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FA-86501 / Procedural level generation constraints / Open access

Chunk seed derivation: Negative chunks reuse positive seeds · case 01

Terrain at chunk -1 repeats chunk 1.

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

ROOT CAUSE

Negative coordinates map to the same code as their positive mirror.

VERIFIED REPAIR

Restore `-v * 2 - 1` at the zigzag negatives step.

Unsuccessful approach: Odd codes offset by one collide with the next negative coordinate.

Case contract

Coordinates are zigzag encoded (v >= 0 -> 2v, v < 0 -> -2v-1). seed = (world_seed*1000003 mod 2^32) xor (zz(cx)*73856093 mod 2^32) xor (zz(cy)*19349663 mod 2^32). The generator state is seed, or 1 when seed is 0, advanced by three 32-bit xorshift rounds (<<13, >>17, <<5, each left shift masked to 32 bits). Returns [seed, state % 100].

Why this case matters

Procedural generators silently emit unplayable or unfair levels when a single constraint check uses the wrong boundary, axis, neighborhood or update order; the defect is visible in exact generated geometry.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(world_seed, cx, cy):
    def zz(v):
        return v * 2 if v >= 0 else -v * 2
    s = (world_seed * 1000003) & 0xFFFFFFFF
    s ^= (zz(cx) * 73856093) & 0xFFFFFFFF
    s ^= (zz(cy) * 19349663) & 0xFFFFFFFF
    x = s or 1
    for _ in range(3):
        x ^= (x << 13) & 0xFFFFFFFF
        x ^= x >> 17
        x ^= (x << 5) & 0xFFFFFFFF
    return [s, x % 100]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [42, -1, 1], [78162845, 87]),
  ('regression zigzag negatives #2', [1099511627779, -1, 155], [1640198686, 4]),
  ('regression zigzag negatives #3', [1099511627779, -374, 0], [3628518806, 87]),
  ('regression zigzag negatives #4', [0, 2, -3], [341807727, 72]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [0, 0, 3], [116097978, 65])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [1099511627779, -1, 155], [1640198686, 4]),
  ('regression zigzag negatives #2', [1099511627779, -374, 0], [3628518806, 87]),
  ('regression zigzag negatives #3', [0, 2, -3], [341807727, 72]),
  ('regression zigzag negatives #4', [1099511627779, -1, 3], [44085550, 32]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [0, 0, 3], [116097978, 65])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [0, 2, -3], [341807727, 72]),
  ('regression zigzag negatives #2', [1099511627779, -1, 3], [44085550, 32]),
  ('regression zigzag negatives #3', [2147483655, -2, 0], [2371752642, 72]),
  ('regression zigzag negatives #4', [1, -2, 1], [259333738, 52]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [0, 0, 3], [116097978, 65])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [2147483655, -2, 0], [2371752642, 72]),
  ('regression zigzag negatives #2', [1, -2, 1], [259333738, 52]),
  ('regression zigzag negatives #3', [42, -1, -303], [3206913376, 67]),
  ('regression zigzag negatives #4', [17601332116874, -1, 1], [1431083645, 95]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [1099511627779, 1, 1], [179220301, 79])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [42, -1, -303], [3206913376, 67]),
  ('regression zigzag negatives #2', [17601332116874, -1, 1], [1431083645, 95]),
  ('regression zigzag negatives #3', [1099511627779, -1, 0], [72037012, 17]),
  ('regression zigzag negatives #4', [0, -22, 3], [3147860005, 93]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [42, 1, 0], [172832324, 53])]]
for label, args, expected in cases[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 fixtureActualExpectedOutcome
negative neighbour chunk #1[172832324, 53][115747491, 56]Failed
regression zigzag negatives #1[134461306, 50][78162845, 87]Failed
regression zigzag negatives #2[1835560697, 73][1640198686, 4]Failed
regression zigzag negatives #3[3707517301, 91][3628518806, 87]Failed
regression zigzag negatives #4[393237198, 82][341807727, 72]Failed
origin with zero seed #1[0, 61][0, 61]Passed
mirrored coordinates #1[294384589, 20][294384589, 20]Passed
control #1[116097978, 65][116097978, 65]Passed

SHA-256 / 5df387e6c9cc79dc48fb00ae571021515ff8d23f43742c048bf32ea16092a13e

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(world_seed, cx, cy):
    def zz(v):
        return v * 2 if v >= 0 else abs(v) * 2 + 1
    s = (world_seed * 1000003) & 0xFFFFFFFF
    s ^= (zz(cx) * 73856093) & 0xFFFFFFFF
    s ^= (zz(cy) * 19349663) & 0xFFFFFFFF
    x = s or 1
    for _ in range(3):
        x ^= (x << 13) & 0xFFFFFFFF
        x ^= x >> 17
        x ^= (x << 5) & 0xFFFFFFFF
    return [s, x % 100]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [42, -1, 1], [78162845, 87]),
  ('regression zigzag negatives #2', [1099511627779, -1, 155], [1640198686, 4]),
  ('regression zigzag negatives #3', [1099511627779, -374, 0], [3628518806, 87]),
  ('regression zigzag negatives #4', [0, 2, -3], [341807727, 72]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [0, 0, 3], [116097978, 65])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [1099511627779, -1, 155], [1640198686, 4]),
  ('regression zigzag negatives #2', [1099511627779, -374, 0], [3628518806, 87]),
  ('regression zigzag negatives #3', [0, 2, -3], [341807727, 72]),
  ('regression zigzag negatives #4', [1099511627779, -1, 3], [44085550, 32]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [0, 0, 3], [116097978, 65])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [0, 2, -3], [341807727, 72]),
  ('regression zigzag negatives #2', [1099511627779, -1, 3], [44085550, 32]),
  ('regression zigzag negatives #3', [2147483655, -2, 0], [2371752642, 72]),
  ('regression zigzag negatives #4', [1, -2, 1], [259333738, 52]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [0, 0, 3], [116097978, 65])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [2147483655, -2, 0], [2371752642, 72]),
  ('regression zigzag negatives #2', [1, -2, 1], [259333738, 52]),
  ('regression zigzag negatives #3', [42, -1, -303], [3206913376, 67]),
  ('regression zigzag negatives #4', [17601332116874, -1, 1], [1431083645, 95]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [1099511627779, 1, 1], [179220301, 79])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [42, -1, -303], [3206913376, 67]),
  ('regression zigzag negatives #2', [17601332116874, -1, 1], [1431083645, 95]),
  ('regression zigzag negatives #3', [1099511627779, -1, 0], [72037012, 17]),
  ('regression zigzag negatives #4', [0, -22, 3], [3147860005, 93]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [42, 1, 0], [172832324, 53])]]
for label, args, expected in cases[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 fixtureActualExpectedOutcome
negative neighbour chunk #1[263455721, 65][115747491, 56]Failed
regression zigzag negatives #1[234521303, 90][78162845, 87]Failed
regression zigzag negatives #2[1754356564, 21][1640198686, 4]Failed
regression zigzag negatives #3[3776262864, 85][3628518806, 87]Failed
regression zigzag negatives #4[428414253, 50][341807727, 72]Failed
origin with zero seed #1[0, 61][0, 61]Passed
mirrored coordinates #1[294384589, 20][294384589, 20]Passed
control #1[116097978, 65][116097978, 65]Passed

SHA-256 / 4ef0e760d4134a84b5b94357c9e3916413495047466a63e30e199d7049ab1e98

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(world_seed, cx, cy):
    def zz(v):
        return v * 2 if v >= 0 else -v * 2 - 1
    s = (world_seed * 1000003) & 0xFFFFFFFF
    s ^= (zz(cx) * 73856093) & 0xFFFFFFFF
    s ^= (zz(cy) * 19349663) & 0xFFFFFFFF
    x = s or 1
    for _ in range(3):
        x ^= (x << 13) & 0xFFFFFFFF
        x ^= x >> 17
        x ^= (x << 5) & 0xFFFFFFFF
    return [s, x % 100]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [42, -1, 1], [78162845, 87]),
  ('regression zigzag negatives #2', [1099511627779, -1, 155], [1640198686, 4]),
  ('regression zigzag negatives #3', [1099511627779, -374, 0], [3628518806, 87]),
  ('regression zigzag negatives #4', [0, 2, -3], [341807727, 72]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [0, 0, 3], [116097978, 65])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [1099511627779, -1, 155], [1640198686, 4]),
  ('regression zigzag negatives #2', [1099511627779, -374, 0], [3628518806, 87]),
  ('regression zigzag negatives #3', [0, 2, -3], [341807727, 72]),
  ('regression zigzag negatives #4', [1099511627779, -1, 3], [44085550, 32]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [0, 0, 3], [116097978, 65])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [0, 2, -3], [341807727, 72]),
  ('regression zigzag negatives #2', [1099511627779, -1, 3], [44085550, 32]),
  ('regression zigzag negatives #3', [2147483655, -2, 0], [2371752642, 72]),
  ('regression zigzag negatives #4', [1, -2, 1], [259333738, 52]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [0, 0, 3], [116097978, 65])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [2147483655, -2, 0], [2371752642, 72]),
  ('regression zigzag negatives #2', [1, -2, 1], [259333738, 52]),
  ('regression zigzag negatives #3', [42, -1, -303], [3206913376, 67]),
  ('regression zigzag negatives #4', [17601332116874, -1, 1], [1431083645, 95]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [1099511627779, 1, 1], [179220301, 79])],
 [('negative neighbour chunk #1', [42, -1, 0], [115747491, 56]),
  ('regression zigzag negatives #1', [42, -1, -303], [3206913376, 67]),
  ('regression zigzag negatives #2', [17601332116874, -1, 1], [1431083645, 95]),
  ('regression zigzag negatives #3', [1099511627779, -1, 0], [72037012, 17]),
  ('regression zigzag negatives #4', [0, -22, 3], [3147860005, 93]),
  ('origin with zero seed #1', [0, 0, 0], [0, 61]),
  ('mirrored coordinates #1', [7, 3, 5], [294384589, 20]),
  ('control #1', [42, 1, 0], [172832324, 53])]]
for label, args, expected in cases[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 fixtureActualExpectedOutcome
negative neighbour chunk #1[115747491, 56][115747491, 56]Passed
regression zigzag negatives #1[78162845, 87][78162845, 87]Passed
regression zigzag negatives #2[1640198686, 4][1640198686, 4]Passed
regression zigzag negatives #3[3628518806, 87][3628518806, 87]Passed
regression zigzag negatives #4[341807727, 72][341807727, 72]Passed
origin with zero seed #1[0, 61][0, 61]Passed
mirrored coordinates #1[294384589, 20][294384589, 20]Passed
control #1[116097978, 65][116097978, 65]Passed

SHA-256 / 5e684957fa8ad4cdc4836094dd66abe1f271388b2d1a5600051846d7bcc1c29d

Verification & scope

Deterministic toy contract stipulated for this model; integer or exact arithmetic only, not a reproduction of any specific game engine. 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 2026-09-29T14:50:50.147349+00:00.

Case digest / a75f6ba1114ac21c277c349156b74e3fa9f83b5d0ade90716610fac24456df45