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FA-55286 / Astronomical coordinate conventions / Open access

Catalog propagated covariance: Backward epoch propagation negates the catalog covariance itself · case 01

The adapter reports an incorrect backward cross covariance while other fields remain valid.

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

ROOT CAUSE

Backward epoch propagation negates the catalog covariance itself. Faulty expression: d['pm']+d['t']*d['m']

VERIFIED REPAIR

Preserve the declared model convention at this site: d['pm']-d['t']*d['m']

Unsuccessful approach: A partial convention repair still uses -d['pm']-d['t']*d['m']

Case contract

A catalog supplies position variance p, proper-motion variance m, covariance pm, elapsed interval t, acceleration variance a, and angular scale s. Under the stipulated independent acceleration model position covariance is p+2*t*pm+t*t*m+t**4*a/4. Motion output excludes acceleration unless explicitly requested. Output fields are defined by: position_variance = d['p']+2*d['t']*d['pm']+d['t']**2*d['m']+d['t']**4*d['a']/4; position_motion_covariance = d['pm']+d['t']*d['m']; acceleration_position_variance = d['t']**4*d['a']/4; motion_variance = d['m']+d['t']**2*d['a']; scaled_motion_covariance = (d['pm']+d['t']*d['m'])*d['s']**2; backward_cross_covariance = d['pm']-d['t']*d['m']

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 {'position_variance': d['p']+2*d['t']*d['pm']+d['t']**2*d['m']+d['t']**4*d['a']/4, 'position_motion_covariance': d['pm']+d['t']*d['m'], 'acceleration_position_variance': d['t']**4*d['a']/4, 'motion_variance': d['m']+d['t']**2*d['a'], 'scaled_motion_covariance': (d['pm']+d['t']*d['m'])*d['s']**2, 'backward_cross_covariance': d['pm']+d['t']*d['m']}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = {1: [({'p': 4, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 68.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 4, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 4.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 4, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 16.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 4, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 22.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 4, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 56.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 4, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 91.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 2: [({'p': 5, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 69.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 5, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 5.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 5, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 17.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 5, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 23.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 5, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 57.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 5, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 92.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 3: [({'p': 6, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 70.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 6, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 6.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 6, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 18.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 6, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 24.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 6, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 58.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 6, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 93.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 4: [({'p': 7, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 71.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 7, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 7.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 7, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 19.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 7, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 25.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 7, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 59.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 7, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 94.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 5: [({'p': 8, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 72.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 8, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 8.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 8, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 20.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 8, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 26.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 8, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 60.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 8, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 95.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})]}
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{'acceleration_position_variance': 40.5, 'backward_cross_covariance': 7, 'motion_variance': 20, 'position_motion_covariance': 7, 'position_variance': 68.5, 'scaled_motion_covariance': 28}{'acceleration_position_variance': 40.5, 'backward_cross_covariance': -5, 'motion_variance': 20, 'position_motion_covariance': 7, 'position_variance': 68.5, 'scaled_motion_covariance': 28}Failed
astronomical fixture 1{'acceleration_position_variance': 0.0, 'backward_cross_covariance': 1, 'motion_variance': 2, 'position_motion_covariance': 1, 'position_variance': 4.0, 'scaled_motion_covariance': 4}{'acceleration_position_variance': 0.0, 'backward_cross_covariance': 1, 'motion_variance': 2, 'position_motion_covariance': 1, 'position_variance': 4.0, 'scaled_motion_covariance': 4}Passed
astronomical fixture 2{'acceleration_position_variance': 8.0, 'backward_cross_covariance': -3, 'motion_variance': 10, 'position_motion_covariance': -3, 'position_variance': 16.0, 'scaled_motion_covariance': -12}{'acceleration_position_variance': 8.0, 'backward_cross_covariance': 5, 'motion_variance': 10, 'position_motion_covariance': -3, 'position_variance': 16.0, 'scaled_motion_covariance': -12}Failed
astronomical fixture 3{'acceleration_position_variance': 0.0, 'backward_cross_covariance': 6, 'motion_variance': 2, 'position_motion_covariance': 6, 'position_variance': 22.0, 'scaled_motion_covariance': 24}{'acceleration_position_variance': 0.0, 'backward_cross_covariance': -6, 'motion_variance': 2, 'position_motion_covariance': 6, 'position_variance': 22.0, 'scaled_motion_covariance': 24}Failed
astronomical fixture 4{'acceleration_position_variance': 40.5, 'backward_cross_covariance': 5, 'motion_variance': 20, 'position_motion_covariance': 5, 'position_variance': 56.5, 'scaled_motion_covariance': 45}{'acceleration_position_variance': 40.5, 'backward_cross_covariance': -7, 'motion_variance': 20, 'position_motion_covariance': 5, 'position_variance': 56.5, 'scaled_motion_covariance': 45}Failed
astronomical fixture 5{'acceleration_position_variance': 81.0, 'backward_cross_covariance': 1, 'motion_variance': 36, 'position_motion_covariance': 1, 'position_variance': 91.0, 'scaled_motion_covariance': 4}{'acceleration_position_variance': 81.0, 'backward_cross_covariance': 1, 'motion_variance': 36, 'position_motion_covariance': 1, 'position_variance': 91.0, 'scaled_motion_covariance': 4}Passed

SHA-256 / 04b9bcec1571c31f6eb66c7cc793d1411807e857d96f7f62c84c7219f9aafb17

2 / The unsuccessful fix

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

N = 1
observations = []
def solve(d):
    return {'position_variance': d['p']+2*d['t']*d['pm']+d['t']**2*d['m']+d['t']**4*d['a']/4, 'position_motion_covariance': d['pm']+d['t']*d['m'], 'acceleration_position_variance': d['t']**4*d['a']/4, 'motion_variance': d['m']+d['t']**2*d['a'], 'scaled_motion_covariance': (d['pm']+d['t']*d['m'])*d['s']**2, 'backward_cross_covariance': -d['pm']-d['t']*d['m']}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = {1: [({'p': 4, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 68.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 4, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 4.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 4, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 16.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 4, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 22.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 4, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 56.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 4, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 91.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 2: [({'p': 5, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 69.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 5, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 5.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 5, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 17.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 5, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 23.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 5, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 57.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 5, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 92.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 3: [({'p': 6, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 70.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 6, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 6.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 6, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 18.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 6, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 24.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 6, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 58.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 6, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 93.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 4: [({'p': 7, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 71.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 7, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 7.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 7, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 19.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 7, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 25.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 7, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 59.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 7, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 94.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 5: [({'p': 8, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 72.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 8, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 8.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 8, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 20.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 8, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 26.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 8, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 60.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 8, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 95.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})]}
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{'acceleration_position_variance': 40.5, 'backward_cross_covariance': -7, 'motion_variance': 20, 'position_motion_covariance': 7, 'position_variance': 68.5, 'scaled_motion_covariance': 28}{'acceleration_position_variance': 40.5, 'backward_cross_covariance': -5, 'motion_variance': 20, 'position_motion_covariance': 7, 'position_variance': 68.5, 'scaled_motion_covariance': 28}Failed
astronomical fixture 1{'acceleration_position_variance': 0.0, 'backward_cross_covariance': -1, 'motion_variance': 2, 'position_motion_covariance': 1, 'position_variance': 4.0, 'scaled_motion_covariance': 4}{'acceleration_position_variance': 0.0, 'backward_cross_covariance': 1, 'motion_variance': 2, 'position_motion_covariance': 1, 'position_variance': 4.0, 'scaled_motion_covariance': 4}Failed
astronomical fixture 2{'acceleration_position_variance': 8.0, 'backward_cross_covariance': 3, 'motion_variance': 10, 'position_motion_covariance': -3, 'position_variance': 16.0, 'scaled_motion_covariance': -12}{'acceleration_position_variance': 8.0, 'backward_cross_covariance': 5, 'motion_variance': 10, 'position_motion_covariance': -3, 'position_variance': 16.0, 'scaled_motion_covariance': -12}Failed
astronomical fixture 3{'acceleration_position_variance': 0.0, 'backward_cross_covariance': -6, 'motion_variance': 2, 'position_motion_covariance': 6, 'position_variance': 22.0, 'scaled_motion_covariance': 24}{'acceleration_position_variance': 0.0, 'backward_cross_covariance': -6, 'motion_variance': 2, 'position_motion_covariance': 6, 'position_variance': 22.0, 'scaled_motion_covariance': 24}Passed
astronomical fixture 4{'acceleration_position_variance': 40.5, 'backward_cross_covariance': -5, 'motion_variance': 20, 'position_motion_covariance': 5, 'position_variance': 56.5, 'scaled_motion_covariance': 45}{'acceleration_position_variance': 40.5, 'backward_cross_covariance': -7, 'motion_variance': 20, 'position_motion_covariance': 5, 'position_variance': 56.5, 'scaled_motion_covariance': 45}Failed
astronomical fixture 5{'acceleration_position_variance': 81.0, 'backward_cross_covariance': -1, 'motion_variance': 36, 'position_motion_covariance': 1, 'position_variance': 91.0, 'scaled_motion_covariance': 4}{'acceleration_position_variance': 81.0, 'backward_cross_covariance': 1, 'motion_variance': 36, 'position_motion_covariance': 1, 'position_variance': 91.0, 'scaled_motion_covariance': 4}Failed

SHA-256 / c6658d66fdf1a25d509d5cb39daf50fee9661e00595daa6bd3dbe70d6e9c432a

3 / The verified repair

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

N = 1
observations = []
def solve(d):
    return {'position_variance': d['p']+2*d['t']*d['pm']+d['t']**2*d['m']+d['t']**4*d['a']/4, 'position_motion_covariance': d['pm']+d['t']*d['m'], 'acceleration_position_variance': d['t']**4*d['a']/4, 'motion_variance': d['m']+d['t']**2*d['a'], 'scaled_motion_covariance': (d['pm']+d['t']*d['m'])*d['s']**2, 'backward_cross_covariance': d['pm']-d['t']*d['m']}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = {1: [({'p': 4, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 68.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 4, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 4.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 4, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 16.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 4, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 22.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 4, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 56.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 4, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 91.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 2: [({'p': 5, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 69.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 5, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 5.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 5, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 17.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 5, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 23.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 5, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 57.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 5, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 92.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 3: [({'p': 6, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 70.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 6, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 6.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 6, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 18.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 6, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 24.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 6, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 58.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 6, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 93.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 4: [({'p': 7, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 71.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 7, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 7.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 7, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 19.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 7, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 25.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 7, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 59.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 7, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 94.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})], 5: [({'p': 8, 'm': 2, 'pm': 1, 't': 3, 'a': 2, 's': 2}, {'position_variance': 72.5, 'position_motion_covariance': 7, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 28, 'backward_cross_covariance': -5}), ({'p': 8, 'm': 2, 'pm': 1, 't': 0, 'a': 2, 's': 2}, {'position_variance': 8.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1}), ({'p': 8, 'm': 2, 'pm': 1, 't': -2, 'a': 2, 's': 2}, {'position_variance': 20.0, 'position_motion_covariance': -3, 'acceleration_position_variance': 8.0, 'motion_variance': 10, 'scaled_motion_covariance': -12, 'backward_cross_covariance': 5}), ({'p': 8, 'm': 2, 'pm': 0, 't': 3, 'a': 0, 's': 2}, {'position_variance': 26.0, 'position_motion_covariance': 6, 'acceleration_position_variance': 0.0, 'motion_variance': 2, 'scaled_motion_covariance': 24, 'backward_cross_covariance': -6}), ({'p': 8, 'm': 2, 'pm': -1, 't': 3, 'a': 2, 's': -3}, {'position_variance': 60.5, 'position_motion_covariance': 5, 'acceleration_position_variance': 40.5, 'motion_variance': 20, 'scaled_motion_covariance': 45, 'backward_cross_covariance': -7}), ({'p': 8, 'm': 0, 'pm': 1, 't': 3, 'a': 4, 's': 2}, {'position_variance': 95.0, 'position_motion_covariance': 1, 'acceleration_position_variance': 81.0, 'motion_variance': 36, 'scaled_motion_covariance': 4, 'backward_cross_covariance': 1})]}
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{'acceleration_position_variance': 40.5, 'backward_cross_covariance': -5, 'motion_variance': 20, 'position_motion_covariance': 7, 'position_variance': 68.5, 'scaled_motion_covariance': 28}{'acceleration_position_variance': 40.5, 'backward_cross_covariance': -5, 'motion_variance': 20, 'position_motion_covariance': 7, 'position_variance': 68.5, 'scaled_motion_covariance': 28}Passed
astronomical fixture 1{'acceleration_position_variance': 0.0, 'backward_cross_covariance': 1, 'motion_variance': 2, 'position_motion_covariance': 1, 'position_variance': 4.0, 'scaled_motion_covariance': 4}{'acceleration_position_variance': 0.0, 'backward_cross_covariance': 1, 'motion_variance': 2, 'position_motion_covariance': 1, 'position_variance': 4.0, 'scaled_motion_covariance': 4}Passed
astronomical fixture 2{'acceleration_position_variance': 8.0, 'backward_cross_covariance': 5, 'motion_variance': 10, 'position_motion_covariance': -3, 'position_variance': 16.0, 'scaled_motion_covariance': -12}{'acceleration_position_variance': 8.0, 'backward_cross_covariance': 5, 'motion_variance': 10, 'position_motion_covariance': -3, 'position_variance': 16.0, 'scaled_motion_covariance': -12}Passed
astronomical fixture 3{'acceleration_position_variance': 0.0, 'backward_cross_covariance': -6, 'motion_variance': 2, 'position_motion_covariance': 6, 'position_variance': 22.0, 'scaled_motion_covariance': 24}{'acceleration_position_variance': 0.0, 'backward_cross_covariance': -6, 'motion_variance': 2, 'position_motion_covariance': 6, 'position_variance': 22.0, 'scaled_motion_covariance': 24}Passed
astronomical fixture 4{'acceleration_position_variance': 40.5, 'backward_cross_covariance': -7, 'motion_variance': 20, 'position_motion_covariance': 5, 'position_variance': 56.5, 'scaled_motion_covariance': 45}{'acceleration_position_variance': 40.5, 'backward_cross_covariance': -7, 'motion_variance': 20, 'position_motion_covariance': 5, 'position_variance': 56.5, 'scaled_motion_covariance': 45}Passed
astronomical fixture 5{'acceleration_position_variance': 81.0, 'backward_cross_covariance': 1, 'motion_variance': 36, 'position_motion_covariance': 1, 'position_variance': 91.0, 'scaled_motion_covariance': 4}{'acceleration_position_variance': 81.0, 'backward_cross_covariance': 1, 'motion_variance': 36, 'position_motion_covariance': 1, 'position_variance': 91.0, 'scaled_motion_covariance': 4}Passed

SHA-256 / ffe50ef2f9f298c08b9b504616fbc2a6bfe3f197fef251c370969a2ff5e50c34

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 2026-09-29T14:45:56.410104+00:00.

Case digest / 0d114d576073667ef2bff6440fd484e471d471467860243bce55627befd4faae