FA-47061 / Bounded deques / Open access

Ring gap insertion reads backing array order as logical order · case 01

Ring gap insertion reads backing array order as logical order.

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

ROOT CAUSE

Ring gap insertion reads backing array order as logical order.

VERIFIED REPAIR

Restore the documented logical gather invariant in ring-gap-insert.

Unsuccessful approach: The partial repair still applies the incorrect transition to an admitted boundary or multi-element case.

Case contract

Insert into a valid logical gap by moving the shorter ring side, preferring the right side on equal cost. Left shifts decrement physical head; reconstruct live arc and clear all vacant slots. Full ring rejects atomically.

Why this case matters

Controlled bounded deque implementation model with explicit storage and lifecycle observations.

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(x):
    slots,head,size,gap,value=x
    cap=len(slots)
    if size==cap:return [slots,head,size,False,0]
    logical=slots[:size]
    left=gap<size-gap
    origin=(head-1)%cap if left else head
    inserted=logical[:gap]+[value]+logical[gap:]
    result=[None]*cap
    for i,v in enumerate(inserted):result[(origin+i)%cap]=v
    count=size+1
    moved=gap if left else size-gap
    return [result,origin,count,True,moved]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('0', solve([[N,N+1,N+2,None,None],0,3,1,N+3]), {1: [[4, 2, 3, None, 1], 4, 4, True, 1], 2: [[5, 3, 4, None, 2], 4, 4, True, 1], 3: [[6, 4, 5, None, 3], 4, 4, True, 1], 4: [[7, 5, 6, None, 4], 4, 4, True, 1], 5: [[8, 6, 7, None, 5], 4, 4, True, 1]}[N])
check('1', solve([[N+2,None,None,N,N+1],3,3,1,N+3]), {1: [[3, None, 1, 4, 2], 2, 4, True, 1], 2: [[4, None, 2, 5, 3], 2, 4, True, 1], 3: [[5, None, 3, 6, 4], 2, 4, True, 1], 4: [[6, None, 4, 7, 5], 2, 4, True, 1], 5: [[7, None, 5, 8, 6], 2, 4, True, 1]}[N])
check('2', solve([[N,N+1,N+2,N+3,None,None],0,4,2,N+4]), {1: [[1, 2, 5, 3, 4, None], 0, 5, True, 2], 2: [[2, 3, 6, 4, 5, None], 0, 5, True, 2], 3: [[3, 4, 7, 5, 6, None], 0, 5, True, 2], 4: [[4, 5, 8, 6, 7, None], 0, 5, True, 2], 5: [[5, 6, 9, 7, 8, None], 0, 5, True, 2]}[N])
check('3', solve([[None,N,N+1,None],1,2,2,N+2]), {1: [[None, 1, 2, 3], 1, 3, True, 0], 2: [[None, 2, 3, 4], 1, 3, True, 0], 3: [[None, 3, 4, 5], 1, 3, True, 0], 4: [[None, 4, 5, 6], 1, 3, True, 0], 5: [[None, 5, 6, 7], 1, 3, True, 0]}[N])
check('4', solve([[N,N+1],0,2,1,N+2]), {1: [[1, 2], 0, 2, False, 0], 2: [[2, 3], 0, 2, False, 0], 3: [[3, 4], 0, 2, False, 0], 4: [[4, 5], 0, 2, False, 0], 5: [[5, 6], 0, 2, False, 0]}[N])
check('5', solve([[None,None,None],2,0,0,N]), {1: [[None, None, 1], 2, 1, True, 0], 2: [[None, None, 2], 2, 1, True, 0], 3: [[None, None, 3], 2, 1, True, 0], 4: [[None, None, 4], 2, 1, True, 0], 5: [[None, None, 5], 2, 1, True, 0]}[N])
check('6', solve([[N+1,None,None,N],3,2,0,N+2]), {1: [[2, None, 3, 1], 2, 3, True, 0], 2: [[3, None, 4, 2], 2, 3, True, 0], 3: [[4, None, 5, 3], 2, 3, True, 0], 4: [[5, None, 6, 4], 2, 3, True, 0], 5: [[6, None, 7, 5], 2, 3, True, 0]}[N])
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
0	`[[4, 2, 3, None, 1], 4, 4, True, 1]`	`[[4, 2, 3, None, 1], 4, 4, True, 1]`	Passed
1	`[[None, None, 3, 4, None], 2, 4, True, 1]`	`[[3, None, 1, 4, 2], 2, 4, True, 1]`	Failed
2	`[[1, 2, 5, 3, 4, None], 0, 5, True, 2]`	`[[1, 2, 5, 3, 4, None], 0, 5, True, 2]`	Passed
3	`[[None, None, 1, 3], 1, 3, True, 0]`	`[[None, 1, 2, 3], 1, 3, True, 0]`	Failed
4	`[[1, 2], 0, 2, False, 0]`	`[[1, 2], 0, 2, False, 0]`	Passed
5	`[[None, None, 1], 2, 1, True, 0]`	`[[None, None, 1], 2, 1, True, 0]`	Passed
6	`[[None, None, 3, 2], 2, 3, True, 0]`	`[[2, None, 3, 1], 2, 3, True, 0]`	Failed

SHA-256 / 520388e7262bc72f683c610d77eb6fdffe2c90ec98aab54fb0087acbc222dfa6

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(x):
    slots,head,size,gap,value=x
    cap=len(slots)
    if size==cap:return [slots,head,size,False,0]
    logical=[slots[(head+i)%cap] for i in range(size)] if head==0 else slots[:size]
    left=gap<size-gap
    origin=(head-1)%cap if left else head
    inserted=logical[:gap]+[value]+logical[gap:]
    result=[None]*cap
    for i,v in enumerate(inserted):result[(origin+i)%cap]=v
    count=size+1
    moved=gap if left else size-gap
    return [result,origin,count,True,moved]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('0', solve([[N,N+1,N+2,None,None],0,3,1,N+3]), {1: [[4, 2, 3, None, 1], 4, 4, True, 1], 2: [[5, 3, 4, None, 2], 4, 4, True, 1], 3: [[6, 4, 5, None, 3], 4, 4, True, 1], 4: [[7, 5, 6, None, 4], 4, 4, True, 1], 5: [[8, 6, 7, None, 5], 4, 4, True, 1]}[N])
check('1', solve([[N+2,None,None,N,N+1],3,3,1,N+3]), {1: [[3, None, 1, 4, 2], 2, 4, True, 1], 2: [[4, None, 2, 5, 3], 2, 4, True, 1], 3: [[5, None, 3, 6, 4], 2, 4, True, 1], 4: [[6, None, 4, 7, 5], 2, 4, True, 1], 5: [[7, None, 5, 8, 6], 2, 4, True, 1]}[N])
check('2', solve([[N,N+1,N+2,N+3,None,None],0,4,2,N+4]), {1: [[1, 2, 5, 3, 4, None], 0, 5, True, 2], 2: [[2, 3, 6, 4, 5, None], 0, 5, True, 2], 3: [[3, 4, 7, 5, 6, None], 0, 5, True, 2], 4: [[4, 5, 8, 6, 7, None], 0, 5, True, 2], 5: [[5, 6, 9, 7, 8, None], 0, 5, True, 2]}[N])
check('3', solve([[None,N,N+1,None],1,2,2,N+2]), {1: [[None, 1, 2, 3], 1, 3, True, 0], 2: [[None, 2, 3, 4], 1, 3, True, 0], 3: [[None, 3, 4, 5], 1, 3, True, 0], 4: [[None, 4, 5, 6], 1, 3, True, 0], 5: [[None, 5, 6, 7], 1, 3, True, 0]}[N])
check('4', solve([[N,N+1],0,2,1,N+2]), {1: [[1, 2], 0, 2, False, 0], 2: [[2, 3], 0, 2, False, 0], 3: [[3, 4], 0, 2, False, 0], 4: [[4, 5], 0, 2, False, 0], 5: [[5, 6], 0, 2, False, 0]}[N])
check('5', solve([[None,None,None],2,0,0,N]), {1: [[None, None, 1], 2, 1, True, 0], 2: [[None, None, 2], 2, 1, True, 0], 3: [[None, None, 3], 2, 1, True, 0], 4: [[None, None, 4], 2, 1, True, 0], 5: [[None, None, 5], 2, 1, True, 0]}[N])
check('6', solve([[N+1,None,None,N],3,2,0,N+2]), {1: [[2, None, 3, 1], 2, 3, True, 0], 2: [[3, None, 4, 2], 2, 3, True, 0], 3: [[4, None, 5, 3], 2, 3, True, 0], 4: [[5, None, 6, 4], 2, 3, True, 0], 5: [[6, None, 7, 5], 2, 3, True, 0]}[N])
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
0	`[[4, 2, 3, None, 1], 4, 4, True, 1]`	`[[4, 2, 3, None, 1], 4, 4, True, 1]`	Passed
1	`[[None, None, 3, 4, None], 2, 4, True, 1]`	`[[3, None, 1, 4, 2], 2, 4, True, 1]`	Failed
2	`[[1, 2, 5, 3, 4, None], 0, 5, True, 2]`	`[[1, 2, 5, 3, 4, None], 0, 5, True, 2]`	Passed
3	`[[None, None, 1, 3], 1, 3, True, 0]`	`[[None, 1, 2, 3], 1, 3, True, 0]`	Failed
4	`[[1, 2], 0, 2, False, 0]`	`[[1, 2], 0, 2, False, 0]`	Passed
5	`[[None, None, 1], 2, 1, True, 0]`	`[[None, None, 1], 2, 1, True, 0]`	Passed
6	`[[None, None, 3, 2], 2, 3, True, 0]`	`[[2, None, 3, 1], 2, 3, True, 0]`	Failed

SHA-256 / 4a6cac0a825a4597887aa3b14a0b5c7159c58634392bb1617a4e5f23364d3734

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(x):
    slots,head,size,gap,value=x
    cap=len(slots)
    if size==cap:return [slots,head,size,False,0]
    logical=[slots[(head+i)%cap] for i in range(size)]
    left=gap<size-gap
    origin=(head-1)%cap if left else head
    inserted=logical[:gap]+[value]+logical[gap:]
    result=[None]*cap
    for i,v in enumerate(inserted):result[(origin+i)%cap]=v
    count=size+1
    moved=gap if left else size-gap
    return [result,origin,count,True,moved]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('0', solve([[N,N+1,N+2,None,None],0,3,1,N+3]), {1: [[4, 2, 3, None, 1], 4, 4, True, 1], 2: [[5, 3, 4, None, 2], 4, 4, True, 1], 3: [[6, 4, 5, None, 3], 4, 4, True, 1], 4: [[7, 5, 6, None, 4], 4, 4, True, 1], 5: [[8, 6, 7, None, 5], 4, 4, True, 1]}[N])
check('1', solve([[N+2,None,None,N,N+1],3,3,1,N+3]), {1: [[3, None, 1, 4, 2], 2, 4, True, 1], 2: [[4, None, 2, 5, 3], 2, 4, True, 1], 3: [[5, None, 3, 6, 4], 2, 4, True, 1], 4: [[6, None, 4, 7, 5], 2, 4, True, 1], 5: [[7, None, 5, 8, 6], 2, 4, True, 1]}[N])
check('2', solve([[N,N+1,N+2,N+3,None,None],0,4,2,N+4]), {1: [[1, 2, 5, 3, 4, None], 0, 5, True, 2], 2: [[2, 3, 6, 4, 5, None], 0, 5, True, 2], 3: [[3, 4, 7, 5, 6, None], 0, 5, True, 2], 4: [[4, 5, 8, 6, 7, None], 0, 5, True, 2], 5: [[5, 6, 9, 7, 8, None], 0, 5, True, 2]}[N])
check('3', solve([[None,N,N+1,None],1,2,2,N+2]), {1: [[None, 1, 2, 3], 1, 3, True, 0], 2: [[None, 2, 3, 4], 1, 3, True, 0], 3: [[None, 3, 4, 5], 1, 3, True, 0], 4: [[None, 4, 5, 6], 1, 3, True, 0], 5: [[None, 5, 6, 7], 1, 3, True, 0]}[N])
check('4', solve([[N,N+1],0,2,1,N+2]), {1: [[1, 2], 0, 2, False, 0], 2: [[2, 3], 0, 2, False, 0], 3: [[3, 4], 0, 2, False, 0], 4: [[4, 5], 0, 2, False, 0], 5: [[5, 6], 0, 2, False, 0]}[N])
check('5', solve([[None,None,None],2,0,0,N]), {1: [[None, None, 1], 2, 1, True, 0], 2: [[None, None, 2], 2, 1, True, 0], 3: [[None, None, 3], 2, 1, True, 0], 4: [[None, None, 4], 2, 1, True, 0], 5: [[None, None, 5], 2, 1, True, 0]}[N])
check('6', solve([[N+1,None,None,N],3,2,0,N+2]), {1: [[2, None, 3, 1], 2, 3, True, 0], 2: [[3, None, 4, 2], 2, 3, True, 0], 3: [[4, None, 5, 3], 2, 3, True, 0], 4: [[5, None, 6, 4], 2, 3, True, 0], 5: [[6, None, 7, 5], 2, 3, True, 0]}[N])
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
0	`[[4, 2, 3, None, 1], 4, 4, True, 1]`	`[[4, 2, 3, None, 1], 4, 4, True, 1]`	Passed
1	`[[3, None, 1, 4, 2], 2, 4, True, 1]`	`[[3, None, 1, 4, 2], 2, 4, True, 1]`	Passed
2	`[[1, 2, 5, 3, 4, None], 0, 5, True, 2]`	`[[1, 2, 5, 3, 4, None], 0, 5, True, 2]`	Passed
3	`[[None, 1, 2, 3], 1, 3, True, 0]`	`[[None, 1, 2, 3], 1, 3, True, 0]`	Passed
4	`[[1, 2], 0, 2, False, 0]`	`[[1, 2], 0, 2, False, 0]`	Passed
5	`[[None, None, 1], 2, 1, True, 0]`	`[[None, None, 1], 2, 1, True, 0]`	Passed
6	`[[2, None, 3, 1], 2, 3, True, 0]`	`[[2, None, 3, 1], 2, 3, True, 0]`	Passed

SHA-256 / d1f9a30d28beb8d9c00c662342ed9547e57ee692af4106f83e6850829d0d4fad

Verification & scope

Offline finite deterministic model; no claim of production implementation or concurrent memory-model conformance. 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 2026-09-29T14:44:37.866307+00:00.

Case digest / b3ac9626605c2975db1d5b08e7abd53eac6cc0ae1604cd22bba6bcbb0092ec0d