FA-88511 / Mesh topology invariants / Open access

Loop subdivision inserts a vertex per face instead of per edge · case 01

Vertex predictions are low for every mesh.

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

ROOT CAUSE

The update uses V+F although Loop inserts one vertex on each edge.

VERIFIED REPAIR

Use V = V + E.

Unsuccessful approach: Adding both E and F imports the Catmull-Clark face point.

Case contract

Input [V,E,arities,B,scheme,levels]: counts of a mesh with face sizes "arities" and B boundary edges. Loop (triangles only, else "non-triangle"): V+=E, E=2E+3F, F=4F. Catmull-Clark: V=V+E+F, E=2E+sum(arities), faces become sum(arities) quads. Boundary edges double per level. Return [V,E,F,B,V-E+F] after the given levels.

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):
    V,E,ar,B,scheme,levels=x
    ar=list(ar)
    for _ in range(levels):
        F=len(ar)
        S=sum(ar)
        if scheme=='loop':
            if any(n!=3 for n in ar): return 'non-triangle'
            V,E,ar=V+F,2*E+3*F,[3]*(4*F)
        else:
            V,E,ar=V+E+F,2*E+S,[4]*S
        B=2*B
    return [V,E,len(ar),B,V-E+len(ar)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['tetra loop 1', [[4, 6, [3, 3, 3, 3], 0, 'loop', 1]], [10, 24, 16, 0, 2]], ['tetra loop 2', [[4, 6, [3, 3, 3, 3], 0, 'loop', 2]], [34, 96, 64, 0, 2]], ['disk loop', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 1]], [16, 35, 20, 10, 1]], ['disk loop 2', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 2]], [51, 130, 80, 20, 1]], ['cube cc', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [26, 48, 24, 0, 2]], ['cube cc 2', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 2]], [98, 192, 96, 0, 2]], ['single quad cc', [[4, 4, [4], 4, 'cc', 2]], [25, 40, 16, 16, 1]]], [['tetra loop 2', [[4, 6, [3, 3, 3, 3], 0, 'loop', 2]], [34, 96, 64, 0, 2]], ['cube cc', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [26, 48, 24, 0, 2]], ['cube cc 2', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 2]], [98, 192, 96, 0, 2]], ['single quad cc', [[4, 4, [4], 4, 'cc', 2]], [25, 40, 16, 16, 1]], ['quad and triangle cc', [[5, 6, [4, 3], 4, 'cc', 1]], [13, 19, 7, 8, 1]], ['quad and triangle cc 2', [[5, 6, [4, 3], 4, 'cc', 2]], [39, 66, 28, 16, 1]], ['pentagon cc', [[5, 5, [5], 5, 'cc', 1]], [11, 15, 5, 10, 1]]], [['disk loop', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 1]], [16, 35, 20, 10, 1]], ['quad and triangle cc', [[5, 6, [4, 3], 4, 'cc', 1]], [13, 19, 7, 8, 1]], ['quad and triangle cc 2', [[5, 6, [4, 3], 4, 'cc', 2]], [39, 66, 28, 16, 1]], ['pentagon cc', [[5, 5, [5], 5, 'cc', 1]], [11, 15, 5, 10, 1]], ['quad under loop', [[4, 4, [4], 4, 'loop', 1]], 'non-triangle'], ['triangle then quad under loop', [[5, 6, [3, 4], 4, 'loop', 1]], 'non-triangle'], ['zero levels', [[4, 6, [3, 3, 3, 3], 0, 'loop', 0]], [4, 6, 4, 0, 2]]], [['disk loop 2', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 2]], [51, 130, 80, 20, 1]], ['quad under loop', [[4, 4, [4], 4, 'loop', 1]], 'non-triangle'], ['triangle then quad under loop', [[5, 6, [3, 4], 4, 'loop', 1]], 'non-triangle'], ['zero levels', [[4, 6, [3, 3, 3, 3], 0, 'loop', 0]], [4, 6, 4, 0, 2]], ['torus cc', [[9, 18, [4, 4, 4, 4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [36, 72, 36, 0, 0]], ['triangle tetra cc', [[4, 6, [3, 3, 3, 3], 0, 'cc', 1]], [14, 24, 12, 0, 2]], ['single triangle loop 3', [[3, 3, [3], 3, 'loop', 3]], [45, 108, 64, 24, 1]]], [['tetra loop 1', [[4, 6, [3, 3, 3, 3], 0, 'loop', 1]], [10, 24, 16, 0, 2]], ['tetra loop 2', [[4, 6, [3, 3, 3, 3], 0, 'loop', 2]], [34, 96, 64, 0, 2]], ['disk loop', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 1]], [16, 35, 20, 10, 1]], ['zero levels', [[4, 6, [3, 3, 3, 3], 0, 'loop', 0]], [4, 6, 4, 0, 2]], ['torus cc', [[9, 18, [4, 4, 4, 4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [36, 72, 36, 0, 0]], ['triangle tetra cc', [[4, 6, [3, 3, 3, 3], 0, 'cc', 1]], [14, 24, 12, 0, 2]], ['single triangle loop 3', [[3, 3, [3], 3, 'loop', 3]], [45, 108, 64, 24, 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
tetra loop 1	`[8, 24, 16, 0, 0]`	`[10, 24, 16, 0, 2]`	Failed
tetra loop 2	`[24, 96, 64, 0, -8]`	`[34, 96, 64, 0, 2]`	Failed
disk loop	`[11, 35, 20, 10, -4]`	`[16, 35, 20, 10, 1]`	Failed
disk loop 2	`[31, 130, 80, 20, -19]`	`[51, 130, 80, 20, 1]`	Failed
cube cc	`[26, 48, 24, 0, 2]`	`[26, 48, 24, 0, 2]`	Passed
cube cc 2	`[98, 192, 96, 0, 2]`	`[98, 192, 96, 0, 2]`	Passed
single quad cc	`[25, 40, 16, 16, 1]`	`[25, 40, 16, 16, 1]`	Passed

SHA-256 / 1e225d6cb6429b113b4e73781602ce41b2912bd03bf490bed8e4e2f8b94350b9

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    V,E,ar,B,scheme,levels=x
    ar=list(ar)
    for _ in range(levels):
        F=len(ar)
        S=sum(ar)
        if scheme=='loop':
            if any(n!=3 for n in ar): return 'non-triangle'
            V,E,ar=V+E+F,2*E+3*F,[3]*(4*F)
        else:
            V,E,ar=V+E+F,2*E+S,[4]*S
        B=2*B
    return [V,E,len(ar),B,V-E+len(ar)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['tetra loop 1', [[4, 6, [3, 3, 3, 3], 0, 'loop', 1]], [10, 24, 16, 0, 2]], ['tetra loop 2', [[4, 6, [3, 3, 3, 3], 0, 'loop', 2]], [34, 96, 64, 0, 2]], ['disk loop', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 1]], [16, 35, 20, 10, 1]], ['disk loop 2', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 2]], [51, 130, 80, 20, 1]], ['cube cc', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [26, 48, 24, 0, 2]], ['cube cc 2', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 2]], [98, 192, 96, 0, 2]], ['single quad cc', [[4, 4, [4], 4, 'cc', 2]], [25, 40, 16, 16, 1]]], [['tetra loop 2', [[4, 6, [3, 3, 3, 3], 0, 'loop', 2]], [34, 96, 64, 0, 2]], ['cube cc', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [26, 48, 24, 0, 2]], ['cube cc 2', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 2]], [98, 192, 96, 0, 2]], ['single quad cc', [[4, 4, [4], 4, 'cc', 2]], [25, 40, 16, 16, 1]], ['quad and triangle cc', [[5, 6, [4, 3], 4, 'cc', 1]], [13, 19, 7, 8, 1]], ['quad and triangle cc 2', [[5, 6, [4, 3], 4, 'cc', 2]], [39, 66, 28, 16, 1]], ['pentagon cc', [[5, 5, [5], 5, 'cc', 1]], [11, 15, 5, 10, 1]]], [['disk loop', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 1]], [16, 35, 20, 10, 1]], ['quad and triangle cc', [[5, 6, [4, 3], 4, 'cc', 1]], [13, 19, 7, 8, 1]], ['quad and triangle cc 2', [[5, 6, [4, 3], 4, 'cc', 2]], [39, 66, 28, 16, 1]], ['pentagon cc', [[5, 5, [5], 5, 'cc', 1]], [11, 15, 5, 10, 1]], ['quad under loop', [[4, 4, [4], 4, 'loop', 1]], 'non-triangle'], ['triangle then quad under loop', [[5, 6, [3, 4], 4, 'loop', 1]], 'non-triangle'], ['zero levels', [[4, 6, [3, 3, 3, 3], 0, 'loop', 0]], [4, 6, 4, 0, 2]]], [['disk loop 2', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 2]], [51, 130, 80, 20, 1]], ['quad under loop', [[4, 4, [4], 4, 'loop', 1]], 'non-triangle'], ['triangle then quad under loop', [[5, 6, [3, 4], 4, 'loop', 1]], 'non-triangle'], ['zero levels', [[4, 6, [3, 3, 3, 3], 0, 'loop', 0]], [4, 6, 4, 0, 2]], ['torus cc', [[9, 18, [4, 4, 4, 4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [36, 72, 36, 0, 0]], ['triangle tetra cc', [[4, 6, [3, 3, 3, 3], 0, 'cc', 1]], [14, 24, 12, 0, 2]], ['single triangle loop 3', [[3, 3, [3], 3, 'loop', 3]], [45, 108, 64, 24, 1]]], [['tetra loop 1', [[4, 6, [3, 3, 3, 3], 0, 'loop', 1]], [10, 24, 16, 0, 2]], ['tetra loop 2', [[4, 6, [3, 3, 3, 3], 0, 'loop', 2]], [34, 96, 64, 0, 2]], ['disk loop', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 1]], [16, 35, 20, 10, 1]], ['zero levels', [[4, 6, [3, 3, 3, 3], 0, 'loop', 0]], [4, 6, 4, 0, 2]], ['torus cc', [[9, 18, [4, 4, 4, 4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [36, 72, 36, 0, 0]], ['triangle tetra cc', [[4, 6, [3, 3, 3, 3], 0, 'cc', 1]], [14, 24, 12, 0, 2]], ['single triangle loop 3', [[3, 3, [3], 3, 'loop', 3]], [45, 108, 64, 24, 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
tetra loop 1	`[14, 24, 16, 0, 6]`	`[10, 24, 16, 0, 2]`	Failed
tetra loop 2	`[54, 96, 64, 0, 22]`	`[34, 96, 64, 0, 2]`	Failed
disk loop	`[21, 35, 20, 10, 6]`	`[16, 35, 20, 10, 1]`	Failed
disk loop 2	`[76, 130, 80, 20, 26]`	`[51, 130, 80, 20, 1]`	Failed
cube cc	`[26, 48, 24, 0, 2]`	`[26, 48, 24, 0, 2]`	Passed
cube cc 2	`[98, 192, 96, 0, 2]`	`[98, 192, 96, 0, 2]`	Passed
single quad cc	`[25, 40, 16, 16, 1]`	`[25, 40, 16, 16, 1]`	Passed

SHA-256 / 0eb833786970055e5e25871c27a1f2b104e8f43a3158b1ee4c5ae6d301c8e9e7

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    V,E,ar,B,scheme,levels=x
    ar=list(ar)
    for _ in range(levels):
        F=len(ar)
        S=sum(ar)
        if scheme=='loop':
            if any(n!=3 for n in ar): return 'non-triangle'
            V,E,ar=V+E,2*E+3*F,[3]*(4*F)
        else:
            V,E,ar=V+E+F,2*E+S,[4]*S
        B=2*B
    return [V,E,len(ar),B,V-E+len(ar)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['tetra loop 1', [[4, 6, [3, 3, 3, 3], 0, 'loop', 1]], [10, 24, 16, 0, 2]], ['tetra loop 2', [[4, 6, [3, 3, 3, 3], 0, 'loop', 2]], [34, 96, 64, 0, 2]], ['disk loop', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 1]], [16, 35, 20, 10, 1]], ['disk loop 2', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 2]], [51, 130, 80, 20, 1]], ['cube cc', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [26, 48, 24, 0, 2]], ['cube cc 2', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 2]], [98, 192, 96, 0, 2]], ['single quad cc', [[4, 4, [4], 4, 'cc', 2]], [25, 40, 16, 16, 1]]], [['tetra loop 2', [[4, 6, [3, 3, 3, 3], 0, 'loop', 2]], [34, 96, 64, 0, 2]], ['cube cc', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [26, 48, 24, 0, 2]], ['cube cc 2', [[8, 12, [4, 4, 4, 4, 4, 4], 0, 'cc', 2]], [98, 192, 96, 0, 2]], ['single quad cc', [[4, 4, [4], 4, 'cc', 2]], [25, 40, 16, 16, 1]], ['quad and triangle cc', [[5, 6, [4, 3], 4, 'cc', 1]], [13, 19, 7, 8, 1]], ['quad and triangle cc 2', [[5, 6, [4, 3], 4, 'cc', 2]], [39, 66, 28, 16, 1]], ['pentagon cc', [[5, 5, [5], 5, 'cc', 1]], [11, 15, 5, 10, 1]]], [['disk loop', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 1]], [16, 35, 20, 10, 1]], ['quad and triangle cc', [[5, 6, [4, 3], 4, 'cc', 1]], [13, 19, 7, 8, 1]], ['quad and triangle cc 2', [[5, 6, [4, 3], 4, 'cc', 2]], [39, 66, 28, 16, 1]], ['pentagon cc', [[5, 5, [5], 5, 'cc', 1]], [11, 15, 5, 10, 1]], ['quad under loop', [[4, 4, [4], 4, 'loop', 1]], 'non-triangle'], ['triangle then quad under loop', [[5, 6, [3, 4], 4, 'loop', 1]], 'non-triangle'], ['zero levels', [[4, 6, [3, 3, 3, 3], 0, 'loop', 0]], [4, 6, 4, 0, 2]]], [['disk loop 2', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 2]], [51, 130, 80, 20, 1]], ['quad under loop', [[4, 4, [4], 4, 'loop', 1]], 'non-triangle'], ['triangle then quad under loop', [[5, 6, [3, 4], 4, 'loop', 1]], 'non-triangle'], ['zero levels', [[4, 6, [3, 3, 3, 3], 0, 'loop', 0]], [4, 6, 4, 0, 2]], ['torus cc', [[9, 18, [4, 4, 4, 4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [36, 72, 36, 0, 0]], ['triangle tetra cc', [[4, 6, [3, 3, 3, 3], 0, 'cc', 1]], [14, 24, 12, 0, 2]], ['single triangle loop 3', [[3, 3, [3], 3, 'loop', 3]], [45, 108, 64, 24, 1]]], [['tetra loop 1', [[4, 6, [3, 3, 3, 3], 0, 'loop', 1]], [10, 24, 16, 0, 2]], ['tetra loop 2', [[4, 6, [3, 3, 3, 3], 0, 'loop', 2]], [34, 96, 64, 0, 2]], ['disk loop', [[6, 10, [3, 3, 3, 3, 3], 5, 'loop', 1]], [16, 35, 20, 10, 1]], ['zero levels', [[4, 6, [3, 3, 3, 3], 0, 'loop', 0]], [4, 6, 4, 0, 2]], ['torus cc', [[9, 18, [4, 4, 4, 4, 4, 4, 4, 4, 4], 0, 'cc', 1]], [36, 72, 36, 0, 0]], ['triangle tetra cc', [[4, 6, [3, 3, 3, 3], 0, 'cc', 1]], [14, 24, 12, 0, 2]], ['single triangle loop 3', [[3, 3, [3], 3, 'loop', 3]], [45, 108, 64, 24, 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
tetra loop 1	`[10, 24, 16, 0, 2]`	`[10, 24, 16, 0, 2]`	Passed
tetra loop 2	`[34, 96, 64, 0, 2]`	`[34, 96, 64, 0, 2]`	Passed
disk loop	`[16, 35, 20, 10, 1]`	`[16, 35, 20, 10, 1]`	Passed
disk loop 2	`[51, 130, 80, 20, 1]`	`[51, 130, 80, 20, 1]`	Passed
cube cc	`[26, 48, 24, 0, 2]`	`[26, 48, 24, 0, 2]`	Passed
cube cc 2	`[98, 192, 96, 0, 2]`	`[98, 192, 96, 0, 2]`	Passed
single quad cc	`[25, 40, 16, 16, 1]`	`[25, 40, 16, 16, 1]`	Passed

SHA-256 / 2f4a0522fb3e21854a957e1ca583c933a084f31c72c55b9f36ba5fbd2965699b

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 2026-09-29T14:51:08.830461+00:00.

Case digest / 4940aa882f4e1eb96817087c0da5edfad56214ec0da4dc96f9227fd32b6ca4ba