Public evidence

math.linear_algebra

Machine-checked verification for calculation version 1.12.0 · tier engine_tested

Open calculator API Capability Download JSON CVP Manifest MCP Review policy

Historical evidence. This page is version 1.12.0 · engine_tested · 54/54 tests. It is not the current published claim. Current evidence is 1.14.0 (58/58 tests) · latest.

CVP status

Order 2–4 square for factorizations / solve. Cholesky is SPD only (A = LLᵀ). lstsq accepts rectangular A with rows,cols ∈ [2,4]. Discovery pages lock one mode. Not a CAS, not Statistics regression, not large-matrix server compute.

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 1.12.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.vector-la-cvp
Verification revision: 2026-09-22.vector-la-cvp · 5/5 property · digest 81ab2bb4ee85

Verification

Legacy regression suite: 54/54 calculation tests · Production surface contract 5/5

Three distinct suites: educational examples on the calculator page, this legacy regression count (54/54 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 · 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 2026-09-22.vector-la-cvp. Calculation version 1.12.0. CVP 1.0.0-proposed. 20/20 golden · 8/8 CVP boundary · 14/14 invalid · 5/5 property · 6/6 cross-interface · 2/2 cross-calculator · 5/5 CVP contract. digest 81ab2bb4ee85.

Golden (oracle-backed) 20 / 20
CVP boundary 8 / 8
Invalid input 14 / 14
Property 5 / 5
Cross-interface 6 / 6
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 1/1 (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:ed31b7d82f3338cc39164fca6ad7c137855bf27eb8a720232078bda719af1d97

Input schema sha256:e4850ca416c7f13aaf6cee9812ad0faae69e431bab6d712a3f7614e4738613ba · Output schema sha256:6e6216e8a012c7844acf14941b4b7dc9ceef5296ad5e2274ecdca757cad16f8b · 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: 2026-09-22
Evidence core: 2026-09-22T07:23:03.591Z
Rendered: 2026-09-22T07:25:48.832Z
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
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

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: Matrix, determinant, inverse, and rank notation for order 2–4
CRC / Strang — Ax=b, LU/QR/SVD, four fundamental subspaces
Supports: Gaussian / Cramer solve; det; Gauss–Jordan inverse; rank-nullity; Householder QR; singular values; RREF; κ₁(A)

Legacy published checks

Golden and boundary cases from the legacy Evidence JSON. CVP property, metamorphic, round-trip, O3, and interface statuses/checks are recorded in the Manifest — including SKIP when a suite has no vectors.

ID	Kind	Status	Detail
`golden-2x2`	golden	pass	Expected x=2 · y=1 · status=unique · Actual mode=solve · order=2 · x=2 · y=1 · status=unique · det=1 · formula=Ax = b · algorithm=inverse_multiply
`golden-det`	golden	pass	Expected det=-2 · Actual mode=det · order=2 · det=-2 · formula=det(A) · algorithm=gaussian_elimination · numeric_backend=float64 · singularity_threshold=1e-15 · condition_number=21
`golden-rank-singular`	golden	pass	Expected rank=1 · det=0 · Actual mode=rank · order=2 · rank=1 · det=0 · formula=rank via Gaussian elimination · algorithm=gaussian_elimination · numeric_backend=float64 · singularity_threshold=1e-15
`golden-identity-3`	golden	pass	Expected x=4 · y=5 · z=6 · Actual mode=solve · order=3 · x=4 · y=5 · z=6 · status=unique · det=1 · formula=Ax = b
`golden-eigen-diag`	golden	pass	Expected {"eigenvalues":[{"re":2,"im":0},{"re":3,"im":0}]} · Actual mode=eigen · order=2 · eigenvectors_found=2 · residual_max=0 · char_poly=x² − 5x + 6 · solver=quadratic · det=6 · formula=det(A − λI) = 0
`golden-lu`	golden	pass	Expected singular=false · Actual mode=lu · order=2 · singular=false · reconstruction_residual=0 · det=1 · formula=PA = LU · algorithm=lu_partial_pivoting · numeric_backend=float64
`golden-qr`	golden	pass	Expected reconstruction_residual=0 · Actual mode=qr · order=2 · reconstruction_residual=0 · orthogonality_residual=0 · formula=A = QR (Householder) · algorithm=householder_qr · numeric_backend=float64 · singularity_threshold=1e-15
`golden-chol`	golden	pass	Expected reconstruction_residual=0 · Actual mode=chol · order=2 · reconstruction_residual=0 · formula=A = LLᵀ · algorithm=cholesky · numeric_backend=float64 · singularity_threshold=1e-15 · condition_number=9
`golden-chol-3`	golden	pass	Expected reconstruction_residual=0 · Actual mode=chol · order=3 · reconstruction_residual=0 · formula=A = LLᵀ · algorithm=cholesky · numeric_backend=float64 · singularity_threshold=1e-15 · condition_number=10209.3611111
`golden-svd-diag`	golden	pass	Expected rank_numeric=2 · Actual mode=svd · order=2 · rank_numeric=2 · reconstruction_residual=0 · u_orthogonality_residual=0 · v_orthogonality_residual=0 · formula=A = U Σ Vᵀ · algorithm=ata_jacobi + Av/σ
`golden-svd-sym`	golden	pass	Expected reconstruction_residual=0 · Actual mode=svd · order=2 · rank_numeric=2 · reconstruction_residual=0 · u_orthogonality_residual=0 · v_orthogonality_residual=0 · formula=A = U Σ Vᵀ · algorithm=ata_jacobi + Av/σ
`golden-lstsq-unique`	golden	pass	Expected x=2 · y=1 · residual_2=0 · Actual mode=lstsq · order=2 · rows=2 · cols=2 · x=2 · y=1 · residual_2=0 · rank_numeric=2
`golden-lstsq-rank1`	golden	pass	Expected x=0.04 · y=0.08 · rank_numeric=1 · Actual mode=lstsq · order=2 · rows=2 · cols=2 · x=0.04 · y=0.08 · residual_2=0.894427191 · rank_numeric=1
`golden-lstsq-rect-over`	golden	pass	Expected x=0.6666666666666666 · y=0.5 · method=qr · status=unique · Actual mode=lstsq · order=2 · rows=3 · cols=2 · x=0.666666666667 · y=0.5 · residual_2=0.408248290464 · rank_numeric=2
`golden-lstsq-rect-under`	golden	pass	Expected x=3 · y=4 · z=0 · method=svd_pinv · status=min_norm · Actual mode=lstsq · order=3 · rows=2 · cols=3 · x=3 · y=4 · z=0 · residual_2=0
`golden-rref-rank1`	golden	pass	Expected rank=1 · Actual mode=rref · order=2 · rank=1 · det=0 · formula=Gauss–Jordan → RREF · algorithm=gauss_jordan_rref · numeric_backend=float64 · singularity_threshold=1e-15
`golden-rref-full`	golden	pass	Expected rank=2 · Actual mode=rref · order=2 · rank=2 · det=-2 · formula=Gauss–Jordan → RREF · algorithm=gauss_jordan_rref · numeric_backend=float64 · singularity_threshold=1e-15
`golden-condition-diag`	golden	pass	Expected condition_number=2 · infinite=false · status=finite · Actual mode=condition · order=2 · condition_number=2 · infinite=false · status=finite · norm=1 · det=2 · formula=κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁
`golden-condition-singular`	golden	pass	Expected infinite=true · status=singular · Actual mode=condition · order=2 · infinite=true · status=singular · norm=1 · det=0 · formula=κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁ · algorithm=one_norm_inverse
`golden-nullspace-rank1`	golden	pass	Expected dimension=1 · rank=1 · trivial=false · Actual mode=nullspace · order=2 · dimension=1 · rank=1 · trivial=false · residual_max=0 · det=0 · formula=dim ker A = n − rank A
`golden-nullspace-full`	golden	pass	Expected dimension=0 · trivial=true · Actual mode=nullspace · order=2 · dimension=0 · rank=2 · trivial=true · residual_max=0 · det=1 · formula=dim ker A = n − rank A
`golden-colspace-rank1`	golden	pass	Expected dimension=1 · rank=1 · trivial=false · Actual mode=columnspace · order=2 · dimension=1 · rank=1 · trivial=false · residual_max=0 · det=0 · formula=dim im A = rank A
`golden-colspace-full`	golden	pass	Expected dimension=2 · trivial=false · Actual mode=columnspace · order=2 · dimension=2 · rank=2 · trivial=false · residual_max=0 · det=1 · formula=dim im A = rank A
`golden-rowspace-rank1`	golden	pass	Expected dimension=1 · rank=1 · trivial=false · Actual mode=rowspace · order=2 · dimension=1 · rank=1 · trivial=false · residual_max=0 · det=0 · formula=dim row A = rank A
`golden-rowspace-full`	golden	pass	Expected dimension=2 · trivial=false · Actual mode=rowspace · order=2 · dimension=2 · rank=2 · trivial=false · residual_max=0 · det=1 · formula=dim row A = rank A
`golden-leftnullspace-rank1`	golden	pass	Expected dimension=1 · rank=1 · trivial=false · Actual mode=leftnullspace · order=2 · dimension=1 · rank=1 · trivial=false · residual_max=0 · det=0 · formula=dim ker Aᵀ = n − rank A
`golden-leftnullspace-full`	golden	pass	Expected dimension=0 · trivial=true · Actual mode=leftnullspace · order=2 · dimension=0 · rank=2 · trivial=true · residual_max=0 · det=1 · formula=dim ker Aᵀ = n − rank A
`boundary-singular`	boundary	pass	Expected SINGULAR_MATRIX · Actual SINGULAR_MATRIX
`boundary-jordan-form`	boundary	pass	Expected INVALID_MODE · Actual INVALID_MODE
`boundary-lstsq-rows`	boundary	pass	Expected INVALID_INPUT · Actual INVALID_INPUT
`boundary-rect-solve`	boundary	pass	Expected INVALID_INPUT · Actual INVALID_INPUT
`boundary-chol-nonsym`	boundary	pass	Expected NOT_SYMMETRIC · Actual NOT_SYMMETRIC
`boundary-chol-indef`	boundary	pass	Expected NOT_POSITIVE_DEFINITE · Actual NOT_POSITIVE_DEFINITE
`property-la-solve`	property	pass	2x+y=5
`property-la-det`	property	pass	det
`property-la-inv`	property	pass	diag inverse
`property-la-rank`	property	pass	rank-1 singular
`property-la-t`	property	pass	transpose
`property-la-qr`	property	pass	QR residual
`property-la-lu`	property	pass	LU residual
`property-la-chol`	property	pass	Cholesky LLᵀ
`property-la-eigen`	property	pass	diag eigen
`property-la-svd`	property	pass	diag SVD
`property-la-svd-sym`	property	pass	sym SVD residual
`property-la-lstsq`	property	pass	lstsq QR unique
`property-la-lstsq-pinv`	property	pass	lstsq SVD min-norm
`property-la-lstsq-rect`	property	pass	rectangular 3×2 QR
`property-la-rref`	property	pass	rank-1 RREF
`property-la-condition`	property	pass	κ₁ diag 2,1
`property-la-nullspace`	property	pass	rank-1 kernel
`property-la-colspace`	property	pass	rank-1 image
`property-la-rowspace`	property	pass	rank-1 rowspace
`property-la-leftnullspace`	property	pass	rank-1 left kernel ≠ right kernel
`property-la-id3`	property	pass	I₃ solve