Public evidence

math.linear_algebra

Machine-checked verification for calculation version 1.14.12 · tier engine_tested

Open calculator API Capability Download JSON CVP Manifest MCP Review policy

Historical evidence. This page is version 1.14.12 · engine_tested · 58/58 tests. It is not the current published claim. Current evidence is 1.14.15 (58/58 tests) · latest.

CVP status

Order 2–4 square for solve/det/inverse/LU/Cholesky. qr, svd, and lstsq accept rectangular A with rows,cols ∈ [2,4]. Cholesky is SPD only (A = LLᵀ). Discovery pages lock one mode. Not a CAS, not Statistics regression, not large-matrix server compute. Scale fixtures cover named operations only; analytic eigen fixtures span 1e-250 through 1e250, and factor diagnostics use rounded published entries. Published subspace diagnostics use scaled QR/SVD projection; exact triangular spectra retain close diagonal values with sensitivity warnings. General 2×2 spectra additionally use 50 exact-input Fraction/Decimal references with an 8-ULP component bound; Strictly symmetric 3–4 matrices use shared Jacobi with 18 dense high-precision references; Nonsymmetric 3–4 uses bounded 256-bit matrix QR; 36 dense actual-input references check clustered/non-normal and complex spectra. Published complex basis conditioning is checked. 46 analytic actual-input fixtures check subnormal 2×2 components and weighted 3–4 cycles. Single directed cycles use exact dyadic edge products and analytic real/complex classification, bypassing the general QR floors only on proven structure. Exact zero-separated 1–2 blocks retain tiny complex spectra; exact nilpotence certifies a sole repeated eigenvalue. Exact square-free characteristic factors of degree ≤ 2 with representable normalized coefficients certify repeated spectra; 42 exact coupled/semisimple real/complex cases and 10 independently computed actual perturbations supplement coverage. Published unresolved basis candidates are omitted. QR rejects nonzero input components at/below its deflation floor. Exact dyadic Sturm real-root counts and exact published binary64 conjugacy are mandatory before and after restoring output scale; mismatches refuse success with NUMERIC_PRECISION_LIMIT. General QR simple-root positions additionally require exact rational disjoint Rouche disks, bounded by max(root-wise 2^-45 component magnitude, 2 component ULPs), separation/4 and |imaginary part|/2. This certifies given-input root positions, not stability under perturbations, correctly rounded components, or all structured solver paths. Three prior approximate golden successes are now explicit refusals because their classification was wrong. Products below fixed 2^-256 precision, features near the 2^-200 deflation floor, broader defective matrices and complete public surface coverage still require review.

Declared partition coverage: PASS · 19/19 declared partitions (solve, det, inverse, multiply, transpose, rank, eigen, lu, chol, qr, svd, lstsq, rref, condition, nullspace, columnspace, rowspace, leftnullspace, invalid-domain)
Versions: Calculation 1.14.12 · CVP protocol 1.0.0-proposed · Evidence 2026-10-09.qr-root-position-disks
Verification revision: 2026-10-09.qr-root-position-disks · 10/10 property · digest 0214764e3fff

Verification

Legacy regression suite: 58/58 calculation tests · Production surface contract 5/5

Three distinct suites: educational examples on the calculator page, this legacy regression count (58/58 golden + boundary), and the CVP verification suite below. They are not the same number. Property, metamorphic, round-trip, O3, and interface statuses/checks are recorded in the CVP Manifest . SKIP (0/0) means that suite has no published vectors.

STALE · Build schema 1.14.15 matches page · production binding STALE · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓

CVP verification suite

Protocol checks from the Evidence Manifest. Distinct from the legacy regression suite above. Verification revision 2026-10-09.qr-root-position-disks. Calculation version 1.14.12. CVP 1.0.0-proposed. 431/431 golden · 8/8 CVP boundary · 68/68 invalid · 10/10 property · 1/1 metamorphic · 48/48 cross-interface · 2/2 cross-calculator · 5/5 CVP contract. digest 0214764e3fff.

Golden (oracle-backed) 431 / 431
CVP boundary 8 / 8
Invalid input 68 / 68
Property 10 / 10
Metamorphic 1 / 1
Cross-interface 48 / 48
Cross-calculator 2 / 2
CVP contract 5 / 5
Interfaces PASS · UI (SSR) / REST / MCP
Interface claim ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 10/10 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Verified scope all published modes
Assurance Core

release artifacts aligned PASS
schema output required declared PASS
schema valid input executes PASS
engine success output validates schema PASS
rest success payload validates schema PASS

Fingerprint sha256:5a8cc900e1d6ad9ee887c2303f1783d9673bad715a123b0993eb47ccbaa1d71d

Input schema sha256:e8b4a633b22958a4c7ac2a3ce0a1e3bc5c1e48b2523e127fcf6933cb3adf5b10 · Output schema sha256:d6f6aac76f8a733d543dc39a3a76af35781c1eac035f5c9420f352a4fb980e15 · Build-time SHA-256 digests prove published artifacts agree with each other (release_integrity.scope=workspace). Production attestation GETs canonical URLs twice (public/cached view and origin), including Capability HTML, via npm run attest:production --write (published at /.well-known/calculatorx-production-attestation.json) and fails if the public Capability page advertises a different calculation_version. This is artifact consistency, not an independent immutable release log.

Build & scope

Engine identity and declared limitations.

Tool: linear-system
Engine: linear-algebra
Engine build: 2026-10-14
Evidence core: 2026-10-14T04:17:16.233Z
Rendered: 2026-10-16T12:29:55.878Z
Last reviewed: —

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
qr, svd, and lstsq accept rectangular m×n with m,n ∈ [2,4]; solve / det / inverse / LU / Cholesky 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

Sources

Standards and references supporting this tool specification. Methods are listed separately — they are algorithms, not bibliographic sources.

ISO 80000-2:2019 Quantities and units — Mathematics
Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
CRC / Strang — Ax=b, LU/QR/SVD, four fundamental subspaces
Supports: LU + partial pivoting solve; det; Gauss–Jordan inverse; rank-nullity; Householder QR; one-sided Jacobi SVD; RREF; κ₁(A)
NIST DLMF §3.2 — Linear Algebra — §3.2
Supports: Gaussian elimination, LU decomposition, conditioning, eigenvalues, and eigenvectors.
MIT OpenCourseWare — The Four Fundamental Subspaces
Supports: Column space, row space, nullspace, left nullspace, and their dimensions.
MIT OpenCourseWare — Orthogonal Matrices and Gram-Schmidt
Supports: Orthogonal matrices and QR factorization; a mathematical reference, not a claim that this implementation uses Gram-Schmidt.
MIT OpenCourseWare — Singular Value Decomposition
Supports: Singular-value decomposition and its relationship to orthogonal bases and matrix rank.
MIT OpenCourseWare — Projection Matrices and Least Squares
Supports: Least-squares minimization, projections, and normal-equation relationships.
NIST DLMF §1.2 — Elementary Algebra — Determinants and matrices; systems of linear equations
Supports: Determinants, inverses, matrix identities, and linear-system relationships.
NumPy — numpy.linalg.cholesky
Supports: Cholesky factorization A = LLᴴ (or UᴴU) for Hermitian positive-definite matrices, with the real symmetric case A = LLᵀ. This states the factorization and domain, not CalculatorX implementation details. Mathematical reference only; not validation or endorsement of this implementation.

Verification disclosure

Aggregate outcomes, scope and artifact commitments are public. Complete test inputs, expected values, execution records and oracle infrastructure are retained privately. This summary does not independently reproduce the full verification run.