FAILURE MAP
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FA-88426 / Mesh topology invariants / Open access

Vertex is called interior when any link vertex closes the fan · case 01

Boundary vertices with more than one incident triangle are reported interior.

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

ROOT CAUSE

The interior test uses any(degree==2) instead of all.

THE FAILURE

The interior test uses any(degree==2) instead of all.

Unsuccessful approach: Counting at least three incident triangles also labels wide boundary fans interior.

Case contract

Input: triangle faces. For each referenced vertex v, its link is the set of opposite edges of incident triangles. v is "nonmanifold" if a link vertex has degree above 2 or the link graph is disconnected, "interior" if every link vertex has degree 2, else "boundary". Return {vertex string: class}.

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
    link={}
    for f in faces:
        for i in range(3):
            v=f[i]
            a=f[(i+1)%3]
            b=f[(i+2)%3]
            link.setdefault(v,[]).append((a,b))
    out={}
    for v in sorted(link):
        es=link[v]
        deg={}
        for a,b in es:
            deg[a]=deg.get(a,0)+1
            deg[b]=deg.get(b,0)+1
        if any(d>2 for d in deg.values()):
            out[str(v)]='nonmanifold'
            continue
        adj={}
        for a,b in es:
            adj.setdefault(a,[]).append(b)
            adj.setdefault(b,[]).append(a)
        start=next(iter(adj))
        stack=[start]
        seen={start}
        while stack:
            u=stack.pop()
            for w in adj.get(u,[]):
                if w not in seen:
                    seen.add(w)
                    stack.append(w)
        if len(seen)!=len(adj): out[str(v)]='nonmanifold'
        elif any(d==2 for d in deg.values()): out[str(v)]='interior'
        else: out[str(v)]='boundary'
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior'}], ['disk', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], {'0': 'interior', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary'}], ['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['bowtie', [[[0, 1, 2], [0, 3, 4]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['book', [[[0, 1, 2], [1, 0, 3], [0, 1, 4]]], {'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['octahedron', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior'}], ['torus', [[[0, 3, 4], [0, 4, 1], [1, 4, 5], [1, 5, 2], [2, 5, 3], [2, 3, 0], [3, 6, 7], [3, 7, 4], [4, 7, 8], [4, 8, 5], [5, 8, 6], [5, 6, 3], [6, 0, 1], [6, 1, 7], [7, 1, 2], [7, 2, 8], [8, 2, 0], [8, 0, 6]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}]], [['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['book', [[[0, 1, 2], [1, 0, 3], [0, 1, 4]]], {'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['octahedron', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior'}], ['torus', [[[0, 3, 4], [0, 4, 1], [1, 4, 5], [1, 5, 2], [2, 5, 3], [2, 3, 0], [3, 6, 7], [3, 7, 4], [4, 7, 8], [4, 8, 5], [5, 8, 6], [5, 6, 3], [6, 0, 1], [6, 1, 7], [7, 1, 2], [7, 2, 8], [8, 2, 0], [8, 0, 6]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}], ['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': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary'}], ['triangle grid', [[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'interior', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}], ['double disk pinch', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1], [0, 6, 7], [0, 7, 8], [0, 8, 6]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}]], [['tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior'}], ['torus', [[[0, 3, 4], [0, 4, 1], [1, 4, 5], [1, 5, 2], [2, 5, 3], [2, 3, 0], [3, 6, 7], [3, 7, 4], [4, 7, 8], [4, 8, 5], [5, 8, 6], [5, 6, 3], [6, 0, 1], [6, 1, 7], [7, 1, 2], [7, 2, 8], [8, 2, 0], [8, 0, 6]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}], ['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': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary'}], ['triangle grid', [[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'interior', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}], ['double disk pinch', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1], [0, 6, 7], [0, 7, 8], [0, 8, 6]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}], ['single triangle', [[[0, 1, 2]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary'}], ['two triangles sharing edge', [[[0, 1, 2], [2, 1, 3]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary'}]], [['tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior'}], ['disk', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], {'0': 'interior', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary'}], ['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['bowtie', [[[0, 1, 2], [0, 3, 4]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['book', [[[0, 1, 2], [1, 0, 3], [0, 1, 4]]], {'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['triangle grid', [[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'interior', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}], ['two triangles sharing edge', [[[0, 1, 2], [2, 1, 3]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary'}]], [['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['bowtie', [[[0, 1, 2], [0, 3, 4]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['book', [[[0, 1, 2], [1, 0, 3], [0, 1, 4]]], {'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['octahedron', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior'}], ['torus', [[[0, 3, 4], [0, 4, 1], [1, 4, 5], [1, 5, 2], [2, 5, 3], [2, 3, 0], [3, 6, 7], [3, 7, 4], [4, 7, 8], [4, 8, 5], [5, 8, 6], [5, 6, 3], [6, 0, 1], [6, 1, 7], [7, 1, 2], [7, 2, 8], [8, 2, 0], [8, 0, 6]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}], ['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': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary'}], ['double disk pinch', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1], [0, 6, 7], [0, 7, 8], [0, 8, 6]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}]]]
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
tetrahedron{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior'}{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior'}Passed
disk{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior'}{'0': 'interior', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary'}Failed
fan{'0': 'interior', '1': 'boundary', '2': 'interior', '3': 'interior', '4': 'boundary'}{'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}Failed
bowtie{'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}{'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}Passed
book{'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}{'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}Passed
octahedron{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior'}{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior'}Passed
torus{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}Passed

SHA-256 / aae0595f907b4c8243f9c60d2c69e5dc8a2164d81348af6ce4a34a9a3b8ff9f3

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
    link={}
    for f in faces:
        for i in range(3):
            v=f[i]
            a=f[(i+1)%3]
            b=f[(i+2)%3]
            link.setdefault(v,[]).append((a,b))
    out={}
    for v in sorted(link):
        es=link[v]
        deg={}
        for a,b in es:
            deg[a]=deg.get(a,0)+1
            deg[b]=deg.get(b,0)+1
        if any(d>2 for d in deg.values()):
            out[str(v)]='nonmanifold'
            continue
        adj={}
        for a,b in es:
            adj.setdefault(a,[]).append(b)
            adj.setdefault(b,[]).append(a)
        start=next(iter(adj))
        stack=[start]
        seen={start}
        while stack:
            u=stack.pop()
            for w in adj.get(u,[]):
                if w not in seen:
                    seen.add(w)
                    stack.append(w)
        if len(seen)!=len(adj): out[str(v)]='nonmanifold'
        elif len(es)>=3: out[str(v)]='interior'
        else: out[str(v)]='boundary'
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior'}], ['disk', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], {'0': 'interior', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary'}], ['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['bowtie', [[[0, 1, 2], [0, 3, 4]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['book', [[[0, 1, 2], [1, 0, 3], [0, 1, 4]]], {'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['octahedron', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior'}], ['torus', [[[0, 3, 4], [0, 4, 1], [1, 4, 5], [1, 5, 2], [2, 5, 3], [2, 3, 0], [3, 6, 7], [3, 7, 4], [4, 7, 8], [4, 8, 5], [5, 8, 6], [5, 6, 3], [6, 0, 1], [6, 1, 7], [7, 1, 2], [7, 2, 8], [8, 2, 0], [8, 0, 6]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}]], [['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['book', [[[0, 1, 2], [1, 0, 3], [0, 1, 4]]], {'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['octahedron', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior'}], ['torus', [[[0, 3, 4], [0, 4, 1], [1, 4, 5], [1, 5, 2], [2, 5, 3], [2, 3, 0], [3, 6, 7], [3, 7, 4], [4, 7, 8], [4, 8, 5], [5, 8, 6], [5, 6, 3], [6, 0, 1], [6, 1, 7], [7, 1, 2], [7, 2, 8], [8, 2, 0], [8, 0, 6]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}], ['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': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary'}], ['triangle grid', [[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'interior', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}], ['double disk pinch', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1], [0, 6, 7], [0, 7, 8], [0, 8, 6]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}]], [['tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior'}], ['torus', [[[0, 3, 4], [0, 4, 1], [1, 4, 5], [1, 5, 2], [2, 5, 3], [2, 3, 0], [3, 6, 7], [3, 7, 4], [4, 7, 8], [4, 8, 5], [5, 8, 6], [5, 6, 3], [6, 0, 1], [6, 1, 7], [7, 1, 2], [7, 2, 8], [8, 2, 0], [8, 0, 6]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}], ['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': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary'}], ['triangle grid', [[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'interior', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}], ['double disk pinch', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1], [0, 6, 7], [0, 7, 8], [0, 8, 6]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}], ['single triangle', [[[0, 1, 2]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary'}], ['two triangles sharing edge', [[[0, 1, 2], [2, 1, 3]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary'}]], [['tetrahedron', [[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior'}], ['disk', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], {'0': 'interior', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary'}], ['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['bowtie', [[[0, 1, 2], [0, 3, 4]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['book', [[[0, 1, 2], [1, 0, 3], [0, 1, 4]]], {'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['triangle grid', [[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'interior', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}], ['two triangles sharing edge', [[[0, 1, 2], [2, 1, 3]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary'}]], [['fan', [[[0, 1, 2], [0, 2, 3], [0, 3, 4]]], {'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['bowtie', [[[0, 1, 2], [0, 3, 4]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['book', [[[0, 1, 2], [1, 0, 3], [0, 1, 4]]], {'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}], ['octahedron', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior'}], ['torus', [[[0, 3, 4], [0, 4, 1], [1, 4, 5], [1, 5, 2], [2, 5, 3], [2, 3, 0], [3, 6, 7], [3, 7, 4], [4, 7, 8], [4, 8, 5], [5, 8, 6], [5, 6, 3], [6, 0, 1], [6, 1, 7], [7, 1, 2], [7, 2, 8], [8, 2, 0], [8, 0, 6]]], {'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}], ['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': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary'}], ['double disk pinch', [[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1], [0, 6, 7], [0, 7, 8], [0, 8, 6]]], {'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary', '6': 'boundary', '7': 'boundary', '8': 'boundary'}]]]
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
tetrahedron{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior'}{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior'}Passed
disk{'0': 'interior', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary'}{'0': 'interior', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary', '5': 'boundary'}Passed
fan{'0': 'interior', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}{'0': 'boundary', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}Failed
bowtie{'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}{'0': 'nonmanifold', '1': 'boundary', '2': 'boundary', '3': 'boundary', '4': 'boundary'}Passed
book{'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}{'0': 'nonmanifold', '1': 'nonmanifold', '2': 'boundary', '3': 'boundary', '4': 'boundary'}Passed
octahedron{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior'}{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior'}Passed
torus{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}{'0': 'interior', '1': 'interior', '2': 'interior', '3': 'interior', '4': 'interior', '5': 'interior', '6': 'interior', '7': 'interior', '8': 'interior'}Passed

SHA-256 / 74544a4fae395d59ede27c3edf843f86ec16db041a74434f3b693d0770330bbe

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.

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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 2026-09-29T14:51:08.128897+00:00.

Case digest / 4b68c6df87019548730ba9584d5e9a88c31545e24af47140a802d4241293786e