FA-86706 / Procedural level generation constraints / Open access

Heightmap smoothing and biome classes: Map edges sink into the sea · case 01

Border cells are averaged as if missing neighbours were zero.

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

ROOT CAUSE

The window sum is always divided by nine.

VERIFIED REPAIR

Restore `v = total // cnt` at the edge normalisation step.

Unsuccessful approach: Counting the centre twice still biases edge cells.

Case contract

Each cell height becomes floor(mean) of the in-bounds cells in its 3x3 window (input values only). Class: '~' when v < sea, 's' beach when v < sea+2, 'g' when v < mountain, otherwise '^'. Returns strings per row.

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(heights, sea, mountain):
    h = len(heights)
    w = len(heights[0])
    out = []
    for r in range(h):
        row = []
        for c in range(w):
            total = 0
            cnt = 0
            for rr in range(max(0, r - 1), min(h, r + 2)):
                for cc in range(max(0, c - 1), min(w, c + 2)):
                    total += heights[rr][cc]
                    cnt += 1
            v = total // 9
            if v < sea:
                row.append('~')
            elif v < sea + 2:
                row.append('s')
            elif v < mountain:
                row.append('g')
            else:
                row.append('^')
        out.append(''.join(row))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[('corner cell average #1', [[[9, 9], [9, 0]], 3, 8], ['gg', 'gg']),
  ('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1', [[[0, 9], [7, 2]], 2, 11], ['gg', 'gg']),
  ('regression edge normalisation #2', [[[10, 11, 6, 11, 0], [9, 0, 9, 4, 4]], 5, 7], ['^^ss~', '^^ss~']),
  ('regression edge normalisation #3', [[[9], [8]], 6, 6], ['^', '^']),
  ('control #1', [[[3], [4], [0]], 4, 10], ['~', '~', '~']),
  ('control #2', [[[9, 5, 6], [9, 0, 4]], 6, 10], ['~~~', '~~~'])],
 [('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1', [[[0, 9], [7, 2]], 2, 11], ['gg', 'gg']),
  ('regression edge normalisation #2', [[[10, 11, 6, 11, 0], [9, 0, 9, 4, 4]], 5, 7], ['^^ss~', '^^ss~']),
  ('regression edge normalisation #3', [[[9], [8]], 6, 6], ['^', '^']),
  ('regression edge normalisation #4',
   [[[5, 4, 2, 6, 4], [12, 10, 8, 2, 4], [4, 10, 11, 2, 6]], 2, 8],
   ['ggggg', 'ggggg', '^^ggs']),
  ('control #1', [[[5, 3, 0, 8]], 6, 7], ['~~~~']),
  ('control #2', [[[4], [0]], 6, 11], ['~', '~'])],
 [('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('regression edge normalisation #1', [[[10, 11, 6, 11, 0], [9, 0, 9, 4, 4]], 5, 7], ['^^ss~', '^^ss~']),
  ('regression edge normalisation #2', [[[9], [8]], 6, 6], ['^', '^']),
  ('regression edge normalisation #3',
   [[[5, 4, 2, 6, 4], [12, 10, 8, 2, 4], [4, 10, 11, 2, 6]], 2, 8],
   ['ggggg', 'ggggg', '^^ggs']),
  ('regression edge normalisation #4', [[[6], [4], [8], [4], [8]], 5, 9], ['s', 's', 's', 's', 's']),
  ('control #1', [[[3, 2], [6, 0], [0, 5]], 4, 7], ['~~', '~~', '~~']),
  ('control #2', [[[3], [0]], 5, 10], ['~', '~'])],
 [('corner cell average #1', [[[9, 9], [9, 0]], 3, 8], ['gg', 'gg']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1',
   [[[5, 4, 2, 6, 4], [12, 10, 8, 2, 4], [4, 10, 11, 2, 6]], 2, 8],
   ['ggggg', 'ggggg', '^^ggs']),
  ('regression edge normalisation #2', [[[6], [4], [8], [4], [8]], 5, 9], ['s', 's', 's', 's', 's']),
  ('regression edge normalisation #3',
   [[[8, 6, 12, 10, 8], [6, 1, 1, 7, 5], [1, 6, 12, 6, 0], [3, 8, 3, 6, 0], [1, 7, 3, 2, 0]], 6, 9],
   ['~~sss', '~~sss', '~~~~~', '~~~~~', '~~~~~']),
  ('regression edge normalisation #4', [[[10, 11], [1, 0]], 3, 9], ['gg', 'gg']),
  ('control #1', [[[1], [10], [0], [2]], 6, 10], ['~', '~', '~', '~']),
  ('control #2', [[[0]], 6, 9], ['~'])],
 [('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1',
   [[[8, 6, 12, 10, 8], [6, 1, 1, 7, 5], [1, 6, 12, 6, 0], [3, 8, 3, 6, 0], [1, 7, 3, 2, 0]], 6, 9],
   ['~~sss', '~~sss', '~~~~~', '~~~~~', '~~~~~']),
  ('fault site edge normalisation #1', [[[5, 5], [10, 3], [7, 0]], 2, 10], ['gg', 'gg', 'gg']),
  ('regression edge normalisation #2', [[[10, 11], [1, 0]], 3, 9], ['gg', 'gg']),
  ('regression edge normalisation #3', [[[9, 7]], 2, 6], ['^^']),
  ('control #1', [[[1, 0]], 2, 8], ['~~']),
  ('control #2', [[[8], [2], [5], [2], [6]], 6, 8], ['~', '~', '~', '~', '~'])]]
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 fixture	Actual	Expected	Outcome
corner cell average #1	`['ss', 'ss']`	`['gg', 'gg']`	Failed
beach band edge #1	`['~']`	`['g']`	Failed
single row #1	`['~~~']`	`['^s^']`	Failed
regression edge normalisation #1	`['ss', 'ss']`	`['gg', 'gg']`	Failed
regression edge normalisation #2	`['~s~~~', '~s~~~']`	`['^^ss~', '^^ss~']`	Failed
regression edge normalisation #3	`['~', '~']`	`['^', '^']`	Failed
control #1	`['~', '~', '~']`	`['~', '~', '~']`	Passed
control #2	`['~~~', '~~~']`	`['~~~', '~~~']`	Passed

SHA-256 / f5f5ebd138e16489cc89b33eea4692d47831c55900dd7c1580f9d8e33e318339

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(heights, sea, mountain):
    h = len(heights)
    w = len(heights[0])
    out = []
    for r in range(h):
        row = []
        for c in range(w):
            total = 0
            cnt = 0
            for rr in range(max(0, r - 1), min(h, r + 2)):
                for cc in range(max(0, c - 1), min(w, c + 2)):
                    total += heights[rr][cc]
                    cnt += 1
            v = total // (cnt + 1)
            if v < sea:
                row.append('~')
            elif v < sea + 2:
                row.append('s')
            elif v < mountain:
                row.append('g')
            else:
                row.append('^')
        out.append(''.join(row))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[('corner cell average #1', [[[9, 9], [9, 0]], 3, 8], ['gg', 'gg']),
  ('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1', [[[0, 9], [7, 2]], 2, 11], ['gg', 'gg']),
  ('regression edge normalisation #2', [[[10, 11, 6, 11, 0], [9, 0, 9, 4, 4]], 5, 7], ['^^ss~', '^^ss~']),
  ('regression edge normalisation #3', [[[9], [8]], 6, 6], ['^', '^']),
  ('control #1', [[[3], [4], [0]], 4, 10], ['~', '~', '~']),
  ('control #2', [[[9, 5, 6], [9, 0, 4]], 6, 10], ['~~~', '~~~'])],
 [('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1', [[[0, 9], [7, 2]], 2, 11], ['gg', 'gg']),
  ('regression edge normalisation #2', [[[10, 11, 6, 11, 0], [9, 0, 9, 4, 4]], 5, 7], ['^^ss~', '^^ss~']),
  ('regression edge normalisation #3', [[[9], [8]], 6, 6], ['^', '^']),
  ('regression edge normalisation #4',
   [[[5, 4, 2, 6, 4], [12, 10, 8, 2, 4], [4, 10, 11, 2, 6]], 2, 8],
   ['ggggg', 'ggggg', '^^ggs']),
  ('control #1', [[[5, 3, 0, 8]], 6, 7], ['~~~~']),
  ('control #2', [[[4], [0]], 6, 11], ['~', '~'])],
 [('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('regression edge normalisation #1', [[[10, 11, 6, 11, 0], [9, 0, 9, 4, 4]], 5, 7], ['^^ss~', '^^ss~']),
  ('regression edge normalisation #2', [[[9], [8]], 6, 6], ['^', '^']),
  ('regression edge normalisation #3',
   [[[5, 4, 2, 6, 4], [12, 10, 8, 2, 4], [4, 10, 11, 2, 6]], 2, 8],
   ['ggggg', 'ggggg', '^^ggs']),
  ('regression edge normalisation #4', [[[6], [4], [8], [4], [8]], 5, 9], ['s', 's', 's', 's', 's']),
  ('control #1', [[[3, 2], [6, 0], [0, 5]], 4, 7], ['~~', '~~', '~~']),
  ('control #2', [[[3], [0]], 5, 10], ['~', '~'])],
 [('corner cell average #1', [[[9, 9], [9, 0]], 3, 8], ['gg', 'gg']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1',
   [[[5, 4, 2, 6, 4], [12, 10, 8, 2, 4], [4, 10, 11, 2, 6]], 2, 8],
   ['ggggg', 'ggggg', '^^ggs']),
  ('regression edge normalisation #2', [[[6], [4], [8], [4], [8]], 5, 9], ['s', 's', 's', 's', 's']),
  ('regression edge normalisation #3',
   [[[8, 6, 12, 10, 8], [6, 1, 1, 7, 5], [1, 6, 12, 6, 0], [3, 8, 3, 6, 0], [1, 7, 3, 2, 0]], 6, 9],
   ['~~sss', '~~sss', '~~~~~', '~~~~~', '~~~~~']),
  ('regression edge normalisation #4', [[[10, 11], [1, 0]], 3, 9], ['gg', 'gg']),
  ('control #1', [[[1], [10], [0], [2]], 6, 10], ['~', '~', '~', '~']),
  ('control #2', [[[0]], 6, 9], ['~'])],
 [('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1',
   [[[8, 6, 12, 10, 8], [6, 1, 1, 7, 5], [1, 6, 12, 6, 0], [3, 8, 3, 6, 0], [1, 7, 3, 2, 0]], 6, 9],
   ['~~sss', '~~sss', '~~~~~', '~~~~~', '~~~~~']),
  ('fault site edge normalisation #1', [[[5, 5], [10, 3], [7, 0]], 2, 10], ['gg', 'gg', 'gg']),
  ('regression edge normalisation #2', [[[10, 11], [1, 0]], 3, 9], ['gg', 'gg']),
  ('regression edge normalisation #3', [[[9, 7]], 2, 6], ['^^']),
  ('control #1', [[[1, 0]], 2, 8], ['~~']),
  ('control #2', [[[8], [2], [5], [2], [6]], 6, 8], ['~', '~', '~', '~', '~'])]]
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 fixture	Actual	Expected	Outcome
corner cell average #1	`['gg', 'gg']`	`['gg', 'gg']`	Passed
beach band edge #1	`['~']`	`['g']`	Failed
single row #1	`['sss']`	`['^s^']`	Failed
regression edge normalisation #1	`['ss', 'ss']`	`['gg', 'gg']`	Failed
regression edge normalisation #2	`['sss~~', 'sss~~']`	`['^^ss~', '^^ss~']`	Failed
regression edge normalisation #3	`['~', '~']`	`['^', '^']`	Failed
control #1	`['~', '~', '~']`	`['~', '~', '~']`	Passed
control #2	`['~~~', '~~~']`	`['~~~', '~~~']`	Passed

SHA-256 / e6f1b6f4ce070f60189dbc38fa4305d434f0124d43ce8022e5ad01597ff64916

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(heights, sea, mountain):
    h = len(heights)
    w = len(heights[0])
    out = []
    for r in range(h):
        row = []
        for c in range(w):
            total = 0
            cnt = 0
            for rr in range(max(0, r - 1), min(h, r + 2)):
                for cc in range(max(0, c - 1), min(w, c + 2)):
                    total += heights[rr][cc]
                    cnt += 1
            v = total // cnt
            if v < sea:
                row.append('~')
            elif v < sea + 2:
                row.append('s')
            elif v < mountain:
                row.append('g')
            else:
                row.append('^')
        out.append(''.join(row))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[('corner cell average #1', [[[9, 9], [9, 0]], 3, 8], ['gg', 'gg']),
  ('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1', [[[0, 9], [7, 2]], 2, 11], ['gg', 'gg']),
  ('regression edge normalisation #2', [[[10, 11, 6, 11, 0], [9, 0, 9, 4, 4]], 5, 7], ['^^ss~', '^^ss~']),
  ('regression edge normalisation #3', [[[9], [8]], 6, 6], ['^', '^']),
  ('control #1', [[[3], [4], [0]], 4, 10], ['~', '~', '~']),
  ('control #2', [[[9, 5, 6], [9, 0, 4]], 6, 10], ['~~~', '~~~'])],
 [('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1', [[[0, 9], [7, 2]], 2, 11], ['gg', 'gg']),
  ('regression edge normalisation #2', [[[10, 11, 6, 11, 0], [9, 0, 9, 4, 4]], 5, 7], ['^^ss~', '^^ss~']),
  ('regression edge normalisation #3', [[[9], [8]], 6, 6], ['^', '^']),
  ('regression edge normalisation #4',
   [[[5, 4, 2, 6, 4], [12, 10, 8, 2, 4], [4, 10, 11, 2, 6]], 2, 8],
   ['ggggg', 'ggggg', '^^ggs']),
  ('control #1', [[[5, 3, 0, 8]], 6, 7], ['~~~~']),
  ('control #2', [[[4], [0]], 6, 11], ['~', '~'])],
 [('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('regression edge normalisation #1', [[[10, 11, 6, 11, 0], [9, 0, 9, 4, 4]], 5, 7], ['^^ss~', '^^ss~']),
  ('regression edge normalisation #2', [[[9], [8]], 6, 6], ['^', '^']),
  ('regression edge normalisation #3',
   [[[5, 4, 2, 6, 4], [12, 10, 8, 2, 4], [4, 10, 11, 2, 6]], 2, 8],
   ['ggggg', 'ggggg', '^^ggs']),
  ('regression edge normalisation #4', [[[6], [4], [8], [4], [8]], 5, 9], ['s', 's', 's', 's', 's']),
  ('control #1', [[[3, 2], [6, 0], [0, 5]], 4, 7], ['~~', '~~', '~~']),
  ('control #2', [[[3], [0]], 5, 10], ['~', '~'])],
 [('corner cell average #1', [[[9, 9], [9, 0]], 3, 8], ['gg', 'gg']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1',
   [[[5, 4, 2, 6, 4], [12, 10, 8, 2, 4], [4, 10, 11, 2, 6]], 2, 8],
   ['ggggg', 'ggggg', '^^ggs']),
  ('regression edge normalisation #2', [[[6], [4], [8], [4], [8]], 5, 9], ['s', 's', 's', 's', 's']),
  ('regression edge normalisation #3',
   [[[8, 6, 12, 10, 8], [6, 1, 1, 7, 5], [1, 6, 12, 6, 0], [3, 8, 3, 6, 0], [1, 7, 3, 2, 0]], 6, 9],
   ['~~sss', '~~sss', '~~~~~', '~~~~~', '~~~~~']),
  ('regression edge normalisation #4', [[[10, 11], [1, 0]], 3, 9], ['gg', 'gg']),
  ('control #1', [[[1], [10], [0], [2]], 6, 10], ['~', '~', '~', '~']),
  ('control #2', [[[0]], 6, 9], ['~'])],
 [('beach band edge #1', [[[5]], 3, 9], ['g']),
  ('single row #1', [[[0, 12, 0]], 3, 6], ['^s^']),
  ('regression edge normalisation #1',
   [[[8, 6, 12, 10, 8], [6, 1, 1, 7, 5], [1, 6, 12, 6, 0], [3, 8, 3, 6, 0], [1, 7, 3, 2, 0]], 6, 9],
   ['~~sss', '~~sss', '~~~~~', '~~~~~', '~~~~~']),
  ('fault site edge normalisation #1', [[[5, 5], [10, 3], [7, 0]], 2, 10], ['gg', 'gg', 'gg']),
  ('regression edge normalisation #2', [[[10, 11], [1, 0]], 3, 9], ['gg', 'gg']),
  ('regression edge normalisation #3', [[[9, 7]], 2, 6], ['^^']),
  ('control #1', [[[1, 0]], 2, 8], ['~~']),
  ('control #2', [[[8], [2], [5], [2], [6]], 6, 8], ['~', '~', '~', '~', '~'])]]
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 fixture	Actual	Expected	Outcome
corner cell average #1	`['gg', 'gg']`	`['gg', 'gg']`	Passed
beach band edge #1	`['g']`	`['g']`	Passed
single row #1	`['^s^']`	`['^s^']`	Passed
regression edge normalisation #1	`['gg', 'gg']`	`['gg', 'gg']`	Passed
regression edge normalisation #2	`['^^ss~', '^^ss~']`	`['^^ss~', '^^ss~']`	Passed
regression edge normalisation #3	`['^', '^']`	`['^', '^']`	Passed
control #1	`['~', '~', '~']`	`['~', '~', '~']`	Passed
control #2	`['~~~', '~~~']`	`['~~~', '~~~']`	Passed

SHA-256 / 34359ce6ad60c3cb3d142339dd41cb21cb052d567b73e342014aefb8beeabd59

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 2026-09-29T14:50:52.075435+00:00.

Case digest / 9181de45ed19d049a623bf79ab57e4ca082a3fd58da246c0a40ea692103718dc