FA-50026 / PCM sample encoding / Open access

Pcm pdm box decimation: bipolar level · case 01

Density output remains unsigned instead of centered PCM.

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

ROOT CAUSE

Density output remains unsigned instead of centered PCM.

VERIFIED REPAIR

Use the stated conversion contract at bipolar level.

Unsuccessful approach: Centering against packet length introduces packet-size-dependent bias.

Case contract

Bounded one-bit-density to PCM adapter using nonoverlapping box windows; zero/one map to -1/+1 and incomplete input bits carry across packets.

Why this case matters

Offline PCM interchange uses these exact bounded packet and sample representation decisions.

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(bits, factor, carry):
    combined=list(carry)+list(bits)
    complete=len(combined)//factor
    samples=[]
    for group in range(complete):
        window=combined[group*factor:(group+1)*factor]
        amplitude=sum(window)
        samples.append(amplitude)
    return [samples,combined[complete*factor:]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('oracle 1', solve(*([1, 0, 1, 1, 0, 0], 3, [])), [[1, -1], []])
check('oracle 2', solve(*([1, 0, 0], 4, [1, 1])), [[2], [0]])
check('oracle 3', solve(*([1, 1, 1, 1], 2, [])), [[2, 2], []])
check('oracle 4', solve(*([], 4, [1])), [[], [1]])
check('oracle 5', solve(*([0, 0, 0, 0], 4, [])), [[-4], []])
check('oracle 6', solve(*([1], 1, [])), [[1], []])
check('oracle 7', solve(*([0, 1, 0, 1, 1], 3, [0])), [[-1, 1], []])
if N == 1: check('variant packet 1', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1], []])
if N == 2: check('variant packet 2', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1], []])
if N == 3: check('variant packet 3', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1, 1, -1], []])
if N == 4: check('variant packet 4', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1, 1, -1, 1, -1], []])
if N == 5: check('variant packet 5', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -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
oracle 1	`[[2, 1], []]`	`[[1, -1], []]`	Failed
oracle 2	`[[3], [0]]`	`[[2], [0]]`	Failed
oracle 3	`[[2, 2], []]`	`[[2, 2], []]`	Passed
oracle 4	`[[], [1]]`	`[[], [1]]`	Passed
oracle 5	`[[0], []]`	`[[-4], []]`	Failed
oracle 6	`[[1], []]`	`[[1], []]`	Passed
oracle 7	`[[1, 2], []]`	`[[-1, 1], []]`	Failed
variant packet 1	`[[2, 1, 2, 1], []]`	`[[1, -1, 1, -1], []]`	Failed

SHA-256 / 758806df9372172911f7a3893ea4c98b163bf808622ae32e12aa24a620527e09

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(bits, factor, carry):
    combined=list(carry)+list(bits)
    complete=len(combined)//factor
    samples=[]
    for group in range(complete):
        window=combined[group*factor:(group+1)*factor]
        amplitude=2*sum(window)-len(bits)
        samples.append(amplitude)
    return [samples,combined[complete*factor:]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('oracle 1', solve(*([1, 0, 1, 1, 0, 0], 3, [])), [[1, -1], []])
check('oracle 2', solve(*([1, 0, 0], 4, [1, 1])), [[2], [0]])
check('oracle 3', solve(*([1, 1, 1, 1], 2, [])), [[2, 2], []])
check('oracle 4', solve(*([], 4, [1])), [[], [1]])
check('oracle 5', solve(*([0, 0, 0, 0], 4, [])), [[-4], []])
check('oracle 6', solve(*([1], 1, [])), [[1], []])
check('oracle 7', solve(*([0, 1, 0, 1, 1], 3, [0])), [[-1, 1], []])
if N == 1: check('variant packet 1', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1], []])
if N == 2: check('variant packet 2', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1], []])
if N == 3: check('variant packet 3', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1, 1, -1], []])
if N == 4: check('variant packet 4', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1, 1, -1, 1, -1], []])
if N == 5: check('variant packet 5', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -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
oracle 1	`[[-2, -4], []]`	`[[1, -1], []]`	Failed
oracle 2	`[[3], [0]]`	`[[2], [0]]`	Failed
oracle 3	`[[0, 0], []]`	`[[2, 2], []]`	Failed
oracle 4	`[[], [1]]`	`[[], [1]]`	Passed
oracle 5	`[[-4], []]`	`[[-4], []]`	Passed
oracle 6	`[[1], []]`	`[[1], []]`	Passed
oracle 7	`[[-3, -1], []]`	`[[-1, 1], []]`	Failed
variant packet 1	`[[-8, -10, -8, -10], []]`	`[[1, -1, 1, -1], []]`	Failed

SHA-256 / 0e0531130091807e8d3125ab69134d9bf5bf1d3c4f3062c284dedc7c8e616825

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(bits, factor, carry):
    combined=list(carry)+list(bits)
    complete=len(combined)//factor
    samples=[]
    for group in range(complete):
        window=combined[group*factor:(group+1)*factor]
        amplitude=2*sum(window)-factor
        samples.append(amplitude)
    return [samples,combined[complete*factor:]]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('oracle 1', solve(*([1, 0, 1, 1, 0, 0], 3, [])), [[1, -1], []])
check('oracle 2', solve(*([1, 0, 0], 4, [1, 1])), [[2], [0]])
check('oracle 3', solve(*([1, 1, 1, 1], 2, [])), [[2, 2], []])
check('oracle 4', solve(*([], 4, [1])), [[], [1]])
check('oracle 5', solve(*([0, 0, 0, 0], 4, [])), [[-4], []])
check('oracle 6', solve(*([1], 1, [])), [[1], []])
check('oracle 7', solve(*([0, 1, 0, 1, 1], 3, [0])), [[-1, 1], []])
if N == 1: check('variant packet 1', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1], []])
if N == 2: check('variant packet 2', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1], []])
if N == 3: check('variant packet 3', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1, 1, -1], []])
if N == 4: check('variant packet 4', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1, 1, -1, 1, -1], []])
if N == 5: check('variant packet 5', solve(*[[1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0], 3, []]), [[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -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
oracle 1	`[[1, -1], []]`	`[[1, -1], []]`	Passed
oracle 2	`[[2], [0]]`	`[[2], [0]]`	Passed
oracle 3	`[[2, 2], []]`	`[[2, 2], []]`	Passed
oracle 4	`[[], [1]]`	`[[], [1]]`	Passed
oracle 5	`[[-4], []]`	`[[-4], []]`	Passed
oracle 6	`[[1], []]`	`[[1], []]`	Passed
oracle 7	`[[-1, 1], []]`	`[[-1, 1], []]`	Passed
variant packet 1	`[[1, -1, 1, -1], []]`	`[[1, -1, 1, -1], []]`	Passed

SHA-256 / 9deafad8a450b297595496d730f837772102dfc7c8bc912048f35912026ad2a4

Verification & scope

Stipulated finite model; no claim of hardware timing or full file-format 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 2026-09-29T14:45:05.427872+00:00.

Case digest / 6a03307e49e6129178eaf537b2984054b30047d2506ca81aa2f6097bf6ff9eeb