{"abstract":"Intermediate stages exceed the limit, so later stages see values the hardware could never produce.","category":"Digital signal filters","checks":7,"contract":"Input [gain, sections, limit, samples]; each section [b0, b1, b2, a1, a2] is a direct-form-I biquad (a0 = 1) with its own state. The input is scaled by gain, each section output is clamped to [-limit, limit] and the clamped value is both stored as that section's output state and fed to the next section. Return exact fraction strings.","contract_signature":"x","evaluation_group":"w2-digital_signal_filters-sos-cascade-saturation","failed_approach":"The attempted repair clamps the first section only.","family":"w2-digital_signal_filters-sos-cascade-saturation-per-stage-clamp","id":"FA-91631","implementations":{"attempt":{"sha256":"c8a3ccdd99eef769c584b7a6e4669aafc6d09835beb5c7adc56e108478f736d1","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    gain, secs, lim, xs = Fraction(x[0]), x[1], Fraction(x[2]), x[3]\n    coefs = [[Fraction(v) for v in s] for s in secs]\n    states = [[Fraction(0)] * 4 for _ in coefs]\n    out = []\n    for v in xs:\n        u = gain * v\n        for k, (b0, b1, b2, a1, a2) in enumerate(coefs):\n            st = states[k]\n            y = b0 * u + b1 * st[0] + b2 * st[1] - a1 * st[2] - a2 * st[3]\n            y = max(-lim, min(lim, y)) if k == 0 else y\n            st[:] = [u, st[0], y, st[2]]\n            u = y\n        out.append(str(u))\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression: three sections', ['1/2', [['1', '1', '1', '-1/2', '1/4'], ['2', '0', '1', '0', '1/4'], ['1', '0', '0', '-1/2', '0']], '5', [3, 0, -3, 1]], ['3', '5', '19/4', '9/8']], ['regression: random cascade 3', ['1', [['1', '0', '0', '-1/2', '0'], ['2', '0', '1', '0', '1/4'], ['1', '-1', '0', '1/4', '0']], '2', [-1, 0, -3, 0, 0]], ['-2', '3/2', '-11/8', '11/32', '-11/128']], ['repair check: two sections saturating', ['1', [['2', '0', '1', '0', '1/4'], ['3', '0', '0', '0', '0']], '4', [2, 2, -1, 0]], ['4', '4', '-3', '3']], ['control: gain before saturation', ['3', [['1', '0', '0', '-1/2', '0']], '2', [1, 1, 1]], ['2', '2', '2']], ['control: random cascade 0', ['1/2', [['1', '1', '1', '-1/2', '1/4']], '100', [3, -1, 0, 2, -3, 3]], ['3/2', '7/4', '3/2', '13/16', '-15/32', '9/16']], ['control: random cascade 2', ['1/2', [['2', '0', '1', '0', '1/4']], '3', [3, 0]], ['3', '0']], ['control: random cascade 4', ['2', [['1', '-1', '0', '1/4', '0']], '2', [-3, -2, 2, 1, -2]], ['-2', '2', '2', '-2', '-2']]], [['regression: random cascade 7', ['1', [['1', '-1', '0', '1/4', '0'], ['1', '0', '0', '-1/2', '0']], '2', [1, 3, 1]], ['1', '2', '-1']], ['regression: random cascade 10', ['1', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0']], '2', [-3, -1, 0]], ['-2', '1/2', '7/8']], ['repair check: random cascade 1', ['3', [['2', '0', '1', '0', '1/4'], ['1', '-1', '0', '1/4', '0'], ['1', '-1', '0', '1/4', '0']], '5', [0, 2, -3, 0, 0, -2]], ['0', '5', '-5', '5', '-5', '5']], ['control: random cascade 6', ['3', [['1', '1', '1', '-1/2', '1/4']], '3', [-1, 2, 0, 1, 1, -1]], ['-3', '3/2', '3', '3', '3', '3']], ['control: random cascade 8', ['1', [['1/2', '1/2', '0', '0', '0'], ['1', '0', '0', '-1/2', '0']], '100', [-3, 3, -2, -2, 1]], ['-3/2', '-3/4', '1/8', '-31/16', '-47/32']], ['control: random cascade 9', ['3', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0']], '100', [-2, 1]], ['-6', '15/2']], ['control: random cascade 13', ['1/2', [['2', '0', '1', '0', '1/4'], ['1/2', '1/2', '0', '0', '0']], '3', [-3, -3, 3, 1]], ['-3/2', '-3', '-3/8', '5/4']]], [['regression: random cascade 12', ['1/2', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0'], ['3', '0', '0', '0', '0']], '2', [-1, 3, 3, -2, 1, 1]], ['-3/2', '2', '27/32', '-2', '2', '0']], ['regression: random cascade 14', ['1', [['3', '0', '0', '0', '0'], ['1', '0', '0', '-1/2', '0']], '5', [-2, 3]], ['-5', '5/2']], ['regression: random cascade 7', ['1', [['1', '-1', '0', '1/4', '0'], ['1', '0', '0', '-1/2', '0']], '2', [1, 3, 1]], ['1', '2', '-1']], ['control: random cascade 15', ['3', [['3', '0', '0', '0', '0']], '100', [-3, 0, 2]], ['-27', '0', '18']], ['control: random cascade 16', ['1/2', [['1', '-1', '0', '1/4', '0']], '100', [0, 1, -3, 0, -3, 1]], ['0', '1/2', '-17/8', '65/32', '-257/128', '1281/512']], ['control: random cascade 20', ['3', [['1/2', '1/2', '0', '0', '0']], '5', [-1, -3, -3]], ['-3/2', '-5', '-5']], ['control: random cascade 21', ['1', [['1', '0', '0', '-1/2', '0'], ['1', '1', '1', '-1/2', '1/4']], '100', [3, 0, 2, 2, 2, 0]], ['3', '6', '19/2', '87/8', '103/8', '101/8']]], [['regression: random cascade 19', ['2', [['1', '0', '0', '-1/2', '0']], '5', [3, -3, 0, 0]], ['5', '-7/2', '-7/4', '-7/8']], ['regression: random cascade 22', ['1/2', [['3', '0', '0', '0', '0'], ['1', '0', '0', '-1/2', '0']], '2', [-1, 2, 2, 2, 1]], ['-3/2', '5/4', '2', '2', '2']], ['regression: random cascade 12', ['1/2', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0'], ['3', '0', '0', '0', '0']], '2', [-1, 3, 3, -2, 1, 1]], ['-3/2', '2', '27/32', '-2', '2', '0']], ['control: random cascade 26', ['1', [['1', '-1', '0', '1/4', '0']], '100', [-1, 1, -2, 1]], ['-1', '9/4', '-57/16', '249/64']], ['control: random cascade 28', ['1/2', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0'], ['1/2', '1/2', '0', '0', '0']], '5', [0, 0, -1, -1, -3, -1]], ['0', '0', '-1/4', '-5/16', '-39/64', '-49/256']], ['control: random cascade 31', ['3', [['1', '1', '1', '-1/2', '1/4']], '2', [-3, 0, 0, -2, 0, 1]], ['-2', '-2', '-2', '-2', '-2', '-2']], ['control: random cascade 33', ['3', [['3', '0', '0', '0', '0']], '2', [-1, 3, 3, -3, -1, -1]], ['-2', '2', '2', '-2', '-2', '-2']]], [['regression: random cascade 24', ['1', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0']], '3', [1, 2, -2, -1, -3]], ['1', '5/4', '-3', '1/8', '-53/32']], ['regression: random cascade 25', ['1', [['2', '0', '1', '0', '1/4']], '5', [-2, 3, -2, -2, 2]], ['-4', '5', '-5', '-9/4', '13/4']], ['regression: random cascade 18', ['2', [['1/2', '1/2', '0', '0', '0'], ['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0']], '2', [-2, -3]], ['-2', '1/2']], ['control: random cascade 35', ['1', [['1', '0', '0', '-1/2', '0']], '3', [0, -2]], ['0', '-2']], ['control: random cascade 36', ['2', [['1', '0', '0', '-1/2', '0']], '100', [0, 1, 2, 1, 3]], ['0', '2', '5', '9/2', '33/4']], ['control: random cascade 37', ['3', [['3', '0', '0', '0', '0'], ['1/2', '1/2', '0', '0', '0']], '100', [-2, -1, 0, -1]], ['-9', '-27/2', '-9/2', '-9/2']], ['control: random cascade 38', ['3', [['3', '0', '0', '0', '0'], ['1', '-1', '0', '1/4', '0'], ['1/2', '1/2', '0', '0', '0']], '100', [1, 1, 3, 2, 3]], ['9/2', '27/8', '261/32', '315/128', '-315/512']]]]\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":"eced8fc3b89d1eeb4c8bf905083c5e6300c7b56a0dc1d270790d051658cf7209","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    gain, secs, lim, xs = Fraction(x[0]), x[1], Fraction(x[2]), x[3]\n    coefs = [[Fraction(v) for v in s] for s in secs]\n    states = [[Fraction(0)] * 4 for _ in coefs]\n    out = []\n    for v in xs:\n        u = gain * v\n        for k, (b0, b1, b2, a1, a2) in enumerate(coefs):\n            st = states[k]\n            y = b0 * u + b1 * st[0] + b2 * st[1] - a1 * st[2] - a2 * st[3]\n            st[:] = [u, st[0], y, st[2]]\n            u = y\n        out.append(str(max(-lim, min(lim, u))))\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression: three sections', ['1/2', [['1', '1', '1', '-1/2', '1/4'], ['2', '0', '1', '0', '1/4'], ['1', '0', '0', '-1/2', '0']], '5', [3, 0, -3, 1]], ['3', '5', '19/4', '9/8']], ['regression: random cascade 3', ['1', [['1', '0', '0', '-1/2', '0'], ['2', '0', '1', '0', '1/4'], ['1', '-1', '0', '1/4', '0']], '2', [-1, 0, -3, 0, 0]], ['-2', '3/2', '-11/8', '11/32', '-11/128']], ['repair check: two sections saturating', ['1', [['2', '0', '1', '0', '1/4'], ['3', '0', '0', '0', '0']], '4', [2, 2, -1, 0]], ['4', '4', '-3', '3']], ['control: gain before saturation', ['3', [['1', '0', '0', '-1/2', '0']], '2', [1, 1, 1]], ['2', '2', '2']], ['control: random cascade 0', ['1/2', [['1', '1', '1', '-1/2', '1/4']], '100', [3, -1, 0, 2, -3, 3]], ['3/2', '7/4', '3/2', '13/16', '-15/32', '9/16']], ['control: random cascade 2', ['1/2', [['2', '0', '1', '0', '1/4']], '3', [3, 0]], ['3', '0']], ['control: random cascade 4', ['2', [['1', '-1', '0', '1/4', '0']], '2', [-3, -2, 2, 1, -2]], ['-2', '2', '2', '-2', '-2']]], [['regression: random cascade 7', ['1', [['1', '-1', '0', '1/4', '0'], ['1', '0', '0', '-1/2', '0']], '2', [1, 3, 1]], ['1', '2', '-1']], ['regression: random cascade 10', ['1', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0']], '2', [-3, -1, 0]], ['-2', '1/2', '7/8']], ['repair check: random cascade 1', ['3', [['2', '0', '1', '0', '1/4'], ['1', '-1', '0', '1/4', '0'], ['1', '-1', '0', '1/4', '0']], '5', [0, 2, -3, 0, 0, -2]], ['0', '5', '-5', '5', '-5', '5']], ['control: random cascade 6', ['3', [['1', '1', '1', '-1/2', '1/4']], '3', [-1, 2, 0, 1, 1, -1]], ['-3', '3/2', '3', '3', '3', '3']], ['control: random cascade 8', ['1', [['1/2', '1/2', '0', '0', '0'], ['1', '0', '0', '-1/2', '0']], '100', [-3, 3, -2, -2, 1]], ['-3/2', '-3/4', '1/8', '-31/16', '-47/32']], ['control: random cascade 9', ['3', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0']], '100', [-2, 1]], ['-6', '15/2']], ['control: random cascade 13', ['1/2', [['2', '0', '1', '0', '1/4'], ['1/2', '1/2', '0', '0', '0']], '3', [-3, -3, 3, 1]], ['-3/2', '-3', '-3/8', '5/4']]], [['regression: random cascade 12', ['1/2', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0'], ['3', '0', '0', '0', '0']], '2', [-1, 3, 3, -2, 1, 1]], ['-3/2', '2', '27/32', '-2', '2', '0']], ['regression: random cascade 14', ['1', [['3', '0', '0', '0', '0'], ['1', '0', '0', '-1/2', '0']], '5', [-2, 3]], ['-5', '5/2']], ['regression: random cascade 7', ['1', [['1', '-1', '0', '1/4', '0'], ['1', '0', '0', '-1/2', '0']], '2', [1, 3, 1]], ['1', '2', '-1']], ['control: random cascade 15', ['3', [['3', '0', '0', '0', '0']], '100', [-3, 0, 2]], ['-27', '0', '18']], ['control: random cascade 16', ['1/2', [['1', '-1', '0', '1/4', '0']], '100', [0, 1, -3, 0, -3, 1]], ['0', '1/2', '-17/8', '65/32', '-257/128', '1281/512']], ['control: random cascade 20', ['3', [['1/2', '1/2', '0', '0', '0']], '5', [-1, -3, -3]], ['-3/2', '-5', '-5']], ['control: random cascade 21', ['1', [['1', '0', '0', '-1/2', '0'], ['1', '1', '1', '-1/2', '1/4']], '100', [3, 0, 2, 2, 2, 0]], ['3', '6', '19/2', '87/8', '103/8', '101/8']]], [['regression: random cascade 19', ['2', [['1', '0', '0', '-1/2', '0']], '5', [3, -3, 0, 0]], ['5', '-7/2', '-7/4', '-7/8']], ['regression: random cascade 22', ['1/2', [['3', '0', '0', '0', '0'], ['1', '0', '0', '-1/2', '0']], '2', [-1, 2, 2, 2, 1]], ['-3/2', '5/4', '2', '2', '2']], ['regression: random cascade 12', ['1/2', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0'], ['3', '0', '0', '0', '0']], '2', [-1, 3, 3, -2, 1, 1]], ['-3/2', '2', '27/32', '-2', '2', '0']], ['control: random cascade 26', ['1', [['1', '-1', '0', '1/4', '0']], '100', [-1, 1, -2, 1]], ['-1', '9/4', '-57/16', '249/64']], ['control: random cascade 28', ['1/2', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0'], ['1/2', '1/2', '0', '0', '0']], '5', [0, 0, -1, -1, -3, -1]], ['0', '0', '-1/4', '-5/16', '-39/64', '-49/256']], ['control: random cascade 31', ['3', [['1', '1', '1', '-1/2', '1/4']], '2', [-3, 0, 0, -2, 0, 1]], ['-2', '-2', '-2', '-2', '-2', '-2']], ['control: random cascade 33', ['3', [['3', '0', '0', '0', '0']], '2', [-1, 3, 3, -3, -1, -1]], ['-2', '2', '2', '-2', '-2', '-2']]], [['regression: random cascade 24', ['1', [['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0']], '3', [1, 2, -2, -1, -3]], ['1', '5/4', '-3', '1/8', '-53/32']], ['regression: random cascade 25', ['1', [['2', '0', '1', '0', '1/4']], '5', [-2, 3, -2, -2, 2]], ['-4', '5', '-5', '-9/4', '13/4']], ['regression: random cascade 18', ['2', [['1/2', '1/2', '0', '0', '0'], ['1', '0', '0', '-1/2', '0'], ['1', '-1', '0', '1/4', '0']], '2', [-2, -3]], ['-2', '1/2']], ['control: random cascade 35', ['1', [['1', '0', '0', '-1/2', '0']], '3', [0, -2]], ['0', '-2']], ['control: random cascade 36', ['2', [['1', '0', '0', '-1/2', '0']], '100', [0, 1, 2, 1, 3]], ['0', '2', '5', '9/2', '33/4']], ['control: random cascade 37', ['3', [['3', '0', '0', '0', '0'], ['1/2', '1/2', '0', '0', '0']], '100', [-2, -1, 0, -1]], ['-9', '-27/2', '-9/2', '-9/2']], ['control: random cascade 38', ['3', [['3', '0', '0', '0', '0'], ['1', '-1', '0', '1/4', '0'], ['1/2', '1/2', '0', '0', '0']], '100', [1, 1, 3, 2, 3]], ['9/2', '27/8', '261/32', '315/128', '-315/512']]]]\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-sos-cascade-saturation-per-stage-clamp","generated_at":"2026-09-29T14:51:37.633580+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Cascaded second-order sections with per-stage saturation model fixed-point IIR hardware; state sharing or gain staging slips change clipping.","root_cause":"The per-section saturation is missing and a single clamp is applied at the output.","sha256":"174f47a68299433fb4fe4fc63885bc37ecd96a3b8db0bc88ef27f7c67d1055a5","title":"SOS cascade clamps only the final output · 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":40.889,"exit_code":1,"observations":[{"actual":["3","6","21/4","11/8"],"check":"regression: three sections","expected":["3","5","19/4","9/8"],"passed":false},{"actual":["-2","3/2","-31/8","103/32","-55/128"],"check":"regression: random cascade 3","expected":["-2","3/2","-11/8","11/32","-11/128"],"passed":false},{"actual":["12","12","-3","3"],"check":"repair check: two sections saturating","expected":["4","4","-3","3"],"passed":false},{"actual":["2","2","2"],"check":"control: gain before saturation","expected":["2","2","2"],"passed":true},{"actual":["3/2","7/4","3/2","13/16","-15/32","9/16"],"check":"control: random cascade 0","expected":["3/2","7/4","3/2","13/16","-15/32","9/16"],"passed":true},{"actual":["3","0"],"check":"control: random cascade 2","expected":["3","0"],"passed":true},{"actual":["-2","2","2","-2","-2"],"check":"control: random cascade 4","expected":["-2","2","2","-2","-2"],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: three sections\", \"actual\": [\"3\", \"6\", \"21/4\", \"11/8\"], \"expected\": [\"3\", \"5\", \"19/4\", \"9/8\"], \"passed\": false}, {\"check\": \"regression: random cascade 3\", \"actual\": [\"-2\", \"3/2\", \"-31/8\", \"103/32\", \"-55/128\"], \"expected\": [\"-2\", \"3/2\", \"-11/8\", \"11/32\", \"-11/128\"], \"passed\": false}, {\"check\": \"repair check: two sections saturating\", \"actual\": [\"12\", \"12\", \"-3\", \"3\"], \"expected\": [\"4\", \"4\", \"-3\", \"3\"], \"passed\": false}, {\"check\": \"control: gain before saturation\", \"actual\": [\"2\", \"2\", \"2\"], \"expected\": [\"2\", \"2\", \"2\"], \"passed\": true}, {\"check\": \"control: random cascade 0\", \"actual\": [\"3/2\", \"7/4\", \"3/2\", \"13/16\", \"-15/32\", \"9/16\"], \"expected\": [\"3/2\", \"7/4\", \"3/2\", \"13/16\", \"-15/32\", \"9/16\"], \"passed\": true}, {\"check\": \"control: random cascade 2\", \"actual\": [\"3\", \"0\"], \"expected\": [\"3\", \"0\"], \"passed\": true}, {\"check\": \"control: random cascade 4\", \"actual\": [\"-2\", \"2\", \"2\", \"-2\", \"-2\"], \"expected\": [\"-2\", \"2\", \"2\", \"-2\", \"-2\"], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":42.842,"exit_code":1,"observations":[{"actual":["3","5","5","11/8"],"check":"regression: three sections","expected":["3","5","19/4","9/8"],"passed":false},{"actual":["-2","3/2","-2","2","-115/128"],"check":"regression: random cascade 3","expected":["-2","3/2","-11/8","11/32","-11/128"],"passed":false},{"actual":["4","4","-3","3"],"check":"repair check: two sections saturating","expected":["4","4","-3","3"],"passed":true},{"actual":["2","2","2"],"check":"control: gain before saturation","expected":["2","2","2"],"passed":true},{"actual":["3/2","7/4","3/2","13/16","-15/32","9/16"],"check":"control: random cascade 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{\"check\": \"control: random cascade 0\", \"actual\": [\"3/2\", \"7/4\", \"3/2\", \"13/16\", \"-15/32\", \"9/16\"], \"expected\": [\"3/2\", \"7/4\", \"3/2\", \"13/16\", \"-15/32\", \"9/16\"], \"passed\": true}, {\"check\": \"control: random cascade 2\", \"actual\": [\"3\", \"0\"], \"expected\": [\"3\", \"0\"], \"passed\": true}, {\"check\": \"control: random cascade 4\", \"actual\": [\"-2\", \"2\", \"2\", \"-2\", \"-2\"], \"expected\": [\"-2\", \"2\", \"2\", \"-2\", \"-2\"], \"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."}}