FA-88676 / Mesh topology invariants / Open access

One-ring overwrites a repeated spoke · case 01

Two faces leaving v along the same edge (a flipped neighbour) are merged into a plausible ring.

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

ROOT CAUSE

A second face with the same next corner replaces the first successor.

THE FAILURE

A second face with the same next corner replaces the first successor.

Unsuccessful approach: Rejecting only exact duplicate faces misses flipped neighbours.

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]
                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']], ['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']], ['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'], ['same spoke twice', [[[[0, 1, 2], [0, 1, 3]], 0]], 'nonmanifold'], ['duplicated face', [[[[0, 1, 2], [0, 1, 2], [0, 2, 3]], 0]], 'nonmanifold']], [['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']], ['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']], [['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'], ['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']], ['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']]]]
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
same spoke twice	`[[1, 3], 'boundary']`	`nonmanifold`	Failed

SHA-256 / 1312984bcf712fae7dfe55a1e1ef701f0f0b6ba4c465b10fa5bde82c120c79e0

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 and succ[p]==q: 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']], ['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']], ['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'], ['same spoke twice', [[[[0, 1, 2], [0, 1, 3]], 0]], 'nonmanifold'], ['duplicated face', [[[[0, 1, 2], [0, 1, 2], [0, 2, 3]], 0]], 'nonmanifold']], [['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']], ['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']], [['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'], ['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']], ['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']]]]
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
same spoke twice	`[[1, 3], 'boundary']`	`nonmanifold`	Failed

SHA-256 / ce378d0dfa049d9f407710bebc78635d97edf9656e9fade05cbd0b4ed0251461

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.

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 2026-09-29T14:51:10.347952+00:00.

Case digest / 5eda7053122e86b8a320d7d3d889cf9dfa2e6f93192eeddfab1c7694604d5e9e