{"abstract":"The first output sample skips b1 contributions and the response is delayed by a sample.","category":"Digital signal filters","checks":7,"contract":"Input [[b0, b1, b2, a1, a2], samples, prime] (a0 = 1). Transposed direct form II: y = b0 x + s1; s1 = b1 x - a1 y + s2; s2 = b2 x - a2 y. If prime and samples exist, initialize s1, s2 to the steady state for a constant input equal to the first sample (y_ss = x0 (b0+b1+b2)/(1+a1+a2), \"dc-pole\" if the denominator is 0). Return exact fraction strings.","contract_signature":"x","evaluation_group":"w2-digital_signal_filters-biquad-transposed-df2","failed_approach":"The attempted repair adds both states, double counting the delayed terms.","family":"w2-digital_signal_filters-biquad-transposed-df2-output-state-tap","id":"FA-91381","implementations":{"attempt":{"sha256":"aab868c3ecb947e6fcb0280ea90ca76f87086def7416f97429ab70c366f53358","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    b0, b1, b2, a1, a2 = [Fraction(v) for v in x[0]]\n    xs, prime = x[1], x[2]\n    s1 = s2 = Fraction(0)\n    if prime and xs:\n        den = 1 + a1 + a2\n        if den == 0:\n            return 'dc-pole'\n        x0 = xs[0]\n        yss = x0 * (b0 + b1 + b2) / den\n        s1 = yss - b0 * x0\n        s2 = b2 * x0 - a2 * yss\n    out = []\n    for v in xs:\n        y = b0 * v + s1 + s2\n        s1 = b1 * v - a1 * y + s2\n        s2 = b2 * v - a2 * y\n        out.append(str(y))\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression: primed step lowpass', [['1/4', '1/2', '1/4', '-1/2', '1/4'], [2, 2, 2, 2], True], ['8/3', '8/3', '8/3', '8/3']], ['regression: primed highpass', [['1', '-2', '1', '0', '0'], [3, 3, 1], True], ['0', '0', '-2']], ['regression: unprimed impulse', [['1', '1/2', '1/3', '1/2', '-1/4'], [1, 0, 0, 0], False], ['1', '0', '7/12', '-7/24']], ['control: dc pole', [['1', '0', '0', '-2', '1'], [1, 1], True], 'dc-pole'], ['control: primed empty', [['1', '0', '0', '0', '0'], [], True], []], ['control: random df2t 3', [['1/8', '3/4', '-3/8', '-1/2', '1/8'], [2], False], ['1/4']], ['control: random df2t 11', [['1/8', '1/8', '5/4', '-3/8', '1/2'], [0], True], ['0']]], [['regression: random df2t 0', [['5/4', '1/3', '3/4', '-1/2', '2'], [-1, 2, 4, 4, 1], False], ['-5/4', '37/24', '131/16', '283/32', '-1223/192']], ['regression: random df2t 1', [['-1/4', '-1/2', '0', '-1/2', '-3/8'], [-3, 3, 1, 2], False], ['3/4', '9/8', '-29/32', '-33/32']], ['regression: unprimed impulse', [['1', '1/2', '1/3', '1/2', '-1/4'], [1, 0, 0, 0], False], ['1', '0', '7/12', '-7/24']], ['control: random df2t 12', [['-1/2', '5/4', '-1/4', '5/4', '-1/4'], [0, 0], False], ['0', '0']], ['control: random df2t 14', [['1/3', '2', '1/2', '1/2', '5/4'], [0, -2], True], ['0', '-2/3']], ['control: random df2t 15', [['-1/4', '2', '-1/2', '1/2', '1/2'], [4], False], ['-1']], ['control: random df2t 20', [['5/4', '0', '5/4', '-1/2', '-1/2'], [-3, 1], True], 'dc-pole']], [['regression: random df2t 4', [['-1/2', '-3/8', '0', '0', '1/3'], [-3], True], ['63/32']], ['regression: random df2t 5', [['-3/8', '-1/4', '-1/4', '1/3', '-1/4'], [2], True], ['-21/13']], ['regression: random df2t 1', [['-1/4', '-1/2', '0', '-1/2', '-3/8'], [-3, 3, 1, 2], False], ['3/4', '9/8', '-29/32', '-33/32']], ['control: random df2t 31', [['-1/2', '1/8', '1/2', '-1/2', '-1/4'], [0, -3], False], ['0', '3/2']], ['control: random df2t 35', [['2', '1/8', '1/8', '-1/2', '-1/2'], [2, -3, 1], True], 'dc-pole'], ['control: random df2t 43', [['1/3', '3/4', '5/4', '1/3', '3/4'], [2], False], ['2/3']], ['control: random df2t 48', [['-1/4', '1/2', '-1/2', '-3/8', '-1/2'], [-1], False], ['1/4']]], [['regression: random df2t 7', [['-1/2', '2', '-1/4', '1/2', '2'], [3, 2, -2, 0], True], ['15/14', '11/7', '37/28', '-465/56']], ['regression: random df2t 8', [['-3/8', '2', '1/8', '1/3', '2'], [4, -3, -2, 1, 4], True], ['21/10', '189/40', '-421/40', '-1283/120', '8951/360']], ['regression: random df2t 4', [['-1/2', '-3/8', '0', '0', '1/3'], [-3], True], ['63/32']], ['control: random df2t 49', [['0', '2', '0', '-1/2', '2'], [0], True], ['0']], ['control: dc pole', [['1', '0', '0', '-2', '1'], [1, 1], True], 'dc-pole'], ['control: primed empty', [['1', '0', '0', '0', '0'], [], True], []], ['control: random df2t 3', [['1/8', '3/4', '-3/8', '-1/2', '1/8'], [2], False], ['1/4']]], [['regression: random df2t 10', [['1/8', '1/3', '-1/4', '1/3', '1/8'], [1, -1, 2, 3, 3], False], ['1/8', '1/6', '-233/576', '2429/1728', '18953/41472']], ['regression: random df2t 13', [['-1/4', '-1/4', '1/2', '5/4', '0'], [2, -3, -1, 3, -3], False], ['-1/2', '7/8', '29/32', '-401/128', '1749/512']], ['regression: random df2t 6', [['0', '-1/2', '1/8', '3/4', '3/4'], [4, 4, 2, 4], False], ['0', '-2', '0', '1']], ['control: random df2t 11', [['1/8', '1/8', '5/4', '-3/8', '1/2'], [0], True], ['0']], ['control: random df2t 12', [['-1/2', '5/4', '-1/4', '5/4', '-1/4'], [0, 0], False], ['0', '0']], ['control: random df2t 14', [['1/3', '2', '1/2', '1/2', '5/4'], [0, -2], True], ['0', '-2/3']], ['control: random df2t 15', [['-1/4', '2', '-1/2', '1/2', '1/2'], [4], False], ['-1']]]]\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":"ef42ab61888b9fbb172e26314eaffe09df1f9eb5ee500a961a9504001c2b1e13","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    b0, b1, b2, a1, a2 = [Fraction(v) for v in x[0]]\n    xs, prime = x[1], x[2]\n    s1 = s2 = Fraction(0)\n    if prime and xs:\n        den = 1 + a1 + a2\n        if den == 0:\n            return 'dc-pole'\n        x0 = xs[0]\n        yss = x0 * (b0 + b1 + b2) / den\n        s1 = yss - b0 * x0\n        s2 = b2 * x0 - a2 * yss\n    out = []\n    for v in xs:\n        y = b0 * v + s2\n        s1 = b1 * v - a1 * y + s2\n        s2 = b2 * v - a2 * y\n        out.append(str(y))\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression: primed step lowpass', [['1/4', '1/2', '1/4', '-1/2', '1/4'], [2, 2, 2, 2], True], ['8/3', '8/3', '8/3', '8/3']], ['regression: primed highpass', [['1', '-2', '1', '0', '0'], [3, 3, 1], True], ['0', '0', '-2']], ['regression: unprimed impulse', [['1', '1/2', '1/3', '1/2', '-1/4'], [1, 0, 0, 0], False], ['1', '0', '7/12', '-7/24']], ['control: dc pole', [['1', '0', '0', '-2', '1'], [1, 1], True], 'dc-pole'], ['control: primed empty', [['1', '0', '0', '0', '0'], [], True], []], ['control: random df2t 3', [['1/8', '3/4', '-3/8', '-1/2', '1/8'], [2], False], ['1/4']], ['control: random df2t 11', [['1/8', '1/8', '5/4', '-3/8', '1/2'], [0], True], ['0']]], [['regression: random df2t 0', [['5/4', '1/3', '3/4', '-1/2', '2'], [-1, 2, 4, 4, 1], False], ['-5/4', '37/24', '131/16', '283/32', '-1223/192']], ['regression: random df2t 1', [['-1/4', '-1/2', '0', '-1/2', '-3/8'], [-3, 3, 1, 2], False], ['3/4', '9/8', '-29/32', '-33/32']], ['regression: unprimed impulse', [['1', '1/2', '1/3', '1/2', '-1/4'], [1, 0, 0, 0], False], ['1', '0', '7/12', '-7/24']], ['control: random df2t 12', [['-1/2', '5/4', '-1/4', '5/4', '-1/4'], [0, 0], False], ['0', '0']], ['control: random df2t 14', [['1/3', '2', '1/2', '1/2', '5/4'], [0, -2], True], ['0', '-2/3']], ['control: random df2t 15', [['-1/4', '2', '-1/2', '1/2', '1/2'], [4], False], ['-1']], ['control: random df2t 20', [['5/4', '0', '5/4', '-1/2', '-1/2'], [-3, 1], True], 'dc-pole']], [['regression: random df2t 4', [['-1/2', '-3/8', '0', '0', '1/3'], [-3], True], ['63/32']], ['regression: random df2t 5', [['-3/8', '-1/4', '-1/4', '1/3', '-1/4'], [2], True], ['-21/13']], ['regression: random df2t 1', [['-1/4', '-1/2', '0', '-1/2', '-3/8'], [-3, 3, 1, 2], False], ['3/4', '9/8', '-29/32', '-33/32']], ['control: random df2t 31', [['-1/2', '1/8', '1/2', '-1/2', '-1/4'], [0, -3], False], ['0', '3/2']], ['control: random df2t 35', [['2', '1/8', '1/8', '-1/2', '-1/2'], [2, -3, 1], True], 'dc-pole'], ['control: random df2t 43', [['1/3', '3/4', '5/4', '1/3', '3/4'], [2], False], ['2/3']], ['control: random df2t 48', [['-1/4', '1/2', '-1/2', '-3/8', '-1/2'], [-1], False], ['1/4']]], [['regression: random df2t 7', [['-1/2', '2', '-1/4', '1/2', '2'], [3, 2, -2, 0], True], ['15/14', '11/7', '37/28', '-465/56']], ['regression: random df2t 8', [['-3/8', '2', '1/8', '1/3', '2'], [4, -3, -2, 1, 4], True], ['21/10', '189/40', '-421/40', '-1283/120', '8951/360']], ['regression: random df2t 4', [['-1/2', '-3/8', '0', '0', '1/3'], [-3], True], ['63/32']], ['control: random df2t 49', [['0', '2', '0', '-1/2', '2'], [0], True], ['0']], ['control: dc pole', [['1', '0', '0', '-2', '1'], [1, 1], True], 'dc-pole'], ['control: primed empty', [['1', '0', '0', '0', '0'], [], True], []], ['control: random df2t 3', [['1/8', '3/4', '-3/8', '-1/2', '1/8'], [2], False], ['1/4']]], [['regression: random df2t 10', [['1/8', '1/3', '-1/4', '1/3', '1/8'], [1, -1, 2, 3, 3], False], ['1/8', '1/6', '-233/576', '2429/1728', '18953/41472']], ['regression: random df2t 13', [['-1/4', '-1/4', '1/2', '5/4', '0'], [2, -3, -1, 3, -3], False], ['-1/2', '7/8', '29/32', '-401/128', '1749/512']], ['regression: random df2t 6', [['0', '-1/2', '1/8', '3/4', '3/4'], [4, 4, 2, 4], False], ['0', '-2', '0', '1']], ['control: random df2t 11', [['1/8', '1/8', '5/4', '-3/8', '1/2'], [0], True], ['0']], ['control: random df2t 12', [['-1/2', '5/4', '-1/4', '5/4', '-1/4'], [0, 0], False], ['0', '0']], ['control: random df2t 14', [['1/3', '2', '1/2', '1/2', '5/4'], [0, -2], True], ['0', '-2/3']], ['control: random df2t 15', [['-1/4', '2', '-1/2', '1/2', '1/2'], [4], False], ['-1']]]]\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-biquad-transposed-df2-output-state-tap","generated_at":"2026-09-29T14:51:35.470709+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Transposed DF-II is the default numerical form in filtering libraries, and steady-state priming avoids start-up transients.","root_cause":"The output equation adds s2 instead of s1.","sha256":"3818f68c5b27cf92c6bec76bc1cd9e88a33083990d3c3c17c080b83120a98460","title":"Transposed DF-II reads the output from the second state · 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":42.238,"exit_code":1,"observations":[{"actual":["5/2","59/24","239/96","321/128"],"check":"regression: primed step lowpass","expected":["8/3","8/3","8/3","8/3"],"passed":false},{"actual":["3","3","1"],"check":"regression: primed highpass","expected":["0","0","-2"],"passed":false},{"actual":["1","7/12","7/16","7/192"],"check":"regression: unprimed impulse","expected":["1","0","7/12","-7/24"],"passed":false},{"actual":"dc-pole","check":"control: dc pole","expected":"dc-pole","passed":true},{"actual":[],"check":"control: primed empty","expected":[],"passed":true},{"actual":["1/4"],"check":"control: random df2t 3","expected":["1/4"],"passed":true},{"actual":["0"],"check":"control: random df2t 11","expected":["0"],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: primed step lowpass\", \"actual\": [\"5/2\", \"59/24\", \"239/96\", \"321/128\"], \"expected\": [\"8/3\", \"8/3\", \"8/3\", \"8/3\"], \"passed\": false}, {\"check\": \"regression: primed highpass\", \"actual\": [\"3\", \"3\", \"1\"], \"expected\": [\"0\", \"0\", \"-2\"], \"passed\": false}, {\"check\": \"regression: unprimed impulse\", \"actual\": [\"1\", \"7/12\", \"7/16\", \"7/192\"], \"expected\": [\"1\", \"0\", \"7/12\", \"-7/24\"], \"passed\": false}, {\"check\": \"control: dc pole\", \"actual\": \"dc-pole\", \"expected\": \"dc-pole\", \"passed\": true}, {\"check\": \"control: primed empty\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"control: random df2t 3\", \"actual\": [\"1/4\"], \"expected\": [\"1/4\"], \"passed\": true}, {\"check\": \"control: random df2t 11\", \"actual\": [\"0\"], \"expected\": [\"0\"], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":41.787,"exit_code":1,"observations":[{"actual":["1/3","11/12","37/48","155/192"],"check":"regression: primed step lowpass","expected":["8/3","8/3","8/3","8/3"],"passed":false},{"actual":["6","6","4"],"check":"regression: primed highpass","expected":["0","0","-2"],"passed":false},{"actual":["1","7/12","7/48","7/192"],"check":"regression: unprimed impulse","expected":["1","0","7/12","-7/24"],"passed":false},{"actual":"dc-pole","check":"control: dc pole","expected":"dc-pole","passed":true},{"actual":[],"check":"control: primed empty","expected":[],"passed":true},{"actual":["1/4"],"check":"control: random df2t 3","expected":["1/4"],"passed":true},{"actual":["0"],"check":"control: random df2t 11","expected":["0"],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: primed step lowpass\", \"actual\": [\"1/3\", \"11/12\", \"37/48\", \"155/192\"], \"expected\": [\"8/3\", \"8/3\", \"8/3\", \"8/3\"], \"passed\": false}, {\"check\": \"regression: primed highpass\", \"actual\": [\"6\", \"6\", \"4\"], \"expected\": [\"0\", \"0\", \"-2\"], \"passed\": false}, {\"check\": \"regression: unprimed impulse\", \"actual\": [\"1\", \"7/12\", \"7/48\", \"7/192\"], \"expected\": [\"1\", \"0\", \"7/12\", \"-7/24\"], \"passed\": false}, {\"check\": \"control: dc pole\", \"actual\": \"dc-pole\", \"expected\": \"dc-pole\", \"passed\": true}, {\"check\": \"control: primed empty\", \"actual\": [], \"expected\": [], \"passed\": true}, {\"check\": \"control: random df2t 3\", \"actual\": [\"1/4\"], \"expected\": [\"1/4\"], \"passed\": true}, {\"check\": \"control: random df2t 11\", \"actual\": [\"0\"], \"expected\": [\"0\"], \"passed\": true}], \"passed\": false}\n"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}