, 6], [0, 1, 2]]], [[6, 7, 8, 9, 10], [0, 1, 2]]]], [['quad grid', [[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]]], [[0, 1, 2, 5, 8, 7, 6, 3]]], ['square with hole', [[[0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]]], [[0, 1, 2, 3], [4, 7, 6, 5]]], ['closed tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], []], ['bowtie', [[[0, 1, 2], [0, 3, 4]]], 'ambiguous'], ['two disks', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1], [10, 11, 12], [10, 12, 13], [10, 13, 14]]], [[1, 2, 3, 4, 5], [10, 11, 12, 13, 14]]], ['open cylinder', [[[0, 1, 5], [0, 5, 4], [1, 2, 6], [1, 6, 5], [2, 3, 7], [2, 7, 6], [3, 0, 4], [3, 4, 7]]], [[0, 1, 2, 3], [4, 7, 6, 5]]], ['disk with small far island', [[[5, 6, 7], [5, 7, 8], [5, 8, 9], [5, 9, 10], [5, 10, 6], [0, 1, 2]]], [[6, 7, 8, 9, 10], [0, 1, 2]]]], [['bowtie', [[[0, 1, 2], [0, 3, 4]]], 'ambiguous'], ['two disks', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1], [10, 11, 12], [10, 12, 13], [10, 13, 14]]], [[1, 2, 3, 4, 5], [10, 11, 12, 13, 14]]], ['open cylinder', [[[0, 1, 5], [0, 5, 4], [1, 2, 6], [1, 6, 5], [2, 3, 7], [2, 7, 6], [3, 0, 4], [3, 4, 7]]], [[0, 1, 2, 3], [4, 7, 6, 5]]], ['single triangle', [[[4, 2, 7]]], [[2, 7, 4]]], ['single quad', [[[3, 1, 0, 2]]], [[0, 2, 3, 1]]], ['triangle grid', [[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]]], [[0, 1, 2, 5, 4, 3]]], ['disk with small far island', [[[5, 6, 7], [5, 7, 8], [5, 8, 9], [5, 9, 10], [5, 10, 6], [0, 1, 2]]], [[6, 7, 8, 9, 10], [0, 1, 2]]]], [['disk', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], [[1, 2, 3, 4, 5]]], ['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], [[0, 1, 2, 3, 4]]], ['annulus', [[[0, 3, 4, 1], [1, 4, 5, 2], [2, 5, 3, 0]]], [[0, 2, 1], [3, 4, 5]]], ['single triangle', [[[4, 2, 7]]], [[2, 7, 4]]], ['single quad', [[[3, 1, 0, 2]]], [[0, 2, 3, 1]]], ['triangle grid', [[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]]], [[0, 1, 2, 5, 4, 3]]], ['disk with small far island', [[[5, 6, 7], [5, 7, 8], [5, 8, 9], [5, 9, 10], [5, 10, 6], [0, 1, 2]]], [[6, 7, 8, 9, 10], [0, 1, 2]]]], [['disk', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], [[1, 2, 3, 4, 5]]], ['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], [[0, 1, 2, 3, 4]]], ['annulus', [[[0, 3, 4, 1], [1, 4, 5, 2], [2, 5, 3, 0]]], [[0, 2, 1], [3, 4, 5]]], ['quad grid', [[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]]], [[0, 1, 2, 5, 8, 7, 6, 3]]], ['square with hole', [[[0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]]], [[0, 1, 2, 3], [4, 7, 6, 5]]], ['closed tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], []], ['disk with small far island', [[[5, 6, 7], [5, 7, 8], [5, 8, 9], [5, 9, 10], [5, 10, 6], [0, 1, 2]]], [[6, 7, 8, 9, 10], [0, 1, 2]]]]]
for label, args, expected in fixtures[N-1]:
    check(label, 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
disk[[1, 2, 3, 4, 5]][[1, 2, 3, 4, 5]]Passed
fan[[0, 1, 2, 3, 4]][[0, 1, 2, 3, 4]]Passed
annulus[[0, 2, 1], [3, 4, 5]][[0, 2, 1], [3, 4, 5]]Passed
quad grid[[0, 1, 2, 5, 8, 7, 6, 3]][[0, 1, 2, 5, 8, 7, 6, 3]]Passed
square with hole[[0, 1, 2, 3], [4, 7, 6, 5]][[0, 1, 2, 3], [4, 7, 6, 5]]Passed
closed tetrahedron[][]Passed
disk with small far island[[0, 1, 2], [6, 7, 8, 9, 10]][[6, 7, 8, 9, 10], [0, 1, 2]]Failed

SHA-256 / e4f8fe5e9033825252b8917efb0de28343e6ed8e062457ef520109edf483cb62

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    faces=x
    he=[]
    for f in faces:
        n=len(f)
        for i in range(n):
            he.append((f[i],f[(i+1)%n]))
    hs=set(he)
    nxt={}
    for a,b in he:
        if (b,a) not in hs:
            if a in nxt: return 'ambiguous'
            nxt[a]=b
    loops=[]
    seen=set()
    for s in sorted(nxt):
        if s in seen: continue
        loop=[]
        v=s
        while v not in seen:
            seen.add(v)
            loop.append(v)
            v=nxt[v]
        loops.append(loop)
    return sorted(loops,key=lambda l:(len(l),l[0]))
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['disk', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], [[1, 2, 3, 4, 5]]], ['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], [[0, 1, 2, 3, 4]]], ['annulus', [[[0, 3, 4, 1], [1, 4, 5, 2], [2, 5, 3, 0]]], [[0, 2, 1], [3, 4, 5]]], ['quad grid', [[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]]], [[0, 1, 2, 5, 8, 7, 6, 3]]], ['square with hole', [[[0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]]], [[0, 1, 2, 3], [4, 7, 6, 5]]], ['closed tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], []], ['disk with small far island', [[[5, 6, 7], [5, 7, 8], [5, 8, 9], [5, 9, 10], [5, 10, 6], [0, 1, 2]]], [[6, 7, 8, 9, 10], [0, 1, 2]]]], [['quad grid', [[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]]], [[0, 1, 2, 5, 8, 7, 6, 3]]], ['square with hole', [[[0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]]], [[0, 1, 2, 3], [4, 7, 6, 5]]], ['closed tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], []], ['bowtie', [[[0, 1, 2], [0, 3, 4]]], 'ambiguous'], ['two disks', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1], [10, 11, 12], [10, 12, 13], [10, 13, 14]]], [[1, 2, 3, 4, 5], [10, 11, 12, 13, 14]]], ['open cylinder', [[[0, 1, 5], [0, 5, 4], [1, 2, 6], [1, 6, 5], [2, 3, 7], [2, 7, 6], [3, 0, 4], [3, 4, 7]]], [[0, 1, 2, 3], [4, 7, 6, 5]]], ['disk with small far island', [[[5, 6, 7], [5, 7, 8], [5, 8, 9], [5, 9, 10], [5, 10, 6], [0, 1, 2]]], [[6, 7, 8, 9, 10], [0, 1, 2]]]], [['bowtie', [[[0, 1, 2], [0, 3, 4]]], 'ambiguous'], ['two disks', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1], [10, 11, 12], [10, 12, 13], [10, 13, 14]]], [[1, 2, 3, 4, 5], [10, 11, 12, 13, 14]]], ['open cylinder', [[[0, 1, 5], [0, 5, 4], [1, 2, 6], [1, 6, 5], [2, 3, 7], [2, 7, 6], [3, 0, 4], [3, 4, 7]]], [[0, 1, 2, 3], [4, 7, 6, 5]]], ['single triangle', [[[4, 2, 7]]], [[2, 7, 4]]], ['single quad', [[[3, 1, 0, 2]]], [[0, 2, 3, 1]]], ['triangle grid', [[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]]], [[0, 1, 2, 5, 4, 3]]], ['disk with small far island', [[[5, 6, 7], [5, 7, 8], [5, 8, 9], [5, 9, 10], [5, 10, 6], [0, 1, 2]]], [[6, 7, 8, 9, 10], [0, 1, 2]]]], [['disk', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], [[1, 2, 3, 4, 5]]], ['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], [[0, 1, 2, 3, 4]]], ['annulus', [[[0, 3, 4, 1], [1, 4, 5, 2], [2, 5, 3, 0]]], [[0, 2, 1], [3, 4, 5]]], ['single triangle', [[[4, 2, 7]]], [[2, 7, 4]]], ['single quad', [[[3, 1, 0, 2]]], [[0, 2, 3, 1]]], ['triangle grid', [[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]]], [[0, 1, 2, 5, 4, 3]]], ['disk with small far island', [[[5, 6, 7], [5, 7, 8], [5, 8, 9], [5, 9, 10], [5, 10, 6], [0, 1, 2]]], [[6, 7, 8, 9, 10], [0, 1, 2]]]], [['disk', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], [[1, 2, 3, 4, 5]]], ['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], [[0, 1, 2, 3, 4]]], ['annulus', [[[0, 3, 4, 1], [1, 4, 5, 2], [2, 5, 3, 0]]], [[0, 2, 1], [3, 4, 5]]], ['quad grid', [[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]]], [[0, 1, 2, 5, 8, 7, 6, 3]]], ['square with hole', [[[0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]]], [[0, 1, 2, 3], [4, 7, 6, 5]]], ['closed tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], []], ['disk with small far island', [[[5, 6, 7], [5, 7, 8], [5, 8, 9], [5, 9, 10], [5, 10, 6], [0, 1, 2]]], [[6, 7, 8, 9, 10], [0, 1, 2]]]]]
for label, args, expected in fixtures[N-1]:
    check(label, 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
disk[[1, 2, 3, 4, 5]][[1, 2, 3, 4, 5]]Passed
fan[[0, 1, 2, 3, 4]][[0, 1, 2, 3, 4]]Passed
annulus[[0, 2, 1], [3, 4, 5]][[0, 2, 1], [3, 4, 5]]Passed
quad grid[[0, 1, 2, 5, 8, 7, 6, 3]][[0, 1, 2, 5, 8, 7, 6, 3]]Passed
square with hole[[0, 1, 2, 3], [4, 7, 6, 5]][[0, 1, 2, 3], [4, 7, 6, 5]]Passed
closed tetrahedron[][]Passed
disk with small far island[[0, 1, 2], [6, 7, 8, 9, 10]][[6, 7, 8, 9, 10], [0, 1, 2]]Failed

SHA-256 / 1cfb103baba997f88e8ff9991ebcb4c551251bf5b71d7ebf21985ec4fd95739a

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 7 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.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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Verification & scope

Pure combinatorial teaching model over integer face lists with a stipulated contract; no geometry kernel or file format is implied. 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 2026-09-29T14:51:07.775039+00:00.

Case digest / 898e11426831b136a0bb47d4ef5b198d6c5fd70e4ba7f14ad30c89a9fe016d02