{"abstract":"With a0 = 2 the numerator is never scaled and the response is twice too large.","category":"Digital signal filters","checks":7,"contract":"Input [b, a, eps, maxlen, hold]; normalize by a0 (\"bad-a0\"), compute the impulse response exactly and stop after hold consecutive samples with |h| < eps, counted only once n >= len(b) - 1 (any non-quiet sample resets the count), or at maxlen. Return the kept samples as fraction strings.","contract_signature":"x","evaluation_group":"w2-digital_signal_filters-iir-impulse-truncation","failed_approach":"The attempted repair normalizes b but leaves a unnormalized, so the recursion still uses raw feedback coefficients.","family":"w2-digital_signal_filters-iir-impulse-truncation-normalization-order","id":"FA-91786","implementations":{"attempt":{"sha256":"67906a8c60af4e71496b4bc2f271bfd96747f2db9cec3837efdf703892dc5800","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    b = [Fraction(v) for v in x[0]]\n    a = [Fraction(v) for v in x[1]]\n    eps, maxlen, hold = Fraction(x[2]), x[3], x[4]\n    if not a or a[0] == 0:\n        return 'bad-a0'\n    b = [v / a[0] for v in b]\n    h = []\n    quiet = 0\n    for n in range(maxlen):\n        acc = b[n] if n < len(b) else Fraction(0)\n        for k in range(1, len(a)):\n            if n - k >= 0:\n                acc -= a[k] * h[n - k]\n        h.append(acc)\n        if n >= len(b) - 1 and abs(acc) < eps:\n            quiet += 1\n            if quiet >= hold:\n                break\n        else:\n            quiet = 0\n    return [str(v) for v in h]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression: a0 not one', [['2', '1'], ['2', '-1'], '1/16', 10, 2], ['1', '1', '1/2', '1/4', '1/8', '1/16', '1/32', '1/64']], ['regression: random impulse 0', [['0', '1', '1'], ['2', '1/2', '1/3'], '1/16', 9, 3], ['0', '1/2', '3/8', '-17/96', '-7/384', '157/4608', '-101/18432']], ['regression: random impulse 1', [['1', '1/2', '2'], ['-1', '1/3'], '1/16', 7, 1], ['-1', '-5/6', '-41/18', '-41/54', '-41/162', '-41/486', '-41/1458']], ['control: negative alternating tail', [['1'], ['1', '1/2'], '1/8', 12, 2], ['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']], ['control: exact epsilon boundary', [['1'], ['1', '-1/2'], '1/8', 12, 1], ['1', '1/2', '1/4', '1/8', '1/16']], ['control: zero taps inside fir part', [['1', '0', '0', '1'], ['1', '-1/2'], '1/100', 12, 1], ['1', '1/2', '1/4', '9/8', '9/16', '9/32', '9/64', '9/128', '9/256', '9/512', '9/1024']], ['control: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']]], [['regression: random impulse 3', [['-1', '-1', '1/2'], ['-1', '-1/4'], '1/8', 12, 3], ['1', '3/4', '-11/16', '11/64', '-11/256', '11/1024', '-11/4096']], ['regression: random impulse 4', [['2'], ['-1', '1/4'], '1/10', 9, 1], ['-2', '-1/2', '-1/8', '-1/32']], ['regression: random impulse 1', [['1', '1/2', '2'], ['-1', '1/3'], '1/16', 7, 1], ['-1', '-5/6', '-41/18', '-41/54', '-41/162', '-41/486', '-41/1458']], ['control: bad a0', [['1'], ['0', '1'], '1/8', 5, 1], 'bad-a0'], ['control: sparse echo hold 4', [['1'], ['1', '0', '0', '0', '-1/2'], '1/3', 20, 4], ['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']], ['control: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['control: sparse negative echo hold 3', [['2'], ['1', '0', '0', '1/2'], '1/2', 20, 3], ['2', '0', '0', '-1', '0', '0', '1/2', '0', '0', '-1/4']]], [['regression: random impulse 8', [['1/2', '2', '-1'], ['-1', '1/4'], '1/8', 5, 1], ['-1/2', '-17/8', '15/32', '15/128']], ['regression: random impulse 9', [['1'], ['-1', '1/4'], '1/8', 9, 3], ['-1', '-1/4', '-1/16', '-1/64', '-1/256']], ['regression: random impulse 4', [['2'], ['-1', '1/4'], '1/10', 9, 1], ['-2', '-1/2', '-1/8', '-1/32']], ['control: gap inside numerator', [['1', '0', '1'], ['1'], '1/2', 8, 1], ['1', '0', '1', '0']], ['control: gap inside numerator with pole', [['1', '0', '0', '2'], ['1', '1/4'], '1/8', 10, 1], ['1', '-1/4', '1/16', '127/64', '-127/256', '127/1024']], ['control: two gaps in numerator', [['1', '0', '0', '-1'], ['1'], '1/2', 8, 2], ['1', '0', '0', '-1', '0', '0']], ['control: random impulse 2', [['0', '1/2', '-1'], ['1', '1/2'], '1/4', 8, 3], ['0', '1/2', '-5/4', '5/8', '-5/16', '5/32', '-5/64', '5/128']]], [['regression: random impulse 11', [['-1', '0'], ['2', '-1/2', '-1/2'], '1/10', 6, 2], ['-1/2', '-1/8', '-5/32', '-9/128', '-29/512']], ['regression: random impulse 13', [['2', '0', '2'], ['-1', '-1/4'], '1/4', 11, 3], ['-2', '1/2', '-17/8', '17/32', '-17/128', '17/512', '-17/2048']], ['regression: random impulse 8', [['1/2', '2', '-1'], ['-1', '1/4'], '1/8', 5, 1], ['-1/2', '-17/8', '15/32', '15/128']], ['control: random impulse 6', [['-1', '1', '-1'], ['1', '1/2', '1/2'], '1/16', 9, 3], ['-1', '3/2', '-5/4', '-1/8', '11/16', '-9/32', '-13/64', '31/128', '-5/256']], ['control: random impulse 7', [['-1', '1', '-1'], ['1', '1/2'], '1/16', 11, 3], ['-1', '3/2', '-7/4', '7/8', '-7/16', '7/32', '-7/64', '7/128', '-7/256', '7/512']], ['control: random impulse 12', [['1/2', '-1'], ['1', '1/2'], '1/4', 11, 1], ['1/2', '-5/4', '5/8', '-5/16', '5/32']], ['control: random impulse 14', [['1'], ['1', '-1/4', '1/4'], '1/10', 8, 3], ['1', '1/4', '-3/16', '-7/64', '5/256', '33/1024', '13/4096']]], [['regression: random impulse 17', [['1/2', '1'], ['2', '1/2'], '1/8', 12, 3], ['1/4', '7/16', '-7/64', '7/256', '-7/1024']], ['regression: random impulse 18', [['1', '-1', '0'], ['2', '1/4', '0'], '1/16', 5, 2], ['1/2', '-9/16', '9/128', '-9/1024', '9/8192']], ['regression: random impulse 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['control: random impulse 15', [['2'], ['1', '0'], '1/8', 11, 1], ['2', '0']], ['control: random impulse 19', [['1/2', '-1', '0'], ['1', '1/3', '0'], '1/8', 9, 2], ['1/2', '-7/6', '7/18', '-7/54', '7/162', '-7/486']], ['control: random impulse 21', [['-1'], ['1', '1/4'], '1/16', 12, 1], ['-1', '1/4', '-1/16', '1/64']], ['control: random impulse 22', [['0'], ['-1', '0', '1/4'], '1/16', 11, 1], ['0']]]]\nfor label, args, expected in fixtures[N-1]:\n    check(label, solve(args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"224dd0026439c55b49459513ba62efdc99c080b8dc32d15193f6fdea2d04935f","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    b = [Fraction(v) for v in x[0]]\n    a = [Fraction(v) for v in x[1]]\n    eps, maxlen, hold = Fraction(x[2]), x[3], x[4]\n    if not a or a[0] == 0:\n        return 'bad-a0'\n    a = [v / a[0] for v in a]\n    b = [v / a[0] for v in b]\n    h = []\n    quiet = 0\n    for n in range(maxlen):\n        acc = b[n] if n < len(b) else Fraction(0)\n        for k in range(1, len(a)):\n            if n - k >= 0:\n                acc -= a[k] * h[n - k]\n        h.append(acc)\n        if n >= len(b) - 1 and abs(acc) < eps:\n            quiet += 1\n            if quiet >= hold:\n                break\n        else:\n            quiet = 0\n    return [str(v) for v in h]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression: a0 not one', [['2', '1'], ['2', '-1'], '1/16', 10, 2], ['1', '1', '1/2', '1/4', '1/8', '1/16', '1/32', '1/64']], ['regression: random impulse 0', [['0', '1', '1'], ['2', '1/2', '1/3'], '1/16', 9, 3], ['0', '1/2', '3/8', '-17/96', '-7/384', '157/4608', '-101/18432']], ['regression: random impulse 1', [['1', '1/2', '2'], ['-1', '1/3'], '1/16', 7, 1], ['-1', '-5/6', '-41/18', '-41/54', '-41/162', '-41/486', '-41/1458']], ['control: negative alternating tail', [['1'], ['1', '1/2'], '1/8', 12, 2], ['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']], ['control: exact epsilon boundary', [['1'], ['1', '-1/2'], '1/8', 12, 1], ['1', '1/2', '1/4', '1/8', '1/16']], ['control: zero taps inside fir part', [['1', '0', '0', '1'], ['1', '-1/2'], '1/100', 12, 1], ['1', '1/2', '1/4', '9/8', '9/16', '9/32', '9/64', '9/128', '9/256', '9/512', '9/1024']], ['control: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']]], [['regression: random impulse 3', [['-1', '-1', '1/2'], ['-1', '-1/4'], '1/8', 12, 3], ['1', '3/4', '-11/16', '11/64', '-11/256', '11/1024', '-11/4096']], ['regression: random impulse 4', [['2'], ['-1', '1/4'], '1/10', 9, 1], ['-2', '-1/2', '-1/8', '-1/32']], ['regression: random impulse 1', [['1', '1/2', '2'], ['-1', '1/3'], '1/16', 7, 1], ['-1', '-5/6', '-41/18', '-41/54', '-41/162', '-41/486', '-41/1458']], ['control: bad a0', [['1'], ['0', '1'], '1/8', 5, 1], 'bad-a0'], ['control: sparse echo hold 4', [['1'], ['1', '0', '0', '0', '-1/2'], '1/3', 20, 4], ['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']], ['control: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['control: sparse negative echo hold 3', [['2'], ['1', '0', '0', '1/2'], '1/2', 20, 3], ['2', '0', '0', '-1', '0', '0', '1/2', '0', '0', '-1/4']]], [['regression: random impulse 8', [['1/2', '2', '-1'], ['-1', '1/4'], '1/8', 5, 1], ['-1/2', '-17/8', '15/32', '15/128']], ['regression: random impulse 9', [['1'], ['-1', '1/4'], '1/8', 9, 3], ['-1', '-1/4', '-1/16', '-1/64', '-1/256']], ['regression: random impulse 4', [['2'], ['-1', '1/4'], '1/10', 9, 1], ['-2', '-1/2', '-1/8', '-1/32']], ['control: gap inside numerator', [['1', '0', '1'], ['1'], '1/2', 8, 1], ['1', '0', '1', '0']], ['control: gap inside numerator with pole', [['1', '0', '0', '2'], ['1', '1/4'], '1/8', 10, 1], ['1', '-1/4', '1/16', '127/64', '-127/256', '127/1024']], ['control: two gaps in numerator', [['1', '0', '0', '-1'], ['1'], '1/2', 8, 2], ['1', '0', '0', '-1', '0', '0']], ['control: random impulse 2', [['0', '1/2', '-1'], ['1', '1/2'], '1/4', 8, 3], ['0', '1/2', '-5/4', '5/8', '-5/16', '5/32', '-5/64', '5/128']]], [['regression: random impulse 11', [['-1', '0'], ['2', '-1/2', '-1/2'], '1/10', 6, 2], ['-1/2', '-1/8', '-5/32', '-9/128', '-29/512']], ['regression: random impulse 13', [['2', '0', '2'], ['-1', '-1/4'], '1/4', 11, 3], ['-2', '1/2', '-17/8', '17/32', '-17/128', '17/512', '-17/2048']], ['regression: random impulse 8', [['1/2', '2', '-1'], ['-1', '1/4'], '1/8', 5, 1], ['-1/2', '-17/8', '15/32', '15/128']], ['control: random impulse 6', [['-1', '1', '-1'], ['1', '1/2', '1/2'], '1/16', 9, 3], ['-1', '3/2', '-5/4', '-1/8', '11/16', '-9/32', '-13/64', '31/128', '-5/256']], ['control: random impulse 7', [['-1', '1', '-1'], ['1', '1/2'], '1/16', 11, 3], ['-1', '3/2', '-7/4', '7/8', '-7/16', '7/32', '-7/64', '7/128', '-7/256', '7/512']], ['control: random impulse 12', [['1/2', '-1'], ['1', '1/2'], '1/4', 11, 1], ['1/2', '-5/4', '5/8', '-5/16', '5/32']], ['control: random impulse 14', [['1'], ['1', '-1/4', '1/4'], '1/10', 8, 3], ['1', '1/4', '-3/16', '-7/64', '5/256', '33/1024', '13/4096']]], [['regression: random impulse 17', [['1/2', '1'], ['2', '1/2'], '1/8', 12, 3], ['1/4', '7/16', '-7/64', '7/256', '-7/1024']], ['regression: random impulse 18', [['1', '-1', '0'], ['2', '1/4', '0'], '1/16', 5, 2], ['1/2', '-9/16', '9/128', '-9/1024', '9/8192']], ['regression: random impulse 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['control: random impulse 15', [['2'], ['1', '0'], '1/8', 11, 1], ['2', '0']], ['control: random impulse 19', [['1/2', '-1', '0'], ['1', '1/3', '0'], '1/8', 9, 2], ['1/2', '-7/6', '7/18', '-7/54', '7/162', '-7/486']], ['control: random impulse 21', [['-1'], ['1', '1/4'], '1/16', 12, 1], ['-1', '1/4', '-1/16', '1/64']], ['control: random impulse 22', [['0'], ['-1', '0', '1/4'], '1/16', 11, 1], ['0']]]]\nfor label, args, expected in fixtures[N-1]:\n    check(label, solve(args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"A deterministic bounded teaching model with a stipulated toy contract; exact rational arithmetic or fixed-decimal rounding keeps outputs strict JSON. It is not a production DSP library and claims no standards 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.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"w2-digital_signal_filters-iir-impulse-truncation-normalization-order","generated_at":"2026-09-29T14:51:39.410769+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"FIR approximations of IIR filters truncate the impulse response; wrong stop rules cut real energy or run forever.","root_cause":"a is divided by a0 first, so the numerator is then divided by the new a[0] = 1.","sha256":"96f9df9486153019aaa8ad120bdb9c11c6c3b9d761104db3bcffd21b54cbf27b","title":"Impulse truncation normalizes the denominator before the numerator · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":44.334,"exit_code":1,"observations":[{"actual":["1","3/2","3/2","3/2","3/2","3/2","3/2","3/2","3/2","3/2"],"check":"regression: a0 not one","expected":["1","1","1/2","1/4","1/8","1/16","1/32","1/64"],"passed":false},{"actual":["0","1/2","1/4","-7/24","1/16","19/288","-31/576","17/3456","107/6912"],"check":"regression: random impulse 0","expected":["0","1/2","3/8","-17/96","-7/384","157/4608","-101/18432"],"passed":false},{"actual":["-1","-1/6","-35/18","35/54","-35/162","35/486","-35/1458"],"check":"regression: random impulse 1","expected":["-1","-5/6","-41/18","-41/54","-41/162","-41/486","-41/1458"],"passed":false},{"actual":["1","-1/2","1/4","-1/8","1/16","-1/32"],"check":"control: negative alternating tail","expected":["1","-1/2","1/4","-1/8","1/16","-1/32"],"passed":true},{"actual":["1","1/2","1/4","1/8","1/16"],"check":"control: exact epsilon boundary","expected":["1","1/2","1/4","1/8","1/16"],"passed":true},{"actual":["1","1/2","1/4","9/8","9/16","9/32","9/64","9/128","9/256","9/512","9/1024"],"check":"control: zero taps inside fir part","expected":["1","1/2","1/4","9/8","9/16","9/32","9/64","9/128","9/256","9/512","9/1024"],"passed":true},{"actual":["1","0","-1/4","0","1/16","0"],"check":"control: noisy tail resets hold","expected":["1","0","-1/4","0","1/16","0"],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: a0 not one\", \"actual\": [\"1\", \"3/2\", \"3/2\", \"3/2\", \"3/2\", \"3/2\", \"3/2\", \"3/2\", \"3/2\", \"3/2\"], \"expected\": [\"1\", \"1\", \"1/2\", \"1/4\", \"1/8\", \"1/16\", \"1/32\", \"1/64\"], \"passed\": false}, {\"check\": \"regression: random impulse 0\", \"actual\": [\"0\", \"1/2\", \"1/4\", \"-7/24\", \"1/16\", \"19/288\", \"-31/576\", \"17/3456\", \"107/6912\"], \"expected\": [\"0\", \"1/2\", \"3/8\", \"-17/96\", \"-7/384\", \"157/4608\", \"-101/18432\"], \"passed\": false}, {\"check\": \"regression: random impulse 1\", \"actual\": [\"-1\", \"-1/6\", \"-35/18\", \"35/54\", \"-35/162\", \"35/486\", \"-35/1458\"], \"expected\": [\"-1\", \"-5/6\", \"-41/18\", \"-41/54\", \"-41/162\", \"-41/486\", \"-41/1458\"], \"passed\": false}, {\"check\": \"control: negative alternating tail\", \"actual\": [\"1\", \"-1/2\", \"1/4\", \"-1/8\", \"1/16\", \"-1/32\"], \"expected\": [\"1\", \"-1/2\", \"1/4\", \"-1/8\", \"1/16\", \"-1/32\"], \"passed\": true}, {\"check\": \"control: exact epsilon boundary\", \"actual\": [\"1\", \"1/2\", \"1/4\", \"1/8\", \"1/16\"], \"expected\": [\"1\", \"1/2\", \"1/4\", \"1/8\", \"1/16\"], \"passed\": true}, {\"check\": \"control: zero taps inside fir part\", \"actual\": [\"1\", \"1/2\", \"1/4\", \"9/8\", \"9/16\", \"9/32\", \"9/64\", \"9/128\", \"9/256\", \"9/512\", \"9/1024\"], \"expected\": [\"1\", \"1/2\", \"1/4\", \"9/8\", \"9/16\", \"9/32\", \"9/64\", \"9/128\", \"9/256\", \"9/512\", \"9/1024\"], \"passed\": true}, {\"check\": \"control: noisy tail resets hold\", \"actual\": [\"1\", \"0\", \"-1/4\", \"0\", \"1/16\", \"0\"], \"expected\": [\"1\", \"0\", \"-1/4\", \"0\", \"1/16\", \"0\"], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":45.107,"exit_code":1,"observations":[{"actual":["2","2","1","1/2","1/4","1/8","1/16","1/32","1/64"],"check":"regression: a0 not one","expected":["1","1","1/2","1/4","1/8","1/16","1/32","1/64"],"passed":false},{"actual":["0","1","3/4","-17/48","-7/192","157/2304","-101/9216","-953/110592","587/147456"],"check":"regression: random impulse 0","expected":["0","1/2","3/8","-17/96","-7/384","157/4608","-101/18432"],"passed":false},{"actual":["1","5/6","41/18","41/54","41/162","41/486","41/1458"],"check":"regression: random impulse 1","expected":["-1","-5/6","-41/18","-41/54","-41/162","-41/486","-41/1458"],"passed":false},{"actual":["1","-1/2","1/4","-1/8","1/16","-1/32"],"check":"control: negative alternating tail","expected":["1","-1/2","1/4","-1/8","1/16","-1/32"],"passed":true},{"actual":["1","1/2","1/4","1/8","1/16"],"check":"control: exact epsilon boundary","expected":["1","1/2","1/4","1/8","1/16"],"passed":true},{"actual":["1","1/2","1/4","9/8","9/16","9/32","9/64","9/128","9/256","9/512","9/1024"],"check":"control: zero taps inside fir part","expected":["1","1/2","1/4","9/8","9/16","9/32","9/64","9/128","9/256","9/512","9/1024"],"passed":true},{"actual":["1","0","-1/4","0","1/16","0"],"check":"control: noisy tail resets hold","expected":["1","0","-1/4","0","1/16","0"],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: a0 not one\", \"actual\": [\"2\", \"2\", \"1\", \"1/2\", \"1/4\", \"1/8\", \"1/16\", \"1/32\", \"1/64\"], \"expected\": [\"1\", \"1\", \"1/2\", \"1/4\", \"1/8\", \"1/16\", \"1/32\", \"1/64\"], \"passed\": false}, {\"check\": \"regression: random impulse 0\", \"actual\": [\"0\", \"1\", \"3/4\", \"-17/48\", \"-7/192\", 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