FAILURE MAP
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FA-89621 / Instruction set emulation / Open access

Auxiliary carry ignores the carry-in · case 01

A low digit sum that only exceeds 15 because of the carry-in is not adjusted.

Verified by executionVariant 1 · 10 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The auxiliary carry is computed without the incoming carry.

THE FAILURE

The auxiliary carry is computed without the incoming carry.

Unsuccessful approach: Reading bit 4 of the sum confuses a high-digit bit with a nibble carry.

Case contract

Input [xs, ys, c]: little-endian packed-BCD byte lists and a carry-in. For each byte: s = a+b+c; AF = low-nibble sum (with carry) > 0x0F; CF = s > 0xFF; al = s & 0xFF. Decimal adjust: if (al & 0x0F) > 9 or AF then al += 6 (8-bit wrap); then if the pre-adjust al > 0x99 or CF then al += 0x60 (8-bit wrap) and carry-out = 1, else carry-out = 0. The carry-out feeds the next byte. Return [result bytes, final carry].

Why this case matters

Decimal-adjust instructions are among the hardest to emulate; the flag inputs and the pre-adjust comparison decide the result.

1 / The failure

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

N = 1
observations = []
def solve(*args):
    xs, ys, c = args
    out = []
    for a, b in zip(xs, ys):
        s = a + b + c
        af = int((a & 0x0F) + (b & 0x0F) > 0x0F)
        cf = int(s > 0xFF)
        al = s & 0xFF
        old_al, old_cf = al, cf
        if (al & 0x0F) > 9 or af:
            al = (al + 6) & 0xFF
        if old_al > 0x99 or old_cf:
            al = (al + 0x60) & 0xFF
            c = 1
        else:
            c = 0
        out.append(al)
    return [out, c]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('digits without adjust', [[18, 52], [33, 67], 0], [[51, 119], 0]), ('low digit overflow', [[24, 0], [8, 0], 0], [[38, 0], 0]), ('half carry without low digit above nine', [[9, 16], [9, 0], 0], [[24, 16], 0]), ('carry-in feeds half carry', [[24, 0], [7, 0], 1], [[38, 0], 0]), ('high digit overflow and carry out', [[144, 85], [48, 68], 0], [[32, 0], 1]), ('low adjust pushes high digit', [[149, 0], [5, 0], 0], [[0, 1], 0]), ('binary carry from high digits', [[153, 153], [153, 1], 0], [[152, 1], 1]), ('carry chains across bytes', [[153, 9, 0], [1, 0, 0], 1], [[1, 16, 0], 0]), ('non-decimal low digit', [[143, 0], [7, 2], 0], [[156, 2], 0]), ('high digit exactly A', [[80, 1], [74, 0], 0], [[0, 2], 0])], [('digits without adjust', [[18, 52], [33, 67], 0], [[51, 119], 0]), ('low digit overflow', [[25, 0], [8, 0], 0], [[39, 0], 0]), ('half carry without low digit above nine', [[9, 16], [9, 0], 0], [[24, 16], 0]), ('carry-in feeds half carry', [[40, 0], [7, 0], 1], [[54, 0], 0]), ('high digit overflow and carry out', [[144, 85], [64, 68], 0], [[48, 0], 1]), ('low adjust pushes high digit', [[149, 0], [5, 0], 0], [[0, 1], 0]), ('binary carry from high digits', [[153, 153], [153, 2], 0], [[152, 2], 1]), ('carry chains across bytes', [[153, 9, 0], [1, 0, 0], 1], [[1, 16, 0], 0]), ('non-decimal low digit', [[143, 0], [7, 3], 0], [[156, 3], 0]), ('high digit exactly A', [[80, 1], [74, 0], 0], [[0, 2], 0])], [('digits without adjust', [[18, 52], [33, 67], 0], [[51, 119], 0]), ('low digit overflow', [[24, 0], [8, 0], 0], [[38, 0], 0]), ('half carry without low digit above nine', [[9, 16], [9, 0], 0], [[24, 16], 0]), ('carry-in feeds half carry', [[8, 0], [7, 0], 1], [[22, 0], 0]), ('high digit overflow and carry out', [[144, 85], [32, 68], 0], [[16, 0], 1]), ('low adjust pushes high digit', [[149, 0], [5, 0], 0], [[0, 1], 0]), ('binary carry from high digits', [[153, 153], [153, 3], 0], [[152, 3], 1]), ('carry chains across bytes', [[153, 9, 0], [1, 0, 0], 1], [[1, 16, 0], 0]), ('non-decimal low digit', [[143, 0], [7, 4], 0], [[156, 4], 0]), ('high digit exactly A', [[80, 1], [74, 0], 0], [[0, 2], 0])], [('digits without adjust', [[18, 52], [33, 67], 0], [[51, 119], 0]), ('low digit overflow', [[25, 0], [8, 0], 0], [[39, 0], 0]), ('half carry without low digit above nine', [[9, 16], [9, 0], 0], [[24, 16], 0]), ('carry-in feeds half carry', [[24, 0], [7, 0], 1], [[38, 0], 0]), ('high digit overflow and carry out', [[144, 85], [48, 68], 0], [[32, 0], 1]), ('low adjust pushes high digit', [[149, 0], [5, 0], 0], [[0, 1], 0]), ('binary carry from high digits', [[153, 153], [153, 4], 0], [[152, 4], 1]), ('carry chains across bytes', [[153, 9, 0], [1, 0, 0], 1], [[1, 16, 0], 0]), ('non-decimal low digit', [[143, 0], [7, 5], 0], [[156, 5], 0]), ('high digit exactly A', [[80, 1], [74, 0], 0], [[0, 2], 0])], [('digits without adjust', [[18, 52], [33, 67], 0], [[51, 119], 0]), ('low digit overflow', [[24, 0], [8, 0], 0], [[38, 0], 0]), ('half carry without low digit above nine', [[9, 16], [9, 0], 0], [[24, 16], 0]), ('carry-in feeds half carry', [[40, 0], [7, 0], 1], [[54, 0], 0]), ('high digit overflow and carry out', [[144, 85], [64, 68], 0], [[48, 0], 1]), ('low adjust pushes high digit', [[149, 0], [5, 0], 0], [[0, 1], 0]), ('binary carry from high digits', [[153, 153], [153, 5], 0], [[152, 5], 1]), ('carry chains across bytes', [[153, 9, 0], [1, 0, 0], 1], [[1, 16, 0], 0]), ('non-decimal low digit', [[143, 0], [7, 6], 0], [[156, 6], 0]), ('high digit exactly A', [[80, 1], [74, 0], 0], [[0, 2], 0])]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(*args), 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
digits without adjust[[51, 119], 0][[51, 119], 0]Passed
low digit overflow[[38, 0], 0][[38, 0], 0]Passed
half carry without low digit above nine[[24, 16], 0][[24, 16], 0]Passed
carry-in feeds half carry[[32, 0], 0][[38, 0], 0]Failed
high digit overflow and carry out[[32, 0], 1][[32, 0], 1]Passed
low adjust pushes high digit[[0, 1], 0][[0, 1], 0]Passed
binary carry from high digits[[152, 1], 1][[152, 1], 1]Passed
carry chains across bytes[[1, 16, 0], 0][[1, 16, 0], 0]Passed
non-decimal low digit[[156, 2], 0][[156, 2], 0]Passed
high digit exactly A[[0, 2], 0][[0, 2], 0]Passed

SHA-256 / bfe75f52df819a8860417cc7737f2bbcbfbf4863408c4ec2429b72826d042d84

2 / The unsuccessful fix

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

N = 1
observations = []
def solve(*args):
    xs, ys, c = args
    out = []
    for a, b in zip(xs, ys):
        s = a + b + c
        af = int((s & 0x10) != 0)
        cf = int(s > 0xFF)
        al = s & 0xFF
        old_al, old_cf = al, cf
        if (al & 0x0F) > 9 or af:
            al = (al + 6) & 0xFF
        if old_al > 0x99 or old_cf:
            al = (al + 0x60) & 0xFF
            c = 1
        else:
            c = 0
        out.append(al)
    return [out, c]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('digits without adjust', [[18, 52], [33, 67], 0], [[51, 119], 0]), ('low digit overflow', [[24, 0], [8, 0], 0], [[38, 0], 0]), ('half carry without low digit above nine', [[9, 16], [9, 0], 0], [[24, 16], 0]), ('carry-in feeds half carry', [[24, 0], [7, 0], 1], [[38, 0], 0]), ('high digit overflow and carry out', [[144, 85], [48, 68], 0], [[32, 0], 1]), ('low adjust pushes high digit', [[149, 0], [5, 0], 0], [[0, 1], 0]), ('binary carry from high digits', [[153, 153], [153, 1], 0], [[152, 1], 1]), ('carry chains across bytes', [[153, 9, 0], [1, 0, 0], 1], [[1, 16, 0], 0]), ('non-decimal low digit', [[143, 0], [7, 2], 0], [[156, 2], 0]), ('high digit exactly A', [[80, 1], [74, 0], 0], [[0, 2], 0])], [('digits without adjust', [[18, 52], [33, 67], 0], [[51, 119], 0]), ('low digit overflow', [[25, 0], [8, 0], 0], [[39, 0], 0]), ('half carry without low digit above nine', [[9, 16], [9, 0], 0], [[24, 16], 0]), ('carry-in feeds half carry', [[40, 0], [7, 0], 1], [[54, 0], 0]), ('high digit overflow and carry out', [[144, 85], [64, 68], 0], [[48, 0], 1]), ('low adjust pushes high digit', [[149, 0], [5, 0], 0], [[0, 1], 0]), ('binary carry from high digits', [[153, 153], [153, 2], 0], [[152, 2], 1]), ('carry chains across bytes', [[153, 9, 0], [1, 0, 0], 1], [[1, 16, 0], 0]), ('non-decimal low digit', [[143, 0], [7, 3], 0], [[156, 3], 0]), ('high digit exactly A', [[80, 1], [74, 0], 0], [[0, 2], 0])], [('digits without adjust', [[18, 52], [33, 67], 0], [[51, 119], 0]), ('low digit overflow', [[24, 0], [8, 0], 0], [[38, 0], 0]), ('half carry without low digit above nine', [[9, 16], [9, 0], 0], [[24, 16], 0]), ('carry-in feeds half carry', [[8, 0], [7, 0], 1], [[22, 0], 0]), ('high digit overflow and carry out', [[144, 85], [32, 68], 0], [[16, 0], 1]), ('low adjust pushes high digit', [[149, 0], [5, 0], 0], [[0, 1], 0]), ('binary carry from high digits', [[153, 153], [153, 3], 0], [[152, 3], 1]), ('carry chains across bytes', [[153, 9, 0], [1, 0, 0], 1], [[1, 16, 0], 0]), ('non-decimal low digit', [[143, 0], [7, 4], 0], [[156, 4], 0]), ('high digit exactly A', [[80, 1], [74, 0], 0], [[0, 2], 0])], [('digits without adjust', [[18, 52], [33, 67], 0], [[51, 119], 0]), ('low digit overflow', [[25, 0], [8, 0], 0], [[39, 0], 0]), ('half carry without low digit above nine', [[9, 16], [9, 0], 0], [[24, 16], 0]), ('carry-in feeds half carry', [[24, 0], [7, 0], 1], [[38, 0], 0]), ('high digit overflow and carry out', [[144, 85], [48, 68], 0], [[32, 0], 1]), ('low adjust pushes high digit', [[149, 0], [5, 0], 0], [[0, 1], 0]), ('binary carry from high digits', [[153, 153], [153, 4], 0], [[152, 4], 1]), ('carry chains across bytes', [[153, 9, 0], [1, 0, 0], 1], [[1, 16, 0], 0]), ('non-decimal low digit', [[143, 0], [7, 5], 0], [[156, 5], 0]), ('high digit exactly A', [[80, 1], [74, 0], 0], [[0, 2], 0])], [('digits without adjust', [[18, 52], [33, 67], 0], [[51, 119], 0]), ('low digit overflow', [[24, 0], [8, 0], 0], [[38, 0], 0]), ('half carry without low digit above nine', [[9, 16], [9, 0], 0], [[24, 16], 0]), ('carry-in feeds half carry', [[40, 0], [7, 0], 1], [[54, 0], 0]), ('high digit overflow and carry out', [[144, 85], [64, 68], 0], [[48, 0], 1]), ('low adjust pushes high digit', [[149, 0], [5, 0], 0], [[0, 1], 0]), ('binary carry from high digits', [[153, 153], [153, 5], 0], [[152, 5], 1]), ('carry chains across bytes', [[153, 9, 0], [1, 0, 0], 1], [[1, 16, 0], 0]), ('non-decimal low digit', [[143, 0], [7, 6], 0], [[156, 6], 0]), ('high digit exactly A', [[80, 1], [74, 0], 0], [[0, 2], 0])]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(*args), 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
digits without adjust[[57, 125], 0][[51, 119], 0]Failed
low digit overflow[[32, 0], 0][[38, 0], 0]Failed
half carry without low digit above nine[[24, 22], 0][[24, 16], 0]Failed
carry-in feeds half carry[[32, 0], 0][[38, 0], 0]Failed
high digit overflow and carry out[[32, 0], 1][[32, 0], 1]Passed
low adjust pushes high digit[[0, 1], 0][[0, 1], 0]Passed
binary carry from high digits[[152, 1], 1][[152, 1], 1]Passed
carry chains across bytes[[1, 16, 0], 0][[1, 16, 0], 0]Passed
non-decimal low digit[[156, 2], 0][[156, 2], 0]Passed
high digit exactly A[[0, 2], 0][[0, 2], 0]Passed

SHA-256 / 2bf11ab2dd556c2c9e6a1911a9e75361ef2da04ab15a985b19e4e8f327f9da99

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 10 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

A deterministic bounded teaching model of one emulator rule; the instruction semantics are a stipulated contract inspired by common ISAs and are not a claim of cycle-exact or architectural 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 2026-09-29T14:51:19.316424+00:00.

Case digest / 06f5dd5c3bbf3127f00f120d4dc98f31b942733e73987b0bc8df27f61dd60780