FA-87476 / Collision detection broadphase / Open access

Objects starting exactly on the midline are treated as straddlers · case 01

Boundary-aligned objects pile up in parent nodes and inflate every query.

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

ROOT CAUSE

The right-child test is minx > mid.

THE FAILURE

The right-child test is minx > mid.

Unsuccessful approach: Letting maxx == mid go left assigns objects touching the right half to the left child.

Case contract

solve(objs, cap, max_depth, q): quadtree over [0,16]^2. Insert [id,minx,miny,maxx,maxy] in order. A node splits when it holds more than cap objects and depth<max_depth; children are ordered cy*2+cx. An object goes to a child only if it fits entirely: maxx<mid -> left, minx>=mid -> right (same for y); straddlers stay. On split all node objects are re-inserted. Query returns [sorted ids of objects in nodes whose closed bounds overlap q, total node count, max depth].

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(objs, cap, max_depth, q):
    def make(x, y, s, d):
        return {'x': x, 'y': y, 's': s, 'd': d, 'objs': [], 'kids': None}
    def child_index(n, o):
        h = n['s'] / 2
        mx, my = n['x'] + h, n['y'] + h
        if o[3] < mx:
            cx = 0
        elif o[1] > mx:
            cx = 1
        else:
            return -1
        if o[4] < my:
            cy = 0
        elif o[2] >= my:
            cy = 1
        else:
            return -1
        return cy * 2 + cx
    def split(n):
        h = n['s'] / 2
        n['kids'] = [make(n['x'] + (k % 2) * h, n['y'] + (k // 2) * h, h, n['d'] + 1) for k in range(4)]
        old, n['objs'] = n['objs'], []
        for o in old:
            insert(n, o)
    def insert(n, o):
        if n['kids'] is not None:
            k = child_index(n, o)
            if k >= 0:
                insert(n['kids'][k], o)
                return
        n['objs'].append(o)
        if n['kids'] is None and len(n['objs']) > cap and n['d'] < max_depth:
            split(n)
    root = make(0, 0, 16, 0)
    for o in objs:
        insert(root, o)
    found, count, deep = [], 0, 0
    stack = [root]
    while stack:
        n = stack.pop()
        count += 1
        deep = max(deep, n['d'])
        if n['kids']:
            stack.extend(n['kids'])
        if n['x'] <= q[2] and q[0] <= n['x'] + n['s'] and n['y'] <= q[3] and q[1] <= n['y'] + n['s']:
            found.extend(o[0] for o in n['objs'])
    return [sorted(found), count, deep]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[[[[[8, 7, 2, 8, 5], [41, 12, 8, 14, 11], [5, 2, 6, 2, 9], [38, 10, 5, 12, 5], [9, 0, 3, 1, 5], [24, 4, 0, 6, 3]], 2, 3, [11, 10, 12, 12]], [[5, 8, 41], 5, 1]], [[[[49, 7, 3, 10, 3], [38, 8, 0, 8, 2], [11, 12, 4, 12, 4], [5, 4, 10, 4, 11], [3, 12, 6, 15, 7], [21, 0, 13, 1, 14], [35, 10, 7, 10, 10], [9, 9, 10, 12, 12]], 2, 2, [9, 2, 10, 7]], [[35, 38, 49], 9, 2]], [[[[2, 13, 0, 16, 1], [21, 1, 6, 1, 6], [5, 5, 8, 8, 8], [39, 10, 6, 11, 9]], 2, 2, [14, 1, 19, 6]], [[2, 5, 39], 5, 1]], [[[[28, 10, 6, 13, 7], [40, 6, 0, 9, 2], [9, 1, 14, 3, 16]], 3, 2, [13, 2, 13, 7]], [[9, 28, 40], 1, 0]], [[[[41, 8, 14, 11, 16], [7, 7, 6, 8, 8], [38, 10, 10, 10, 13], [17, 7, 9, 8, 11], [2, 6, 3, 9, 4], [6, 12, 10, 12, 10], [30, 9, 12, 10, 12]], 3, 3, [1, 4, 1, 6]], [[2, 7, 17], 9, 2]], [[[[3, 15, 7, 16, 7], [5, 1, 8, 3, 10], [28, 2, 13, 3, 14], [8, 13, 13, 14, 14], [16, 8, 14, 11, 16], [42, 11, 14, 14, 14], [17, 0, 15, 1, 15]], 1, 1, [3, 9, 5, 14]], [[5, 17, 28], 5, 1]], [[[[40, 9, 14, 12, 15], [14, 9, 1, 10, 2], [37, 8, 7, 10, 7], [45, 7, 10, 7, 13], [11, 4, 1, 6, 4], [18, 10, 14, 11, 16]], 2, 2, [11, 14, 14, 15]], [[18, 40], 5, 1]], [[[[11, 13, 14, 15, 15], [7, 15, 15, 16, 16], [21, 2, 5, 2, 5], [5, 13, 14, 13, 15], [37, 14, 6, 15, 7]], 2, 2, [13, 10, 18, 12]], [[5, 7, 11], 9, 2]]], [[[[[10, 3, 15, 4, 15], [48, 15, 10, 15, 12], [19, 15, 9, 15, 12], [6, 7, 11, 8, 13], [45, 5, 14, 5, 16], [30, 13, 9, 14, 11], [25, 14, 13, 14, 13], [34, 15, 1, 15, 1]], 2, 1, [13, 2, 16, 2]], [[6, 34], 5, 1]], [[[[27, 11, 13, 14, 16], [22, 12, 7, 15, 7]], 3, 1, [2, 4, 7, 5]], [[22, 27], 1, 0]], [[[[2, 13, 13, 14, 13], [39, 12, 5, 12, 7], [33, 14, 5, 15, 7]], 3, 1, [12, 1, 17, 4]], [[2, 33, 39], 1, 0]], [[[[14, 10, 6, 10, 6], [3, 8, 10, 9, 10], [42, 13, 1, 15, 1], [46, 12, 1, 13, 1], [40, 14, 11, 14, 11], [27, 7, 4, 10, 4], [19, 9, 12, 12, 13]], 3, 2, [3, 6, 4, 7]], [[27], 5, 1]], [[[[20, 2, 5, 2, 5], [1, 4, 5, 4, 6], [36, 1, 1, 1, 1], [15, 3, 12, 3, 13], [9, 4, 6, 6, 7], [33, 14, 11, 16, 14], [24, 0, 4, 3, 5], [34, 6, 0, 9, 0]], 1, 3, [4, 4, 5, 5]], [[1, 9, 20, 24, 34, 36], 17, 3]], [[[[1, 4, 0, 5, 3], [17, 10, 15, 12, 16]], 2, 2, [13, 9, 14, 14]], [[1, 17], 1, 0]], [[[[28, 9, 14, 12, 16], [29, 2, 3, 2, 6], [23, 8, 0, 8, 2], [17, 2, 9, 5, 11], [33, 15, 5, 16, 7], [46, 14, 15, 16, 16], [14, 13, 0, 16, 3]], 2, 2, [0, 14, 1, 18]], [[17], 9, 2]], [[[[45, 15, 11, 16, 11], [24, 11, 4, 12, 5], [3, 1, 6, 1, 9], [37, 12, 8, 15, 11], [44, 7, 10, 9, 11], [9, 13, 4, 16, 5], [27, 11, 11, 11, 14]], 1, 3, [7, 13, 9, 15]], [[3, 27, 44], 17, 3]]], [[[[[1, 6, 9, 9, 12], [46, 1, 14, 4, 15], [45, 4, 13, 4, 16], [40, 3, 7, 6, 8], [24, 8, 12, 8, 13], [21, 4, 15, 4, 16], [7, 9, 11, 11, 14], [34, 11, 7, 13, 10]], 2, 2, [1, 9, 4, 14]], [[1, 21, 34, 40, 45, 46], 9, 2]], [[[[27, 0, 2, 3, 3], [21, 0, 12, 2, 14], [38, 0, 2, 2, 5], [2, 0, 5, 3, 5], [31, 10, 9, 10, 12], [20, 10, 2, 11, 5]], 2, 2, [2, 10, 6, 14]], [[21], 9, 2]], [[[[25, 2, 2, 3, 2], [19, 13, 7, 14, 7], [28, 4, 8, 7, 11], [18, 11, 3, 11, 3]], 2, 1, [11, 12, 14, 17]], [[], 5, 1]], [[[[42, 9, 0, 12, 2], [33, 14, 15, 15, 15], [46, 5, 9, 7, 12], [36, 11, 0, 13, 2], [43, 10, 3, 13, 4], [41, 4, 1, 6, 3], [49, 12, 12, 12, 14], [1, 10, 2, 12, 3], [11, 9, 7, 11, 10], [4, 8, 2, 8, 2]], 2, 1, [13, 13, 15, 15]], [[11, 33, 49], 5, 1]], [[[[33, 4, 12, 4, 12], [26, 7, 15, 8, 15], [29, 8, 7, 10, 7], [4, 15, 0, 15, 3], [15, 7, 13, 7, 14], [16, 7, 6, 10, 9], [5, 8, 3, 8, 3], [31, 14, 15, 14, 16], [22, 3, 2, 5, 3]], 3, 1, [3, 7, 5, 11]], [[15, 16, 22, 26, 33], 5, 1]], [[[[30, 9, 5, 12, 7], [25, 13, 6, 13, 6], [20, 0, 3, 1, 5], [10, 11, 13, 11, 15], [33, 6, 3, 9, 6], [17, 9, 9, 10, 11], [45, 3, 9, 4, 11], [5, 10, 4, 11, 6], [43, 8, 9, 9, 12], [39, 7, 15, 10, 16]], 1, 1, [1, 10, 5, 11]], [[33, 39, 45], 5, 1]], [[[[46, 4, 0, 4, 0], [7, 14, 13, 16, 14], [25, 10, 11, 11, 14], [33, 14, 8, 16, 10], [9, 12, 5, 15, 8], [19, 2, 12, 3, 14], [42, 2, 13, 2, 13], [13, 15, 15, 16, 15], [34, 15, 7, 15, 10], [5, 8, 10, 9, 10]], 1, 3, [4, 3, 8, 7]], [[9, 34, 46], 21, 3]], [[[[7, 12, 2, 12, 5], [38, 9, 14, 9, 16], [5, 5, 12, 7, 13], [14, 11, 3, 11, 3], [26, 0, 4, 3, 5], [6, 12, 1, 12, 1], [27, 7, 3, 10, 5], [22, 14, 1, 15, 2]], 2, 2, [5, 10, 10, 13]], [[5, 27, 38], 9, 2]]], [[[[[28, 6, 13, 6, 15], [12, 0, 1, 0, 3], [30, 1, 2, 2, 2], [9, 14, 12, 16, 13], [7, 5, 11, 8, 12]], 1, 3, [8, 2, 12, 3]], [[7], 13, 3]], [[[[46, 13, 3, 16, 4], [35, 8, 3, 9, 3], [29, 9, 8, 12, 9], [9, 4, 15, 6, 16], [32, 2, 6, 3, 6], [23, 1, 1, 2, 3], [49, 9, 11, 10, 12], [27, 2, 5, 5, 8]], 1, 2, [0, 13, 4, 15]], [[9, 27], 17, 2]], [[[[13, 10, 13, 11, 13], [3, 2, 3, 3, 4], [39, 8, 13, 9, 13], [43, 1, 1, 2, 3], [31, 3, 13, 5, 14], [4, 15, 12, 16, 13], [17, 6, 8, 7, 8], [24, 3, 3, 6, 6], [15, 11, 6, 11, 6]], 3, 3, [14, 14, 17, 16]], [[4, 13, 39], 5, 1]], [[[[17, 6, 9, 8, 12], [37, 8, 2, 10, 2], [42, 11, 14, 12, 14], [39, 9, 8, 10, 9], [34, 7, 2, 7, 5], [32, 13, 4, 16, 7], [23, 2, 9, 2, 10], [1, 4, 10, 7, 13], [40, 4, 2, 4, 3], [29, 4, 11, 6, 14]], 2, 2, [0, 9, 3, 13]], [[1, 17, 23, 29], 9, 2]], [[[[17, 1, 10, 4, 12], [11, 9, 3, 9, 4], [36, 1, 13, 3, 15], [49, 3, 14, 5, 16], [7, 8, 0, 10, 0], [39, 1, 15, 2, 16], [23, 1, 12, 2, 15], [8, 6, 14, 8, 16], [43, 11, 13, 11, 16], [29, 11, 8, 12, 9], [32, 7, 5, 9, 8]], 3, 3, [3, 2, 3, 3]], [[8, 32], 9, 2]], [[[[7, 13, 7, 14, 10], [49, 6, 5, 9, 6], [35, 8, 9, 10, 9]], 2, 3, [0, 10, 0, 13]], [[7, 49], 5, 1]], [[[[12, 0, 8, 0, 10], [34, 13, 6, 15, 9], [33, 15, 14, 16, 16]], 2, 2, [12, 8, 16, 9]], [[33, 34], 5, 1]], [[[[12, 4, 4, 5, 4], [36, 13, 8, 13, 10], [9, 15, 1, 16, 2], [43, 9, 11, 12, 14], [17, 7, 15, 7, 15]], 1, 1, [11, 0, 13, 2]], [[9], 5, 1]]], [[[[[49, 11, 1, 11, 4], [4, 9, 14, 12, 16], [3, 1, 4, 2, 7], [11, 10, 4, 10, 5], [25, 6, 12, 9, 12], [26, 11, 14, 13, 15], [36, 3, 14, 6, 16], [18, 3, 1, 3, 2], [6, 15, 5, 16, 7], [31, 3, 11, 3, 12], [40, 2, 6, 4, 9]], 3, 3, [11, 14, 15, 14]], [[4, 25, 26, 40], 5, 1]], [[[[4, 15, 8, 16, 9], [30, 1, 0, 2, 0], [35, 15, 6, 16, 9], [12, 10, 4, 13, 7], [1, 2, 14, 5, 15], [28, 5, 2, 5, 5], [20, 15, 1, 15, 4], [46, 9, 0, 12, 0], [19, 4, 5, 5, 5], [43, 0, 9, 2, 11], [38, 6, 1, 9, 3], [7, 15, 6, 16, 8]], 3, 2, [14, 9, 16, 11]], [[4, 7, 35, 38], 5, 1]], [[[[29, 6, 0, 8, 3], [20, 11, 15, 11, 16], [27, 6, 12, 9, 15], [38, 0, 9, 2, 11], [30, 10, 9, 10, 10], [7, 2, 9, 3, 11], [48, 8, 11, 11, 14], [9, 2, 7, 2, 10]], 1, 2, [1, 4, 2, 6]], [[9, 27, 29], 13, 2]], [[[[27, 4, 14, 7, 14], [36, 6, 1, 8, 4], [10, 8, 9, 10, 11], [15, 13, 4, 13, 5], [24, 9, 5, 12, 7], [48, 11, 0, 14, 1], [20, 5, 6, 7, 9], [47, 1, 11, 1, 12]], 3, 3, [6, 7, 6, 7]], [[20, 36], 5, 1]], [[[[4, 7, 3, 10, 5], [26, 13, 7, 14, 7], [19, 14, 7, 14, 10], [23, 14, 7, 15, 10], [10, 7, 12, 10, 15], [11, 14, 7, 16, 8], [17, 8, 6, 8, 9], [13, 15, 9, 15, 11], [34, 5, 9, 8, 9], [9, 5, 7, 6, 10], [2, 7, 6, 8, 9], [14, 4, 8, 7, 8]], 2, 3, [3, 0, 3, 3]], [[2, 4, 9, 10, 11, 17, 19, 23, 34], 5, 1]], [[[[34, 13, 8, 16, 9], [35, 0, 9, 2, 12], [8, 11, 12, 13, 13], [43, 12, 9, 15, 9], [42, 12, 3, 15, 5], [21, 13, 3, 13, 3], [27, 12, 0, 12, 3], [37, 6, 2, 8, 5], [25, 0, 9, 2, 12]], 2, 3, [10, 6, 10, 11]], [[8, 37, 42], 13, 2]], [[[[2, 6, 3, 8, 4], [12, 0, 8, 0, 8], [38, 9, 13, 11, 16], [4, 4, 1, 4, 3], [36, 14, 3, 15, 3], [48, 8, 14, 9, 16], [22, 4, 3, 5, 5], [3, 10, 9, 10, 12], [26, 6, 6, 7, 6], [11, 4, 15, 6, 16], [18, 9, 3, 12, 4], [27, 12, 1, 14, 2]], 1, 2, [1, 4, 2, 5]], [[2, 22], 21, 2]], [[[[41, 9, 3, 12, 4], [48, 3, 6, 4, 7], [28, 8, 15, 8, 15]], 3, 3, [10, 1, 13, 6]], [[28, 41, 48], 1, 0]]]]
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	`[[5, 8, 41], 5, 1]`	`[[5, 8, 41], 5, 1]`	Passed
case 1	`[[3, 11, 35, 38, 49], 5, 1]`	`[[35, 38, 49], 9, 2]`	Failed
case 2	`[[2, 5, 39], 5, 1]`	`[[2, 5, 39], 5, 1]`	Passed
case 3	`[[9, 28, 40], 1, 0]`	`[[9, 28, 40], 1, 0]`	Passed
case 4	`[[2, 7, 17, 41], 5, 1]`	`[[2, 7, 17], 9, 2]`	Failed
case 5	`[[5, 16, 17, 28], 5, 1]`	`[[5, 17, 28], 5, 1]`	Failed
case 6	`[[18, 37, 40], 5, 1]`	`[[18, 40], 5, 1]`	Failed
case 7	`[[5, 7, 11], 9, 2]`	`[[5, 7, 11], 9, 2]`	Passed

SHA-256 / e4dbdc3097d59b9f4f141e962b47ecc8ed9ff57a8a556af2b23404066f2841ea

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(objs, cap, max_depth, q):
    def make(x, y, s, d):
        return {'x': x, 'y': y, 's': s, 'd': d, 'objs': [], 'kids': None}
    def child_index(n, o):
        h = n['s'] / 2
        mx, my = n['x'] + h, n['y'] + h
        if o[3] <= mx:
            cx = 0
        elif o[1] >= mx:
            cx = 1
        else:
            return -1
        if o[4] < my:
            cy = 0
        elif o[2] >= my:
            cy = 1
        else:
            return -1
        return cy * 2 + cx
    def split(n):
        h = n['s'] / 2
        n['kids'] = [make(n['x'] + (k % 2) * h, n['y'] + (k // 2) * h, h, n['d'] + 1) for k in range(4)]
        old, n['objs'] = n['objs'], []
        for o in old:
            insert(n, o)
    def insert(n, o):
        if n['kids'] is not None:
            k = child_index(n, o)
            if k >= 0:
                insert(n['kids'][k], o)
                return
        n['objs'].append(o)
        if n['kids'] is None and len(n['objs']) > cap and n['d'] < max_depth:
            split(n)
    root = make(0, 0, 16, 0)
    for o in objs:
        insert(root, o)
    found, count, deep = [], 0, 0
    stack = [root]
    while stack:
        n = stack.pop()
        count += 1
        deep = max(deep, n['d'])
        if n['kids']:
            stack.extend(n['kids'])
        if n['x'] <= q[2] and q[0] <= n['x'] + n['s'] and n['y'] <= q[3] and q[1] <= n['y'] + n['s']:
            found.extend(o[0] for o in n['objs'])
    return [sorted(found), count, deep]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[[[[[8, 7, 2, 8, 5], [41, 12, 8, 14, 11], [5, 2, 6, 2, 9], [38, 10, 5, 12, 5], [9, 0, 3, 1, 5], [24, 4, 0, 6, 3]], 2, 3, [11, 10, 12, 12]], [[5, 8, 41], 5, 1]], [[[[49, 7, 3, 10, 3], [38, 8, 0, 8, 2], [11, 12, 4, 12, 4], [5, 4, 10, 4, 11], [3, 12, 6, 15, 7], [21, 0, 13, 1, 14], [35, 10, 7, 10, 10], [9, 9, 10, 12, 12]], 2, 2, [9, 2, 10, 7]], [[35, 38, 49], 9, 2]], [[[[2, 13, 0, 16, 1], [21, 1, 6, 1, 6], [5, 5, 8, 8, 8], [39, 10, 6, 11, 9]], 2, 2, [14, 1, 19, 6]], [[2, 5, 39], 5, 1]], [[[[28, 10, 6, 13, 7], [40, 6, 0, 9, 2], [9, 1, 14, 3, 16]], 3, 2, [13, 2, 13, 7]], [[9, 28, 40], 1, 0]], [[[[41, 8, 14, 11, 16], [7, 7, 6, 8, 8], [38, 10, 10, 10, 13], [17, 7, 9, 8, 11], [2, 6, 3, 9, 4], [6, 12, 10, 12, 10], [30, 9, 12, 10, 12]], 3, 3, [1, 4, 1, 6]], [[2, 7, 17], 9, 2]], [[[[3, 15, 7, 16, 7], [5, 1, 8, 3, 10], [28, 2, 13, 3, 14], [8, 13, 13, 14, 14], [16, 8, 14, 11, 16], [42, 11, 14, 14, 14], [17, 0, 15, 1, 15]], 1, 1, [3, 9, 5, 14]], [[5, 17, 28], 5, 1]], [[[[40, 9, 14, 12, 15], [14, 9, 1, 10, 2], [37, 8, 7, 10, 7], [45, 7, 10, 7, 13], [11, 4, 1, 6, 4], [18, 10, 14, 11, 16]], 2, 2, [11, 14, 14, 15]], [[18, 40], 5, 1]], [[[[11, 13, 14, 15, 15], [7, 15, 15, 16, 16], [21, 2, 5, 2, 5], [5, 13, 14, 13, 15], [37, 14, 6, 15, 7]], 2, 2, [13, 10, 18, 12]], [[5, 7, 11], 9, 2]]], [[[[[10, 3, 15, 4, 15], [48, 15, 10, 15, 12], [19, 15, 9, 15, 12], [6, 7, 11, 8, 13], [45, 5, 14, 5, 16], [30, 13, 9, 14, 11], [25, 14, 13, 14, 13], [34, 15, 1, 15, 1]], 2, 1, [13, 2, 16, 2]], [[6, 34], 5, 1]], [[[[27, 11, 13, 14, 16], [22, 12, 7, 15, 7]], 3, 1, [2, 4, 7, 5]], [[22, 27], 1, 0]], [[[[2, 13, 13, 14, 13], [39, 12, 5, 12, 7], [33, 14, 5, 15, 7]], 3, 1, [12, 1, 17, 4]], [[2, 33, 39], 1, 0]], [[[[14, 10, 6, 10, 6], [3, 8, 10, 9, 10], [42, 13, 1, 15, 1], [46, 12, 1, 13, 1], [40, 14, 11, 14, 11], [27, 7, 4, 10, 4], [19, 9, 12, 12, 13]], 3, 2, [3, 6, 4, 7]], [[27], 5, 1]], [[[[20, 2, 5, 2, 5], [1, 4, 5, 4, 6], [36, 1, 1, 1, 1], [15, 3, 12, 3, 13], [9, 4, 6, 6, 7], [33, 14, 11, 16, 14], [24, 0, 4, 3, 5], [34, 6, 0, 9, 0]], 1, 3, [4, 4, 5, 5]], [[1, 9, 20, 24, 34, 36], 17, 3]], [[[[1, 4, 0, 5, 3], [17, 10, 15, 12, 16]], 2, 2, [13, 9, 14, 14]], [[1, 17], 1, 0]], [[[[28, 9, 14, 12, 16], [29, 2, 3, 2, 6], [23, 8, 0, 8, 2], [17, 2, 9, 5, 11], [33, 15, 5, 16, 7], [46, 14, 15, 16, 16], [14, 13, 0, 16, 3]], 2, 2, [0, 14, 1, 18]], [[17], 9, 2]], [[[[45, 15, 11, 16, 11], [24, 11, 4, 12, 5], [3, 1, 6, 1, 9], [37, 12, 8, 15, 11], [44, 7, 10, 9, 11], [9, 13, 4, 16, 5], [27, 11, 11, 11, 14]], 1, 3, [7, 13, 9, 15]], [[3, 27, 44], 17, 3]]], [[[[[1, 6, 9, 9, 12], [46, 1, 14, 4, 15], [45, 4, 13, 4, 16], [40, 3, 7, 6, 8], [24, 8, 12, 8, 13], [21, 4, 15, 4, 16], [7, 9, 11, 11, 14], [34, 11, 7, 13, 10]], 2, 2, [1, 9, 4, 14]], [[1, 21, 34, 40, 45, 46], 9, 2]], [[[[27, 0, 2, 3, 3], [21, 0, 12, 2, 14], [38, 0, 2, 2, 5], [2, 0, 5, 3, 5], [31, 10, 9, 10, 12], [20, 10, 2, 11, 5]], 2, 2, [2, 10, 6, 14]], [[21], 9, 2]], [[[[25, 2, 2, 3, 2], [19, 13, 7, 14, 7], [28, 4, 8, 7, 11], [18, 11, 3, 11, 3]], 2, 1, [11, 12, 14, 17]], [[], 5, 1]], [[[[42, 9, 0, 12, 2], [33, 14, 15, 15, 15], [46, 5, 9, 7, 12], [36, 11, 0, 13, 2], [43, 10, 3, 13, 4], [41, 4, 1, 6, 3], [49, 12, 12, 12, 14], [1, 10, 2, 12, 3], [11, 9, 7, 11, 10], [4, 8, 2, 8, 2]], 2, 1, [13, 13, 15, 15]], [[11, 33, 49], 5, 1]], [[[[33, 4, 12, 4, 12], [26, 7, 15, 8, 15], [29, 8, 7, 10, 7], [4, 15, 0, 15, 3], [15, 7, 13, 7, 14], [16, 7, 6, 10, 9], [5, 8, 3, 8, 3], [31, 14, 15, 14, 16], [22, 3, 2, 5, 3]], 3, 1, [3, 7, 5, 11]], [[15, 16, 22, 26, 33], 5, 1]], [[[[30, 9, 5, 12, 7], [25, 13, 6, 13, 6], [20, 0, 3, 1, 5], [10, 11, 13, 11, 15], [33, 6, 3, 9, 6], [17, 9, 9, 10, 11], [45, 3, 9, 4, 11], [5, 10, 4, 11, 6], [43, 8, 9, 9, 12], [39, 7, 15, 10, 16]], 1, 1, [1, 10, 5, 11]], [[33, 39, 45], 5, 1]], [[[[46, 4, 0, 4, 0], [7, 14, 13, 16, 14], [25, 10, 11, 11, 14], [33, 14, 8, 16, 10], [9, 12, 5, 15, 8], [19, 2, 12, 3, 14], [42, 2, 13, 2, 13], [13, 15, 15, 16, 15], [34, 15, 7, 15, 10], [5, 8, 10, 9, 10]], 1, 3, [4, 3, 8, 7]], [[9, 34, 46], 21, 3]], [[[[7, 12, 2, 12, 5], [38, 9, 14, 9, 16], [5, 5, 12, 7, 13], [14, 11, 3, 11, 3], [26, 0, 4, 3, 5], [6, 12, 1, 12, 1], [27, 7, 3, 10, 5], [22, 14, 1, 15, 2]], 2, 2, [5, 10, 10, 13]], [[5, 27, 38], 9, 2]]], [[[[[28, 6, 13, 6, 15], [12, 0, 1, 0, 3], [30, 1, 2, 2, 2], [9, 14, 12, 16, 13], [7, 5, 11, 8, 12]], 1, 3, [8, 2, 12, 3]], [[7], 13, 3]], [[[[46, 13, 3, 16, 4], [35, 8, 3, 9, 3], [29, 9, 8, 12, 9], [9, 4, 15, 6, 16], [32, 2, 6, 3, 6], [23, 1, 1, 2, 3], [49, 9, 11, 10, 12], [27, 2, 5, 5, 8]], 1, 2, [0, 13, 4, 15]], [[9, 27], 17, 2]], [[[[13, 10, 13, 11, 13], [3, 2, 3, 3, 4], [39, 8, 13, 9, 13], [43, 1, 1, 2, 3], [31, 3, 13, 5, 14], [4, 15, 12, 16, 13], [17, 6, 8, 7, 8], [24, 3, 3, 6, 6], [15, 11, 6, 11, 6]], 3, 3, [14, 14, 17, 16]], [[4, 13, 39], 5, 1]], [[[[17, 6, 9, 8, 12], [37, 8, 2, 10, 2], [42, 11, 14, 12, 14], [39, 9, 8, 10, 9], [34, 7, 2, 7, 5], [32, 13, 4, 16, 7], [23, 2, 9, 2, 10], [1, 4, 10, 7, 13], [40, 4, 2, 4, 3], [29, 4, 11, 6, 14]], 2, 2, [0, 9, 3, 13]], [[1, 17, 23, 29], 9, 2]], [[[[17, 1, 10, 4, 12], [11, 9, 3, 9, 4], [36, 1, 13, 3, 15], [49, 3, 14, 5, 16], [7, 8, 0, 10, 0], [39, 1, 15, 2, 16], [23, 1, 12, 2, 15], [8, 6, 14, 8, 16], [43, 11, 13, 11, 16], [29, 11, 8, 12, 9], [32, 7, 5, 9, 8]], 3, 3, [3, 2, 3, 3]], [[8, 32], 9, 2]], [[[[7, 13, 7, 14, 10], [49, 6, 5, 9, 6], [35, 8, 9, 10, 9]], 2, 3, [0, 10, 0, 13]], [[7, 49], 5, 1]], [[[[12, 0, 8, 0, 10], [34, 13, 6, 15, 9], [33, 15, 14, 16, 16]], 2, 2, [12, 8, 16, 9]], [[33, 34], 5, 1]], [[[[12, 4, 4, 5, 4], [36, 13, 8, 13, 10], [9, 15, 1, 16, 2], [43, 9, 11, 12, 14], [17, 7, 15, 7, 15]], 1, 1, [11, 0, 13, 2]], [[9], 5, 1]]], [[[[[49, 11, 1, 11, 4], [4, 9, 14, 12, 16], [3, 1, 4, 2, 7], [11, 10, 4, 10, 5], [25, 6, 12, 9, 12], [26, 11, 14, 13, 15], [36, 3, 14, 6, 16], [18, 3, 1, 3, 2], [6, 15, 5, 16, 7], [31, 3, 11, 3, 12], [40, 2, 6, 4, 9]], 3, 3, [11, 14, 15, 14]], [[4, 25, 26, 40], 5, 1]], [[[[4, 15, 8, 16, 9], [30, 1, 0, 2, 0], [35, 15, 6, 16, 9], [12, 10, 4, 13, 7], [1, 2, 14, 5, 15], [28, 5, 2, 5, 5], [20, 15, 1, 15, 4], [46, 9, 0, 12, 0], [19, 4, 5, 5, 5], [43, 0, 9, 2, 11], [38, 6, 1, 9, 3], [7, 15, 6, 16, 8]], 3, 2, [14, 9, 16, 11]], [[4, 7, 35, 38], 5, 1]], [[[[29, 6, 0, 8, 3], [20, 11, 15, 11, 16], [27, 6, 12, 9, 15], [38, 0, 9, 2, 11], [30, 10, 9, 10, 10], [7, 2, 9, 3, 11], [48, 8, 11, 11, 14], [9, 2, 7, 2, 10]], 1, 2, [1, 4, 2, 6]], [[9, 27, 29], 13, 2]], [[[[27, 4, 14, 7, 14], [36, 6, 1, 8, 4], [10, 8, 9, 10, 11], [15, 13, 4, 13, 5], [24, 9, 5, 12, 7], [48, 11, 0, 14, 1], [20, 5, 6, 7, 9], [47, 1, 11, 1, 12]], 3, 3, [6, 7, 6, 7]], [[20, 36], 5, 1]], [[[[4, 7, 3, 10, 5], [26, 13, 7, 14, 7], [19, 14, 7, 14, 10], [23, 14, 7, 15, 10], [10, 7, 12, 10, 15], [11, 14, 7, 16, 8], [17, 8, 6, 8, 9], [13, 15, 9, 15, 11], [34, 5, 9, 8, 9], [9, 5, 7, 6, 10], [2, 7, 6, 8, 9], [14, 4, 8, 7, 8]], 2, 3, [3, 0, 3, 3]], [[2, 4, 9, 10, 11, 17, 19, 23, 34], 5, 1]], [[[[34, 13, 8, 16, 9], [35, 0, 9, 2, 12], [8, 11, 12, 13, 13], [43, 12, 9, 15, 9], [42, 12, 3, 15, 5], [21, 13, 3, 13, 3], [27, 12, 0, 12, 3], [37, 6, 2, 8, 5], [25, 0, 9, 2, 12]], 2, 3, [10, 6, 10, 11]], [[8, 37, 42], 13, 2]], [[[[2, 6, 3, 8, 4], [12, 0, 8, 0, 8], [38, 9, 13, 11, 16], [4, 4, 1, 4, 3], [36, 14, 3, 15, 3], [48, 8, 14, 9, 16], [22, 4, 3, 5, 5], [3, 10, 9, 10, 12], [26, 6, 6, 7, 6], [11, 4, 15, 6, 16], [18, 9, 3, 12, 4], [27, 12, 1, 14, 2]], 1, 2, [1, 4, 2, 5]], [[2, 22], 21, 2]], [[[[41, 9, 3, 12, 4], [48, 3, 6, 4, 7], [28, 8, 15, 8, 15]], 3, 3, [10, 1, 13, 6]], [[28, 41, 48], 1, 0]]]]
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	`[[5, 41], 9, 2]`	`[[5, 8, 41], 5, 1]`	Failed
case 1	`[[3, 11, 35, 49], 5, 1]`	`[[35, 38, 49], 9, 2]`	Failed
case 2	`[[2, 39], 5, 1]`	`[[2, 5, 39], 5, 1]`	Failed
case 3	`[[9, 28, 40], 1, 0]`	`[[9, 28, 40], 1, 0]`	Passed
case 4	`[[2, 7], 9, 2]`	`[[2, 7, 17], 9, 2]`	Failed
case 5	`[[5, 17, 28], 5, 1]`	`[[5, 17, 28], 5, 1]`	Passed
case 6	`[[18, 40], 5, 1]`	`[[18, 40], 5, 1]`	Passed
case 7	`[[5, 7, 11], 9, 2]`	`[[5, 7, 11], 9, 2]`	Passed

SHA-256 / 30550fb5b5f13728f6aa4e126978d6b11fb85c7265025af676c7c50084eac947

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 2026-09-29T14:50:59.176760+00:00.

Case digest / 6905352383d7b71a1d3b3b6b0dd18b2c3401a3f7a085a06065475e025143f7c2