{ "capability_id": "math.numerical.ode", "tool_id": "ordinary-differential-equation", "engine": "ordinary-differential-equation", "calculation_version": "1.1.1", "generated_at": "2026-10-23T01:27:11.653Z", "engine_build": "2026-10-18", "test_count": 9, "passed": 9, "failed": 0, "tests_summary": { "golden": { "passed": 3, "total": 3 }, "boundary": { "passed": 2, "total": 2 }, "property": { "passed": 4, "total": 4 } }, "engine_tested": true, "expert_reviewed": false, "internally_reviewed": false, "review_kind": null, "source_checked": true, "reviewed_by": null, "reviewer_role": null, "last_reviewed": null, "expert_review": { "status": "optional_not_performed", "reviewer": null, "scope": [ "formula", "input semantics", "assumptions", "limitations", "sources", "tests" ], "note": "Domain expert review is optional under CVP (CVP-EXP-01). Absence does not block CVP Verified. See Review policy for supplemental domain assurance." }, "internal_review": { "status": "not_applicable", "reviewer": null }, "page": "https://www.calculatorx.com/calc/math/ordinary-differential-equation", "api": "https://www.calculatorx.com/api/v1/calc/ordinary-differential-equation", "schema_input": "https://www.calculatorx.com/schemas/math.numerical.ode.input.json", "schema_output": "https://www.calculatorx.com/schemas/math.numerical.ode.output.json", "assumptions": [ "The Result Card includes an engine-linked Live graph of numerical y(t) using the same Overview / Focus template as Log (Overview window t0…t1; current point is published y(t1); same Euler/RK4 stepper as REST/MCP). Exact y(t1) is a reference when it differs. Not rk45, not systems, not PDE, not CAS.", "Scalar first-order only", "Polynomial quadrature and baseline use exact dyadic intermediates over binary64 inputs and actual representable time nodes; output remains binary64, not symbolic.", "y, exact and signed difference publish12 significant digits; model inputs and h retain binary64 precision. The graph is sampled to at most257 points, not a complete trajectory export.", "Polynomial degree ≤ 8" ], "limitations": [ "Not rk45 / adaptive / Dormand–Prince", "Not systems of ODEs or second-order IVP", "Not PDE / CAS / symbolic solve", "n ≤ 10000", "t1 = t0 → INVALID_INPUT", "n must be an integer; fractional steps and unknown inputs are rejected", "abs_err is a legacy signed difference before publication rounding; diagnostics names signed/absolute/relative error explicitly", "Fixed-step method completion is not accuracy acceptance; inspect diagnostics and warnings", "Finite decimal text only; aliases must agree; explicit empty modes/fields do not default.", "Polynomial intermediates and quadrature sums use exact binary64-input dyadic arithmetic; final results are binary64 projections.", "Raw input/model fields and h retain binary64 precision; y, exact and error publish12 significant digits. Live graph uses at most257 samples and is not a complete trajectory export." ], "boundary_conditions": [ "Use finite decimal numbers/text; explicit empty fields and modes do not default, aliases must agree, nonzero decimal underflow is rejected.", "n must be an integer in 1–10000; fractional steps and unrelated input fields are rejected.", "diagnostics reports signed/absolute/relative difference from the closed-form reference and step stability; accuracy_status=not_assessed is not a tolerance pass.", "Unrepresentable time grid, zero/nonfinite h, nonfinite step/reference/error or nonzero analytic underflow → NUMERICAL_RANGE_ERROR", "t1 = t0 → INVALID_INPUT", "n < 1 or n > 10000 → INVALID_INPUT", "unknown method (including rk45) → INVALID_MODE" ], "sources": [ { "title": "ISO 80000-2:2019 Quantities and units — Mathematics", "url": "https://www.iso.org/standard/64973.html", "supports": "Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.", "kind": "standard", "publisher": "International Organization for Standardization", "reference_scope": "all" }, { "title": "CRC Standard Mathematical Tables — Euler and classical RK4", "supports": "Forward Euler y ← y+h f; classical RK4 four-stage step; exact y0 e^{kΔt} for y′=ky", "kind": "reference", "reference_scope": "model", "oracle_class": "O1" }, { "title": "Gilbert Strang, Calculus — 16.2 Differential Equations", "section": "Section 16.2, Numerical Methods (PDF page 13)", "url": "https://ocw.mit.edu/ans7870/textbooks/Strang/Edited/Calculus/16.1-16.2.pdf", "supports": "Forward Euler and classical four-stage fourth-order Runge–Kutta for first-order initial-value problems, together with step-size accuracy and stability discussion. Mathematical reference only; not validation or endorsement of this implementation.", "kind": "reference", "publisher": "Massachusetts Institute of Technology, OpenCourseWare" } ], "methods": [], "fingerprint": "sha256:561c36f3d0381a72bbd6f3e64fcc2ed76606765e189a55b5a1329fa99a328f31", "tier": "engine_tested", "verification": { "tests": { "passed": 9, "total": 9, "golden": 3, "boundary": 2, "property": 4 }, "contract_checks": { "passed": 5, "total": 5 } }, "evidence_core_generated_at": "2026-10-18T12:51:28.122Z", "rendered_at": "2026-10-23T01:27:11.653Z", "release_integrity": { "status": "verified", "scope": "workspace", "production_attestation_required": true, "calculation_version": "1.1.1", "artifacts": { "engine": "1.1.1", "page": "1.1.1", "capability": "1.1.1", "evidence": "1.1.1", "input_schema": "1.1.1", "output_schema": "1.1.1", "mcp_contract": "1.1.1" }, "digests": { "input_schema_sha256": "76f7d15fb6093a1cfca27199d6571322d789a599ac7012371c89573ef2159f22", "output_schema_sha256": "89d59e3d5b954d05402cc590fd56c6aedbe6f972553fd3a9441d862632f62cf1", "evidence_sha256": "77444970d9ea60b81b72346b7ea9ca5fa1e8f0ecd0c9f2c779eb37c0f6698dfd", "evidence_core_sha256": "77444970d9ea60b81b72346b7ea9ca5fa1e8f0ecd0c9f2c779eb37c0f6698dfd", "capability_sha256": "52de705c84a87e2e892671d3611d4a61f3657e78838ba88f02c284dcd2d36e1a", "engine_sha256": "a664ae25cd2134de7de43a99352933043215473589aec38c9416508150cb1008", "mcp_contract_sha256": "8defa770ea0b34118e3444b698842353896edfffa88b4ef1c34898f3f5378790" }, "contract_checks": { "total": 5, "passed": 5, "failed": 0, "items": [ { "id": "release-artifacts-aligned", "status": "pass", "detail": "1.1.1" }, { "id": "schema-output-required-declared", "status": "pass" }, { "id": "schema-valid-input-executes", "status": "pass" }, { "id": "engine-success-output-validates-schema", "status": "pass" }, { "id": "rest-success-payload-validates-schema", "status": "pass" } ] }, "report": "https://www.calculatorx.com/.well-known/calculatorx-release-integrity.json" }, "tests": [], "disclosure": { "policy": "calculatorx-public-evidence-v1", "full_validation": "private", "representation": "summary" } }