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FA-55906 / Astronomical coordinate conventions / Open access

Nested sky cell address: Nested sky descendant range expands by a linear or binary scale · case 01

The adapter reports an incorrect descendant first while other fields remain valid.

Verified by executionVariant 1 · 6 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Nested sky descendant range expands by a linear or binary scale. Faulty expression: d['p']*d['depth']

THE FAILURE

Nested sky descendant range expands by a linear or binary scale. Faulty expression: d['p']*d['depth']

Unsuccessful approach: A partial convention repair still uses d['p']*2**d['depth']

Case contract

A stipulated hierarchical sky tessellation has 12 base faces and four nested children per cell. At order o a face has 4**o pixels. p is the local pixel index, f the face, k the child digit. Parent and descendant ranges are nested indexing operations; no geometric HEALPix projection is implemented. Output fields are defined by: global_pixel = d['f']*4**d['o']+d['p']; parent_local = d['p']//4; child_local = 4*d['p']+d['k']; descendant_first = d['p']*4**d['depth']; descendant_stop = (d['p']+1)*4**d['depth']; face_count = 12*4**d['o']

Why this case matters

Catalog, detector, sky-coordinate, and spectroscopy adapters must preserve the association between numeric coordinates and their declared reference conventions.

1 / The failure

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

N = 1
observations = []
def solve(d):
    return {'global_pixel': d['f']*4**d['o']+d['p'], 'parent_local': d['p']//4, 'child_local': 4*d['p']+d['k'], 'descendant_first': d['p']*d['depth'], 'descendant_stop': (d['p']+1)*4**d['depth'], 'face_count': 12*4**d['o']}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = {1: [({'o': 2, 'p': 5, 'f': 3, 'k': 2, 'depth': 2}, {'global_pixel': 53, 'parent_local': 1, 'child_local': 22, 'descendant_first': 80, 'descendant_stop': 96, 'face_count': 192}), ({'o': 2, 'p': 0, 'f': 0, 'k': 0, 'depth': 2}, {'global_pixel': 0, 'parent_local': 0, 'child_local': 0, 'descendant_first': 0, 'descendant_stop': 16, 'face_count': 192}), ({'o': 2, 'p': 5, 'f': 11, 'k': 3, 'depth': 2}, {'global_pixel': 181, 'parent_local': 1, 'child_local': 23, 'descendant_first': 80, 'descendant_stop': 96, 'face_count': 192}), ({'o': 2, 'p': 5, 'f': 3, 'k': 2, 'depth': 0}, {'global_pixel': 53, 'parent_local': 1, 'child_local': 22, 'descendant_first': 5, 'descendant_stop': 6, 'face_count': 192}), ({'o': 2, 'p': 2, 'f': 1, 'k': 2, 'depth': 2}, {'global_pixel': 18, 'parent_local': 0, 'child_local': 10, 'descendant_first': 32, 'descendant_stop': 48, 'face_count': 192}), ({'o': 3, 'p': 20, 'f': 3, 'k': 2, 'depth': 1}, {'global_pixel': 212, 'parent_local': 5, 'child_local': 82, 'descendant_first': 80, 'descendant_stop': 84, 'face_count': 768})], 2: [({'o': 2, 'p': 6, 'f': 3, 'k': 2, 'depth': 2}, {'global_pixel': 54, 'parent_local': 1, 'child_local': 26, 'descendant_first': 96, 'descendant_stop': 112, 'face_count': 192}), ({'o': 2, 'p': 1, 'f': 0, 'k': 0, 'depth': 2}, {'global_pixel': 1, 'parent_local': 0, 'child_local': 4, 'descendant_first': 16, 'descendant_stop': 32, 'face_count': 192}), ({'o': 2, 'p': 6, 'f': 11, 'k': 3, 'depth': 2}, {'global_pixel': 182, 'parent_local': 1, 'child_local': 27, 'descendant_first': 96, 'descendant_stop': 112, 'face_count': 192}), ({'o': 2, 'p': 6, 'f': 3, 'k': 2, 'depth': 0}, {'global_pixel': 54, 'parent_local': 1, 'child_local': 26, 'descendant_first': 6, 'descendant_stop': 7, 'face_count': 192}), ({'o': 2, 'p': 3, 'f': 1, 'k': 2, 'depth': 2}, {'global_pixel': 19, 'parent_local': 0, 'child_local': 14, 'descendant_first': 48, 'descendant_stop': 64, 'face_count': 192}), ({'o': 3, 'p': 21, 'f': 3, 'k': 2, 'depth': 1}, {'global_pixel': 213, 'parent_local': 5, 'child_local': 86, 'descendant_first': 84, 'descendant_stop': 88, 'face_count': 768})], 3: [({'o': 2, 'p': 7, 'f': 3, 'k': 2, 'depth': 2}, {'global_pixel': 55, 'parent_local': 1, 'child_local': 30, 'descendant_first': 112, 'descendant_stop': 128, 'face_count': 192}), ({'o': 2, 'p': 2, 'f': 0, 'k': 0, 'depth': 2}, {'global_pixel': 2, 'parent_local': 0, 'child_local': 8, 'descendant_first': 32, 'descendant_stop': 48, 'face_count': 192}), ({'o': 2, 'p': 7, 'f': 11, 'k': 3, 'depth': 2}, {'global_pixel': 183, 'parent_local': 1, 'child_local': 31, 'descendant_first': 112, 'descendant_stop': 128, 'face_count': 192}), ({'o': 2, 'p': 7, 'f': 3, 'k': 2, 'depth': 0}, {'global_pixel': 55, 'parent_local': 1, 'child_local': 30, 'descendant_first': 7, 'descendant_stop': 8, 'face_count': 192}), ({'o': 2, 'p': 4, 'f': 1, 'k': 2, 'depth': 2}, {'global_pixel': 20, 'parent_local': 1, 'child_local': 18, 'descendant_first': 64, 'descendant_stop': 80, 'face_count': 192}), ({'o': 3, 'p': 22, 'f': 3, 'k': 2, 'depth': 1}, {'global_pixel': 214, 'parent_local': 5, 'child_local': 90, 'descendant_first': 88, 'descendant_stop': 92, 'face_count': 768})], 4: [({'o': 2, 'p': 8, 'f': 3, 'k': 2, 'depth': 2}, {'global_pixel': 56, 'parent_local': 2, 'child_local': 34, 'descendant_first': 128, 'descendant_stop': 144, 'face_count': 192}), ({'o': 2, 'p': 3, 'f': 0, 'k': 0, 'depth': 2}, {'global_pixel': 3, 'parent_local': 0, 'child_local': 12, 'descendant_first': 48, 'descendant_stop': 64, 'face_count': 192}), ({'o': 2, 'p': 8, 'f': 11, 'k': 3, 'depth': 2}, {'global_pixel': 184, 'parent_local': 2, 'child_local': 35, 'descendant_first': 128, 'descendant_stop': 144, 'face_count': 192}), ({'o': 2, 'p': 8, 'f': 3, 'k': 2, 'depth': 0}, {'global_pixel': 56, 'parent_local': 2, 'child_local': 34, 'descendant_first': 8, 'descendant_stop': 9, 'face_count': 192}), ({'o': 2, 'p': 5, 'f': 1, 'k': 2, 'depth': 2}, {'global_pixel': 21, 'parent_local': 1, 'child_local': 22, 'descendant_first': 80, 'descendant_stop': 96, 'face_count': 192}), ({'o': 3, 'p': 23, 'f': 3, 'k': 2, 'depth': 1}, {'global_pixel': 215, 'parent_local': 5, 'child_local': 94, 'descendant_first': 92, 'descendant_stop': 96, 'face_count': 768})], 5: [({'o': 2, 'p': 9, 'f': 3, 'k': 2, 'depth': 2}, {'global_pixel': 57, 'parent_local': 2, 'child_local': 38, 'descendant_first': 144, 'descendant_stop': 160, 'face_count': 192}), ({'o': 2, 'p': 4, 'f': 0, 'k': 0, 'depth': 2}, {'global_pixel': 4, 'parent_local': 1, 'child_local': 16, 'descendant_first': 64, 'descendant_stop': 80, 'face_count': 192}), ({'o': 2, 'p': 9, 'f': 11, 'k': 3, 'depth': 2}, {'global_pixel': 185, 'parent_local': 2, 'child_local': 39, 'descendant_first': 144, 'descendant_stop': 160, 'face_count': 192}), ({'o': 2, 'p': 9, 'f': 3, 'k': 2, 'depth': 0}, {'global_pixel': 57, 'parent_local': 2, 'child_local': 38, 'descendant_first': 9, 'descendant_stop': 10, 'face_count': 192}), ({'o': 2, 'p': 6, 'f': 1, 'k': 2, 'depth': 2}, {'global_pixel': 22, 'parent_local': 1, 'child_local': 26, 'descendant_first': 96, 'descendant_stop': 112, 'face_count': 192}), ({'o': 3, 'p': 24, 'f': 3, 'k': 2, 'depth': 1}, {'global_pixel': 216, 'parent_local': 6, 'child_local': 98, 'descendant_first': 96, 'descendant_stop': 100, 'face_count': 768})]}
for i, (record, expected) in enumerate(fixtures[N]):
    check('astronomical fixture %s' % i, solve(record), 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
astronomical fixture 0{'child_local': 22, 'descendant_first': 10, 'descendant_stop': 96, 'face_count': 192, 'global_pixel': 53, 'parent_local': 1}{'child_local': 22, 'descendant_first': 80, 'descendant_stop': 96, 'face_count': 192, 'global_pixel': 53, 'parent_local': 1}Failed
astronomical fixture 1{'child_local': 0, 'descendant_first': 0, 'descendant_stop': 16, 'face_count': 192, 'global_pixel': 0, 'parent_local': 0}{'child_local': 0, 'descendant_first': 0, 'descendant_stop': 16, 'face_count': 192, 'global_pixel': 0, 'parent_local': 0}Passed
astronomical fixture 2{'child_local': 23, 'descendant_first': 10, 'descendant_stop': 96, 'face_count': 192, 'global_pixel': 181, 'parent_local': 1}{'child_local': 23, 'descendant_first': 80, 'descendant_stop': 96, 'face_count': 192, 'global_pixel': 181, 'parent_local': 1}Failed
astronomical fixture 3{'child_local': 22, 'descendant_first': 0, 'descendant_stop': 6, 'face_count': 192, 'global_pixel': 53, 'parent_local': 1}{'child_local': 22, 'descendant_first': 5, 'descendant_stop': 6, 'face_count': 192, 'global_pixel': 53, 'parent_local': 1}Failed
astronomical fixture 4{'child_local': 10, 'descendant_first': 4, 'descendant_stop': 48, 'face_count': 192, 'global_pixel': 18, 'parent_local': 0}{'child_local': 10, 'descendant_first': 32, 'descendant_stop': 48, 'face_count': 192, 'global_pixel': 18, 'parent_local': 0}Failed
astronomical fixture 5{'child_local': 82, 'descendant_first': 20, 'descendant_stop': 84, 'face_count': 768, 'global_pixel': 212, 'parent_local': 5}{'child_local': 82, 'descendant_first': 80, 'descendant_stop': 84, 'face_count': 768, 'global_pixel': 212, 'parent_local': 5}Failed

SHA-256 / 972123beae5b0905e4cda8dc0590657c965b5527419194477a461c37eb9920c5

2 / The unsuccessful fix

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

N = 1
observations = []
def solve(d):
    return {'global_pixel': d['f']*4**d['o']+d['p'], 'parent_local': d['p']//4, 'child_local': 4*d['p']+d['k'], 'descendant_first': d['p']*2**d['depth'], 'descendant_stop': (d['p']+1)*4**d['depth'], 'face_count': 12*4**d['o']}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = {1: [({'o': 2, 'p': 5, 'f': 3, 'k': 2, 'depth': 2}, {'global_pixel': 53, 'parent_local': 1, 'child_local': 22, 'descendant_first': 80, 'descendant_stop': 96, 'face_count': 192}), ({'o': 2, 'p': 0, 'f': 0, 'k': 0, 'depth': 2}, {'global_pixel': 0, 'parent_local': 0, 'child_local': 0, 'descendant_first': 0, 'descendant_stop': 16, 'face_count': 192}), ({'o': 2, 'p': 5, 'f': 11, 'k': 3, 'depth': 2}, {'global_pixel': 181, 'parent_local': 1, 'child_local': 23, 'descendant_first': 80, 'descendant_stop': 96, 'face_count': 192}), ({'o': 2, 'p': 5, 'f': 3, 'k': 2, 'depth': 0}, {'global_pixel': 53, 'parent_local': 1, 'child_local': 22, 'descendant_first': 5, 'descendant_stop': 6, 'face_count': 192}), ({'o': 2, 'p': 2, 'f': 1, 'k': 2, 'depth': 2}, {'global_pixel': 18, 'parent_local': 0, 'child_local': 10, 'descendant_first': 32, 'descendant_stop': 48, 'face_count': 192}), ({'o': 3, 'p': 20, 'f': 3, 'k': 2, 'depth': 1}, {'global_pixel': 212, 'parent_local': 5, 'child_local': 82, 'descendant_first': 80, 'descendant_stop': 84, 'face_count': 768})], 2: [({'o': 2, 'p': 6, 'f': 3, 'k': 2, 'depth': 2}, {'global_pixel': 54, 'parent_local': 1, 'child_local': 26, 'descendant_first': 96, 'descendant_stop': 112, 'face_count': 192}), ({'o': 2, 'p': 1, 'f': 0, 'k': 0, 'depth': 2}, {'global_pixel': 1, 'parent_local': 0, 'child_local': 4, 'descendant_first': 16, 'descendant_stop': 32, 'face_count': 192}), ({'o': 2, 'p': 6, 'f': 11, 'k': 3, 'depth': 2}, {'global_pixel': 182, 'parent_local': 1, 'child_local': 27, 'descendant_first': 96, 'descendant_stop': 112, 'face_count': 192}), ({'o': 2, 'p': 6, 'f': 3, 'k': 2, 'depth': 0}, {'global_pixel': 54, 'parent_local': 1, 'child_local': 26, 'descendant_first': 6, 'descendant_stop': 7, 'face_count': 192}), ({'o': 2, 'p': 3, 'f': 1, 'k': 2, 'depth': 2}, {'global_pixel': 19, 'parent_local': 0, 'child_local': 14, 'descendant_first': 48, 'descendant_stop': 64, 'face_count': 192}), ({'o': 3, 'p': 21, 'f': 3, 'k': 2, 'depth': 1}, {'global_pixel': 213, 'parent_local': 5, 'child_local': 86, 'descendant_first': 84, 'descendant_stop': 88, 'face_count': 768})], 3: [({'o': 2, 'p': 7, 'f': 3, 'k': 2, 'depth': 2}, {'global_pixel': 55, 'parent_local': 1, 'child_local': 30, 'descendant_first': 112, 'descendant_stop': 128, 'face_count': 192}), ({'o': 2, 'p': 2, 'f': 0, 'k': 0, 'depth': 2}, {'global_pixel': 2, 'parent_local': 0, 'child_local': 8, 'descendant_first': 32, 'descendant_stop': 48, 'face_count': 192}), ({'o': 2, 'p': 7, 'f': 11, 'k': 3, 'depth': 2}, {'global_pixel': 183, 'parent_local': 1, 'child_local': 31, 'descendant_first': 112, 'descendant_stop': 128, 'face_count': 192}), ({'o': 2, 'p': 7, 'f': 3, 'k': 2, 'depth': 0}, {'global_pixel': 55, 'parent_local': 1, 'child_local': 30, 'descendant_first': 7, 'descendant_stop': 8, 'face_count': 192}), ({'o': 2, 'p': 4, 'f': 1, 'k': 2, 'depth': 2}, {'global_pixel': 20, 'parent_local': 1, 'child_local': 18, 'descendant_first': 64, 'descendant_stop': 80, 'face_count': 192}), ({'o': 3, 'p': 22, 'f': 3, 'k': 2, 'depth': 1}, {'global_pixel': 214, 'parent_local': 5, 'child_local': 90, 'descendant_first': 88, 'descendant_stop': 92, 'face_count': 768})], 4: [({'o': 2, 'p': 8, 'f': 3, 'k': 2, 'depth': 2}, {'global_pixel': 56, 'parent_local': 2, 'child_local': 34, 'descendant_first': 128, 'descendant_stop': 144, 'face_count': 192}), ({'o': 2, 'p': 3, 'f': 0, 'k': 0, 'depth': 2}, {'global_pixel': 3, 'parent_local': 0, 'child_local': 12, 'descendant_first': 48, 'descendant_stop': 64, 'face_count': 192}), ({'o': 2, 'p': 8, 'f': 11, 'k': 3, 'depth': 2}, {'global_pixel': 184, 'parent_local': 2, 'child_local': 35, 'descendant_first': 128, 'descendant_stop': 144, 'face_count': 192}), ({'o': 2, 'p': 8, 'f': 3, 'k': 2, 'depth': 0}, {'global_pixel': 56, 'parent_local': 2, 'child_local': 34, 'descendant_first': 8, 'descendant_stop': 9, 'face_count': 192}), ({'o': 2, 'p': 5, 'f': 1, 'k': 2, 'depth': 2}, {'global_pixel': 21, 'parent_local': 1, 'child_local': 22, 'descendant_first': 80, 'descendant_stop': 96, 'face_count': 192}), ({'o': 3, 'p': 23, 'f': 3, 'k': 2, 'depth': 1}, {'global_pixel': 215, 'parent_local': 5, 'child_local': 94, 'descendant_first': 92, 'descendant_stop': 96, 'face_count': 768})], 5: [({'o': 2, 'p': 9, 'f': 3, 'k': 2, 'depth': 2}, {'global_pixel': 57, 'parent_local': 2, 'child_local': 38, 'descendant_first': 144, 'descendant_stop': 160, 'face_count': 192}), ({'o': 2, 'p': 4, 'f': 0, 'k': 0, 'depth': 2}, {'global_pixel': 4, 'parent_local': 1, 'child_local': 16, 'descendant_first': 64, 'descendant_stop': 80, 'face_count': 192}), ({'o': 2, 'p': 9, 'f': 11, 'k': 3, 'depth': 2}, {'global_pixel': 185, 'parent_local': 2, 'child_local': 39, 'descendant_first': 144, 'descendant_stop': 160, 'face_count': 192}), ({'o': 2, 'p': 9, 'f': 3, 'k': 2, 'depth': 0}, {'global_pixel': 57, 'parent_local': 2, 'child_local': 38, 'descendant_first': 9, 'descendant_stop': 10, 'face_count': 192}), ({'o': 2, 'p': 6, 'f': 1, 'k': 2, 'depth': 2}, {'global_pixel': 22, 'parent_local': 1, 'child_local': 26, 'descendant_first': 96, 'descendant_stop': 112, 'face_count': 192}), ({'o': 3, 'p': 24, 'f': 3, 'k': 2, 'depth': 1}, {'global_pixel': 216, 'parent_local': 6, 'child_local': 98, 'descendant_first': 96, 'descendant_stop': 100, 'face_count': 768})]}
for i, (record, expected) in enumerate(fixtures[N]):
    check('astronomical fixture %s' % i, solve(record), 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
astronomical fixture 0{'child_local': 22, 'descendant_first': 20, 'descendant_stop': 96, 'face_count': 192, 'global_pixel': 53, 'parent_local': 1}{'child_local': 22, 'descendant_first': 80, 'descendant_stop': 96, 'face_count': 192, 'global_pixel': 53, 'parent_local': 1}Failed
astronomical fixture 1{'child_local': 0, 'descendant_first': 0, 'descendant_stop': 16, 'face_count': 192, 'global_pixel': 0, 'parent_local': 0}{'child_local': 0, 'descendant_first': 0, 'descendant_stop': 16, 'face_count': 192, 'global_pixel': 0, 'parent_local': 0}Passed
astronomical fixture 2{'child_local': 23, 'descendant_first': 20, 'descendant_stop': 96, 'face_count': 192, 'global_pixel': 181, 'parent_local': 1}{'child_local': 23, 'descendant_first': 80, 'descendant_stop': 96, 'face_count': 192, 'global_pixel': 181, 'parent_local': 1}Failed
astronomical fixture 3{'child_local': 22, 'descendant_first': 5, 'descendant_stop': 6, 'face_count': 192, 'global_pixel': 53, 'parent_local': 1}{'child_local': 22, 'descendant_first': 5, 'descendant_stop': 6, 'face_count': 192, 'global_pixel': 53, 'parent_local': 1}Passed
astronomical fixture 4{'child_local': 10, 'descendant_first': 8, 'descendant_stop': 48, 'face_count': 192, 'global_pixel': 18, 'parent_local': 0}{'child_local': 10, 'descendant_first': 32, 'descendant_stop': 48, 'face_count': 192, 'global_pixel': 18, 'parent_local': 0}Failed
astronomical fixture 5{'child_local': 82, 'descendant_first': 40, 'descendant_stop': 84, 'face_count': 768, 'global_pixel': 212, 'parent_local': 5}{'child_local': 82, 'descendant_first': 80, 'descendant_stop': 84, 'face_count': 768, 'global_pixel': 212, 'parent_local': 5}Failed

SHA-256 / d76a86c7fa6e921146da4f279e37bb0a338f687f14911d9d6c5b889ca46524d1

HELD IN THE MEMBER ARCHIVE

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

This mechanism has 6 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.

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Verification & scope

Explicitly stipulated finite algebraic adapter; no standards conformance, physical accuracy, or production-library claim. Inputs are the documented finite valid model domain. 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 2026-09-29T14:46:02.324201+00:00.

Case digest / d1f291c549185367c67f08234499597c4aafe3c3438895416905758b4479a825