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See Review policy for supplemental domain assurance." }, "internal_review": { "status": "not_applicable", "reviewer": null }, "page": "https://www.calculatorx.com/calc/math/linear-system", "api": "https://www.calculatorx.com/api/v1/calc/linear-system", "schema_input": "https://www.calculatorx.com/schemas/math.linear_algebra.input.json", "schema_output": "https://www.calculatorx.com/schemas/math.linear_algebra.output.json", "assumptions": [ "Real matrices", "Order ≤ 4 square for solve/det/inverse/factorizations", "lstsq also A ∈ R^{m×n} with m,n ∈ [2,4]", "Planar ℝ² figure when 2×2; other shapes omit it", "Not a CAS" ], "limitations": [ "Order ≤ 4 (local small-matrix seed)", "Singular inverse → SINGULAR_MATRIX", "lstsq accepts rectangular m×n with m,n ∈ [2,4]; other modes stay square", "Cholesky is SPD only (NOT_SYMMETRIC / NOT_POSITIVE_DEFINITE)", "Not Statistics regression / large-matrix server compute", "Not 2-norm κ₂ as a product / CAS / Jordan form" ], "boundary_conditions": [ 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"basis": [] }, "expectWarning": null, "actual": { "mode": "leftnullspace", "order": 2, "A": [ [ 1, 0 ], [ 0, 1 ] ], "basis": [], "dimension": 0, "rank": 2, "trivial": true, "residual_max": 0, "free_columns": [], "pivot_columns": [ 1, 2 ], "rref": [ [ 1, 0 ], [ 0, 1 ] ], "det": 1, "formula": "dim ker Aᵀ = n − rank A", "algorithm": "gauss_jordan_rref + transpose + free_variables", "numeric_backend": "float64", "singularity_threshold": 1e-15, "condition_number": 1, "warnings": [] }, "tol": null }, { "id": "boundary-singular", "ok": true, "kind": "boundary", "inputs": { "mode": "inverse", "order": 2, "a11": 1, "a12": 2, "a21": 2, "a22": 4 }, "expectError": "SINGULAR_MATRIX", "code": "SINGULAR_MATRIX", "error": "Matrix is singular; inverse does not exist." }, { "id": "boundary-jordan-form", "ok": true, "kind": "boundary", "inputs": { "mode": "jordan-form", "a11": 1, "a12": 0, "a21": 0, "a22": 1 }, "expectError": "INVALID_MODE", "code": "INVALID_MODE", "error": "mode must be solve, det, inverse, rank, multiply, transpose, eigen, lu, chol, qr, svd, lstsq, rref, condition, nullspace, columnspace, rowspace, or leftnullspace." }, { "id": "boundary-lstsq-rows", "ok": true, "kind": "boundary", "inputs": { "mode": "lstsq", "rows": 5, "cols": 2, "a11": 1, "a12": 0, "a21": 0, "a22": 1, "b1": 1, "b2": 0 }, "expectError": "INVALID_INPUT", "code": "INVALID_INPUT", "error": "rows must be an integer 2–4." }, { "id": "boundary-rect-solve", "ok": true, "kind": "boundary", "inputs": { "mode": "solve", "rows": 3, "cols": 2, "a11": 1, "a12": 1, "a21": 1, "a22": 2, "a31": 1, "a32": 3, "b1": 1, "b2": 2, "b3": 2 }, "expectError": "INVALID_INPUT", "code": "INVALID_INPUT", "error": "solve/det/inverse/factorizations need a square matrix. Rectangular A is lstsq-only." }, { "id": "boundary-chol-nonsym", "ok": true, "kind": "boundary", "inputs": { "mode": "chol", "order": 2, "a11": 1, "a12": 2, "a21": 0, "a22": 1 }, "expectError": "NOT_SYMMETRIC", "code": "NOT_SYMMETRIC", "error": "Cholesky needs a symmetric matrix. Asymmetric A is NOT_SYMMETRIC." }, { "id": "boundary-chol-indef", "ok": true, "kind": "boundary", "inputs": { "mode": "chol", "order": 2, "a11": 1, "a12": 0, "a21": 0, "a22": -1 }, "expectError": "NOT_POSITIVE_DEFINITE", "code": "NOT_POSITIVE_DEFINITE", "error": "Cholesky needs a positive-definite matrix. Nonpositive pivot is NOT_POSITIVE_DEFINITE." }, { "id": "property-la-solve", "ok": true, "kind": "property", "detail": "2x+y=5", "expect": 2, "actual": 2 }, { "id": "property-la-det", "ok": true, "kind": "property", "detail": "det", "expect": -2, "actual": -2 }, { "id": "property-la-inv", "ok": true, "kind": "property", "detail": "diag inverse", "expect": 0.5, "actual": 0.5 }, { "id": "property-la-rank", "ok": true, "kind": "property", "detail": "rank-1 singular", "expect": 1, "actual": 1 }, { "id": "property-la-t", "ok": true, "kind": "property", "detail": "transpose", "expect": 3, "actual": 3 }, { "id": "property-la-qr", "ok": true, "kind": "property", "detail": "QR residual", "expect": 0, "actual": 0 }, { "id": "property-la-lu", "ok": true, "kind": "property", "detail": "LU residual", "expect": 0, "actual": 0 }, { "id": "property-la-chol", "ok": true, "kind": "property", "detail": "Cholesky LLᵀ", "expect": 2, "actual": 2 }, { "id": "property-la-eigen", "ok": true, "kind": "property", "detail": "diag eigen", "expect": 2, "actual": 2 }, { "id": "property-la-svd", "ok": true, "kind": "property", "detail": "diag SVD", "expect": 2, "actual": 2 }, { "id": "property-la-svd-sym", "ok": true, "kind": "property", "detail": "sym SVD residual", "expect": 0, "actual": 0 }, { "id": "property-la-lstsq", "ok": true, "kind": "property", "detail": "lstsq QR unique", "expect": 2, "actual": 2 }, { "id": "property-la-lstsq-pinv", "ok": true, "kind": "property", "detail": "lstsq SVD min-norm", "expect": 0.04, "actual": 0.04 }, { "id": "property-la-lstsq-rect", "ok": true, "kind": "property", "detail": "rectangular 3×2 QR", "expect": 0.6666666666666666, "actual": 0.666666666667 }, { "id": "property-la-rref", "ok": true, "kind": "property", "detail": "rank-1 RREF", "expect": 1, "actual": 1 }, { "id": "property-la-condition", "ok": true, "kind": "property", "detail": "κ₁ diag 2,1", "expect": 2, "actual": 2 }, { "id": "property-la-nullspace", "ok": true, "kind": "property", "detail": "rank-1 kernel", "expect": 1, "actual": 1 }, { "id": "property-la-colspace", "ok": true, "kind": "property", "detail": "rank-1 image", "expect": 1, "actual": 1 }, { "id": "property-la-rowspace", "ok": true, "kind": "property", "detail": "rank-1 rowspace", "expect": 1, "actual": 1 }, { "id": "property-la-leftnullspace", "ok": true, "kind": "property", "detail": "rank-1 left kernel ≠ right kernel", "expect": 1, "actual": 1 }, { "id": "property-la-id3", "ok": true, "kind": "property", "detail": "I₃ solve", "expect": 9, "actual": 9 } ], "tier": "engine_tested", "verification": { "tests": { "passed": 54, "total": 54, "golden": 27, "boundary": 6, "property": 21 }, "contract_checks": { "passed": 5, "total": 5 } }, "evidence_core_generated_at": "2026-09-22T07:23:03.591Z", "rendered_at": "2026-09-22T07:25:48.832Z", "release_integrity": { "status": "verified", "scope": "workspace", "production_attestation_required": true, "calculation_version": "1.12.0", "artifacts": { "engine": "1.12.0", "page": "1.12.0", "capability": "1.12.0", "evidence": "1.12.0", "input_schema": "1.12.0", "output_schema": "1.12.0", "mcp_contract": "1.12.0" }, "digests": { "input_schema_sha256": "e4850ca416c7f13aaf6cee9812ad0faae69e431bab6d712a3f7614e4738613ba", "output_schema_sha256": "6e6216e8a012c7844acf14941b4b7dc9ceef5296ad5e2274ecdca757cad16f8b", "evidence_sha256": "2f1cbade00bfc05f771d6fc98db7456dd09e0d16fedf03f0979b5c79d5ae55b9", "evidence_core_sha256": "2f1cbade00bfc05f771d6fc98db7456dd09e0d16fedf03f0979b5c79d5ae55b9", "capability_sha256": "e6d0b05d448a7051ae73ea542e733ff7867f7d6bca34ee0713b44eac25265ffb", "engine_sha256": "925bd3b9c373f747a10f30cedc6352b3f419cf91144ae6744c9c9adcd703033c", "mcp_contract_sha256": "b2189f75ddce2d16459887371b0aa48a211716dcc26f04f13868a4277e0aa56b" }, "contract_checks": { "total": 5, "passed": 5, "failed": 0, "items": [ { "id": "release-artifacts-aligned", "status": "pass", "detail": "1.12.0" }, { "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" } }