FA-88701 / Mesh topology invariants / Open access

Euler-Poincare check ignores inner rings · case 01

Any solid with a ring loop is flagged as broken.

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

ROOT CAUSE

The relation is checked as V-E+F-L instead of V-E+F-(L-F).

VERIFIED REPAIR

Check V - E + F - (L - F) = 2(S - G).

Unsuccessful approach: Dropping the genus term rejects every solid with a through-hole.

Case contract

Input: list of Euler operator names applied to counts [V,E,F,L,S,G] starting at zero. Deltas: mvfs(+V,+F,+L,+S), mev(+V,+E), mef(+E,+F,+L), kemr(-E,+L), mekr(+E,-L), kfmrh(-F,+G) and their inverses kvfs, kev, kef, mfkrh. After each op: an unknown name returns ["unknown", i]; a negative count or fewer loops than faces returns ["invalid", i]; a violated Euler-Poincare relation V-E+F-(L-F)=2(S-G) returns ["broken-invariant", i]. Return the final counts.

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):
    ops=x
    V=E=Fc=L=S=G=0
    D={'mvfs':(1,0,1,1,1,0),'kvfs':(-1,0,-1,-1,-1,0),'mev':(1,1,0,0,0,0),'kev':(-1,-1,0,0,0,0),
       'mef':(0,1,1,1,0,0),'kef':(0,-1,-1,-1,0,0),'kemr':(0,-1,0,1,0,0),'mekr':(0,1,0,-1,0,0),
       'kfmrh':(0,0,-1,0,0,1),'mfkrh':(0,0,1,0,0,-1)}
    for i,op in enumerate(ops):
        if op not in D: return ['unknown',i]
        d=D[op]
        V,E,Fc,L,S,G=V+d[0],E+d[1],Fc+d[2],L+d[3],S+d[4],G+d[5]
        if min(V,E,Fc,L,S,G)<0 or L<Fc: return ['invalid',i]
        if V-E+Fc-L!=2*(S-G): return ['broken-invariant',i]
    return [V,E,Fc,L,S,G]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['triangle lamina', [['mvfs', 'mev', 'mev', 'mef']], [3, 3, 2, 2, 1, 0]], ['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['square with hole ring', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr']], [7, 7, 3, 4, 1, 0]], ['torus handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh']], [7, 7, 2, 4, 1, 1]], ['undo handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mfkrh']], [7, 7, 3, 4, 1, 0]], ['ring then remake edge', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'kemr', 'mekr']], [4, 4, 2, 2, 1, 0]], ['kill edge on empty', [['mvfs', 'kev']], ['invalid', 1]]], [['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['undo handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mfkrh']], [7, 7, 3, 4, 1, 0]], ['ring then remake edge', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'kemr', 'mekr']], [4, 4, 2, 2, 1, 0]], ['kill edge on empty', [['mvfs', 'kev']], ['invalid', 1]], ['unknown op', [['mvfs', 'mev', 'split']], ['unknown', 2]], ['uppercase op', [['mvfs', 'MEV']], ['unknown', 1]], ['loop deficit', [['mvfs', 'mev', 'mekr']], ['invalid', 2]]], [['square with hole ring', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr']], [7, 7, 3, 4, 1, 0]], ['unknown op', [['mvfs', 'mev', 'split']], ['unknown', 2]], ['uppercase op', [['mvfs', 'MEV']], ['unknown', 1]], ['loop deficit', [['mvfs', 'mev', 'mekr']], ['invalid', 2]], ['two shells', [['mvfs', 'mev', 'mvfs', 'mev', 'mev', 'mef']], [5, 4, 3, 3, 2, 0]], ['face without loop', [['mvfs', 'mfkrh']], ['invalid', 1]], ['hole into lamina', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mev']], [7, 7, 2, 4, 1, 1]]], [['triangle lamina', [['mvfs', 'mev', 'mev', 'mef']], [3, 3, 2, 2, 1, 0]], ['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['torus handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh']], [7, 7, 2, 4, 1, 1]], ['two shells', [['mvfs', 'mev', 'mvfs', 'mev', 'mev', 'mef']], [5, 4, 3, 3, 2, 0]], ['face without loop', [['mvfs', 'mfkrh']], ['invalid', 1]], ['build and kill', [['mvfs', 'mev', 'mev', 'mef', 'kef', 'kev', 'kev', 'kvfs']], [0, 0, 0, 0, 0, 0]], ['hole into lamina', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mev']], [7, 7, 2, 4, 1, 1]]], [['triangle lamina', [['mvfs', 'mev', 'mev', 'mef']], [3, 3, 2, 2, 1, 0]], ['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['square with hole ring', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr']], [7, 7, 3, 4, 1, 0]], ['torus handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh']], [7, 7, 2, 4, 1, 1]], ['undo handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mfkrh']], [7, 7, 3, 4, 1, 0]], ['ring then remake edge', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'kemr', 'mekr']], [4, 4, 2, 2, 1, 0]], ['hole into lamina', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mev']], [7, 7, 2, 4, 1, 1]]]]
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
triangle lamina	`['broken-invariant', 0]`	`[3, 3, 2, 2, 1, 0]`	Failed
tetrahedron	`['broken-invariant', 0]`	`[4, 6, 4, 4, 1, 0]`	Failed
square with hole ring	`['broken-invariant', 0]`	`[7, 7, 3, 4, 1, 0]`	Failed
torus handle	`['broken-invariant', 0]`	`[7, 7, 2, 4, 1, 1]`	Failed
undo handle	`['broken-invariant', 0]`	`[7, 7, 3, 4, 1, 0]`	Failed
ring then remake edge	`['broken-invariant', 0]`	`[4, 4, 2, 2, 1, 0]`	Failed
kill edge on empty	`['broken-invariant', 0]`	`['invalid', 1]`	Failed

SHA-256 / 135dd16a7ab64695f939048ffd82de1ff9dc2faa394dac72019798da7ebe90d1

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    ops=x
    V=E=Fc=L=S=G=0
    D={'mvfs':(1,0,1,1,1,0),'kvfs':(-1,0,-1,-1,-1,0),'mev':(1,1,0,0,0,0),'kev':(-1,-1,0,0,0,0),
       'mef':(0,1,1,1,0,0),'kef':(0,-1,-1,-1,0,0),'kemr':(0,-1,0,1,0,0),'mekr':(0,1,0,-1,0,0),
       'kfmrh':(0,0,-1,0,0,1),'mfkrh':(0,0,1,0,0,-1)}
    for i,op in enumerate(ops):
        if op not in D: return ['unknown',i]
        d=D[op]
        V,E,Fc,L,S,G=V+d[0],E+d[1],Fc+d[2],L+d[3],S+d[4],G+d[5]
        if min(V,E,Fc,L,S,G)<0 or L<Fc: return ['invalid',i]
        if V-E+2*Fc-L!=2*S: return ['broken-invariant',i]
    return [V,E,Fc,L,S,G]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['triangle lamina', [['mvfs', 'mev', 'mev', 'mef']], [3, 3, 2, 2, 1, 0]], ['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['square with hole ring', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr']], [7, 7, 3, 4, 1, 0]], ['torus handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh']], [7, 7, 2, 4, 1, 1]], ['undo handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mfkrh']], [7, 7, 3, 4, 1, 0]], ['ring then remake edge', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'kemr', 'mekr']], [4, 4, 2, 2, 1, 0]], ['kill edge on empty', [['mvfs', 'kev']], ['invalid', 1]]], [['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['undo handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mfkrh']], [7, 7, 3, 4, 1, 0]], ['ring then remake edge', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'kemr', 'mekr']], [4, 4, 2, 2, 1, 0]], ['kill edge on empty', [['mvfs', 'kev']], ['invalid', 1]], ['unknown op', [['mvfs', 'mev', 'split']], ['unknown', 2]], ['uppercase op', [['mvfs', 'MEV']], ['unknown', 1]], ['loop deficit', [['mvfs', 'mev', 'mekr']], ['invalid', 2]]], [['square with hole ring', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr']], [7, 7, 3, 4, 1, 0]], ['unknown op', [['mvfs', 'mev', 'split']], ['unknown', 2]], ['uppercase op', [['mvfs', 'MEV']], ['unknown', 1]], ['loop deficit', [['mvfs', 'mev', 'mekr']], ['invalid', 2]], ['two shells', [['mvfs', 'mev', 'mvfs', 'mev', 'mev', 'mef']], [5, 4, 3, 3, 2, 0]], ['face without loop', [['mvfs', 'mfkrh']], ['invalid', 1]], ['hole into lamina', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mev']], [7, 7, 2, 4, 1, 1]]], [['triangle lamina', [['mvfs', 'mev', 'mev', 'mef']], [3, 3, 2, 2, 1, 0]], ['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['torus handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh']], [7, 7, 2, 4, 1, 1]], ['two shells', [['mvfs', 'mev', 'mvfs', 'mev', 'mev', 'mef']], [5, 4, 3, 3, 2, 0]], ['face without loop', [['mvfs', 'mfkrh']], ['invalid', 1]], ['build and kill', [['mvfs', 'mev', 'mev', 'mef', 'kef', 'kev', 'kev', 'kvfs']], [0, 0, 0, 0, 0, 0]], ['hole into lamina', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mev']], [7, 7, 2, 4, 1, 1]]], [['triangle lamina', [['mvfs', 'mev', 'mev', 'mef']], [3, 3, 2, 2, 1, 0]], ['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['square with hole ring', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr']], [7, 7, 3, 4, 1, 0]], ['torus handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh']], [7, 7, 2, 4, 1, 1]], ['undo handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mfkrh']], [7, 7, 3, 4, 1, 0]], ['ring then remake edge', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'kemr', 'mekr']], [4, 4, 2, 2, 1, 0]], ['hole into lamina', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mev']], [7, 7, 2, 4, 1, 1]]]]
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
triangle lamina	`[3, 3, 2, 2, 1, 0]`	`[3, 3, 2, 2, 1, 0]`	Passed
tetrahedron	`[4, 6, 4, 4, 1, 0]`	`[4, 6, 4, 4, 1, 0]`	Passed
square with hole ring	`[7, 7, 3, 4, 1, 0]`	`[7, 7, 3, 4, 1, 0]`	Passed
torus handle	`['broken-invariant', 10]`	`[7, 7, 2, 4, 1, 1]`	Failed
undo handle	`['broken-invariant', 10]`	`[7, 7, 3, 4, 1, 0]`	Failed
ring then remake edge	`[4, 4, 2, 2, 1, 0]`	`[4, 4, 2, 2, 1, 0]`	Passed
kill edge on empty	`['invalid', 1]`	`['invalid', 1]`	Passed

SHA-256 / a7eae164c257ef60f4535b8fe05c424c02abd9db5042977895b50f8d32852ba6

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    ops=x
    V=E=Fc=L=S=G=0
    D={'mvfs':(1,0,1,1,1,0),'kvfs':(-1,0,-1,-1,-1,0),'mev':(1,1,0,0,0,0),'kev':(-1,-1,0,0,0,0),
       'mef':(0,1,1,1,0,0),'kef':(0,-1,-1,-1,0,0),'kemr':(0,-1,0,1,0,0),'mekr':(0,1,0,-1,0,0),
       'kfmrh':(0,0,-1,0,0,1),'mfkrh':(0,0,1,0,0,-1)}
    for i,op in enumerate(ops):
        if op not in D: return ['unknown',i]
        d=D[op]
        V,E,Fc,L,S,G=V+d[0],E+d[1],Fc+d[2],L+d[3],S+d[4],G+d[5]
        if min(V,E,Fc,L,S,G)<0 or L<Fc: return ['invalid',i]
        if V-E+Fc-(L-Fc)!=2*(S-G): return ['broken-invariant',i]
    return [V,E,Fc,L,S,G]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['triangle lamina', [['mvfs', 'mev', 'mev', 'mef']], [3, 3, 2, 2, 1, 0]], ['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['square with hole ring', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr']], [7, 7, 3, 4, 1, 0]], ['torus handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh']], [7, 7, 2, 4, 1, 1]], ['undo handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mfkrh']], [7, 7, 3, 4, 1, 0]], ['ring then remake edge', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'kemr', 'mekr']], [4, 4, 2, 2, 1, 0]], ['kill edge on empty', [['mvfs', 'kev']], ['invalid', 1]]], [['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['undo handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mfkrh']], [7, 7, 3, 4, 1, 0]], ['ring then remake edge', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'kemr', 'mekr']], [4, 4, 2, 2, 1, 0]], ['kill edge on empty', [['mvfs', 'kev']], ['invalid', 1]], ['unknown op', [['mvfs', 'mev', 'split']], ['unknown', 2]], ['uppercase op', [['mvfs', 'MEV']], ['unknown', 1]], ['loop deficit', [['mvfs', 'mev', 'mekr']], ['invalid', 2]]], [['square with hole ring', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr']], [7, 7, 3, 4, 1, 0]], ['unknown op', [['mvfs', 'mev', 'split']], ['unknown', 2]], ['uppercase op', [['mvfs', 'MEV']], ['unknown', 1]], ['loop deficit', [['mvfs', 'mev', 'mekr']], ['invalid', 2]], ['two shells', [['mvfs', 'mev', 'mvfs', 'mev', 'mev', 'mef']], [5, 4, 3, 3, 2, 0]], ['face without loop', [['mvfs', 'mfkrh']], ['invalid', 1]], ['hole into lamina', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mev']], [7, 7, 2, 4, 1, 1]]], [['triangle lamina', [['mvfs', 'mev', 'mev', 'mef']], [3, 3, 2, 2, 1, 0]], ['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['torus handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh']], [7, 7, 2, 4, 1, 1]], ['two shells', [['mvfs', 'mev', 'mvfs', 'mev', 'mev', 'mef']], [5, 4, 3, 3, 2, 0]], ['face without loop', [['mvfs', 'mfkrh']], ['invalid', 1]], ['build and kill', [['mvfs', 'mev', 'mev', 'mef', 'kef', 'kev', 'kev', 'kvfs']], [0, 0, 0, 0, 0, 0]], ['hole into lamina', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mev']], [7, 7, 2, 4, 1, 1]]], [['triangle lamina', [['mvfs', 'mev', 'mev', 'mef']], [3, 3, 2, 2, 1, 0]], ['tetrahedron', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mef', 'mef']], [4, 6, 4, 4, 1, 0]], ['square with hole ring', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr']], [7, 7, 3, 4, 1, 0]], ['torus handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh']], [7, 7, 2, 4, 1, 1]], ['undo handle', [['mvfs', 'mev', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mfkrh']], [7, 7, 3, 4, 1, 0]], ['ring then remake edge', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'kemr', 'mekr']], [4, 4, 2, 2, 1, 0]], ['hole into lamina', [['mvfs', 'mev', 'mev', 'mef', 'mev', 'mev', 'mev', 'mef', 'kemr', 'kfmrh', 'mev']], [7, 7, 2, 4, 1, 1]]]]
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
triangle lamina	`[3, 3, 2, 2, 1, 0]`	`[3, 3, 2, 2, 1, 0]`	Passed
tetrahedron	`[4, 6, 4, 4, 1, 0]`	`[4, 6, 4, 4, 1, 0]`	Passed
square with hole ring	`[7, 7, 3, 4, 1, 0]`	`[7, 7, 3, 4, 1, 0]`	Passed
torus handle	`[7, 7, 2, 4, 1, 1]`	`[7, 7, 2, 4, 1, 1]`	Passed
undo handle	`[7, 7, 3, 4, 1, 0]`	`[7, 7, 3, 4, 1, 0]`	Passed
ring then remake edge	`[4, 4, 2, 2, 1, 0]`	`[4, 4, 2, 2, 1, 0]`	Passed
kill edge on empty	`['invalid', 1]`	`['invalid', 1]`	Passed

SHA-256 / 3e6d8424ff44c6e8fd8b077545842e3e93092907e124fe4ef612d5ed81ca1668

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 2026-09-29T14:51:10.601820+00:00.

Case digest / 81a189f429c42b393304efe4c078be28776bf7830704073f72bea0b7465929a5