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math.linear_algebra

Machine-checked verification for calculation version 1.10.0 · tier engine_tested

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Verification

44/44 calculation tests · Production surface contract 3/3

  • Golden tests 23 / 23
  • Boundary tests 2 / 2
  • Property tests 19 / 19
  • Production surface contract 3 / 3
    • }
  • Engine tested Yes
  • Source checked Yes
  • Artifact integrity PASS
  • Production attestation Embedded snapshot: STALE · Last attested schema matched 1.10.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
  • Semantic contract PASS
  • Evidence attested at 2026-09-18T21:41:55.460Z
  • Last production check 2026-09-18T21:41:55.460Z
  • Public / cache PASS
  • Origin PASS
  • Supplemental domain review Not performed
  • Named expert review Not performed
  • release artifacts aligned PASS
  • schema valid input executes PASS
  • engine success output validates schema PASS

Fingerprint sha256:be9faeb0dd90fd44f168e0b886ef014f8f10a8beec55387af17b3c692592b29c

Input schema sha256:ed209544d5848c991c7b1a8438f084152cac2c05d88b6ed5b1144fe148ae2746 · Output schema sha256:b23ad2c9b4032d97dc75b76daa7987cca6d6e247dc4f600aaf797da8ebd6fa56 · 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
2026-09-19
Evidence core
2026-09-19T13:08:30.598Z
Rendered
2026-09-19T13:13:12.836Z
Last reviewed
—

Assumptions

  • Real matrices
  • Order ≤ 4
  • Not a CAS

Limitations

  • Order ≤ 4 (local small-matrix seed)
  • Singular inverse → SINGULAR_MATRIX
  • Not rectangular / 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.

Published checks

Golden, boundary, and property cases included in this evidence build.

IDKindStatusDetail
golden-2x2goldenpassExpected 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-detgoldenpassExpected 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-singulargoldenpassExpected 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-3goldenpassExpected 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-diaggoldenpassExpected {"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-lugoldenpassExpected 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-qrgoldenpassExpected 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-svd-diaggoldenpassExpected 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-symgoldenpassExpected 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-uniquegoldenpassExpected x=2 · y=1 · residual_2=0 · Actual mode=lstsq · order=2 · x=2 · y=1 · residual_2=0 · rank_numeric=2 · method=qr · status=unique
golden-lstsq-rank1goldenpassExpected x=0.04 · y=0.08 · rank_numeric=1 · Actual mode=lstsq · order=2 · x=0.04 · y=0.08 · residual_2=0.894427191 · rank_numeric=1 · method=svd_pinv · status=min_norm
golden-rref-rank1goldenpassExpected 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-fullgoldenpassExpected 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-diaggoldenpassExpected 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-singulargoldenpassExpected 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-rank1goldenpassExpected 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-fullgoldenpassExpected 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-rank1goldenpassExpected 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-fullgoldenpassExpected 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-rank1goldenpassExpected 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-fullgoldenpassExpected 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-rank1goldenpassExpected 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-fullgoldenpassExpected 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-singularboundarypassExpected SINGULAR_MATRIX · Actual SINGULAR_MATRIX
boundary-jordan-formboundarypassExpected INVALID_MODE · Actual INVALID_MODE
property-la-solvepropertypass2x+y=5
property-la-detpropertypassdet
property-la-invpropertypassdiag inverse
property-la-rankpropertypassrank-1 singular
property-la-tpropertypasstranspose
property-la-qrpropertypassQR residual
property-la-lupropertypassLU residual
property-la-eigenpropertypassdiag eigen
property-la-svdpropertypassdiag SVD
property-la-svd-sympropertypasssym SVD residual
property-la-lstsqpropertypasslstsq QR unique
property-la-lstsq-pinvpropertypasslstsq SVD min-norm
property-la-rrefpropertypassrank-1 RREF
property-la-conditionpropertypassκ₁ diag 2,1
property-la-nullspacepropertypassrank-1 kernel
property-la-colspacepropertypassrank-1 image
property-la-rowspacepropertypassrank-1 rowspace
property-la-leftnullspacepropertypassrank-1 left kernel ≠ right kernel
property-la-id3propertypassI₃ solve