FA-88651 / Mesh topology invariants / Open access

Edge flip swaps which face slot receives which new triangle · case 01

Face indices referenced elsewhere now point at the other triangle.

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

ROOT CAUSE

The first triangle slot receives the second new triangle and vice versa.

VERIFIED REPAIR

Replace t1 in place with [c,a,d] and t2 with [d,b,c].

Unsuccessful approach: Deleting both faces and appending new ones reorders the face list.

Case contract

Input [faces, a, b] for oriented triangles. The flip needs a triangle holding half-edge a->b (apex c) and one holding b->a (apex d), else "boundary"; exactly two faces may contain both a and b, else "non-manifold"; c==d is "degenerate"; an existing undirected edge c-d is "edge-exists". Otherwise replace the first triangle in place by [c,a,d] and the second by [d,b,c] and return the face list.

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,a,b=x
    fs=[list(f) for f in faces]
    t1=t2=None
    for i,f in enumerate(fs):
        for j in range(3):
            if f[j]==a and f[(j+1)%3]==b: t1=(i,f[(j+2)%3])
            if f[j]==b and f[(j+1)%3]==a: t2=(i,f[(j+2)%3])
    if t1 is None or t2 is None: return 'boundary'
    cnt=sum(1 for f in fs if a in f and b in f)
    if cnt!=2: return 'non-manifold'
    c,d=t1[1],t2[1]
    if c==d: return 'degenerate'
    for f in fs:
        for j in range(3):
            if {f[j],f[(j+1)%3]}=={c,d}: return 'edge-exists'
    fs[t1[0]]=[d,b,c]
    fs[t2[0]]=[c,a,d]
    return fs
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['square diagonal', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [[0, 1, 3], [3, 2, 0]]], ['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['rim edge', [[[[0, 1, 2], [2, 1, 3]], 0, 1]], 'boundary'], ['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], 'edge-exists'], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['book spine', [[[[0, 1, 2], [1, 0, 3], [0, 1, 4], [1, 0, 5]], 0, 1]], 'non-manifold']], [['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['book spine', [[[[0, 1, 2], [1, 0, 3], [0, 1, 4], [1, 0, 5]], 0, 1]], 'non-manifold'], ['folded pair', [[[[0, 1, 2], [1, 0, 2]], 0, 1]], 'degenerate'], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['grid interior edge', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 1, 4]], [[0, 1, 5], [0, 4, 3], [1, 2, 5], [5, 4, 0]]]], [['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['grid interior edge', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 1, 4]], [[0, 1, 5], [0, 4, 3], [1, 2, 5], [5, 4, 0]]], ['fan spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0, 3]], [[0, 1, 2], [2, 3, 4], [4, 0, 2]]], ['rotated storage', [[[[2, 0, 1], [3, 1, 0], [2, 3, 4]], 0, 1]], 'edge-exists'], ['diagonal closing edge', [[[[0, 1, 2], [1, 0, 3], [3, 4, 2]], 0, 1]], 'edge-exists']], [['square diagonal', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [[0, 1, 3], [3, 2, 0]]], ['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['rotated storage', [[[[2, 0, 1], [3, 1, 0], [2, 3, 4]], 0, 1]], 'edge-exists'], ['diagonal closing edge', [[[[0, 1, 2], [1, 0, 3], [3, 4, 2]], 0, 1]], 'edge-exists'], ['diagonal stored reversed', [[[[0, 1, 2], [1, 0, 3], [3, 2, 4]], 0, 1]], 'edge-exists']], [['square diagonal', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [[0, 1, 3], [3, 2, 0]]], ['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['rim edge', [[[[0, 1, 2], [2, 1, 3]], 0, 1]], 'boundary'], ['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], 'edge-exists'], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['grid interior edge', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 1, 4]], [[0, 1, 5], [0, 4, 3], [1, 2, 5], [5, 4, 0]]]]]
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 fixture	Actual	Expected	Outcome
square diagonal	`[[3, 2, 0], [0, 1, 3]]`	`[[0, 1, 3], [3, 2, 0]]`	Failed
square diagonal reversed	`[[3, 2, 0], [0, 1, 3]]`	`[[0, 1, 3], [3, 2, 0]]`	Failed
rim edge	`boundary`	`boundary`	Passed
disk spoke	`[[3, 0, 1], [1, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]]`	`[[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]`	Failed
tetrahedron edge	`edge-exists`	`edge-exists`	Passed
octahedron edge	`[[4, 1, 2], [0, 2, 3], [0, 3, 4], [2, 0, 4], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]`	`[[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]`	Failed
book spine	`non-manifold`	`non-manifold`	Passed

SHA-256 / 3def6376dfaf5b174c4103113fbcf853f34013212e7f99f3ff0ff06cc4cf4574

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,a,b=x
    fs=[list(f) for f in faces]
    t1=t2=None
    for i,f in enumerate(fs):
        for j in range(3):
            if f[j]==a and f[(j+1)%3]==b: t1=(i,f[(j+2)%3])
            if f[j]==b and f[(j+1)%3]==a: t2=(i,f[(j+2)%3])
    if t1 is None or t2 is None: return 'boundary'
    cnt=sum(1 for f in fs if a in f and b in f)
    if cnt!=2: return 'non-manifold'
    c,d=t1[1],t2[1]
    if c==d: return 'degenerate'
    for f in fs:
        for j in range(3):
            if {f[j],f[(j+1)%3]}=={c,d}: return 'edge-exists'
    fs=[f for k,f in enumerate(fs) if k not in (t1[0],t2[0])]+[[c,a,d],[d,b,c]]
    return fs
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['square diagonal', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [[0, 1, 3], [3, 2, 0]]], ['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['rim edge', [[[[0, 1, 2], [2, 1, 3]], 0, 1]], 'boundary'], ['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], 'edge-exists'], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['book spine', [[[[0, 1, 2], [1, 0, 3], [0, 1, 4], [1, 0, 5]], 0, 1]], 'non-manifold']], [['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['book spine', [[[[0, 1, 2], [1, 0, 3], [0, 1, 4], [1, 0, 5]], 0, 1]], 'non-manifold'], ['folded pair', [[[[0, 1, 2], [1, 0, 2]], 0, 1]], 'degenerate'], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['grid interior edge', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 1, 4]], [[0, 1, 5], [0, 4, 3], [1, 2, 5], [5, 4, 0]]]], [['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['grid interior edge', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 1, 4]], [[0, 1, 5], [0, 4, 3], [1, 2, 5], [5, 4, 0]]], ['fan spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0, 3]], [[0, 1, 2], [2, 3, 4], [4, 0, 2]]], ['rotated storage', [[[[2, 0, 1], [3, 1, 0], [2, 3, 4]], 0, 1]], 'edge-exists'], ['diagonal closing edge', [[[[0, 1, 2], [1, 0, 3], [3, 4, 2]], 0, 1]], 'edge-exists']], [['square diagonal', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [[0, 1, 3], [3, 2, 0]]], ['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['rotated storage', [[[[2, 0, 1], [3, 1, 0], [2, 3, 4]], 0, 1]], 'edge-exists'], ['diagonal closing edge', [[[[0, 1, 2], [1, 0, 3], [3, 4, 2]], 0, 1]], 'edge-exists'], ['diagonal stored reversed', [[[[0, 1, 2], [1, 0, 3], [3, 2, 4]], 0, 1]], 'edge-exists']], [['square diagonal', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [[0, 1, 3], [3, 2, 0]]], ['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['rim edge', [[[[0, 1, 2], [2, 1, 3]], 0, 1]], 'boundary'], ['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], 'edge-exists'], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['grid interior edge', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 1, 4]], [[0, 1, 5], [0, 4, 3], [1, 2, 5], [5, 4, 0]]]]]
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 fixture	Actual	Expected	Outcome
square diagonal	`[[0, 1, 3], [3, 2, 0]]`	`[[0, 1, 3], [3, 2, 0]]`	Passed
square diagonal reversed	`[[3, 2, 0], [0, 1, 3]]`	`[[0, 1, 3], [3, 2, 0]]`	Failed
rim edge	`boundary`	`boundary`	Passed
disk spoke	`[[0, 3, 4], [0, 4, 5], [0, 5, 1], [3, 0, 1], [1, 2, 3]]`	`[[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]`	Failed
tetrahedron edge	`edge-exists`	`edge-exists`	Passed
octahedron edge	`[[0, 2, 3], [0, 3, 4], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4], [2, 0, 4], [4, 1, 2]]`	`[[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]`	Failed
book spine	`non-manifold`	`non-manifold`	Passed

SHA-256 / d5c168a5b98f14f8165cfc9f69d4aca5a9dd4b8b41616e16ea5f797191fc18e8

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    faces,a,b=x
    fs=[list(f) for f in faces]
    t1=t2=None
    for i,f in enumerate(fs):
        for j in range(3):
            if f[j]==a and f[(j+1)%3]==b: t1=(i,f[(j+2)%3])
            if f[j]==b and f[(j+1)%3]==a: t2=(i,f[(j+2)%3])
    if t1 is None or t2 is None: return 'boundary'
    cnt=sum(1 for f in fs if a in f and b in f)
    if cnt!=2: return 'non-manifold'
    c,d=t1[1],t2[1]
    if c==d: return 'degenerate'
    for f in fs:
        for j in range(3):
            if {f[j],f[(j+1)%3]}=={c,d}: return 'edge-exists'
    fs[t1[0]]=[c,a,d]
    fs[t2[0]]=[d,b,c]
    return fs
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['square diagonal', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [[0, 1, 3], [3, 2, 0]]], ['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['rim edge', [[[[0, 1, 2], [2, 1, 3]], 0, 1]], 'boundary'], ['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], 'edge-exists'], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['book spine', [[[[0, 1, 2], [1, 0, 3], [0, 1, 4], [1, 0, 5]], 0, 1]], 'non-manifold']], [['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['book spine', [[[[0, 1, 2], [1, 0, 3], [0, 1, 4], [1, 0, 5]], 0, 1]], 'non-manifold'], ['folded pair', [[[[0, 1, 2], [1, 0, 2]], 0, 1]], 'degenerate'], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['grid interior edge', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 1, 4]], [[0, 1, 5], [0, 4, 3], [1, 2, 5], [5, 4, 0]]]], [['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['grid interior edge', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 1, 4]], [[0, 1, 5], [0, 4, 3], [1, 2, 5], [5, 4, 0]]], ['fan spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0, 3]], [[0, 1, 2], [2, 3, 4], [4, 0, 2]]], ['rotated storage', [[[[2, 0, 1], [3, 1, 0], [2, 3, 4]], 0, 1]], 'edge-exists'], ['diagonal closing edge', [[[[0, 1, 2], [1, 0, 3], [3, 4, 2]], 0, 1]], 'edge-exists']], [['square diagonal', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [[0, 1, 3], [3, 2, 0]]], ['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['octahedron edge', [[[[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, 1]], [[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['rotated storage', [[[[2, 0, 1], [3, 1, 0], [2, 3, 4]], 0, 1]], 'edge-exists'], ['diagonal closing edge', [[[[0, 1, 2], [1, 0, 3], [3, 4, 2]], 0, 1]], 'edge-exists'], ['diagonal stored reversed', [[[[0, 1, 2], [1, 0, 3], [3, 2, 4]], 0, 1]], 'edge-exists']], [['square diagonal', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [[0, 1, 3], [3, 2, 0]]], ['square diagonal reversed', [[[[0, 1, 2], [2, 1, 3]], 2, 1]], [[0, 1, 3], [3, 2, 0]]], ['rim edge', [[[[0, 1, 2], [2, 1, 3]], 0, 1]], 'boundary'], ['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 2]], [[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], 'edge-exists'], ['grid diagonal', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 0, 4]], [[1, 4, 3], [3, 0, 1], [1, 2, 5], [1, 5, 4]]], ['grid interior edge', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4]], 1, 4]], [[0, 1, 5], [0, 4, 3], [1, 2, 5], [5, 4, 0]]]]]
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 fixture	Actual	Expected	Outcome
square diagonal	`[[0, 1, 3], [3, 2, 0]]`	`[[0, 1, 3], [3, 2, 0]]`	Passed
square diagonal reversed	`[[0, 1, 3], [3, 2, 0]]`	`[[0, 1, 3], [3, 2, 0]]`	Passed
rim edge	`boundary`	`boundary`	Passed
disk spoke	`[[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]`	`[[1, 2, 3], [3, 0, 1], [0, 3, 4], [0, 4, 5], [0, 5, 1]]`	Passed
tetrahedron edge	`edge-exists`	`edge-exists`	Passed
octahedron edge	`[[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]`	`[[2, 0, 4], [0, 2, 3], [0, 3, 4], [4, 1, 2], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]]`	Passed
book spine	`non-manifold`	`non-manifold`	Passed

SHA-256 / 4bdca1d2aeea08d1e97c0ad023746860cfef4888ccdad55b3e49f582e08e94d4

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 2026-09-29T14:51:10.128779+00:00.

Case digest / 2bfbc7072d7eae4420fe9fa2f9d47d8039e550886af30f01ab65a7fc046d0dce