{"abstract":"The exact forward difference newton basis result violates the stated contract at left edge extraction.","category":"Numerics","checks":8,"contract":"Input nonempty integer sequence f(0)..f(n); return binomial-basis coefficients Delta^k f(0).","contract_signature":"x","evaluation_group":"s3-numerics-forward-difference-newton-basis","failed_approach":"The partial repair row[len(row)//2] still violates the left edge extraction invariant.","family":"s3-numerics-forward-difference-newton-basis-left-edge-extraction","id":"FA-14721","implementations":{"attempt":{"sha256":"2ed284d9b15461d25488ed192dd13a8268a521bd08115cc109e4cc3c0896cd9e","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport itertools\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    row=x[:];out=[]\n    for _ in range(len(x)):\n     if not row:break\n     out.append(row[len(row)//2])\n     row=[row[i+1]-row[i] for i in range(len(row)-1)]\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([-1, 0], [-1, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([0], [0]), ([1], [1]), ([-1, -1], [-1, 0]), ([-1, 1], [-1, 2]), ([0, -1], [0, -1])], [([-1, 1], [-1, 2]), ([0, 1], [0, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([-1, 0, 1], [-1, 1, 0]), ([-1, 1, -1], [-1, 2, -4]), ([-1, 1, 0], [-1, 2, -3]), ([-1, 1, 1], [-1, 2, -2])], [([0, -1], [0, -1]), ([-1, -1, 0], [-1, 0, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([1, 0, 0], [1, -1, 1]), ([1, 0, 1], [1, -1, 2]), ([1, 1, -1], [1, 0, -2]), ([1, 1, 0], [1, 0, -1])], [([0, 1], [0, 1]), ([-1, 0, 0], [-1, 1, -1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([-1, 0, 0, -1], [-1, 1, -1, 0]), ([-1, 0, 0, 0], [-1, 1, -1, 1]), ([-1, 0, 0, 1], [-1, 1, -1, 2]), ([-1, 0, 1, -1], [-1, 1, 0, -3])], [([1, -1], [1, -2]), ([-1, 1, 0], [-1, 2, -3]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([0, -1, -1, 1], [0, -1, 1, 1]), ([0, -1, 0, -1], [0, -1, 2, -4]), ([0, -1, 0, 0], [0, -1, 2, -3]), ([0, -1, 0, 1], [0, -1, 2, -2])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"explicit oracle %d\" % i, 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":"3474eb637f9f8b40aa977905c9a0de087afdccaa4aaaeab018f6d2fb05c56db8","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport itertools\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    row=x[:];out=[]\n    for _ in range(len(x)):\n     if not row:break\n     out.append(row[-1])\n     row=[row[i+1]-row[i] for i in range(len(row)-1)]\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([-1, 0], [-1, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([0], [0]), ([1], [1]), ([-1, -1], [-1, 0]), ([-1, 1], [-1, 2]), ([0, -1], [0, -1])], [([-1, 1], [-1, 2]), ([0, 1], [0, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([-1, 0, 1], [-1, 1, 0]), ([-1, 1, -1], [-1, 2, -4]), ([-1, 1, 0], [-1, 2, -3]), ([-1, 1, 1], [-1, 2, -2])], [([0, -1], [0, -1]), ([-1, -1, 0], [-1, 0, 1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([1, 0, 0], [1, -1, 1]), ([1, 0, 1], [1, -1, 2]), ([1, 1, -1], [1, 0, -2]), ([1, 1, 0], [1, 0, -1])], [([0, 1], [0, 1]), ([-1, 0, 0], [-1, 1, -1]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([-1, 0, 0, -1], [-1, 1, -1, 0]), ([-1, 0, 0, 0], [-1, 1, -1, 1]), ([-1, 0, 0, 1], [-1, 1, -1, 2]), ([-1, 0, 1, -1], [-1, 1, 0, -3])], [([1, -1], [1, -2]), ([-1, 1, 0], [-1, 2, -3]), ([-1], [-1]), ([1, 1, 1, 1, 1], [1, 0, 0, 0, 0]), ([0, -1, -1, 1], [0, -1, 1, 1]), ([0, -1, 0, -1], [0, -1, 2, -4]), ([0, -1, 0, 0], [0, -1, 2, -3]), ([0, -1, 0, 1], [0, -1, 2, -2])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"explicit oracle %d\" % i, 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. Inputs are restricted to the explicit contract; this is not a production algebra library. 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":"s3-numerics-forward-difference-newton-basis-left-edge-extraction","generated_at":"2026-09-29T14:39:19.781670+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Exact discrete arithmetic with observable algorithmic state; no floating point approximation is used.","root_cause":"The left edge extraction step uses row[-1] instead of row[0].","sha256":"f7c6c040f89edf17e3dbf5cf2f4a5582dc66d5520d08dc6882aa3fa3016a3521","title":"Forward difference newton basis: left edge extraction · 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":45.909,"exit_code":1,"observations":[{"actual":[0,1],"check":"explicit oracle 0","expected":[-1,1],"passed":false},{"actual":[-1],"check":"explicit oracle 1","expected":[-1],"passed":true},{"actual":[1,0,0,0,0],"check":"explicit oracle 2","expected":[1,0,0,0,0],"passed":true},{"actual":[0],"check":"explicit oracle 3","expected":[0],"passed":true},{"actual":[1],"check":"explicit oracle 4","expected":[1],"passed":true},{"actual":[-1,0],"check":"explicit oracle 5","expected":[-1,0],"passed":true},{"actual":[1,2],"check":"explicit oracle 6","expected":[-1,2],"passed":false},{"actual":[-1,-1],"check":"explicit oracle 7","expected":[0,-1],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [0, 1], \"expected\": [-1, 1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [-1], \"expected\": [-1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [1, 0, 0, 0, 0], \"expected\": [1, 0, 0, 0, 0], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [0], \"expected\": [0], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [1], \"expected\": [1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [-1, 0], \"expected\": [-1, 0], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [1, 2], \"expected\": [-1, 2], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [-1, -1], \"expected\": [0, -1], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.278,"exit_code":1,"observations":[{"actual":[0,1],"check":"explicit oracle 0","expected":[-1,1],"passed":false},{"actual":[-1],"check":"explicit oracle 1","expected":[-1],"passed":true},{"actual":[1,0,0,0,0],"check":"explicit oracle 2","expected":[1,0,0,0,0],"passed":true},{"actual":[0],"check":"explicit oracle 3","expected":[0],"passed":true},{"actual":[1],"check":"explicit oracle 4","expected":[1],"passed":true},{"actual":[-1,0],"check":"explicit oracle 5","expected":[-1,0],"passed":true},{"actual":[1,2],"check":"explicit oracle 6","expected":[-1,2],"passed":false},{"actual":[-1,-1],"check":"explicit oracle 7","expected":[0,-1],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [0, 1], \"expected\": [-1, 1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [-1], \"expected\": [-1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [1, 0, 0, 0, 0], \"expected\": [1, 0, 0, 0, 0], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [0], \"expected\": [0], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [1], \"expected\": [1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [-1, 0], \"expected\": [-1, 0], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [1, 2], \"expected\": [-1, 2], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [-1, -1], \"expected\": [0, -1], \"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."}}