FA-40906 / Heap invariants / Open access

Calendar queue interval upper bounds are exclusive next-bucket starts · case 01

The bounded calendar placement certificate reports an incorrect upper bounds.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Calendar queue interval upper bounds are exclusive next-bucket starts.

VERIFIED REPAIR

Derive upper bounds using [o+(v+1)*w for v in virtual] under the stated bounded certificate contract.

Unsuccessful approach: The local patch uses [o+(v+1)*w-1 for v in virtual] and still violates the stated relation.

Case contract

A calendar priority queue uses integer tick origin O, positive bucket width W, and B buckets. Events [id,tick] map to virtual bucket floor((tick-O)/W), then physical index modulo B, with absolute interval boundaries retained. Report virtual and physical placements, lower and upper bounds, same-cycle eligibility relative to cursor virtual bucket, and exact-boundary events.

Why this case matters

This isolates an internal heap representation or priority-structure invariant using deterministic finite records.

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(d):
    a=d['events']; o=d['origin']; w=d['width']; b=d['buckets']; cursor=d['cursor']; virtual=[(x[1]-o)//w for x in a]
    return {'virtual': virtual,
    'physical': [v%b for v in virtual],
    'lower_bounds': [o+v*w for v in virtual],
    'upper_bounds': [o+v*w for v in virtual],
    'cycle_eligible': [x[0] for x,v in zip(a,virtual) if cursor<=v<cursor+b],
    'boundaries': [x[0] for x in a if (x[1]-o)%w==0]}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 4, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 2, 2, 4, 6], 'physical': [2, 2, 2, 4, 1], 'lower_bounds': [12, 12, 12, 20, 28], 'upper_bounds': [16, 16, 16, 24, 32], 'cycle_eligible': ['3', '4'], 'boundaries': ['0', '4']})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 5, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 2, 3, 4], 'physical': [1, 1, 2, 3, 4], 'lower_bounds': [9, 9, 14, 19, 24], 'upper_bounds': [14, 14, 19, 24, 29], 'cycle_eligible': ['3', '4'], 'boundaries': ['2']})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 6, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 1, 3, 4], 'physical': [1, 1, 1, 3, 4], 'lower_bounds': [10, 10, 10, 22, 28], 'upper_bounds': [16, 16, 16, 28, 34], 'cycle_eligible': ['3', '4'], 'boundaries': ['3', '4']})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 7, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 1, 2, 3], 'physical': [1, 1, 1, 2, 3], 'lower_bounds': [11, 11, 11, 18, 25], 'upper_bounds': [18, 18, 18, 25, 32], 'cycle_eligible': ['4'], 'boundaries': []})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 8, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 1, 2, 3], 'physical': [1, 1, 1, 2, 3], 'lower_bounds': [12, 12, 12, 20, 28], 'upper_bounds': [20, 20, 20, 28, 36], 'cycle_eligible': ['4'], 'boundaries': ['0', '4']})]][N-1]
check('regression certificate 1', solve(cases[0][0]), cases[0][1])
check('regression certificate 2', solve(cases[1][0]), cases[1][1])
check('regression certificate 3', solve(cases[2][0]), cases[2][1])
check('regression certificate 4', solve(cases[3][0]), cases[3][1])
check('regression certificate 5', solve(cases[4][0]), cases[4][1])
check('regression certificate 6', solve(cases[5][0]), cases[5][1])
check('variant-dependent certificate', solve(cases[6][0]), cases[6][1])
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
regression certificate 1	`{'boundaries': [], 'cycle_eligible': [], 'lower_bounds': [], 'physical': [], 'upper_bounds': [], 'virtual': []}`	`{'boundaries': [], 'cycle_eligible': [], 'lower_bounds': [], 'physical': [], 'upper_bounds': [], 'virtual': []}`	Passed
regression certificate 2	`{'boundaries': ['0', '2', '3'], 'cycle_eligible': ['0', '1', '2'], 'lower_bounds': [3, 3, 7, 19], 'physical': [0, 0, 1, 0], 'upper_bounds': [3, 3, 7, 19], 'virtual': [0, 0, 1, 4]}`	`{'boundaries': ['0', '2', '3'], 'cycle_eligible': ['0', '1', '2'], 'lower_bounds': [3, 3, 7, 19], 'physical': [0, 0, 1, 0], 'upper_bounds': [7, 7, 11, 23], 'virtual': [0, 0, 1, 4]}`	Failed
regression certificate 3	`{'boundaries': ['1', '3', '4', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'physical': [2, 3, 3, 0, 1, 0], 'upper_bounds': [1, 4, 4, 7, 10, 19], 'virtual': [-2, -1, -1, 0, 1, 4]}`	`{'boundaries': ['1', '3', '4', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'physical': [2, 3, 3, 0, 1, 0], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'virtual': [-2, -1, -1, 0, 1, 4]}`	Failed
regression certificate 4	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'physical': [1, 2, 2, 0, 1, 2], 'upper_bounds': [7, 12, 12, 17, 22, 27], 'virtual': [1, 2, 2, 3, 4, 5]}`	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'physical': [1, 2, 2, 0, 1, 2], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'virtual': [1, 2, 2, 3, 4, 5]}`	Failed
regression certificate 5	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'physical': [0, 1, 1, 0, 0, 0], 'upper_bounds': [-3, -1, -1, 1, 1, 5], 'virtual': [-2, -1, -1, 0, 0, 2]}`	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'physical': [0, 1, 1, 0, 0, 0], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'virtual': [-2, -1, -1, 0, 0, 2]}`	Failed
regression certificate 6	`{'boundaries': ['1', '3', '4'], 'cycle_eligible': ['1', '2', '3'], 'lower_bounds': [10, 13, 13, 22, 28], 'physical': [2, 3, 3, 1, 3], 'upper_bounds': [10, 13, 13, 22, 28], 'virtual': [2, 3, 3, 6, 8]}`	`{'boundaries': ['1', '3', '4'], 'cycle_eligible': ['1', '2', '3'], 'lower_bounds': [10, 13, 13, 22, 28], 'physical': [2, 3, 3, 1, 3], 'upper_bounds': [13, 16, 16, 25, 31], 'virtual': [2, 3, 3, 6, 8]}`	Failed
variant-dependent certificate	`{'boundaries': ['0', '4'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [12, 12, 12, 20, 28], 'physical': [2, 2, 2, 4, 1], 'upper_bounds': [12, 12, 12, 20, 28], 'virtual': [2, 2, 2, 4, 6]}`	`{'boundaries': ['0', '4'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [12, 12, 12, 20, 28], 'physical': [2, 2, 2, 4, 1], 'upper_bounds': [16, 16, 16, 24, 32], 'virtual': [2, 2, 2, 4, 6]}`	Failed

SHA-256 / f3d196ff93150e1fc2cf16483d5e5b6043336a8a0752a6ec3e3fbb2c347da01c

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(d):
    a=d['events']; o=d['origin']; w=d['width']; b=d['buckets']; cursor=d['cursor']; virtual=[(x[1]-o)//w for x in a]
    return {'virtual': virtual,
    'physical': [v%b for v in virtual],
    'lower_bounds': [o+v*w for v in virtual],
    'upper_bounds': [o+(v+1)*w-1 for v in virtual],
    'cycle_eligible': [x[0] for x,v in zip(a,virtual) if cursor<=v<cursor+b],
    'boundaries': [x[0] for x in a if (x[1]-o)%w==0]}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 4, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 2, 2, 4, 6], 'physical': [2, 2, 2, 4, 1], 'lower_bounds': [12, 12, 12, 20, 28], 'upper_bounds': [16, 16, 16, 24, 32], 'cycle_eligible': ['3', '4'], 'boundaries': ['0', '4']})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 5, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 2, 3, 4], 'physical': [1, 1, 2, 3, 4], 'lower_bounds': [9, 9, 14, 19, 24], 'upper_bounds': [14, 14, 19, 24, 29], 'cycle_eligible': ['3', '4'], 'boundaries': ['2']})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 6, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 1, 3, 4], 'physical': [1, 1, 1, 3, 4], 'lower_bounds': [10, 10, 10, 22, 28], 'upper_bounds': [16, 16, 16, 28, 34], 'cycle_eligible': ['3', '4'], 'boundaries': ['3', '4']})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 7, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 1, 2, 3], 'physical': [1, 1, 1, 2, 3], 'lower_bounds': [11, 11, 11, 18, 25], 'upper_bounds': [18, 18, 18, 25, 32], 'cycle_eligible': ['4'], 'boundaries': []})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 8, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 1, 2, 3], 'physical': [1, 1, 1, 2, 3], 'lower_bounds': [12, 12, 12, 20, 28], 'upper_bounds': [20, 20, 20, 28, 36], 'cycle_eligible': ['4'], 'boundaries': ['0', '4']})]][N-1]
check('regression certificate 1', solve(cases[0][0]), cases[0][1])
check('regression certificate 2', solve(cases[1][0]), cases[1][1])
check('regression certificate 3', solve(cases[2][0]), cases[2][1])
check('regression certificate 4', solve(cases[3][0]), cases[3][1])
check('regression certificate 5', solve(cases[4][0]), cases[4][1])
check('regression certificate 6', solve(cases[5][0]), cases[5][1])
check('variant-dependent certificate', solve(cases[6][0]), cases[6][1])
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
regression certificate 1	`{'boundaries': [], 'cycle_eligible': [], 'lower_bounds': [], 'physical': [], 'upper_bounds': [], 'virtual': []}`	`{'boundaries': [], 'cycle_eligible': [], 'lower_bounds': [], 'physical': [], 'upper_bounds': [], 'virtual': []}`	Passed
regression certificate 2	`{'boundaries': ['0', '2', '3'], 'cycle_eligible': ['0', '1', '2'], 'lower_bounds': [3, 3, 7, 19], 'physical': [0, 0, 1, 0], 'upper_bounds': [6, 6, 10, 22], 'virtual': [0, 0, 1, 4]}`	`{'boundaries': ['0', '2', '3'], 'cycle_eligible': ['0', '1', '2'], 'lower_bounds': [3, 3, 7, 19], 'physical': [0, 0, 1, 0], 'upper_bounds': [7, 7, 11, 23], 'virtual': [0, 0, 1, 4]}`	Failed
regression certificate 3	`{'boundaries': ['1', '3', '4', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'physical': [2, 3, 3, 0, 1, 0], 'upper_bounds': [3, 6, 6, 9, 12, 21], 'virtual': [-2, -1, -1, 0, 1, 4]}`	`{'boundaries': ['1', '3', '4', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'physical': [2, 3, 3, 0, 1, 0], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'virtual': [-2, -1, -1, 0, 1, 4]}`	Failed
regression certificate 4	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'physical': [1, 2, 2, 0, 1, 2], 'upper_bounds': [11, 16, 16, 21, 26, 31], 'virtual': [1, 2, 2, 3, 4, 5]}`	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'physical': [1, 2, 2, 0, 1, 2], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'virtual': [1, 2, 2, 3, 4, 5]}`	Failed
regression certificate 5	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'physical': [0, 1, 1, 0, 0, 0], 'upper_bounds': [-2, 0, 0, 2, 2, 6], 'virtual': [-2, -1, -1, 0, 0, 2]}`	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'physical': [0, 1, 1, 0, 0, 0], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'virtual': [-2, -1, -1, 0, 0, 2]}`	Failed
regression certificate 6	`{'boundaries': ['1', '3', '4'], 'cycle_eligible': ['1', '2', '3'], 'lower_bounds': [10, 13, 13, 22, 28], 'physical': [2, 3, 3, 1, 3], 'upper_bounds': [12, 15, 15, 24, 30], 'virtual': [2, 3, 3, 6, 8]}`	`{'boundaries': ['1', '3', '4'], 'cycle_eligible': ['1', '2', '3'], 'lower_bounds': [10, 13, 13, 22, 28], 'physical': [2, 3, 3, 1, 3], 'upper_bounds': [13, 16, 16, 25, 31], 'virtual': [2, 3, 3, 6, 8]}`	Failed
variant-dependent certificate	`{'boundaries': ['0', '4'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [12, 12, 12, 20, 28], 'physical': [2, 2, 2, 4, 1], 'upper_bounds': [15, 15, 15, 23, 31], 'virtual': [2, 2, 2, 4, 6]}`	`{'boundaries': ['0', '4'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [12, 12, 12, 20, 28], 'physical': [2, 2, 2, 4, 1], 'upper_bounds': [16, 16, 16, 24, 32], 'virtual': [2, 2, 2, 4, 6]}`	Failed

SHA-256 / acffe76d8e325be9f7b979ac8f128370e9acc55352dcb7cf79c9759d357e9fb1

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(d):
    a=d['events']; o=d['origin']; w=d['width']; b=d['buckets']; cursor=d['cursor']; virtual=[(x[1]-o)//w for x in a]
    return {'virtual': virtual,
    'physical': [v%b for v in virtual],
    'lower_bounds': [o+v*w for v in virtual],
    'upper_bounds': [o+(v+1)*w for v in virtual],
    'cycle_eligible': [x[0] for x,v in zip(a,virtual) if cursor<=v<cursor+b],
    'boundaries': [x[0] for x in a if (x[1]-o)%w==0]}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 4, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 2, 2, 4, 6], 'physical': [2, 2, 2, 4, 1], 'lower_bounds': [12, 12, 12, 20, 28], 'upper_bounds': [16, 16, 16, 24, 32], 'cycle_eligible': ['3', '4'], 'boundaries': ['0', '4']})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 5, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 2, 3, 4], 'physical': [1, 1, 2, 3, 4], 'lower_bounds': [9, 9, 14, 19, 24], 'upper_bounds': [14, 14, 19, 24, 29], 'cycle_eligible': ['3', '4'], 'boundaries': ['2']})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 6, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 1, 3, 4], 'physical': [1, 1, 1, 3, 4], 'lower_bounds': [10, 10, 10, 22, 28], 'upper_bounds': [16, 16, 16, 28, 34], 'cycle_eligible': ['3', '4'], 'boundaries': ['3', '4']})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 7, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 1, 2, 3], 'physical': [1, 1, 1, 2, 3], 'lower_bounds': [11, 11, 11, 18, 25], 'upper_bounds': [18, 18, 18, 25, 32], 'cycle_eligible': ['4'], 'boundaries': []})], [({'origin': 0, 'width': 2, 'buckets': 4, 'cursor': 0, 'events': []}, {'virtual': [], 'physical': [], 'lower_bounds': [], 'upper_bounds': [], 'cycle_eligible': [], 'boundaries': []}), ({'origin': 3, 'width': 4, 'buckets': 4, 'cursor': 0, 'events': [['0', 3], ['1', 6], ['2', 7], ['3', 19]]}, {'virtual': [0, 0, 1, 4], 'physical': [0, 0, 1, 0], 'lower_bounds': [3, 3, 7, 19], 'upper_bounds': [7, 7, 11, 23], 'cycle_eligible': ['0', '1', '2'], 'boundaries': ['0', '2', '3']}), ({'origin': 7, 'width': 3, 'buckets': 4, 'cursor': -1, 'events': [['0', 2], ['1', 4], ['2', 6], ['3', 7], ['4', 10], ['5', 19]]}, {'virtual': [-2, -1, -1, 0, 1, 4], 'physical': [2, 3, 3, 0, 1, 0], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '4', '5']}), ({'origin': 2, 'width': 5, 'buckets': 3, 'cursor': 2, 'events': [['0', 11], ['1', 12], ['2', 16], ['3', 17], ['4', 26], ['5', 27]]}, {'virtual': [1, 2, 2, 3, 4, 5], 'physical': [1, 2, 2, 0, 1, 2], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'cycle_eligible': ['1', '2', '3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 1, 'width': 2, 'buckets': 2, 'cursor': 0, 'events': [['0', -2], ['1', -1], ['2', 0], ['3', 1], ['4', 2], ['5', 5]]}, {'virtual': [-2, -1, -1, 0, 0, 2], 'physical': [0, 1, 1, 0, 0, 0], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'cycle_eligible': ['3', '4'], 'boundaries': ['1', '3', '5']}), ({'origin': 4, 'width': 3, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [2, 3, 3, 6, 8], 'physical': [2, 3, 3, 1, 3], 'lower_bounds': [10, 13, 13, 22, 28], 'upper_bounds': [13, 16, 16, 25, 31], 'cycle_eligible': ['1', '2', '3'], 'boundaries': ['1', '3', '4']}), ({'origin': 4, 'width': 8, 'buckets': 5, 'cursor': 3, 'events': [['0', 12], ['1', 13], ['2', 14], ['3', 22], ['4', 28]]}, {'virtual': [1, 1, 1, 2, 3], 'physical': [1, 1, 1, 2, 3], 'lower_bounds': [12, 12, 12, 20, 28], 'upper_bounds': [20, 20, 20, 28, 36], 'cycle_eligible': ['4'], 'boundaries': ['0', '4']})]][N-1]
check('regression certificate 1', solve(cases[0][0]), cases[0][1])
check('regression certificate 2', solve(cases[1][0]), cases[1][1])
check('regression certificate 3', solve(cases[2][0]), cases[2][1])
check('regression certificate 4', solve(cases[3][0]), cases[3][1])
check('regression certificate 5', solve(cases[4][0]), cases[4][1])
check('regression certificate 6', solve(cases[5][0]), cases[5][1])
check('variant-dependent certificate', solve(cases[6][0]), cases[6][1])
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
regression certificate 1	`{'boundaries': [], 'cycle_eligible': [], 'lower_bounds': [], 'physical': [], 'upper_bounds': [], 'virtual': []}`	`{'boundaries': [], 'cycle_eligible': [], 'lower_bounds': [], 'physical': [], 'upper_bounds': [], 'virtual': []}`	Passed
regression certificate 2	`{'boundaries': ['0', '2', '3'], 'cycle_eligible': ['0', '1', '2'], 'lower_bounds': [3, 3, 7, 19], 'physical': [0, 0, 1, 0], 'upper_bounds': [7, 7, 11, 23], 'virtual': [0, 0, 1, 4]}`	`{'boundaries': ['0', '2', '3'], 'cycle_eligible': ['0', '1', '2'], 'lower_bounds': [3, 3, 7, 19], 'physical': [0, 0, 1, 0], 'upper_bounds': [7, 7, 11, 23], 'virtual': [0, 0, 1, 4]}`	Passed
regression certificate 3	`{'boundaries': ['1', '3', '4', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'physical': [2, 3, 3, 0, 1, 0], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'virtual': [-2, -1, -1, 0, 1, 4]}`	`{'boundaries': ['1', '3', '4', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [1, 4, 4, 7, 10, 19], 'physical': [2, 3, 3, 0, 1, 0], 'upper_bounds': [4, 7, 7, 10, 13, 22], 'virtual': [-2, -1, -1, 0, 1, 4]}`	Passed
regression certificate 4	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'physical': [1, 2, 2, 0, 1, 2], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'virtual': [1, 2, 2, 3, 4, 5]}`	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['1', '2', '3', '4'], 'lower_bounds': [7, 12, 12, 17, 22, 27], 'physical': [1, 2, 2, 0, 1, 2], 'upper_bounds': [12, 17, 17, 22, 27, 32], 'virtual': [1, 2, 2, 3, 4, 5]}`	Passed
regression certificate 5	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'physical': [0, 1, 1, 0, 0, 0], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'virtual': [-2, -1, -1, 0, 0, 2]}`	`{'boundaries': ['1', '3', '5'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [-3, -1, -1, 1, 1, 5], 'physical': [0, 1, 1, 0, 0, 0], 'upper_bounds': [-1, 1, 1, 3, 3, 7], 'virtual': [-2, -1, -1, 0, 0, 2]}`	Passed
regression certificate 6	`{'boundaries': ['1', '3', '4'], 'cycle_eligible': ['1', '2', '3'], 'lower_bounds': [10, 13, 13, 22, 28], 'physical': [2, 3, 3, 1, 3], 'upper_bounds': [13, 16, 16, 25, 31], 'virtual': [2, 3, 3, 6, 8]}`	`{'boundaries': ['1', '3', '4'], 'cycle_eligible': ['1', '2', '3'], 'lower_bounds': [10, 13, 13, 22, 28], 'physical': [2, 3, 3, 1, 3], 'upper_bounds': [13, 16, 16, 25, 31], 'virtual': [2, 3, 3, 6, 8]}`	Passed
variant-dependent certificate	`{'boundaries': ['0', '4'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [12, 12, 12, 20, 28], 'physical': [2, 2, 2, 4, 1], 'upper_bounds': [16, 16, 16, 24, 32], 'virtual': [2, 2, 2, 4, 6]}`	`{'boundaries': ['0', '4'], 'cycle_eligible': ['3', '4'], 'lower_bounds': [12, 12, 12, 20, 28], 'physical': [2, 2, 2, 4, 1], 'upper_bounds': [16, 16, 16, 24, 32], 'virtual': [2, 2, 2, 4, 6]}`	Passed

SHA-256 / 9aa638854a6d93de418090410ab0087869a58608d9f0fac24fd3e0914ce51e4d

Verification & scope

A stipulated offline diagnostic model; it does not implement a production allocator, concurrency protocol, or complete heap library. 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 2026-09-29T14:43:35.566336+00:00.

Case digest / e670bac4e7a1ce7942e3159975c5305fa40d6b47a9d7ff8c99f98a4eaf4bff33