FA-88661 / Mesh topology invariants / Open access

One-ring is walked clockwise · case 01

Neighbour rings come out in reverse angular order.

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

ROOT CAUSE

The successor map is built from previous corner to next corner.

VERIFIED REPAIR

Map the next corner p to the previous corner q.

Unsuccessful approach: Reversing the finished ring changes its start vertex and still breaks the fan start rule.

Case contract

Input [faces, v] with consistently oriented polygons. Each face containing v maps its next corner p to its previous corner q (counter-clockwise order around v). A repeated p is "nonmanifold"; no incident face is "isolated". If exactly one p has no predecessor the ring is a boundary fan starting there; more than one is "nonmanifold"; otherwise start at the smallest neighbour. Walk successors; a walk that does not visit every neighbour is "nonmanifold". Return [ring, "boundary"|"interior"].

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,v=x
    succ={}
    for f in faces:
        n=len(f)
        for j in range(n):
            if f[j]==v:
                p,q=f[(j-1)%n],f[(j+1)%n]
                if p in succ: return 'nonmanifold'
                succ[p]=q
    if not succ: return 'isolated'
    starts=[p for p in succ if p not in succ.values()]
    if len(starts)>1: return 'nonmanifold'
    s=starts[0] if starts else min(succ)
    ring=[s]
    u=s
    while u in succ and succ[u]!=s:
        u=succ[u]
        ring.append(u)
    if len(ring)!=len(set(succ)|set(succ.values())): return 'nonmanifold'
    return [ring,'boundary' if starts else 'interior']
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['disk centre', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0]], [[1, 2, 3, 4, 5], 'interior']], ['disk rim vertex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 2]], [[3, 0, 1], 'boundary']], ['fan apex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0]], [[1, 2, 3, 4], 'boundary']], ['fan middle rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 3]], [[4, 0, 2], 'boundary']], ['octahedron top', [[[[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, 3, 4], 'interior']], ['octahedron equator', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 2]], [[0, 1, 5, 3], 'interior']], ['bowtie pinch', [[[[0, 1, 2], [0, 3, 4]], 0]], 'nonmanifold']], [['disk rim vertex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 2]], [[3, 0, 1], 'boundary']], ['octahedron top', [[[[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, 3, 4], 'interior']], ['octahedron equator', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 2]], [[0, 1, 5, 3], 'interior']], ['bowtie pinch', [[[[0, 1, 2], [0, 3, 4]], 0]], 'nonmanifold'], ['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'], ['quad grid centre', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 4]], [[1, 5, 7, 3], 'interior']], ['quad grid edge', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 1]], [[2, 4, 0], 'boundary']]], [['fan apex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0]], [[1, 2, 3, 4], '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'], ['quad grid centre', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 4]], [[1, 5, 7, 3], 'interior']], ['quad grid edge', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 1]], [[2, 4, 0], 'boundary']], ['isolated vertex', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 9]], 'isolated'], ['flipped neighbour', [[[[0, 1, 2], [0, 3, 2]], 0]], 'nonmanifold'], ['same spoke twice', [[[[0, 1, 2], [0, 1, 3]], 0]], 'nonmanifold']], [['fan middle rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 3]], [[4, 0, 2], 'boundary']], ['isolated vertex', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 9]], 'isolated'], ['flipped neighbour', [[[[0, 1, 2], [0, 3, 2]], 0]], 'nonmanifold'], ['same spoke twice', [[[[0, 1, 2], [0, 1, 3]], 0]], 'nonmanifold'], ['duplicated face', [[[[0, 1, 2], [0, 1, 2], [0, 2, 3]], 0]], 'nonmanifold'], ['cube corner', [[[[0, 3, 2, 1], [4, 5, 6, 7], [0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]], 6]], [[2, 7, 5], 'interior']], ['torus vertex', [[[[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]], 4]], [[0, 3, 7, 8, 5, 1], 'interior']]], [['disk centre', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0]], [[1, 2, 3, 4, 5], 'interior']], ['disk rim vertex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 2]], [[3, 0, 1], 'boundary']], ['fan apex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0]], [[1, 2, 3, 4], 'boundary']], ['octahedron top', [[[[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, 3, 4], 'interior']], ['duplicated face', [[[[0, 1, 2], [0, 1, 2], [0, 2, 3]], 0]], 'nonmanifold'], ['cube corner', [[[[0, 3, 2, 1], [4, 5, 6, 7], [0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]], 6]], [[2, 7, 5], 'interior']], ['torus vertex', [[[[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]], 4]], [[0, 3, 7, 8, 5, 1], 'interior']]]]
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
disk centre	`[[1, 5, 4, 3, 2], 'interior']`	`[[1, 2, 3, 4, 5], 'interior']`	Failed
disk rim vertex	`[[1, 0, 3], 'boundary']`	`[[3, 0, 1], 'boundary']`	Failed
fan apex	`[[4, 3, 2, 1], 'boundary']`	`[[1, 2, 3, 4], 'boundary']`	Failed
fan middle rim	`[[2, 0, 4], 'boundary']`	`[[4, 0, 2], 'boundary']`	Failed
octahedron top	`[[1, 4, 3, 2], 'interior']`	`[[1, 2, 3, 4], 'interior']`	Failed
octahedron equator	`[[0, 3, 5, 1], 'interior']`	`[[0, 1, 5, 3], 'interior']`	Failed
bowtie pinch	`nonmanifold`	`nonmanifold`	Passed

SHA-256 / 56a71173179f42c2f2b0c180e448b073394cc88da3c1e5e498006c31decde791

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,v=x
    succ={}
    for f in faces:
        n=len(f)
        for j in range(n):
            if f[j]==v:
                p,q=f[(j+1)%n],f[(j-1)%n]
                if p in succ: return 'nonmanifold'
                succ[p]=q
    if not succ: return 'isolated'
    starts=[p for p in succ if p not in succ.values()]
    if len(starts)>1: return 'nonmanifold'
    s=starts[0] if starts else min(succ)
    ring=[s]
    u=s
    while u in succ and succ[u]!=s:
        u=succ[u]
        ring.append(u)
    if len(ring)!=len(set(succ)|set(succ.values())): return 'nonmanifold'
    return [ring[::-1],'boundary' if starts else 'interior']
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['disk centre', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0]], [[1, 2, 3, 4, 5], 'interior']], ['disk rim vertex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 2]], [[3, 0, 1], 'boundary']], ['fan apex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0]], [[1, 2, 3, 4], 'boundary']], ['fan middle rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 3]], [[4, 0, 2], 'boundary']], ['octahedron top', [[[[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, 3, 4], 'interior']], ['octahedron equator', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 2]], [[0, 1, 5, 3], 'interior']], ['bowtie pinch', [[[[0, 1, 2], [0, 3, 4]], 0]], 'nonmanifold']], [['disk rim vertex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 2]], [[3, 0, 1], 'boundary']], ['octahedron top', [[[[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, 3, 4], 'interior']], ['octahedron equator', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 2]], [[0, 1, 5, 3], 'interior']], ['bowtie pinch', [[[[0, 1, 2], [0, 3, 4]], 0]], 'nonmanifold'], ['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'], ['quad grid centre', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 4]], [[1, 5, 7, 3], 'interior']], ['quad grid edge', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 1]], [[2, 4, 0], 'boundary']]], [['fan apex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0]], [[1, 2, 3, 4], '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'], ['quad grid centre', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 4]], [[1, 5, 7, 3], 'interior']], ['quad grid edge', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 1]], [[2, 4, 0], 'boundary']], ['isolated vertex', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 9]], 'isolated'], ['flipped neighbour', [[[[0, 1, 2], [0, 3, 2]], 0]], 'nonmanifold'], ['same spoke twice', [[[[0, 1, 2], [0, 1, 3]], 0]], 'nonmanifold']], [['fan middle rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 3]], [[4, 0, 2], 'boundary']], ['isolated vertex', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 9]], 'isolated'], ['flipped neighbour', [[[[0, 1, 2], [0, 3, 2]], 0]], 'nonmanifold'], ['same spoke twice', [[[[0, 1, 2], [0, 1, 3]], 0]], 'nonmanifold'], ['duplicated face', [[[[0, 1, 2], [0, 1, 2], [0, 2, 3]], 0]], 'nonmanifold'], ['cube corner', [[[[0, 3, 2, 1], [4, 5, 6, 7], [0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]], 6]], [[2, 7, 5], 'interior']], ['torus vertex', [[[[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]], 4]], [[0, 3, 7, 8, 5, 1], 'interior']]], [['disk centre', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0]], [[1, 2, 3, 4, 5], 'interior']], ['disk rim vertex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 2]], [[3, 0, 1], 'boundary']], ['fan apex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0]], [[1, 2, 3, 4], 'boundary']], ['octahedron top', [[[[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, 3, 4], 'interior']], ['duplicated face', [[[[0, 1, 2], [0, 1, 2], [0, 2, 3]], 0]], 'nonmanifold'], ['cube corner', [[[[0, 3, 2, 1], [4, 5, 6, 7], [0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]], 6]], [[2, 7, 5], 'interior']], ['torus vertex', [[[[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]], 4]], [[0, 3, 7, 8, 5, 1], 'interior']]]]
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
disk centre	`[[5, 4, 3, 2, 1], 'interior']`	`[[1, 2, 3, 4, 5], 'interior']`	Failed
disk rim vertex	`[[1, 0, 3], 'boundary']`	`[[3, 0, 1], 'boundary']`	Failed
fan apex	`[[4, 3, 2, 1], 'boundary']`	`[[1, 2, 3, 4], 'boundary']`	Failed
fan middle rim	`[[2, 0, 4], 'boundary']`	`[[4, 0, 2], 'boundary']`	Failed
octahedron top	`[[4, 3, 2, 1], 'interior']`	`[[1, 2, 3, 4], 'interior']`	Failed
octahedron equator	`[[3, 5, 1, 0], 'interior']`	`[[0, 1, 5, 3], 'interior']`	Failed
bowtie pinch	`nonmanifold`	`nonmanifold`	Passed

SHA-256 / 064d259b7e308f5fcfb2f64ca76d4d4dcabf9d336c9f8d1361adddfeeda8949d

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,v=x
    succ={}
    for f in faces:
        n=len(f)
        for j in range(n):
            if f[j]==v:
                p,q=f[(j+1)%n],f[(j-1)%n]
                if p in succ: return 'nonmanifold'
                succ[p]=q
    if not succ: return 'isolated'
    starts=[p for p in succ if p not in succ.values()]
    if len(starts)>1: return 'nonmanifold'
    s=starts[0] if starts else min(succ)
    ring=[s]
    u=s
    while u in succ and succ[u]!=s:
        u=succ[u]
        ring.append(u)
    if len(ring)!=len(set(succ)|set(succ.values())): return 'nonmanifold'
    return [ring,'boundary' if starts else 'interior']
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['disk centre', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0]], [[1, 2, 3, 4, 5], 'interior']], ['disk rim vertex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 2]], [[3, 0, 1], 'boundary']], ['fan apex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0]], [[1, 2, 3, 4], 'boundary']], ['fan middle rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 3]], [[4, 0, 2], 'boundary']], ['octahedron top', [[[[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, 3, 4], 'interior']], ['octahedron equator', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 2]], [[0, 1, 5, 3], 'interior']], ['bowtie pinch', [[[[0, 1, 2], [0, 3, 4]], 0]], 'nonmanifold']], [['disk rim vertex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 2]], [[3, 0, 1], 'boundary']], ['octahedron top', [[[[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, 3, 4], 'interior']], ['octahedron equator', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 2]], [[0, 1, 5, 3], 'interior']], ['bowtie pinch', [[[[0, 1, 2], [0, 3, 4]], 0]], 'nonmanifold'], ['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'], ['quad grid centre', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 4]], [[1, 5, 7, 3], 'interior']], ['quad grid edge', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 1]], [[2, 4, 0], 'boundary']]], [['fan apex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0]], [[1, 2, 3, 4], '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'], ['quad grid centre', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 4]], [[1, 5, 7, 3], 'interior']], ['quad grid edge', [[[[0, 1, 4, 3], [1, 2, 5, 4], [3, 4, 7, 6], [4, 5, 8, 7]], 1]], [[2, 4, 0], 'boundary']], ['isolated vertex', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 9]], 'isolated'], ['flipped neighbour', [[[[0, 1, 2], [0, 3, 2]], 0]], 'nonmanifold'], ['same spoke twice', [[[[0, 1, 2], [0, 1, 3]], 0]], 'nonmanifold']], [['fan middle rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 3]], [[4, 0, 2], 'boundary']], ['isolated vertex', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 9]], 'isolated'], ['flipped neighbour', [[[[0, 1, 2], [0, 3, 2]], 0]], 'nonmanifold'], ['same spoke twice', [[[[0, 1, 2], [0, 1, 3]], 0]], 'nonmanifold'], ['duplicated face', [[[[0, 1, 2], [0, 1, 2], [0, 2, 3]], 0]], 'nonmanifold'], ['cube corner', [[[[0, 3, 2, 1], [4, 5, 6, 7], [0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]], 6]], [[2, 7, 5], 'interior']], ['torus vertex', [[[[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]], 4]], [[0, 3, 7, 8, 5, 1], 'interior']]], [['disk centre', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0]], [[1, 2, 3, 4, 5], 'interior']], ['disk rim vertex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 2]], [[3, 0, 1], 'boundary']], ['fan apex', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0]], [[1, 2, 3, 4], 'boundary']], ['octahedron top', [[[[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, 3, 4], 'interior']], ['duplicated face', [[[[0, 1, 2], [0, 1, 2], [0, 2, 3]], 0]], 'nonmanifold'], ['cube corner', [[[[0, 3, 2, 1], [4, 5, 6, 7], [0, 1, 5, 4], [1, 2, 6, 5], [2, 3, 7, 6], [3, 0, 4, 7]], 6]], [[2, 7, 5], 'interior']], ['torus vertex', [[[[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]], 4]], [[0, 3, 7, 8, 5, 1], 'interior']]]]
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
disk centre	`[[1, 2, 3, 4, 5], 'interior']`	`[[1, 2, 3, 4, 5], 'interior']`	Passed
disk rim vertex	`[[3, 0, 1], 'boundary']`	`[[3, 0, 1], 'boundary']`	Passed
fan apex	`[[1, 2, 3, 4], 'boundary']`	`[[1, 2, 3, 4], 'boundary']`	Passed
fan middle rim	`[[4, 0, 2], 'boundary']`	`[[4, 0, 2], 'boundary']`	Passed
octahedron top	`[[1, 2, 3, 4], 'interior']`	`[[1, 2, 3, 4], 'interior']`	Passed
octahedron equator	`[[0, 1, 5, 3], 'interior']`	`[[0, 1, 5, 3], 'interior']`	Passed
bowtie pinch	`nonmanifold`	`nonmanifold`	Passed

SHA-256 / ac8b96d9bcbbdc4c2a9e54a70ffa30e4108e93e76de1b0ff40e56edaec94d937

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 2026-09-29T14:51:10.258523+00:00.

Case digest / 18344be5a07ab5466135cf4feb20a21bea7c977148ba55f0ed1998a5444cfa36