FA-49696 / PCM sample encoding / Open access

Pcm onebit pulse density: threshold inclusive · case 01

An exact pulse threshold is delayed by one sample.

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

ROOT CAUSE

An exact pulse threshold is delayed by one sample.

VERIFIED REPAIR

Use the stated conversion contract at threshold inclusive.

Unsuccessful approach: Thresholding only the current level ignores density accumulation.

Case contract

Convert bounded nonnegative PCM levels no greater than denominator into a deterministic one-bit pulse-density stream, preserving fractional accumulator state.

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(samples, denominator, accumulator):
    bits=[]
    for sample in samples:
        accumulator+=sample
        pulse=accumulator>denominator
        if pulse:
            accumulator-=denominator
        bits.append(int(pulse))
    return [bits,accumulator]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('oracle 1', solve(*([1, 2, 3, 4], 5, 0)), [[0, 0, 1, 1], 0])
check('oracle 2', solve(*([5, 0, 5], 5, 0)), [[1, 0, 1], 0])
check('oracle 3', solve(*([2, 2, 2, 2], 3, 1)), [[1, 0, 1, 1], 0])
check('oracle 4', solve(*([], 5, 2)), [[], 2])
check('oracle 5', solve(*([0, 0], 4, 3)), [[0, 0], 3])
check('oracle 6', solve(*([3, 3, 3], 4, 2)), [[1, 1, 0], 3])
check('oracle 7', solve(*([1, 1, 1], 2, 0)), [[0, 1, 0], 1])
if N == 1: check('variant packet 1', solve(*[[1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 2: check('variant packet 2', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 3: check('variant packet 3', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 4: check('variant packet 4', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 5: check('variant packet 5', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
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	`[[0, 0, 1, 0], 5]`	`[[0, 0, 1, 1], 0]`	Failed
oracle 2	`[[0, 0, 1], 5]`	`[[1, 0, 1], 0]`	Failed
oracle 3	`[[0, 1, 1, 0], 3]`	`[[1, 0, 1, 1], 0]`	Failed
oracle 4	`[[], 2]`	`[[], 2]`	Passed
oracle 5	`[[0, 0], 3]`	`[[0, 0], 3]`	Passed
oracle 6	`[[1, 0, 1], 3]`	`[[1, 1, 0], 3]`	Failed
oracle 7	`[[0, 0, 1], 1]`	`[[0, 1, 0], 1]`	Failed
variant packet 1	`[[0, 0, 1, 0, 1, 0, 1, 0], 5]`	`[[0, 0, 1, 1, 0, 0, 1, 1], 0]`	Failed

SHA-256 / 235bb0211424cb2459c5953e3030cee8eda7e8850e48836d52588fe52184a44b

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(samples, denominator, accumulator):
    bits=[]
    for sample in samples:
        accumulator+=sample
        pulse=sample>=denominator
        if pulse:
            accumulator-=denominator
        bits.append(int(pulse))
    return [bits,accumulator]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('oracle 1', solve(*([1, 2, 3, 4], 5, 0)), [[0, 0, 1, 1], 0])
check('oracle 2', solve(*([5, 0, 5], 5, 0)), [[1, 0, 1], 0])
check('oracle 3', solve(*([2, 2, 2, 2], 3, 1)), [[1, 0, 1, 1], 0])
check('oracle 4', solve(*([], 5, 2)), [[], 2])
check('oracle 5', solve(*([0, 0], 4, 3)), [[0, 0], 3])
check('oracle 6', solve(*([3, 3, 3], 4, 2)), [[1, 1, 0], 3])
check('oracle 7', solve(*([1, 1, 1], 2, 0)), [[0, 1, 0], 1])
if N == 1: check('variant packet 1', solve(*[[1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 2: check('variant packet 2', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 3: check('variant packet 3', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 4: check('variant packet 4', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 5: check('variant packet 5', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
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	`[[0, 0, 0, 0], 10]`	`[[0, 0, 1, 1], 0]`	Failed
oracle 2	`[[1, 0, 1], 0]`	`[[1, 0, 1], 0]`	Passed
oracle 3	`[[0, 0, 0, 0], 9]`	`[[1, 0, 1, 1], 0]`	Failed
oracle 4	`[[], 2]`	`[[], 2]`	Passed
oracle 5	`[[0, 0], 3]`	`[[0, 0], 3]`	Passed
oracle 6	`[[0, 0, 0], 11]`	`[[1, 1, 0], 3]`	Failed
oracle 7	`[[0, 0, 0], 3]`	`[[0, 1, 0], 1]`	Failed
variant packet 1	`[[0, 0, 0, 0, 0, 0, 0, 0], 20]`	`[[0, 0, 1, 1, 0, 0, 1, 1], 0]`	Failed

SHA-256 / 084ff08c9f2118a9df4590a02132c3af210fc22aebb75f6fed6c79739e146e38

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(samples, denominator, accumulator):
    bits=[]
    for sample in samples:
        accumulator+=sample
        pulse=accumulator>=denominator
        if pulse:
            accumulator-=denominator
        bits.append(int(pulse))
    return [bits,accumulator]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('oracle 1', solve(*([1, 2, 3, 4], 5, 0)), [[0, 0, 1, 1], 0])
check('oracle 2', solve(*([5, 0, 5], 5, 0)), [[1, 0, 1], 0])
check('oracle 3', solve(*([2, 2, 2, 2], 3, 1)), [[1, 0, 1, 1], 0])
check('oracle 4', solve(*([], 5, 2)), [[], 2])
check('oracle 5', solve(*([0, 0], 4, 3)), [[0, 0], 3])
check('oracle 6', solve(*([3, 3, 3], 4, 2)), [[1, 1, 0], 3])
check('oracle 7', solve(*([1, 1, 1], 2, 0)), [[0, 1, 0], 1])
if N == 1: check('variant packet 1', solve(*[[1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 2: check('variant packet 2', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 3: check('variant packet 3', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 4: check('variant packet 4', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
if N == 5: check('variant packet 5', solve(*[[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4], 5, 0]), [[0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1], 0])
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	`[[0, 0, 1, 1], 0]`	`[[0, 0, 1, 1], 0]`	Passed
oracle 2	`[[1, 0, 1], 0]`	`[[1, 0, 1], 0]`	Passed
oracle 3	`[[1, 0, 1, 1], 0]`	`[[1, 0, 1, 1], 0]`	Passed
oracle 4	`[[], 2]`	`[[], 2]`	Passed
oracle 5	`[[0, 0], 3]`	`[[0, 0], 3]`	Passed
oracle 6	`[[1, 1, 0], 3]`	`[[1, 1, 0], 3]`	Passed
oracle 7	`[[0, 1, 0], 1]`	`[[0, 1, 0], 1]`	Passed
variant packet 1	`[[0, 0, 1, 1, 0, 0, 1, 1], 0]`	`[[0, 0, 1, 1, 0, 0, 1, 1], 0]`	Passed

SHA-256 / a35e3a0b70e13f2156299bc65806267432f0c203f88fd326ca134eab6fc0641d

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 2026-09-29T14:45:02.322865+00:00.

Case digest / 50347abd8f03776a3510ace7f8ecc8da630182a1593407cfd280c48135e46388