FA-87741 / Collision detection broadphase / Open access

Particles in colliding hash buckets are paired regardless of cell · case 01

Distant particles that hash to the same bucket become neighbours.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The exact-cell comparison is missing.

THE FAILURE

The exact-cell comparison is missing.

Unsuccessful approach: Comparing only the x cell still pairs particles in different rows.

Case contract

solve(frames, n): particle hash table of n buckets reused across frames without clearing; a bucket whose stamp is older than the frame is lazily reset on first insert. Cell = floor(x), floor(y); bucket = ((ix*73856093 mod 2^32) xor (iy*19349663 mod 2^32)) mod n. A particle pairs with earlier particles in the same bucket only if they share the exact cell. Per frame return [sorted [min,max] pairs, buckets stamped this frame].

Why this case matters

Broadphase stages decide which object pairs ever reach narrowphase; a wrong boundary, ordering or bookkeeping rule silently drops real contacts or floods the solver with false candidates.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(frames, n):
    table = [[-1, []] for _ in range(n)]
    result = []
    for f, pts in enumerate(frames):
        pairs = []
        for pid, x, y in pts:
            ix, iy = math.floor(x), math.floor(y)
            h = (((ix * 73856093) & 0xFFFFFFFF) ^ ((iy * 19349663) & 0xFFFFFFFF)) % n
            slot = table[h]
            if slot[0] != f:
                slot[0] = f
                slot[1] = []
            for qid, qx, qy in slot[1]:
                if True:
                    pairs.append([min(pid, qid), max(pid, qid)])
            slot[1].append((pid, ix, iy))
        result.append([sorted(pairs), sum(1 for s in table if s[0] == f)])
    return result
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[[[[[[4, -3.0, -2.5], [1, -2.5, -1.0]], [[3, -3.0, 2.5], [8, 1.5, 2.0]], [[3, 0.5, 3.0], [14, -2.5, 0.5]]], 11], [[[], 2], [[], 2], [[], 2]]], [[[[[13, 0.5, -1.5], [9, 1.5, 1.5], [11, 0.0, -3.0], [5, 3.0, 3.0], [15, -3.0, -2.5]], [[11, -0.5, -1.0], [13, 2.5, -3.0]], [[9, 1.0, 2.5], [19, 3.0, 1.5], [2, -2.0, 2.0], [11, -2.5, 3.0], [18, -1.0, -2.0], [16, -1.5, -3.0]]], 11], [[[], 3], [[], 2], [[], 4]]], [[[[[15, 1.0, 3.0], [12, 1.5, 1.0]]], 11], [[[], 2]]], [[[[[9, -1.5, -0.5], [10, -1.5, 0.0], [15, -0.5, 1.5]], [[19, 0.0, 3.0], [6, 1.5, -3.0], [18, 3.0, -1.5], [14, -1.5, -3.0], [7, 1.5, 2.5]]], 11], [[[], 3], [[], 4]]], [[[[[7, 3.0, -0.5], [4, 0.5, -0.5], [3, -2.0, 2.5], [17, 1.0, 1.5]], [[17, 1.5, -2.0], [4, -1.5, 0.0], [8, 1.0, 1.5]], [[7, 2.0, -2.0], [4, -2.5, 2.0]], [[18, 3.0, -3.0], [13, -1.5, 2.0], [17, 2.0, 2.5], [2, 1.5, 2.5]]], 11], [[[], 4], [[], 2], [[], 2], [[], 2]]], [[[[[4, 1.5, 2.5], [13, -1.0, 0.5]], [[18, -1.5, -0.5], [16, 0.0, 3.0], [6, 1.0, -1.5], [1, 0.0, 3.0], [8, -2.5, 2.5], [4, -2.5, 0.5]]], 3], [[[], 2], [[[1, 16]], 3]]], [[[[[1, 2.5, 1.5], [19, 1.0, 0.5], [8, 3.0, -2.5], [10, -3.0, 2.0], [14, 1.0, 2.5]], [[9, 0.0, 2.5], [17, 1.5, 0.0], [14, -1.5, 2.5], [10, 0.5, 2.5]], [[1, 2.5, 1.0], [2, -1.0, -2.5], [14, -3.0, 1.0], [10, -2.0, 1.0], [7, -2.0, 0.5], [9, 2.5, 1.5]], [[18, -0.5, -1.0], [4, -2.0, -1.5], [16, -1.5, 0.0], [6, -2.0, -3.0], [12, 1.5, 1.0], [19, 1.0, -2.5]]], 3], [[[], 2], [[[9, 10]], 3], [[[1, 9]], 3], [[], 3]]], [[[[[12, 1.5, 1.0], [2, 1.0, 1.0]]], 3], [[[[2, 12]], 1]]]], [[[[[[1, -0.5, -1.0], [8, 1.0, 0.5]], [[7, -1.0, -1.0], [1, 0.0, 2.5]], [[11, 1.0, 3.0], [7, 3.0, 2.5], [5, 1.0, -2.5], [10, 3.0, 0.0], [18, -0.5, -1.0]], [[11, -0.5, 2.5], [4, 0.5, 2.0]]], 5], [[[], 2], [[], 2], [[], 3], [[], 2]]], [[[[[1, 0.5, 1.0], [2, 1.5, 0.0], [6, 3.0, 1.0], [3, 2.5, 2.5], [12, -1.0, -2.0]], [[2, 0.5, -2.0], [5, 2.5, -1.5]], [[11, -3.0, 3.0], [1, 0.0, 1.5], [7, 1.0, 2.5], [15, -2.0, 2.5], [2, -1.5, 3.0], [4, 2.5, 2.0]]], 11], [[[], 3], [[], 2], [[], 4]]], [[[[[5, -2.5, 2.5], [18, -3.0, -1.5], [15, -0.5, -1.0], [12, 1.0, -2.0], [9, 0.0, 1.5]], [[10, -1.5, 1.0], [17, -2.0, -1.5], [13, 3.0, 1.5], [14, 0.5, 0.5], [8, 3.0, 2.0], [2, -0.5, -3.0]]], 3], [[[], 3], [[], 2]]], [[[[[10, 2.0, 0.0], [11, 3.0, -1.5], [13, 2.5, -3.0], [18, -3.0, -1.0], [12, 3.0, -3.0], [14, 1.5, 2.0]], [[19, 0.5, 0.0], [16, -0.5, -1.5], [10, 0.5, 3.0], [14, 3.0, -2.5]], [[6, -3.0, 1.0], [19, -1.0, 3.0]]], 11], [[[], 6], [[], 4], [[], 2]]], [[[[[9, 2.5, 3.0], [3, -1.5, 1.0]], [[4, -1.5, 2.5], [1, 0.0, 0.5], [19, 2.5, 1.5]], [[16, 3.0, -2.0], [9, 0.5, 1.5]], [[8, 1.0, -1.0], [12, 0.0, -1.5], [6, 0.0, -1.5], [17, -1.5, -2.0], [19, 2.5, -1.5]]], 3], [[[], 2], [[], 2], [[], 1], [[[6, 12]], 2]]], [[[[[6, -1.5, -3.0], [3, 0.0, 2.0]], [[11, 3.0, 2.0], [17, 2.0, 1.5], [12, 0.5, -1.5], [14, 1.5, 0.5], [1, 1.0, -3.0]]], 11], [[[], 2], [[], 5]]], [[[[[10, 1.5, -2.0], [17, 0.0, 3.0]], [[16, -1.5, 0.5], [2, 2.0, 3.0]]], 5], [[[], 2], [[], 2]]], [[[[[16, -1.5, -2.5], [10, -2.0, 0.0], [6, -2.5, 1.0]], [[19, -1.0, 2.5], [18, -0.5, -0.5], [13, 1.0, -0.5], [2, -0.5, 2.5], [9, 2.5, -1.5]], [[6, 0.5, -2.5], [18, 2.0, 3.0], [12, 2.5, 1.0], [17, -1.5, -2.5], [19, -0.5, -0.5], [3, 2.5, 1.0]], [[1, 0.5, 3.0], [10, 1.0, 3.0], [12, 2.0, -1.5], [13, -1.5, -0.5]]], 7], [[[], 2], [[[2, 19]], 3], [[[3, 12]], 4], [[], 3]]]], [[[[[[1, 1.0, -0.5], [7, 0.5, -1.0]]], 7], [[[], 2]]], [[[[[17, 2.5, 1.0], [12, 2.5, -1.5], [13, 1.0, -1.0], [7, -3.0, -2.5]]], 3], [[[], 3]]], [[[[[16, 1.5, -1.5], [3, -1.0, -2.5], [10, -2.0, 3.0]], [[1, -1.5, -0.5], [14, 1.5, 0.5]]], 5], [[[], 3], [[], 2]]], [[[[[1, -2.0, 3.0], [3, 1.0, 0.5], [17, -1.5, -1.0], [12, 0.5, -2.0], [15, -0.5, -1.0], [6, 0.5, 3.0]], [[2, -0.5, -2.5], [14, -3.0, -2.5], [7, -0.5, 0.0], [17, 1.5, 2.0], [12, 3.0, -3.0], [13, 0.0, -1.0]], [[7, -2.5, 0.0], [13, -2.0, 0.0], [14, -1.0, 1.5], [18, -1.0, 2.5], [9, 0.5, 2.5], [4, 2.0, -2.5]]], 3], [[[], 2], [[], 3], [[], 3]]], [[[[[2, 2.0, -3.0], [16, -1.5, -0.5], [10, 3.0, 2.5], [13, -1.0, -1.0], [12, -1.5, -2.0], [19, 3.0, 2.5]], [[12, 2.0, 1.0], [9, 2.0, -1.5]]], 3], [[[[10, 19]], 2], [[], 2]]], [[[[[8, 2.5, -0.5], [16, 0.5, 1.5], [13, 3.0, -1.5], [6, 3.0, -2.0]], [[17, -1.0, 0.0], [10, 2.5, -2.5], [8, 3.0, 3.0], [12, -1.0, -3.0]]], 11], [[[[6, 13]], 3], [[], 4]]], [[[[[15, -0.5, 3.0], [14, -2.5, 0.5], [10, 2.5, 0.5], [11, -0.5, 0.5], [19, 1.0, 1.0]], [[14, 3.0, -2.5], [17, 2.0, 1.0], [5, 1.5, 2.5], [3, 3.0, 0.0], [19, -1.0, -3.0]], [[14, 0.5, -1.0], [4, -0.5, 0.0], [15, 0.5, -1.5]], [[10, -2.5, -1.0], [8, -0.5, 2.0], [9, 0.5, -3.0], [3, 2.5, 1.5], [19, -2.0, -3.0], [1, -1.5, 0.0]]], 3], [[[], 3], [[], 3], [[], 2], [[], 2]]], [[[[[17, 0.5, -1.0], [11, -2.0, 2.5], [8, 2.5, -1.5], [19, -1.0, 2.0], [1, 1.0, 3.0]]], 11], [[[], 4]]]], [[[[[[4, -1.5, -0.5], [9, -1.0, -0.5], [19, -0.5, -1.5], [12, -2.5, 2.5], [13, -1.0, 2.0], [17, 0.0, 1.0]], [[6, 1.0, 2.0], [4, -1.5, -2.0], [15, -1.0, -3.0], [9, -1.5, 1.5], [14, -2.0, -1.5], [18, -3.0, 2.5]]], 3], [[[], 3], [[[4, 14]], 2]]], [[[[[18, -0.5, 1.5], [15, -2.5, -0.5], [1, 1.5, 2.0], [8, 2.5, -1.0]]], 11], [[[], 4]]], [[[[[19, -2.5, 1.5], [1, 1.0, -1.0], [18, -3.0, 2.0], [10, -1.5, 0.0], [11, 0.0, 2.0], [4, -1.0, -1.0]], [[15, -0.5, 3.0], [14, -0.5, -1.5], [4, -1.5, -3.0], [8, 1.5, -2.0], [6, -2.0, 0.0]], [[5, -0.5, 0.0], [9, 1.0, 2.0]], [[15, -1.0, -3.0], [16, 2.0, -2.5], [13, -2.0, -3.0], [14, 1.0, 3.0], [4, 2.0, -2.5], [11, -1.5, 2.0]]], 5], [[[], 3], [[], 3], [[], 2], [[[4, 16]], 2]]], [[[[[9, 3.0, 0.5], [10, 1.0, 2.5], [19, -2.0, -1.5]], [[11, -3.0, 1.0], [13, 1.0, -0.5]], [[13, -2.0, 1.0], [17, 1.5, -0.5]], [[6, -3.0, 3.0], [19, 1.0, 0.5], [15, -1.0, 0.0], [11, 0.0, 0.5], [9, 0.0, -2.0], [4, -0.5, -0.5]]], 5], [[[], 3], [[], 1], [[], 2], [[], 3]]], [[[[[10, -3.0, 0.5], [19, 0.0, 2.5], [15, 0.0, -1.0], [16, -0.5, 3.0], [8, -3.0, 3.0]], [[3, -2.5, 3.0], [12, 2.0, 0.0], [2, 3.0, -2.5]]], 7], [[[], 4], [[], 1]]], [[[[[1, -1.5, 0.5], [4, -1.0, 2.0], [19, 2.5, 1.5]], [[2, 2.0, 2.0], [18, 2.0, 0.0], [19, 0.0, -1.5], [14, -0.5, -2.0]], [[19, 3.0, -2.0], [13, 2.0, -2.5], [1, -3.0, 2.5], [17, 0.5, 2.0], [4, 1.5, -2.5]], [[17, -2.0, -1.0], [16, -3.0, 0.0]]], 11], [[[], 3], [[], 4], [[], 5], [[], 2]]], [[[[[11, 2.5, -1.0], [10, -3.0, -3.0], [16, 0.5, -0.5]], [[7, -1.5, 2.5], [9, 3.0, 2.5]], [[9, -2.5, -2.0], [18, -1.0, 1.0], [11, -2.5, 1.5], [4, 1.5, -3.0]], [[18, 1.5, -1.0], [7, 3.0, -1.0], [10, -2.0, 0.0]]], 3], [[[], 2], [[], 1], [[], 1], [[], 2]]], [[[[[19, 1.5, 0.5], [1, -0.5, 0.5], [14, 2.0, -2.0]], [[7, -1.0, 0.5], [14, 3.0, -3.0]]], 11], [[[], 3], [[], 2]]]], [[[[[[3, 1.5, 1.0], [8, -2.0, 0.0], [9, -1.5, -1.0], [18, 1.5, 2.5]]], 11], [[[], 4]]], [[[[[10, -0.5, 3.0], [5, 2.5, 3.0], [18, 1.5, -3.0], [9, -2.0, 2.5], [16, -3.0, -2.5], [13, -2.0, -2.5]], [[6, 2.0, 0.0], [16, -0.5, 1.5], [10, 2.0, -2.0], [9, 1.5, 0.5], [8, 0.5, 1.0], [1, 1.0, -2.0]], [[6, -1.0, 1.0], [18, 2.0, -3.0], [2, -1.5, 0.0], [1, 1.0, 2.5], [19, -2.5, -0.5], [13, -0.5, 1.0]], [[12, 0.0, 0.5], [10, -3.0, 2.5], [1, 1.0, -0.5], [7, 0.0, 2.5], [18, 0.5, -1.5]]], 11], [[[], 5], [[], 5], [[[6, 13]], 5], [[], 4]]], [[[[[4, -3.0, 0.5], [6, 2.0, 3.0], [13, 0.5, -1.5]], [[12, -0.5, 0.0], [13, 2.5, -2.5], [19, 3.0, -2.0], [3, -0.5, 3.0]], [[11, 0.5, -1.5], [6, -1.0, -1.5], [3, -2.0, 2.0], [4, 2.5, 2.5]], [[19, -1.5, 0.5], [9, -1.0, 3.0], [8, 1.5, -3.0], [4, 2.5, -2.5], [12, -1.5, 0.0], [1, -3.0, -1.0]]], 3], [[[], 2], [[], 2], [[], 2], [[[12, 19]], 3]]], [[[[[18, 0.5, -3.0], [17, 2.0, 2.0], [15, 2.0, 1.0], [7, 1.5, -1.5], [10, 1.5, 2.5]]], 7], [[[], 5]]], [[[[[14, -1.0, 0.5], [17, 0.0, 3.0], [11, -1.0, 1.5]]], 11], [[[], 3]]], [[[[[16, -2.0, 1.0], [14, 0.0, 2.5], [15, -1.5, -0.5], [1, -3.0, 2.0], [18, -2.0, 2.0], [7, -1.0, 3.0]]], 7], [[[], 4]]], [[[[[4, 2.0, 2.5], [7, -1.5, 2.0], [18, 3.0, 3.0]], [[2, -2.0, 0.5], [19, 2.5, -1.0], [8, -2.5, -3.0], [1, -3.0, -0.5], [7, -3.0, 0.0]], [[16, 1.5, -0.5], [7, 0.5, -1.0], [6, -1.5, 1.5], [3, -0.5, -1.5], [10, 1.0, -1.0]]], 7], [[[], 3], [[], 4], [[[10, 16]], 3]]], [[[[[17, 2.0, -2.0], [12, 1.0, -0.5], [3, 1.0, -1.0]], [[13, 0.0, 0.0], [16, -2.0, 0.0]], [[19, 3.0, 3.0], [11, 1.0, 0.5], [12, 2.5, 1.5], [3, -2.5, 2.5]], [[15, 0.5, -0.5], [7, 1.0, 3.0]]], 5], [[[[3, 12]], 2], [[], 1], [[], 3], [[], 2]]]]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("case %d" % i, 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
case 0	`[[[], 2], [[], 2], [[], 2]]`	`[[[], 2], [[], 2], [[], 2]]`	Passed
case 1	`[[[[5, 11], [5, 15], [11, 15]], 3], [[], 2], [[[2, 11], [18, 19]], 4]]`	`[[[], 3], [[], 2], [[], 4]]`	Failed
case 2	`[[[], 2]]`	`[[[], 2]]`	Passed
case 3	`[[[], 3], [[[6, 7]], 4]]`	`[[[], 3], [[], 4]]`	Failed
case 4	`[[[], 4], [[[8, 17]], 2], [[], 2], [[[2, 17], [13, 18]], 2]]`	`[[[], 4], [[], 2], [[], 2], [[], 2]]`	Failed
case 5	`[[[], 2], [[[1, 16], [1, 18], [4, 8], [16, 18]], 3]]`	`[[[], 2], [[[1, 16]], 3]]`	Failed
case 6	`[[[[1, 10], [1, 14], [8, 19], [10, 14]], 2], [[[9, 10]], 3], [[[1, 9], [1, 10], [2, 7], [9, 10]], 3], [[[4, 12], [4, 16], [4, 18], [12, 16], [12, 18], [16, 18]], 3]]`	`[[[], 2], [[[9, 10]], 3], [[[1, 9]], 3], [[], 3]]`	Failed
case 7	`[[[[2, 12]], 1]]`	`[[[[2, 12]], 1]]`	Passed

SHA-256 / 6e6055cf8abe6d5b9ac34e96fdd8864025d55f1f78a7cc363da6bac537c8f468

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(frames, n):
    table = [[-1, []] for _ in range(n)]
    result = []
    for f, pts in enumerate(frames):
        pairs = []
        for pid, x, y in pts:
            ix, iy = math.floor(x), math.floor(y)
            h = (((ix * 73856093) & 0xFFFFFFFF) ^ ((iy * 19349663) & 0xFFFFFFFF)) % n
            slot = table[h]
            if slot[0] != f:
                slot[0] = f
                slot[1] = []
            for qid, qx, qy in slot[1]:
                if qx == ix:
                    pairs.append([min(pid, qid), max(pid, qid)])
            slot[1].append((pid, ix, iy))
        result.append([sorted(pairs), sum(1 for s in table if s[0] == f)])
    return result
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[[[[[[4, -3.0, -2.5], [1, -2.5, -1.0]], [[3, -3.0, 2.5], [8, 1.5, 2.0]], [[3, 0.5, 3.0], [14, -2.5, 0.5]]], 11], [[[], 2], [[], 2], [[], 2]]], [[[[[13, 0.5, -1.5], [9, 1.5, 1.5], [11, 0.0, -3.0], [5, 3.0, 3.0], [15, -3.0, -2.5]], [[11, -0.5, -1.0], [13, 2.5, -3.0]], [[9, 1.0, 2.5], [19, 3.0, 1.5], [2, -2.0, 2.0], [11, -2.5, 3.0], [18, -1.0, -2.0], [16, -1.5, -3.0]]], 11], [[[], 3], [[], 2], [[], 4]]], [[[[[15, 1.0, 3.0], [12, 1.5, 1.0]]], 11], [[[], 2]]], [[[[[9, -1.5, -0.5], [10, -1.5, 0.0], [15, -0.5, 1.5]], [[19, 0.0, 3.0], [6, 1.5, -3.0], [18, 3.0, -1.5], [14, -1.5, -3.0], [7, 1.5, 2.5]]], 11], [[[], 3], [[], 4]]], [[[[[7, 3.0, -0.5], [4, 0.5, -0.5], [3, -2.0, 2.5], [17, 1.0, 1.5]], [[17, 1.5, -2.0], [4, -1.5, 0.0], [8, 1.0, 1.5]], [[7, 2.0, -2.0], [4, -2.5, 2.0]], [[18, 3.0, -3.0], [13, -1.5, 2.0], [17, 2.0, 2.5], [2, 1.5, 2.5]]], 11], [[[], 4], [[], 2], [[], 2], [[], 2]]], [[[[[4, 1.5, 2.5], [13, -1.0, 0.5]], [[18, -1.5, -0.5], [16, 0.0, 3.0], [6, 1.0, -1.5], [1, 0.0, 3.0], [8, -2.5, 2.5], [4, -2.5, 0.5]]], 3], [[[], 2], [[[1, 16]], 3]]], [[[[[1, 2.5, 1.5], [19, 1.0, 0.5], [8, 3.0, -2.5], [10, -3.0, 2.0], [14, 1.0, 2.5]], [[9, 0.0, 2.5], [17, 1.5, 0.0], [14, -1.5, 2.5], [10, 0.5, 2.5]], [[1, 2.5, 1.0], [2, -1.0, -2.5], [14, -3.0, 1.0], [10, -2.0, 1.0], [7, -2.0, 0.5], [9, 2.5, 1.5]], [[18, -0.5, -1.0], [4, -2.0, -1.5], [16, -1.5, 0.0], [6, -2.0, -3.0], [12, 1.5, 1.0], [19, 1.0, -2.5]]], 3], [[[], 2], [[[9, 10]], 3], [[[1, 9]], 3], [[], 3]]], [[[[[12, 1.5, 1.0], [2, 1.0, 1.0]]], 3], [[[[2, 12]], 1]]]], [[[[[[1, -0.5, -1.0], [8, 1.0, 0.5]], [[7, -1.0, -1.0], [1, 0.0, 2.5]], [[11, 1.0, 3.0], [7, 3.0, 2.5], [5, 1.0, -2.5], [10, 3.0, 0.0], [18, -0.5, -1.0]], [[11, -0.5, 2.5], [4, 0.5, 2.0]]], 5], [[[], 2], [[], 2], [[], 3], [[], 2]]], [[[[[1, 0.5, 1.0], [2, 1.5, 0.0], [6, 3.0, 1.0], [3, 2.5, 2.5], [12, -1.0, -2.0]], [[2, 0.5, -2.0], [5, 2.5, -1.5]], [[11, -3.0, 3.0], [1, 0.0, 1.5], [7, 1.0, 2.5], [15, -2.0, 2.5], [2, -1.5, 3.0], [4, 2.5, 2.0]]], 11], [[[], 3], [[], 2], [[], 4]]], [[[[[5, -2.5, 2.5], [18, -3.0, -1.5], [15, -0.5, -1.0], [12, 1.0, -2.0], [9, 0.0, 1.5]], [[10, -1.5, 1.0], [17, -2.0, -1.5], [13, 3.0, 1.5], [14, 0.5, 0.5], [8, 3.0, 2.0], [2, -0.5, -3.0]]], 3], [[[], 3], [[], 2]]], [[[[[10, 2.0, 0.0], [11, 3.0, -1.5], [13, 2.5, -3.0], [18, -3.0, -1.0], [12, 3.0, -3.0], [14, 1.5, 2.0]], [[19, 0.5, 0.0], [16, -0.5, -1.5], [10, 0.5, 3.0], [14, 3.0, -2.5]], [[6, -3.0, 1.0], [19, -1.0, 3.0]]], 11], [[[], 6], [[], 4], [[], 2]]], [[[[[9, 2.5, 3.0], [3, -1.5, 1.0]], [[4, -1.5, 2.5], [1, 0.0, 0.5], [19, 2.5, 1.5]], [[16, 3.0, -2.0], [9, 0.5, 1.5]], [[8, 1.0, -1.0], [12, 0.0, -1.5], [6, 0.0, -1.5], [17, -1.5, -2.0], [19, 2.5, -1.5]]], 3], [[[], 2], [[], 2], [[], 1], [[[6, 12]], 2]]], [[[[[6, -1.5, -3.0], [3, 0.0, 2.0]], [[11, 3.0, 2.0], [17, 2.0, 1.5], [12, 0.5, -1.5], [14, 1.5, 0.5], [1, 1.0, -3.0]]], 11], [[[], 2], [[], 5]]], [[[[[10, 1.5, -2.0], [17, 0.0, 3.0]], [[16, -1.5, 0.5], [2, 2.0, 3.0]]], 5], [[[], 2], [[], 2]]], [[[[[16, -1.5, -2.5], [10, -2.0, 0.0], [6, -2.5, 1.0]], [[19, -1.0, 2.5], [18, -0.5, -0.5], [13, 1.0, -0.5], [2, -0.5, 2.5], [9, 2.5, -1.5]], [[6, 0.5, -2.5], [18, 2.0, 3.0], [12, 2.5, 1.0], [17, -1.5, -2.5], [19, -0.5, -0.5], [3, 2.5, 1.0]], [[1, 0.5, 3.0], [10, 1.0, 3.0], [12, 2.0, -1.5], [13, -1.5, -0.5]]], 7], [[[], 2], [[[2, 19]], 3], [[[3, 12]], 4], [[], 3]]]], [[[[[[1, 1.0, -0.5], [7, 0.5, -1.0]]], 7], [[[], 2]]], [[[[[17, 2.5, 1.0], [12, 2.5, -1.5], [13, 1.0, -1.0], [7, -3.0, -2.5]]], 3], [[[], 3]]], [[[[[16, 1.5, -1.5], [3, -1.0, -2.5], [10, -2.0, 3.0]], [[1, -1.5, -0.5], [14, 1.5, 0.5]]], 5], [[[], 3], [[], 2]]], [[[[[1, -2.0, 3.0], [3, 1.0, 0.5], [17, -1.5, -1.0], [12, 0.5, -2.0], [15, -0.5, -1.0], [6, 0.5, 3.0]], [[2, -0.5, -2.5], [14, -3.0, -2.5], [7, -0.5, 0.0], [17, 1.5, 2.0], [12, 3.0, -3.0], [13, 0.0, -1.0]], [[7, -2.5, 0.0], [13, -2.0, 0.0], [14, -1.0, 1.5], [18, -1.0, 2.5], [9, 0.5, 2.5], [4, 2.0, -2.5]]], 3], [[[], 2], [[], 3], [[], 3]]], [[[[[2, 2.0, -3.0], [16, -1.5, -0.5], [10, 3.0, 2.5], [13, -1.0, -1.0], [12, -1.5, -2.0], [19, 3.0, 2.5]], [[12, 2.0, 1.0], [9, 2.0, -1.5]]], 3], [[[[10, 19]], 2], [[], 2]]], [[[[[8, 2.5, -0.5], [16, 0.5, 1.5], [13, 3.0, -1.5], [6, 3.0, -2.0]], [[17, -1.0, 0.0], [10, 2.5, -2.5], [8, 3.0, 3.0], [12, -1.0, -3.0]]], 11], [[[[6, 13]], 3], [[], 4]]], [[[[[15, -0.5, 3.0], [14, -2.5, 0.5], [10, 2.5, 0.5], [11, -0.5, 0.5], [19, 1.0, 1.0]], [[14, 3.0, -2.5], [17, 2.0, 1.0], [5, 1.5, 2.5], [3, 3.0, 0.0], [19, -1.0, -3.0]], [[14, 0.5, -1.0], [4, -0.5, 0.0], [15, 0.5, -1.5]], [[10, -2.5, -1.0], [8, -0.5, 2.0], [9, 0.5, -3.0], [3, 2.5, 1.5], [19, -2.0, -3.0], [1, -1.5, 0.0]]], 3], [[[], 3], [[], 3], [[], 2], [[], 2]]], [[[[[17, 0.5, -1.0], [11, -2.0, 2.5], [8, 2.5, -1.5], [19, -1.0, 2.0], [1, 1.0, 3.0]]], 11], [[[], 4]]]], [[[[[[4, -1.5, -0.5], [9, -1.0, -0.5], [19, -0.5, -1.5], [12, -2.5, 2.5], [13, -1.0, 2.0], [17, 0.0, 1.0]], [[6, 1.0, 2.0], [4, -1.5, -2.0], [15, -1.0, -3.0], [9, -1.5, 1.5], [14, -2.0, -1.5], [18, -3.0, 2.5]]], 3], [[[], 3], [[[4, 14]], 2]]], [[[[[18, -0.5, 1.5], [15, -2.5, -0.5], [1, 1.5, 2.0], [8, 2.5, -1.0]]], 11], [[[], 4]]], [[[[[19, -2.5, 1.5], [1, 1.0, -1.0], [18, -3.0, 2.0], [10, -1.5, 0.0], [11, 0.0, 2.0], [4, -1.0, -1.0]], [[15, -0.5, 3.0], [14, -0.5, -1.5], [4, -1.5, -3.0], [8, 1.5, -2.0], [6, -2.0, 0.0]], [[5, -0.5, 0.0], [9, 1.0, 2.0]], [[15, -1.0, -3.0], [16, 2.0, -2.5], [13, -2.0, -3.0], [14, 1.0, 3.0], [4, 2.0, -2.5], [11, -1.5, 2.0]]], 5], [[[], 3], [[], 3], [[], 2], [[[4, 16]], 2]]], [[[[[9, 3.0, 0.5], [10, 1.0, 2.5], [19, -2.0, -1.5]], [[11, -3.0, 1.0], [13, 1.0, -0.5]], [[13, -2.0, 1.0], [17, 1.5, -0.5]], [[6, -3.0, 3.0], [19, 1.0, 0.5], [15, -1.0, 0.0], [11, 0.0, 0.5], [9, 0.0, -2.0], [4, -0.5, -0.5]]], 5], [[[], 3], [[], 1], [[], 2], [[], 3]]], [[[[[10, -3.0, 0.5], [19, 0.0, 2.5], [15, 0.0, -1.0], [16, -0.5, 3.0], [8, -3.0, 3.0]], [[3, -2.5, 3.0], [12, 2.0, 0.0], [2, 3.0, -2.5]]], 7], [[[], 4], [[], 1]]], [[[[[1, -1.5, 0.5], [4, -1.0, 2.0], [19, 2.5, 1.5]], [[2, 2.0, 2.0], [18, 2.0, 0.0], [19, 0.0, -1.5], [14, -0.5, -2.0]], [[19, 3.0, -2.0], [13, 2.0, -2.5], [1, -3.0, 2.5], [17, 0.5, 2.0], [4, 1.5, -2.5]], [[17, -2.0, -1.0], [16, -3.0, 0.0]]], 11], [[[], 3], [[], 4], [[], 5], [[], 2]]], [[[[[11, 2.5, -1.0], [10, -3.0, -3.0], [16, 0.5, -0.5]], [[7, -1.5, 2.5], [9, 3.0, 2.5]], [[9, -2.5, -2.0], [18, -1.0, 1.0], [11, -2.5, 1.5], [4, 1.5, -3.0]], [[18, 1.5, -1.0], [7, 3.0, -1.0], [10, -2.0, 0.0]]], 3], [[[], 2], [[], 1], [[], 1], [[], 2]]], [[[[[19, 1.5, 0.5], [1, -0.5, 0.5], [14, 2.0, -2.0]], [[7, -1.0, 0.5], [14, 3.0, -3.0]]], 11], [[[], 3], [[], 2]]]], [[[[[[3, 1.5, 1.0], [8, -2.0, 0.0], [9, -1.5, -1.0], [18, 1.5, 2.5]]], 11], [[[], 4]]], [[[[[10, -0.5, 3.0], [5, 2.5, 3.0], [18, 1.5, -3.0], [9, -2.0, 2.5], [16, -3.0, -2.5], [13, -2.0, -2.5]], [[6, 2.0, 0.0], [16, -0.5, 1.5], [10, 2.0, -2.0], [9, 1.5, 0.5], [8, 0.5, 1.0], [1, 1.0, -2.0]], [[6, -1.0, 1.0], [18, 2.0, -3.0], [2, -1.5, 0.0], [1, 1.0, 2.5], [19, -2.5, -0.5], [13, -0.5, 1.0]], [[12, 0.0, 0.5], [10, -3.0, 2.5], [1, 1.0, -0.5], [7, 0.0, 2.5], [18, 0.5, -1.5]]], 11], [[[], 5], [[], 5], [[[6, 13]], 5], [[], 4]]], [[[[[4, -3.0, 0.5], [6, 2.0, 3.0], [13, 0.5, -1.5]], [[12, -0.5, 0.0], [13, 2.5, -2.5], [19, 3.0, -2.0], [3, -0.5, 3.0]], [[11, 0.5, -1.5], [6, -1.0, -1.5], [3, -2.0, 2.0], [4, 2.5, 2.5]], [[19, -1.5, 0.5], [9, -1.0, 3.0], [8, 1.5, -3.0], [4, 2.5, -2.5], [12, -1.5, 0.0], [1, -3.0, -1.0]]], 3], [[[], 2], [[], 2], [[], 2], [[[12, 19]], 3]]], [[[[[18, 0.5, -3.0], [17, 2.0, 2.0], [15, 2.0, 1.0], [7, 1.5, -1.5], [10, 1.5, 2.5]]], 7], [[[], 5]]], [[[[[14, -1.0, 0.5], [17, 0.0, 3.0], [11, -1.0, 1.5]]], 11], [[[], 3]]], [[[[[16, -2.0, 1.0], [14, 0.0, 2.5], [15, -1.5, -0.5], [1, -3.0, 2.0], [18, -2.0, 2.0], [7, -1.0, 3.0]]], 7], [[[], 4]]], [[[[[4, 2.0, 2.5], [7, -1.5, 2.0], [18, 3.0, 3.0]], [[2, -2.0, 0.5], [19, 2.5, -1.0], [8, -2.5, -3.0], [1, -3.0, -0.5], [7, -3.0, 0.0]], [[16, 1.5, -0.5], [7, 0.5, -1.0], [6, -1.5, 1.5], [3, -0.5, -1.5], [10, 1.0, -1.0]]], 7], [[[], 3], [[], 4], [[[10, 16]], 3]]], [[[[[17, 2.0, -2.0], [12, 1.0, -0.5], [3, 1.0, -1.0]], [[13, 0.0, 0.0], [16, -2.0, 0.0]], [[19, 3.0, 3.0], [11, 1.0, 0.5], [12, 2.5, 1.5], [3, -2.5, 2.5]], [[15, 0.5, -0.5], [7, 1.0, 3.0]]], 5], [[[[3, 12]], 2], [[], 1], [[], 3], [[], 2]]]]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("case %d" % i, 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
case 0	`[[[], 2], [[], 2], [[], 2]]`	`[[[], 2], [[], 2], [[], 2]]`	Passed
case 1	`[[[], 3], [[], 2], [[], 4]]`	`[[[], 3], [[], 2], [[], 4]]`	Passed
case 2	`[[[], 2]]`	`[[[], 2]]`	Passed
case 3	`[[[], 3], [[[6, 7]], 4]]`	`[[[], 3], [[], 4]]`	Failed
case 4	`[[[], 4], [[[8, 17]], 2], [[], 2], [[], 2]]`	`[[[], 4], [[], 2], [[], 2], [[], 2]]`	Failed
case 5	`[[[], 2], [[[1, 16], [4, 8]], 3]]`	`[[[], 2], [[[1, 16]], 3]]`	Failed
case 6	`[[[], 2], [[[9, 10]], 3], [[[1, 9]], 3], [[[4, 16]], 3]]`	`[[[], 2], [[[9, 10]], 3], [[[1, 9]], 3], [[], 3]]`	Failed
case 7	`[[[[2, 12]], 1]]`	`[[[[2, 12]], 1]]`	Passed

SHA-256 / 1b8f9f81e366c91b5818bf5856d7ba0955a07ffaa3415eb5a289b2fdcc60b41d

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 8 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

A deterministic bounded teaching model with stipulated toy conventions; not a production physics engine or spatial index. 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 2026-09-29T14:51:01.682329+00:00.

Case digest / d1f328eee621ef59a8d254f77e6450754ba8b366fa09d363dc33d93f6bd087eb