FA-88591 / Mesh topology invariants / Open access

Welding removes every repeated corner, not only consecutive ones · case 01

Polygons that legitimately revisit a vertex lose corners.

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

ROOT CAUSE

Corners are skipped whenever they already appear anywhere in the face.

VERIFIED REPAIR

Skip a corner only when it equals the previous kept corner.

Unsuccessful approach: Comparing against the first corner instead of the previous one misses runs in the middle.

Case contract

Input [points, faces, eps]. Points weld when floor(c/eps+0.5) agrees on every coordinate; the welded vertex takes the next new id and is represented by its lowest original index. Remap faces, merge consecutive equal corners (also across the wrap), drop faces with fewer than 3 corners, rotate each face so its smallest id comes first (orientation kept) and drop later copies of an identical rotated face. Return [representatives, faces].

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):
    pts,faces,eps=x
    key_to_new={}
    remap=[]
    reps=[]
    for i,p in enumerate(pts):
        k=tuple(math.floor(c/eps+0.5) for c in p)
        if k not in key_to_new:
            key_to_new[k]=len(reps)
            reps.append(i)
        remap.append(key_to_new[k])
    out=[]
    seen=set()
    for f in faces:
        g=[remap[v] for v in f]
        h=[]
        for v in g:
            if v not in h: h.append(v)
        while len(h)>1 and h[0]==h[-1]: h.pop()
        if len(h)<3: continue
        m=h.index(min(h))
        c=tuple(h[m:]+h[:m])
        if c in seen: continue
        seen.add(c)
        out.append(list(c))
    return [reps,out]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['near duplicate corner', [[[[0, 0], [1, 0], [0, 1], [1.02, 0.01]], [[0, 1, 2], [3, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['negative coordinates', [[[[-0.7, 0], [0.3, 0], [0, 1], [1, 1]], [[0, 2, 3], [1, 2, 3]], 1.0]], [[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]], ['half grid step', [[[[0.5, 0], [2.5, 0], [1.5, 1], [0.4, 0], [2.6, 0]], [[0, 1, 2], [3, 4, 2]], 1.0]], [[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]], ['collapsing triangle', [[[[0, 0], [0.01, 0], [1, 1], [2, 2]], [[0, 1, 2], [1, 2, 3]], 0.1]], [[0, 2, 3], [[0, 1, 2]]]], ['rotated duplicate', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [1, 2, 0], [2, 1, 0]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]]], [['collapsing triangle', [[[[0, 0], [0.01, 0], [1, 1], [2, 2]], [[0, 1, 2], [1, 2, 3]], 0.1]], [[0, 2, 3], [[0, 1, 2]]]], ['rotated duplicate', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [1, 2, 0], [2, 1, 0]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['wrap-around duplicate', [[[[0, 0], [1, 0], [1, 1], [0.01, 0.01]], [[0, 1, 2, 3], [3, 0, 1, 2, 3]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['trailing repeats', [[[[0, 0], [1, 0], [1, 1], [0, 1], [0.02, 0]], [[0, 1, 2, 4, 4], [0, 2, 3]], 0.1]], [[0, 1, 2, 3], [[0, 1, 2], [0, 2, 3]]]], ['three points weld to one', [[[[5, 5], [5.01, 5], [4.99, 5.01], [6, 5], [5, 6]], [[3, 0, 4], [1, 3, 4], [2, 4, 3]], 0.1]], [[0, 3, 4], [[0, 2, 1], [0, 1, 2]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['two-corner collapse', [[[[0, 0], [0, 0.01], [3, 3], [3.01, 3]], [[0, 2, 1, 3]], 0.1]], [[0, 2], [[0, 1, 0, 1]]]]], [['trailing repeats', [[[[0, 0], [1, 0], [1, 1], [0, 1], [0.02, 0]], [[0, 1, 2, 4, 4], [0, 2, 3]], 0.1]], [[0, 1, 2, 3], [[0, 1, 2], [0, 2, 3]]]], ['three points weld to one', [[[[5, 5], [5.01, 5], [4.99, 5.01], [6, 5], [5, 6]], [[3, 0, 4], [1, 3, 4], [2, 4, 3]], 0.1]], [[0, 3, 4], [[0, 2, 1], [0, 1, 2]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['opposite orientation kept', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [0, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['late lowest representative', [[[[3, 3], [1, 1], [3.04, 3], [1, 0], [0, 0]], [[1, 3, 0], [4, 3, 2]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2], [0, 3, 2]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]], ['two-corner collapse', [[[[0, 0], [0, 0.01], [3, 3], [3.01, 3]], [[0, 2, 1, 3]], 0.1]], [[0, 2], [[0, 1, 0, 1]]]]], [['near duplicate corner', [[[[0, 0], [1, 0], [0, 1], [1.02, 0.01]], [[0, 1, 2], [3, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['negative coordinates', [[[[-0.7, 0], [0.3, 0], [0, 1], [1, 1]], [[0, 2, 3], [1, 2, 3]], 1.0]], [[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]], ['half grid step', [[[[0.5, 0], [2.5, 0], [1.5, 1], [0.4, 0], [2.6, 0]], [[0, 1, 2], [3, 4, 2]], 1.0]], [[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]], ['opposite orientation kept', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [0, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['late lowest representative', [[[[3, 3], [1, 1], [3.04, 3], [1, 0], [0, 0]], [[1, 3, 0], [4, 3, 2]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2], [0, 3, 2]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]], ['two-corner collapse', [[[[0, 0], [0, 0.01], [3, 3], [3.01, 3]], [[0, 2, 1, 3]], 0.1]], [[0, 2], [[0, 1, 0, 1]]]]], [['negative coordinates', [[[[-0.7, 0], [0.3, 0], [0, 1], [1, 1]], [[0, 2, 3], [1, 2, 3]], 1.0]], [[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]], ['half grid step', [[[[0.5, 0], [2.5, 0], [1.5, 1], [0.4, 0], [2.6, 0]], [[0, 1, 2], [3, 4, 2]], 1.0]], [[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]], ['collapsing triangle', [[[[0, 0], [0.01, 0], [1, 1], [2, 2]], [[0, 1, 2], [1, 2, 3]], 0.1]], [[0, 2, 3], [[0, 1, 2]]]], ['rotated duplicate', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [1, 2, 0], [2, 1, 0]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['wrap-around duplicate', [[[[0, 0], [1, 0], [1, 1], [0.01, 0.01]], [[0, 1, 2, 3], [3, 0, 1, 2, 3]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]]]]
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
near duplicate corner	`[[0, 1, 2], [[0, 1, 2]]]`	`[[0, 1, 2], [[0, 1, 2]]]`	Passed
negative coordinates	`[[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]`	`[[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]`	Passed
half grid step	`[[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]`	`[[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]`	Passed
collapsing triangle	`[[0, 2, 3], [[0, 1, 2]]]`	`[[0, 2, 3], [[0, 1, 2]]]`	Passed
rotated duplicate	`[[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]`	`[[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]`	Passed
non-consecutive repeat kept	`[[0, 1, 2, 3], [[0, 1, 3, 2]]]`	`[[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]`	Failed
bowed quad two repeats	`[[0, 1, 3, 4], [[0, 1, 2, 3]]]`	`[[0, 1, 3, 4], [[0, 1, 2, 3]]]`	Passed

SHA-256 / 8cbc68506c3e35482a437c66ba68b29806d9c08a35f86a197dc910ba98c7b51a

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    pts,faces,eps=x
    key_to_new={}
    remap=[]
    reps=[]
    for i,p in enumerate(pts):
        k=tuple(math.floor(c/eps+0.5) for c in p)
        if k not in key_to_new:
            key_to_new[k]=len(reps)
            reps.append(i)
        remap.append(key_to_new[k])
    out=[]
    seen=set()
    for f in faces:
        g=[remap[v] for v in f]
        h=[]
        for v in g:
            if not h or h[0]!=v: h.append(v)
        while len(h)>1 and h[0]==h[-1]: h.pop()
        if len(h)<3: continue
        m=h.index(min(h))
        c=tuple(h[m:]+h[:m])
        if c in seen: continue
        seen.add(c)
        out.append(list(c))
    return [reps,out]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['near duplicate corner', [[[[0, 0], [1, 0], [0, 1], [1.02, 0.01]], [[0, 1, 2], [3, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['negative coordinates', [[[[-0.7, 0], [0.3, 0], [0, 1], [1, 1]], [[0, 2, 3], [1, 2, 3]], 1.0]], [[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]], ['half grid step', [[[[0.5, 0], [2.5, 0], [1.5, 1], [0.4, 0], [2.6, 0]], [[0, 1, 2], [3, 4, 2]], 1.0]], [[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]], ['collapsing triangle', [[[[0, 0], [0.01, 0], [1, 1], [2, 2]], [[0, 1, 2], [1, 2, 3]], 0.1]], [[0, 2, 3], [[0, 1, 2]]]], ['rotated duplicate', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [1, 2, 0], [2, 1, 0]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]]], [['collapsing triangle', [[[[0, 0], [0.01, 0], [1, 1], [2, 2]], [[0, 1, 2], [1, 2, 3]], 0.1]], [[0, 2, 3], [[0, 1, 2]]]], ['rotated duplicate', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [1, 2, 0], [2, 1, 0]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['wrap-around duplicate', [[[[0, 0], [1, 0], [1, 1], [0.01, 0.01]], [[0, 1, 2, 3], [3, 0, 1, 2, 3]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['trailing repeats', [[[[0, 0], [1, 0], [1, 1], [0, 1], [0.02, 0]], [[0, 1, 2, 4, 4], [0, 2, 3]], 0.1]], [[0, 1, 2, 3], [[0, 1, 2], [0, 2, 3]]]], ['three points weld to one', [[[[5, 5], [5.01, 5], [4.99, 5.01], [6, 5], [5, 6]], [[3, 0, 4], [1, 3, 4], [2, 4, 3]], 0.1]], [[0, 3, 4], [[0, 2, 1], [0, 1, 2]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['two-corner collapse', [[[[0, 0], [0, 0.01], [3, 3], [3.01, 3]], [[0, 2, 1, 3]], 0.1]], [[0, 2], [[0, 1, 0, 1]]]]], [['trailing repeats', [[[[0, 0], [1, 0], [1, 1], [0, 1], [0.02, 0]], [[0, 1, 2, 4, 4], [0, 2, 3]], 0.1]], [[0, 1, 2, 3], [[0, 1, 2], [0, 2, 3]]]], ['three points weld to one', [[[[5, 5], [5.01, 5], [4.99, 5.01], [6, 5], [5, 6]], [[3, 0, 4], [1, 3, 4], [2, 4, 3]], 0.1]], [[0, 3, 4], [[0, 2, 1], [0, 1, 2]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['opposite orientation kept', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [0, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['late lowest representative', [[[[3, 3], [1, 1], [3.04, 3], [1, 0], [0, 0]], [[1, 3, 0], [4, 3, 2]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2], [0, 3, 2]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]], ['two-corner collapse', [[[[0, 0], [0, 0.01], [3, 3], [3.01, 3]], [[0, 2, 1, 3]], 0.1]], [[0, 2], [[0, 1, 0, 1]]]]], [['near duplicate corner', [[[[0, 0], [1, 0], [0, 1], [1.02, 0.01]], [[0, 1, 2], [3, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['negative coordinates', [[[[-0.7, 0], [0.3, 0], [0, 1], [1, 1]], [[0, 2, 3], [1, 2, 3]], 1.0]], [[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]], ['half grid step', [[[[0.5, 0], [2.5, 0], [1.5, 1], [0.4, 0], [2.6, 0]], [[0, 1, 2], [3, 4, 2]], 1.0]], [[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]], ['opposite orientation kept', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [0, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['late lowest representative', [[[[3, 3], [1, 1], [3.04, 3], [1, 0], [0, 0]], [[1, 3, 0], [4, 3, 2]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2], [0, 3, 2]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]], ['two-corner collapse', [[[[0, 0], [0, 0.01], [3, 3], [3.01, 3]], [[0, 2, 1, 3]], 0.1]], [[0, 2], [[0, 1, 0, 1]]]]], [['negative coordinates', [[[[-0.7, 0], [0.3, 0], [0, 1], [1, 1]], [[0, 2, 3], [1, 2, 3]], 1.0]], [[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]], ['half grid step', [[[[0.5, 0], [2.5, 0], [1.5, 1], [0.4, 0], [2.6, 0]], [[0, 1, 2], [3, 4, 2]], 1.0]], [[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]], ['collapsing triangle', [[[[0, 0], [0.01, 0], [1, 1], [2, 2]], [[0, 1, 2], [1, 2, 3]], 0.1]], [[0, 2, 3], [[0, 1, 2]]]], ['rotated duplicate', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [1, 2, 0], [2, 1, 0]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['wrap-around duplicate', [[[[0, 0], [1, 0], [1, 1], [0.01, 0.01]], [[0, 1, 2, 3], [3, 0, 1, 2, 3]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]]]]
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
near duplicate corner	`[[0, 1, 2], [[0, 1, 2]]]`	`[[0, 1, 2], [[0, 1, 2]]]`	Passed
negative coordinates	`[[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]`	`[[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]`	Passed
half grid step	`[[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]`	`[[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]`	Passed
collapsing triangle	`[[0, 2, 3], [[0, 1, 2]]]`	`[[0, 2, 3], [[0, 1, 2]]]`	Passed
rotated duplicate	`[[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]`	`[[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]`	Passed
non-consecutive repeat kept	`[[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]`	`[[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]`	Passed
bowed quad two repeats	`[[0, 1, 3, 4], [[0, 1, 1, 2, 3]]]`	`[[0, 1, 3, 4], [[0, 1, 2, 3]]]`	Failed

SHA-256 / 9b4b173d901a6ebfe40bd72380c0adef04eea80ae877846036b58e80f170b3ef

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    pts,faces,eps=x
    key_to_new={}
    remap=[]
    reps=[]
    for i,p in enumerate(pts):
        k=tuple(math.floor(c/eps+0.5) for c in p)
        if k not in key_to_new:
            key_to_new[k]=len(reps)
            reps.append(i)
        remap.append(key_to_new[k])
    out=[]
    seen=set()
    for f in faces:
        g=[remap[v] for v in f]
        h=[]
        for v in g:
            if not h or h[-1]!=v: h.append(v)
        while len(h)>1 and h[0]==h[-1]: h.pop()
        if len(h)<3: continue
        m=h.index(min(h))
        c=tuple(h[m:]+h[:m])
        if c in seen: continue
        seen.add(c)
        out.append(list(c))
    return [reps,out]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['near duplicate corner', [[[[0, 0], [1, 0], [0, 1], [1.02, 0.01]], [[0, 1, 2], [3, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['negative coordinates', [[[[-0.7, 0], [0.3, 0], [0, 1], [1, 1]], [[0, 2, 3], [1, 2, 3]], 1.0]], [[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]], ['half grid step', [[[[0.5, 0], [2.5, 0], [1.5, 1], [0.4, 0], [2.6, 0]], [[0, 1, 2], [3, 4, 2]], 1.0]], [[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]], ['collapsing triangle', [[[[0, 0], [0.01, 0], [1, 1], [2, 2]], [[0, 1, 2], [1, 2, 3]], 0.1]], [[0, 2, 3], [[0, 1, 2]]]], ['rotated duplicate', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [1, 2, 0], [2, 1, 0]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]]], [['collapsing triangle', [[[[0, 0], [0.01, 0], [1, 1], [2, 2]], [[0, 1, 2], [1, 2, 3]], 0.1]], [[0, 2, 3], [[0, 1, 2]]]], ['rotated duplicate', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [1, 2, 0], [2, 1, 0]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['wrap-around duplicate', [[[[0, 0], [1, 0], [1, 1], [0.01, 0.01]], [[0, 1, 2, 3], [3, 0, 1, 2, 3]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['trailing repeats', [[[[0, 0], [1, 0], [1, 1], [0, 1], [0.02, 0]], [[0, 1, 2, 4, 4], [0, 2, 3]], 0.1]], [[0, 1, 2, 3], [[0, 1, 2], [0, 2, 3]]]], ['three points weld to one', [[[[5, 5], [5.01, 5], [4.99, 5.01], [6, 5], [5, 6]], [[3, 0, 4], [1, 3, 4], [2, 4, 3]], 0.1]], [[0, 3, 4], [[0, 2, 1], [0, 1, 2]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['two-corner collapse', [[[[0, 0], [0, 0.01], [3, 3], [3.01, 3]], [[0, 2, 1, 3]], 0.1]], [[0, 2], [[0, 1, 0, 1]]]]], [['trailing repeats', [[[[0, 0], [1, 0], [1, 1], [0, 1], [0.02, 0]], [[0, 1, 2, 4, 4], [0, 2, 3]], 0.1]], [[0, 1, 2, 3], [[0, 1, 2], [0, 2, 3]]]], ['three points weld to one', [[[[5, 5], [5.01, 5], [4.99, 5.01], [6, 5], [5, 6]], [[3, 0, 4], [1, 3, 4], [2, 4, 3]], 0.1]], [[0, 3, 4], [[0, 2, 1], [0, 1, 2]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['opposite orientation kept', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [0, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['late lowest representative', [[[[3, 3], [1, 1], [3.04, 3], [1, 0], [0, 0]], [[1, 3, 0], [4, 3, 2]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2], [0, 3, 2]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]], ['two-corner collapse', [[[[0, 0], [0, 0.01], [3, 3], [3.01, 3]], [[0, 2, 1, 3]], 0.1]], [[0, 2], [[0, 1, 0, 1]]]]], [['near duplicate corner', [[[[0, 0], [1, 0], [0, 1], [1.02, 0.01]], [[0, 1, 2], [3, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['negative coordinates', [[[[-0.7, 0], [0.3, 0], [0, 1], [1, 1]], [[0, 2, 3], [1, 2, 3]], 1.0]], [[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]], ['half grid step', [[[[0.5, 0], [2.5, 0], [1.5, 1], [0.4, 0], [2.6, 0]], [[0, 1, 2], [3, 4, 2]], 1.0]], [[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]], ['opposite orientation kept', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [0, 2, 1]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['late lowest representative', [[[[3, 3], [1, 1], [3.04, 3], [1, 0], [0, 0]], [[1, 3, 0], [4, 3, 2]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2], [0, 3, 2]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]], ['two-corner collapse', [[[[0, 0], [0, 0.01], [3, 3], [3.01, 3]], [[0, 2, 1, 3]], 0.1]], [[0, 2], [[0, 1, 0, 1]]]]], [['negative coordinates', [[[[-0.7, 0], [0.3, 0], [0, 1], [1, 1]], [[0, 2, 3], [1, 2, 3]], 1.0]], [[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]], ['half grid step', [[[[0.5, 0], [2.5, 0], [1.5, 1], [0.4, 0], [2.6, 0]], [[0, 1, 2], [3, 4, 2]], 1.0]], [[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]], ['collapsing triangle', [[[[0, 0], [0.01, 0], [1, 1], [2, 2]], [[0, 1, 2], [1, 2, 3]], 0.1]], [[0, 2, 3], [[0, 1, 2]]]], ['rotated duplicate', [[[[0, 0], [1, 0], [0, 1]], [[0, 1, 2], [1, 2, 0], [2, 1, 0]], 0.1]], [[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]], ['wrap-around duplicate', [[[[0, 0], [1, 0], [1, 1], [0.01, 0.01]], [[0, 1, 2, 3], [3, 0, 1, 2, 3]], 0.1]], [[0, 1, 2], [[0, 1, 2]]]], ['non-consecutive repeat kept', [[[[0, 0], [1, 0], [2, 0], [1, 1], [1.01, 1]], [[0, 1, 3, 2, 4]], 0.1]], [[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]], ['bowed quad two repeats', [[[[0, 0], [1, 0], [1, 0.01], [1, 1], [0, 1]], [[0, 1, 2, 3, 4]], 0.1]], [[0, 1, 3, 4], [[0, 1, 2, 3]]]]]]
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
near duplicate corner	`[[0, 1, 2], [[0, 1, 2]]]`	`[[0, 1, 2], [[0, 1, 2]]]`	Passed
negative coordinates	`[[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]`	`[[0, 1, 2, 3], [[0, 2, 3], [1, 2, 3]]]`	Passed
half grid step	`[[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]`	`[[0, 1, 2, 3], [[0, 1, 2], [1, 2, 3]]]`	Passed
collapsing triangle	`[[0, 2, 3], [[0, 1, 2]]]`	`[[0, 2, 3], [[0, 1, 2]]]`	Passed
rotated duplicate	`[[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]`	`[[0, 1, 2], [[0, 1, 2], [0, 2, 1]]]`	Passed
non-consecutive repeat kept	`[[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]`	`[[0, 1, 2, 3], [[0, 1, 3, 2, 3]]]`	Passed
bowed quad two repeats	`[[0, 1, 3, 4], [[0, 1, 2, 3]]]`	`[[0, 1, 3, 4], [[0, 1, 2, 3]]]`	Passed

SHA-256 / ab04d0828bff4013ba127969b1b216a9b467ea87d7c906866d9d9aba6e05f8a1

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 2026-09-29T14:51:09.589372+00:00.

Case digest / d51353bbd0816fe8c2277751d27460947fe7ee7cfd636b3ae6ba87e82b4a94a3