{"abstract":"The x[n-2] tap always equals x[n-1], so b2 acts on the wrong sample.","category":"Digital signal filters","checks":7,"contract":"Input [[b0, b1, b2, a0, a1, a2], samples]; coefficients are rational strings, normalized by a0 (\"a0-zero\" if a0 == 0). y[n] = b0 x[n] + b1 x[n-1] + b2 x[n-2] - a1 y[n-1] - a2 y[n-2] with zero initial state; return outputs as exact fraction strings.","contract_signature":"x","evaluation_group":"w2-digital_signal_filters-biquad-direct-form-one","failed_approach":"The attempted repair swaps the tuple targets, storing the current sample in x[n-2].","family":"w2-digital_signal_filters-biquad-direct-form-one-input-delay-shift-order","id":"FA-91361","implementations":{"attempt":{"sha256":"553c3beda358cd0ba05dbae8e308a07d6e47a01a44ddfb6b7e1a4abf10988c7e","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, a0, a1, a2 = [Fraction(v) for v in x[0]]\n    if a0 == 0:\n        return 'a0-zero'\n    b0, b1, b2, a1, a2 = b0 / a0, b1 / a0, b2 / a0, a1 / a0, a2 / a0\n    x1 = x2 = y1 = y2 = Fraction(0)\n    out = []\n    for s in x[1]:\n        y = b0 * s + b1 * x1 + b2 * x2 - a1 * y1 - a2 * y2\n        x2, x1 = s, x1\n        y2, y1 = y1, 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: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['regression: random biquad 0', [['1/3', '3/4', '1/2', '-1', '2', '1/3'], [2, 2]], ['-2/3', '-7/2']], ['repair check: a0 not one', [['2', '1', '0', '2', '1', '1/2'], [2, 0, 0, 1]], ['2', '0', '-1/2', '5/4']], ['control: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['control: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['regression: random biquad 1', [['1/8', '-1/4', '3/4', '2', '1/2', '1/8'], [-2, 1, 2, 2, 0, 2]], ['-1/8', '11/32', '-53/64', '223/512', '907/2048', '3019/4096']]], [['regression: random biquad 2', [['1/2', '5/4', '1/2', '-1', '-1/2', '-1/2'], [-2, -2, 3, -3, 1]], ['1', '3', '0', '-11/4', '25/8']], ['regression: random biquad 3', [['-3/8', '5/4', '1/2', '2', '5/4', '1/8'], [-2, 0, -1, 3]], ['3/8', '-95/64', '303/512', '-5999/4096']], ['regression: random biquad 0', [['1/3', '3/4', '1/2', '-1', '2', '1/3'], [2, 2]], ['-2/3', '-7/2']], ['control: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['regression: random biquad 1', [['1/8', '-1/4', '3/4', '2', '1/2', '1/8'], [-2, 1, 2, 2, 0, 2]], ['-1/8', '11/32', '-53/64', '223/512', '907/2048', '3019/4096']]], [['regression: random biquad 5', [['1/2', '0', '-1/2', '-1', '-1/2', '-3/8'], [0, 2, -2, -1, 1]], ['0', '-1', '3/2', '9/8', '-21/8']], ['regression: random biquad 6', [['1/8', '0', '5/4', '1', '-1/2', '5/4'], [1, 1, 1, 0, 3, 2]], ['1/8', '3/16', '21/16', '107/64', '105/128', '-183/128']], ['regression: random biquad 2', [['1/2', '5/4', '1/2', '-1', '-1/2', '-1/2'], [-2, -2, 3, -3, 1]], ['1', '3', '0', '-11/4', '25/8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['control: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['regression: random biquad 3', [['-3/8', '5/4', '1/2', '2', '5/4', '1/8'], [-2, 0, -1, 3]], ['3/8', '-95/64', '303/512', '-5999/4096']]], [['regression: random biquad 9', [['3/4', '1/3', '1/2', '-1', '0', '1/8'], [-1, -1, 2]], ['3/4', '13/12', '-55/96']], ['regression: random biquad 10', [['3/4', '2', '1/2', '1', '-3/8', '1/3'], [2, 2, -3]], ['3/2', '97/16', '579/128']], ['regression: random biquad 4', [['-3/8', '1/3', '-1/2', '2', '1/3', '0'], [-1, 1]], ['3/16', '-37/96']], ['control: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['control: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['regression: random biquad 3', [['-3/8', '5/4', '1/2', '2', '5/4', '1/8'], [-2, 0, -1, 3]], ['3/8', '-95/64', '303/512', '-5999/4096']]], [['regression: random biquad 13', [['1/8', '-1/4', '5/4', '2', '-1/4', '1/2'], [1, -1, 2]], ['1/16', '-23/128', '857/1024']], ['regression: random biquad 14', [['-1/4', '1/8', '-1/4', '1', '5/4', '0'], [-2, -3, -3, -2]], ['1/2', '-1/8', '33/32', '-53/128']], ['regression: random biquad 6', [['1/8', '0', '5/4', '1', '-1/2', '5/4'], [1, 1, 1, 0, 3, 2]], ['1/8', '3/16', '21/16', '107/64', '105/128', '-183/128']], ['control: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['regression: random biquad 4', [['-3/8', '1/3', '-1/2', '2', '1/3', '0'], [-1, 1]], ['3/16', '-37/96']]]]\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":"597e97b4a7873344cb56e73f6b2f37bb9d37992a8d1f6fa14156452faca8336f","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, a0, a1, a2 = [Fraction(v) for v in x[0]]\n    if a0 == 0:\n        return 'a0-zero'\n    b0, b1, b2, a1, a2 = b0 / a0, b1 / a0, b2 / a0, a1 / a0, a2 / a0\n    x1 = x2 = y1 = y2 = Fraction(0)\n    out = []\n    for s in x[1]:\n        y = b0 * s + b1 * x1 + b2 * x2 - a1 * y1 - a2 * y2\n        x1 = s\n        x2 = x1\n        y2, y1 = y1, 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: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['regression: random biquad 0', [['1/3', '3/4', '1/2', '-1', '2', '1/3'], [2, 2]], ['-2/3', '-7/2']], ['repair check: a0 not one', [['2', '1', '0', '2', '1', '1/2'], [2, 0, 0, 1]], ['2', '0', '-1/2', '5/4']], ['control: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['control: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['regression: random biquad 1', [['1/8', '-1/4', '3/4', '2', '1/2', '1/8'], [-2, 1, 2, 2, 0, 2]], ['-1/8', '11/32', '-53/64', '223/512', '907/2048', '3019/4096']]], [['regression: random biquad 2', [['1/2', '5/4', '1/2', '-1', '-1/2', '-1/2'], [-2, -2, 3, -3, 1]], ['1', '3', '0', '-11/4', '25/8']], ['regression: random biquad 3', [['-3/8', '5/4', '1/2', '2', '5/4', '1/8'], [-2, 0, -1, 3]], ['3/8', '-95/64', '303/512', '-5999/4096']], ['regression: random biquad 0', [['1/3', '3/4', '1/2', '-1', '2', '1/3'], [2, 2]], ['-2/3', '-7/2']], ['control: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['regression: random biquad 1', [['1/8', '-1/4', '3/4', '2', '1/2', '1/8'], [-2, 1, 2, 2, 0, 2]], ['-1/8', '11/32', '-53/64', '223/512', '907/2048', '3019/4096']]], [['regression: random biquad 5', [['1/2', '0', '-1/2', '-1', '-1/2', '-3/8'], [0, 2, -2, -1, 1]], ['0', '-1', '3/2', '9/8', '-21/8']], ['regression: random biquad 6', [['1/8', '0', '5/4', '1', '-1/2', '5/4'], [1, 1, 1, 0, 3, 2]], ['1/8', '3/16', '21/16', '107/64', '105/128', '-183/128']], ['regression: random biquad 2', [['1/2', '5/4', '1/2', '-1', '-1/2', '-1/2'], [-2, -2, 3, -3, 1]], ['1', '3', '0', '-11/4', '25/8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['control: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['regression: random biquad 3', [['-3/8', '5/4', '1/2', '2', '5/4', '1/8'], [-2, 0, -1, 3]], ['3/8', '-95/64', '303/512', '-5999/4096']]], [['regression: random biquad 9', [['3/4', '1/3', '1/2', '-1', '0', '1/8'], [-1, -1, 2]], ['3/4', '13/12', '-55/96']], ['regression: random biquad 10', [['3/4', '2', '1/2', '1', '-3/8', '1/3'], [2, 2, -3]], ['3/2', '97/16', '579/128']], ['regression: random biquad 4', [['-3/8', '1/3', '-1/2', '2', '1/3', '0'], [-1, 1]], ['3/16', '-37/96']], ['control: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['control: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['regression: random biquad 3', [['-3/8', '5/4', '1/2', '2', '5/4', '1/8'], [-2, 0, -1, 3]], ['3/8', '-95/64', '303/512', '-5999/4096']]], [['regression: random biquad 13', [['1/8', '-1/4', '5/4', '2', '-1/4', '1/2'], [1, -1, 2]], ['1/16', '-23/128', '857/1024']], ['regression: random biquad 14', [['-1/4', '1/8', '-1/4', '1', '5/4', '0'], [-2, -3, -3, -2]], ['1/2', '-1/8', '33/32', '-53/128']], ['regression: random biquad 6', [['1/8', '0', '5/4', '1', '-1/2', '5/4'], [1, 1, 1, 0, 3, 2]], ['1/8', '3/16', '21/16', '107/64', '105/128', '-183/128']], ['control: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['regression: random biquad 4', [['-3/8', '1/3', '-1/2', '2', '1/3', '0'], [-1, 1]], ['3/16', '-37/96']]]]\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-direct-form-one-input-delay-shift-order","generated_at":"2026-09-29T14:51:35.376881+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"The biquad is the workhorse IIR section; sign, state or normalization slips produce unstable or wrong-gain filters.","root_cause":"The input delay line is shifted with two sequential assignments in the wrong order.","sha256":"7755523d329b4920a5c71ae575a8b218ae1fdfe1ee1ea66154d5906766f7281a","title":"Biquad overwrites x[n-1] before moving it to x[n-2] · 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":44.396,"exit_code":1,"observations":[{"actual":["1","3","5"],"check":"regression: fir only","expected":["1","4","8"],"passed":false},{"actual":["-2/3","-3"],"check":"regression: random biquad 0","expected":["-2/3","-7/2"],"passed":false},{"actual":["2","-1","0","5/4"],"check":"repair check: a0 not one","expected":["2","0","-1/2","5/4"],"passed":false},{"actual":["1","1/2","1/4","1/8"],"check":"control: pure feedback impulse","expected":["1","1/2","1/4","1/8"],"passed":true},{"actual":["1","0","-1/4","0","1/16"],"check":"control: second order feedback","expected":["1","0","-1/4","0","1/16"],"passed":true},{"actual":["0","-9/8"],"check":"control: random biquad 7","expected":["0","-9/8"],"passed":true},{"actual":["-1/8","-21/32","43/64","383/512","1067/2048","-213/4096"],"check":"regression: random biquad 1","expected":["-1/8","11/32","-53/64","223/512","907/2048","3019/4096"],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: fir only\", \"actual\": [\"1\", \"3\", \"5\"], \"expected\": [\"1\", \"4\", \"8\"], \"passed\": false}, {\"check\": \"regression: random biquad 0\", \"actual\": [\"-2/3\", \"-3\"], \"expected\": [\"-2/3\", \"-7/2\"], \"passed\": false}, {\"check\": \"repair check: a0 not one\", \"actual\": [\"2\", \"-1\", \"0\", \"5/4\"], \"expected\": [\"2\", \"0\", \"-1/2\", \"5/4\"], \"passed\": false}, {\"check\": \"control: pure feedback impulse\", \"actual\": [\"1\", \"1/2\", \"1/4\", \"1/8\"], \"expected\": [\"1\", \"1/2\", \"1/4\", \"1/8\"], \"passed\": true}, {\"check\": \"control: second order feedback\", \"actual\": [\"1\", \"0\", \"-1/4\", \"0\", \"1/16\"], \"expected\": [\"1\", \"0\", \"-1/4\", \"0\", \"1/16\"], \"passed\": true}, {\"check\": \"control: random biquad 7\", \"actual\": [\"0\", \"-9/8\"], \"expected\": [\"0\", \"-9/8\"], \"passed\": true}, {\"check\": \"regression: random biquad 1\", \"actual\": [\"-1/8\", \"-21/32\", \"43/64\", \"383/512\", \"1067/2048\", \"-213/4096\"], \"expected\": [\"-1/8\", \"11/32\", \"-53/64\", \"223/512\", \"907/2048\", \"3019/4096\"], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.948,"exit_code":1,"observations":[{"actual":["1","5","9"],"check":"regression: fir only","expected":["1","4","8"],"passed":false},{"actual":["-2/3","-9/2"],"check":"regression: random biquad 0","expected":["-2/3","-7/2"],"passed":false},{"actual":["2","0","-1/2","5/4"],"check":"repair check: a0 not one","expected":["2","0","-1/2","5/4"],"passed":true},{"actual":["1","1/2","1/4","1/8"],"check":"control: pure feedback impulse","expected":["1","1/2","1/4","1/8"],"passed":true},{"actual":["1","0","-1/4","0","1/16"],"check":"control: second order feedback","expected":["1","0","-1/4","0","1/16"],"passed":true},{"actual":["0","-9/8"],"check":"control: random biquad 7","expected":["0","-9/8"],"passed":true},{"actual":["-1/8","-13/32","31/64","271/512","691/2048","31/4096"],"check":"regression: random biquad 1","expected":["-1/8","11/32","-53/64","223/512","907/2048","3019/4096"],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: fir only\", \"actual\": [\"1\", \"5\", \"9\"], \"expected\": [\"1\", \"4\", \"8\"], \"passed\": false}, {\"check\": \"regression: random biquad 0\", \"actual\": [\"-2/3\", \"-9/2\"], \"expected\": [\"-2/3\", \"-7/2\"], \"passed\": false}, {\"check\": \"repair check: a0 not one\", \"actual\": [\"2\", \"0\", \"-1/2\", \"5/4\"], \"expected\": [\"2\", \"0\", \"-1/2\", \"5/4\"], \"passed\": true}, {\"check\": \"control: pure feedback impulse\", \"actual\": [\"1\", \"1/2\", \"1/4\", \"1/8\"], \"expected\": [\"1\", \"1/2\", \"1/4\", \"1/8\"], \"passed\": true}, {\"check\": \"control: second order feedback\", \"actual\": [\"1\", \"0\", \"-1/4\", \"0\", \"1/16\"], \"expected\": [\"1\", \"0\", \"-1/4\", \"0\", \"1/16\"], \"passed\": true}, {\"check\": \"control: random biquad 7\", \"actual\": [\"0\", \"-9/8\"], \"expected\": [\"0\", \"-9/8\"], \"passed\": true}, {\"check\": \"regression: random biquad 1\", \"actual\": [\"-1/8\", \"-13/32\", \"31/64\", \"271/512\", \"691/2048\", \"31/4096\"], \"expected\": [\"-1/8\", \"11/32\", \"-53/64\", \"223/512\", \"907/2048\", \"3019/4096\"], \"passed\": false}], \"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."}}