FA-88481 / Mesh topology invariants / Open access

Edge collapse accepts extra common neighbours · case 01

Collapses that would fuse two faces or pinch the surface are accepted.

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

ROOT CAUSE

The link test only checks that the opposite vertices are common neighbours, not that no other neighbour is shared.

VERIFIED REPAIR

Require the common neighbour set to equal the opposite vertex set.

Unsuccessful approach: Allowing up to two common neighbours is wrong for boundary edges with one opposite vertex.

Case contract

Input [faces, a, b] for a triangle mesh. Collapsing edge (a,b) is checked in order: missing edge -> [false,"no-edge"]; common neighbours of a and b must equal the opposite vertices of the triangles on the edge -> else [false,"link"]; if a and b are both boundary vertices the edge itself must be a boundary edge -> else [false,"boundary"]; a closed mesh with 4 or fewer vertices -> [false,"minimal"]; otherwise [true,"ok"].

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
    adj={}
    opp=[]
    ecount={}
    for f in faces:
        for i in range(3):
            u,v=f[i],f[(i+1)%3]
            adj.setdefault(u,set()).add(v)
            adj.setdefault(v,set()).add(u)
            k=(min(u,v),max(u,v))
            ecount[k]=ecount.get(k,0)+1
        if a in f and b in f:
            opp.append([w for w in f if w!=a and w!=b][0])
    k=(min(a,b),max(a,b))
    if k not in ecount: return [False,'no-edge']
    if not set(opp)<=(adj[a]&adj[b]): return [False,'link']
    bnd=set()
    for (u,v),c in ecount.items():
        if c==1: bnd.update((u,v))
    if a in bnd and b in bnd and ecount[k]!=1: return [False,'boundary']
    if not bnd and len(adj)<=4: return [False,'minimal']
    return [True,'ok']
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 1]], [True, 'ok']], ['disk rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 1, 2]], [True, 'ok']], ['fan interior edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0, 2]], [False, 'boundary']], ['fan missing edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 1, 3]], [False, 'no-edge']], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], [False, 'minimal']], ['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]], [True, 'ok']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']]], [['fan missing edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 1, 3]], [False, 'no-edge']], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], [False, 'minimal']], ['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]], [True, 'ok']], ['octahedron reversed', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 5, 2]], [True, 'ok']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['two triangles shared edge', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [False, 'boundary']], ['torus edge', [[[[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, 1]], [False, 'link']]], [['octahedron reversed', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 5, 2]], [True, 'ok']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['two triangles shared edge', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [False, 'boundary']], ['two triangles rim edge', [[[[0, 1, 2], [2, 1, 3]], 0, 1]], [True, 'ok']], ['torus edge', [[[[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, 1]], [False, 'link']], ['cylinder rung', [[[[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, 5]], [False, 'boundary']], ['strip over-constrained', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1], [1, 3, 4]], 1, 3]], [False, 'link']]], [['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['torus edge', [[[[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, 1]], [False, 'link']], ['cylinder rung', [[[[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, 5]], [False, 'boundary']], ['strip over-constrained', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1], [1, 3, 4]], 1, 3]], [False, 'link']], ['grid diagonal', [[[[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, 4]], [True, 'ok']], ['grid interior spoke', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]], 4, 5]], [True, 'ok']], ['rotated face order', [[[[2, 0, 1], [3, 0, 2]], 0, 2]], [False, 'boundary']]], [['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 1]], [True, 'ok']], ['disk rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 1, 2]], [True, 'ok']], ['fan interior edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0, 2]], [False, 'boundary']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['torus edge', [[[[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, 1]], [False, 'link']], ['grid interior spoke', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]], 4, 5]], [True, 'ok']], ['rotated face order', [[[[2, 0, 1], [3, 0, 2]], 0, 2]], [False, '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 fixture	Actual	Expected	Outcome
disk spoke	`[True, 'ok']`	`[True, 'ok']`	Passed
disk rim	`[True, 'ok']`	`[True, 'ok']`	Passed
fan interior edge	`[False, 'boundary']`	`[False, 'boundary']`	Passed
fan missing edge	`[False, 'no-edge']`	`[False, 'no-edge']`	Passed
tetrahedron edge	`[False, 'minimal']`	`[False, 'minimal']`	Passed
octahedron edge	`[True, 'ok']`	`[True, 'ok']`	Passed
open tetra cap	`[True, 'ok']`	`[False, 'link']`	Failed

SHA-256 / 839c0f9d9a5da1f395e32122b2c2cedd1a2428c8bbe6b37c18f78229414e80d3

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
    adj={}
    opp=[]
    ecount={}
    for f in faces:
        for i in range(3):
            u,v=f[i],f[(i+1)%3]
            adj.setdefault(u,set()).add(v)
            adj.setdefault(v,set()).add(u)
            k=(min(u,v),max(u,v))
            ecount[k]=ecount.get(k,0)+1
        if a in f and b in f:
            opp.append([w for w in f if w!=a and w!=b][0])
    k=(min(a,b),max(a,b))
    if k not in ecount: return [False,'no-edge']
    if len(adj[a]&adj[b])>2: return [False,'link']
    bnd=set()
    for (u,v),c in ecount.items():
        if c==1: bnd.update((u,v))
    if a in bnd and b in bnd and ecount[k]!=1: return [False,'boundary']
    if not bnd and len(adj)<=4: return [False,'minimal']
    return [True,'ok']
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 1]], [True, 'ok']], ['disk rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 1, 2]], [True, 'ok']], ['fan interior edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0, 2]], [False, 'boundary']], ['fan missing edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 1, 3]], [False, 'no-edge']], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], [False, 'minimal']], ['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]], [True, 'ok']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']]], [['fan missing edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 1, 3]], [False, 'no-edge']], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], [False, 'minimal']], ['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]], [True, 'ok']], ['octahedron reversed', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 5, 2]], [True, 'ok']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['two triangles shared edge', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [False, 'boundary']], ['torus edge', [[[[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, 1]], [False, 'link']]], [['octahedron reversed', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 5, 2]], [True, 'ok']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['two triangles shared edge', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [False, 'boundary']], ['two triangles rim edge', [[[[0, 1, 2], [2, 1, 3]], 0, 1]], [True, 'ok']], ['torus edge', [[[[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, 1]], [False, 'link']], ['cylinder rung', [[[[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, 5]], [False, 'boundary']], ['strip over-constrained', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1], [1, 3, 4]], 1, 3]], [False, 'link']]], [['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['torus edge', [[[[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, 1]], [False, 'link']], ['cylinder rung', [[[[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, 5]], [False, 'boundary']], ['strip over-constrained', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1], [1, 3, 4]], 1, 3]], [False, 'link']], ['grid diagonal', [[[[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, 4]], [True, 'ok']], ['grid interior spoke', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]], 4, 5]], [True, 'ok']], ['rotated face order', [[[[2, 0, 1], [3, 0, 2]], 0, 2]], [False, 'boundary']]], [['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 1]], [True, 'ok']], ['disk rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 1, 2]], [True, 'ok']], ['fan interior edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0, 2]], [False, 'boundary']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['torus edge', [[[[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, 1]], [False, 'link']], ['grid interior spoke', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]], 4, 5]], [True, 'ok']], ['rotated face order', [[[[2, 0, 1], [3, 0, 2]], 0, 2]], [False, '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 fixture	Actual	Expected	Outcome
disk spoke	`[True, 'ok']`	`[True, 'ok']`	Passed
disk rim	`[True, 'ok']`	`[True, 'ok']`	Passed
fan interior edge	`[False, 'boundary']`	`[False, 'boundary']`	Passed
fan missing edge	`[False, 'no-edge']`	`[False, 'no-edge']`	Passed
tetrahedron edge	`[False, 'minimal']`	`[False, 'minimal']`	Passed
octahedron edge	`[True, 'ok']`	`[True, 'ok']`	Passed
open tetra cap	`[True, 'ok']`	`[False, 'link']`	Failed

SHA-256 / d0fbc9497cc4b7705a67b44508e904dabb11f26e4c3d9dd102de434b87d45910

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
    adj={}
    opp=[]
    ecount={}
    for f in faces:
        for i in range(3):
            u,v=f[i],f[(i+1)%3]
            adj.setdefault(u,set()).add(v)
            adj.setdefault(v,set()).add(u)
            k=(min(u,v),max(u,v))
            ecount[k]=ecount.get(k,0)+1
        if a in f and b in f:
            opp.append([w for w in f if w!=a and w!=b][0])
    k=(min(a,b),max(a,b))
    if k not in ecount: return [False,'no-edge']
    if (adj[a]&adj[b])!=set(opp): return [False,'link']
    bnd=set()
    for (u,v),c in ecount.items():
        if c==1: bnd.update((u,v))
    if a in bnd and b in bnd and ecount[k]!=1: return [False,'boundary']
    if not bnd and len(adj)<=4: return [False,'minimal']
    return [True,'ok']
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 1]], [True, 'ok']], ['disk rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 1, 2]], [True, 'ok']], ['fan interior edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0, 2]], [False, 'boundary']], ['fan missing edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 1, 3]], [False, 'no-edge']], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], [False, 'minimal']], ['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]], [True, 'ok']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']]], [['fan missing edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 1, 3]], [False, 'no-edge']], ['tetrahedron edge', [[[[0, 2, 1], [0, 1, 3], [1, 2, 3], [0, 3, 2]], 0, 1]], [False, 'minimal']], ['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]], [True, 'ok']], ['octahedron reversed', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 5, 2]], [True, 'ok']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['two triangles shared edge', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [False, 'boundary']], ['torus edge', [[[[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, 1]], [False, 'link']]], [['octahedron reversed', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 1], [5, 2, 1], [5, 3, 2], [5, 4, 3], [5, 1, 4]], 5, 2]], [True, 'ok']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['two triangles shared edge', [[[[0, 1, 2], [2, 1, 3]], 1, 2]], [False, 'boundary']], ['two triangles rim edge', [[[[0, 1, 2], [2, 1, 3]], 0, 1]], [True, 'ok']], ['torus edge', [[[[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, 1]], [False, 'link']], ['cylinder rung', [[[[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, 5]], [False, 'boundary']], ['strip over-constrained', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1], [1, 3, 4]], 1, 3]], [False, 'link']]], [['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['torus edge', [[[[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, 1]], [False, 'link']], ['cylinder rung', [[[[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, 5]], [False, 'boundary']], ['strip over-constrained', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1], [1, 3, 4]], 1, 3]], [False, 'link']], ['grid diagonal', [[[[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, 4]], [True, 'ok']], ['grid interior spoke', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]], 4, 5]], [True, 'ok']], ['rotated face order', [[[[2, 0, 1], [3, 0, 2]], 0, 2]], [False, 'boundary']]], [['disk spoke', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 0, 1]], [True, 'ok']], ['disk rim', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5], [0, 5, 1]], 1, 2]], [True, 'ok']], ['fan interior edge', [[[[0, 1, 2], [0, 2, 3], [0, 3, 4]], 0, 2]], [False, 'boundary']], ['open tetra cap', [[[[0, 1, 2], [0, 2, 3], [0, 3, 1]], 1, 2]], [False, 'link']], ['torus edge', [[[[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, 1]], [False, 'link']], ['grid interior spoke', [[[[0, 1, 4], [0, 4, 3], [1, 2, 5], [1, 5, 4], [3, 4, 7], [3, 7, 6], [4, 5, 8], [4, 8, 7]], 4, 5]], [True, 'ok']], ['rotated face order', [[[[2, 0, 1], [3, 0, 2]], 0, 2]], [False, '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 fixture	Actual	Expected	Outcome
disk spoke	`[True, 'ok']`	`[True, 'ok']`	Passed
disk rim	`[True, 'ok']`	`[True, 'ok']`	Passed
fan interior edge	`[False, 'boundary']`	`[False, 'boundary']`	Passed
fan missing edge	`[False, 'no-edge']`	`[False, 'no-edge']`	Passed
tetrahedron edge	`[False, 'minimal']`	`[False, 'minimal']`	Passed
octahedron edge	`[True, 'ok']`	`[True, 'ok']`	Passed
open tetra cap	`[False, 'link']`	`[False, 'link']`	Passed

SHA-256 / df5b322769fb8a9b7c97ff969e8f4c1ef40abbe8b5be69a26c95e18508cf624f

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 2026-09-29T14:51:08.440365+00:00.

Case digest / 023a3792ef50b0b1715aa9015c47eafcdfc54fa4b320776e095c647c1c3acc6f