{"abstract":"The impulse response loses its two-sample memory: s1 uses the freshly computed s2 from the same 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.","evaluation_group":"w2-digital_signal_filters-biquad-transposed-df2","failed_approach":"The attempted repair drops s2 from the s1 update altogether, turning the section into first order.","family":"w2-digital_signal_filters-biquad-transposed-df2-state-update-ordering","id":"FA-91376","implementations":{"attempt":{"sha256":"f68566bbc6e149ae465333d4ba5d78c7480c2035aac235df46b58415639672a2","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\n        s1 = b1 * v - a1 * y\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: unprimed impulse', [['1', '1/2', '1/3', '1/2', '-1/4'], [1, 0, 0, 0], False], ['1', '0', '7/12', '-7/24']], ['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']], ['repair check: 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']], ['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 4', [['-1/2', '-3/8', '0', '0', '1/3'], [-3], True], ['63/32']]], [['regression: random df2t 6', [['0', '-1/2', '1/8', '3/4', '3/4'], [4, 4, 2, 4], False], ['0', '-2', '0', '1']], ['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: 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 5', [['-3/8', '-1/4', '-1/4', '1/3', '-1/4'], [2], True], ['-21/13']], ['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']]], [['regression: random df2t 9', [['-3/8', '0', '-1/2', '2', '5/4'], [3, -3, 3, 2], False], ['-9/8', '27/8', '-255/32', '399/32']], ['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 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 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'], ['control: random df2t 22', [['1/3', '-3/8', '1/3', '0', '5/4'], [4], True], ['14/27']], ['control: random df2t 31', [['-1/2', '1/8', '1/2', '-1/2', '-1/4'], [0, -3], False], ['0', '3/2']]], [['regression: random df2t 16', [['5/4', '1/8', '1/8', '-3/8', '1/3'], [-2, 2, 3], True], ['-72/23', '43/23', '1011/184']], ['regression: random df2t 18', [['0', '-3/8', '1/3', '1/3', '2'], [4, 4, 2, 0, -3], True], ['-1/20', '-1/20', '-1/20', '7/10', '8/15']], ['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 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']], ['control: random df2t 49', [['0', '2', '0', '-1/2', '2'], [0], True], ['0']]], [['regression: random df2t 21', [['1/2', '1/8', '-1/4', '-1/4', '3/4'], [2, 0, 3, -3], False], ['1', '1/2', '3/8', '-45/32']], ['regression: random df2t 23', [['3/4', '0', '-1/2', '-1/2', '0'], [3, 0, 3], False], ['9/4', '9/8', '21/16']], ['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']], ['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 4', [['-1/2', '-3/8', '0', '0', '1/3'], [-3], True], ['63/32']]]]\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":"cf36836fa51a3bb4a35b23004c233ce7822d58ed7c98d9c0920a99af039d3f14","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\n        s2 = b2 * v - a2 * y\n        s1 = b1 * v - a1 * y + s2\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: unprimed impulse', [['1', '1/2', '1/3', '1/2', '-1/4'], [1, 0, 0, 0], False], ['1', '0', '7/12', '-7/24']], ['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']], ['repair check: 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']], ['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 4', [['-1/2', '-3/8', '0', '0', '1/3'], [-3], True], ['63/32']]], [['regression: random df2t 6', [['0', '-1/2', '1/8', '3/4', '3/4'], [4, 4, 2, 4], False], ['0', '-2', '0', '1']], ['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: 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 5', [['-3/8', '-1/4', '-1/4', '1/3', '-1/4'], [2], True], ['-21/13']], ['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']]], [['regression: random df2t 9', [['-3/8', '0', '-1/2', '2', '5/4'], [3, -3, 3, 2], False], ['-9/8', '27/8', '-255/32', '399/32']], ['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 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 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'], ['control: random df2t 22', [['1/3', '-3/8', '1/3', '0', '5/4'], [4], True], ['14/27']], ['control: random df2t 31', [['-1/2', '1/8', '1/2', '-1/2', '-1/4'], [0, -3], False], ['0', '3/2']]], [['regression: random df2t 16', [['5/4', '1/8', '1/8', '-3/8', '1/3'], [-2, 2, 3], True], ['-72/23', '43/23', '1011/184']], ['regression: random df2t 18', [['0', '-3/8', '1/3', '1/3', '2'], [4, 4, 2, 0, -3], True], ['-1/20', '-1/20', '-1/20', '7/10', '8/15']], ['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 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']], ['control: random df2t 49', [['0', '2', '0', '-1/2', '2'], [0], True], ['0']]], [['regression: random df2t 21', [['1/2', '1/8', '-1/4', '-1/4', '3/4'], [2, 0, 3, -3], False], ['1', '1/2', '3/8', '-45/32']], ['regression: random df2t 23', [['3/4', '0', '-1/2', '-1/2', '0'], [3, 0, 3], False], ['9/4', '9/8', '21/16']], ['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']], ['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 4', [['-1/2', '-3/8', '0', '0', '1/3'], [-3], True], ['63/32']]]]\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"},"fixed":{"sha256":"9de33c5bd3b3823743d646a5aa15f3ee66142c2d8dbfdd1a5bbfae621b7215b9","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\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: unprimed impulse', [['1', '1/2', '1/3', '1/2', '-1/4'], [1, 0, 0, 0], False], ['1', '0', '7/12', '-7/24']], ['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']], ['repair check: 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']], ['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 4', [['-1/2', '-3/8', '0', '0', '1/3'], [-3], True], ['63/32']]], [['regression: random df2t 6', [['0', '-1/2', '1/8', '3/4', '3/4'], [4, 4, 2, 4], False], ['0', '-2', '0', '1']], ['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: 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 5', [['-3/8', '-1/4', '-1/4', '1/3', '-1/4'], [2], True], ['-21/13']], ['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']]], [['regression: random df2t 9', [['-3/8', '0', '-1/2', '2', '5/4'], [3, -3, 3, 2], False], ['-9/8', '27/8', '-255/32', '399/32']], ['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 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 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'], ['control: random df2t 22', [['1/3', '-3/8', '1/3', '0', '5/4'], [4], True], ['14/27']], ['control: random df2t 31', [['-1/2', '1/8', '1/2', '-1/2', '-1/4'], [0, -3], False], ['0', '3/2']]], [['regression: random df2t 16', [['5/4', '1/8', '1/8', '-3/8', '1/3'], [-2, 2, 3], True], ['-72/23', '43/23', '1011/184']], ['regression: random df2t 18', [['0', '-3/8', '1/3', '1/3', '2'], [4, 4, 2, 0, -3], True], ['-1/20', '-1/20', '-1/20', '7/10', '8/15']], ['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 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']], ['control: random df2t 49', [['0', '2', '0', '-1/2', '2'], [0], True], ['0']]], [['regression: random df2t 21', [['1/2', '1/8', '-1/4', '-1/4', '3/4'], [2, 0, 3, -3], False], ['1', '1/2', '3/8', '-45/32']], ['regression: random df2t 23', [['3/4', '0', '-1/2', '-1/2', '0'], [3, 0, 3], False], ['9/4', '9/8', '21/16']], ['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']], ['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 4', [['-1/2', '-3/8', '0', '0', '1/3'], [-3], True], ['63/32']]]]\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-state-update-ordering","generated_at":"2026-09-29T14:51:35.469312+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.","repair":"Update s1 from the previous s2 first, then overwrite s2.","root_cause":"The s2 update is executed before the s1 update, so s1 adds the new rather than the previous s2.","sha256":"e4d1c325fbe5c4e268b160af4eec4a9e2345291ade03721edf510a69950c2d95","title":"Transposed DF-II updates s2 before s1 consumes it · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":41.375,"exit_code":1,"observations":[{"actual":["1","0","0","0"],"check":"regression: unprimed impulse","expected":["1","0","7/12","-7/24"],"passed":false},{"actual":["-5/4","37/24","103/16","917/96","471/64"],"check":"regression: random df2t 0","expected":["-5/4","37/24","131/16","283/32","-1223/192"],"passed":false},{"actual":["8/3","17/6","35/12","71/24"],"check":"repair check: primed step lowpass","expected":["8/3","8/3","8/3","8/3"],"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":["63/32"],"check":"control: random df2t 4","expected":["63/32"],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": 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