CVP verified calculation capability

Linear system / small matrix Calculation Capability

Solve Ax=b for order 2–4, plus det, inverse, rank, LU, Cholesky (SPD A=LLᵀ), QR and SVD (square or rectangular m×n with m,n∈[2,4]), least squares (same rectangular shape), RREF, condition number, nullspace, column space, rowspace, left-nullspace, eigen, multiply, transpose. Same engine as /workspace/matrix. Discovery pages share this capability. Not a CAS. Not Statistics regression.

Math Capability ID math.linear_algebra version 1.14.16 CVP VERIFIED CURRENT Core

Model

2×2 determinant

2×2 determinant: det = a₁₁a₂₂ − a₁₂a₂₁
Solve: PA = LU, then Ly = Pb, Ux = y

Modes: solve · system · axb · linear-system · det · determinant · inverse · inv · multiply · mul · product · transpose · t · rank · matrix-rank · eigen · eig · eigenvalue · eigenvalues · eigenvector · eigenvectors · lu · plu · factor-lu · chol · cholesky · factor-chol · factor-cholesky · llt · llT · qr · householder · factor-qr · svd · singular · singular value · singular values · lstsq · least squares · least-squares · leastsquare · ols · rref · reduced-row-echelon · reduced row echelon · row-reduced · gauss-jordan · condition · cond · kappa · condition number · condition-number · matrix-condition · nullspace · null-space · kernel · ker · null · matrix-nullspace · nullity · columnspace · column-space · colspace · image · im · range · matrix-column-space · col-space · rowspace · row-space · matrix-row-space · row space · leftnullspace · left-nullspace · left-null-space · left nullspace · cokernel · left-kernel · matrix-left-nullspace

Inputs / Outputs

Canonical symbols for this capability. Agent schemas remain the contract.

Symbol	Quantity	Unit	Role
`order`	—	—	input / output
`rows`	—	—	input / output
`cols`	—	—	input / output
`a11`	Finite binary64 decimal	—	input
`a12`	Finite binary64 decimal	—	input
`a13`	Finite binary64 decimal	—	input
`a21`	Finite binary64 decimal	—	input
`a22`	Finite binary64 decimal	—	input
`a23`	Finite binary64 decimal	—	input
`a31`	Finite binary64 decimal	—	input
`a32`	Finite binary64 decimal	—	input
`a33`	Finite binary64 decimal	—	input
`b1`	Finite binary64 decimal	—	input
`b2`	Finite binary64 decimal	—	input
`b3`	Finite binary64 decimal	—	input
`A`	—	—	input / output
`B`	—	—	input / output
`b`	—	—	input
`rhs`	—	—	input / output
`n`	—	—	input
`m`	—	—	input
`b11`	Finite binary64 decimal	—	input
`b12`	Finite binary64 decimal	—	input
`b13`	Finite binary64 decimal	—	input
`a14`	Finite binary64 decimal	—	input
`b14`	Finite binary64 decimal	—	input
`b21`	Finite binary64 decimal	—	input
`b22`	Finite binary64 decimal	—	input
`b23`	Finite binary64 decimal	—	input
`a24`	Finite binary64 decimal	—	input
`b24`	Finite binary64 decimal	—	input
`b31`	Finite binary64 decimal	—	input
`b32`	Finite binary64 decimal	—	input
`b33`	Finite binary64 decimal	—	input
`a34`	Finite binary64 decimal	—	input
`b34`	Finite binary64 decimal	—	input
`b4`	Finite binary64 decimal	—	input
`a41`	Finite binary64 decimal	—	input
`b41`	Finite binary64 decimal	—	input
`a42`	Finite binary64 decimal	—	input
`b42`	Finite binary64 decimal	—	input
`a43`	Finite binary64 decimal	—	input
`b43`	Finite binary64 decimal	—	input
`a44`	Finite binary64 decimal	—	input
`b44`	Finite binary64 decimal	—	input
`x`	—	—	output
`y`	—	—	output
`z`	—	—	output
`residual_2`	—	—	output
`backward_error`	—	—	output
`rref`	—	—	output
`pivot_columns`	—	—	output
`det`	—	—	output
`rank`	—	—	output
`method`	—	—	output
`algorithm`	—	—	output
`numeric_backend`	—	—	output
`condition_number`	—	—	output
`basis`	—	—	output
`dimension`	—	—	output
`singular_values`	—	—	output
`rank_numeric`	—	—	output
`S`	—	—	output
`VT`	—	—	output
`eigenvalues`	—	—	output
`L`	—	—	input / output
`Q`	—	—	output
`inv`	—	—	output
`reconstruction_residual`	—	—	output
`orthogonality_residual`	—	—	output
`residual_max`	—	—	output
`singular`	—	—	output
`eigenvectors`	—	—	output
`U`	—	—	output
`R`	—	—	output
`AT`	—	—	output
`product`	—	—	output
`P`	—	—	output
`vector`	—	—	output
`p`	—	—	output
`char_poly`	—	—	output
`singularity_threshold`	—	—	output
`u_orthogonality_residual`	—	—	output
`v_orthogonality_residual`	—	—	output
`basis_residual`	—	—	output
`projection_residual`	—	—	output
`norm_1`	—	—	output
`inverse_norm_1`	—	—	output
`solver`	—	—	output
`norm`	—	—	output
`w`	—	—	output
`infinite`	—	—	output
`eigenvectors_found`	—	—	output
`trivial`	—	—	output
`free_columns`	—	—	output
`effective`	—	A	input / output
`symmetry_error`	—	—	output
`symmetry_tolerance`	—	—	output
`symmetry_policy`	—	—	output
`relative_reconstruction_residual`	—	—	output
`reconstruction_tolerance`	—	—	output

Trusted Math contract

Inspectable numeric contract for this Capability@Version. Not a rewrite of the engine.

Field	Declaration
`numeric_domain`	real
`exactness`	float64
`algorithm`	small-square seed: LU+partial-pivot solve (not explicit inverse), Gauss–Jordan inverse/RREF, Cholesky LLᵀ (SPD), Householder QR (square or rectangular), one-sided Jacobi SVD (square or rectangular; does not form AᵀA), Faddeev–LeVerrier eigen (P2 vs Hessenberg+QR), QR/SVD lstsq (rectangular m×n), 1-norm κ; scale-aware singularity ε·max(m,n)·‖A‖_max
`valid_domain`	real square order 2–4 for solve/det/inverse/LU/Cholesky; Cholesky requires SPD; qr, svd, and lstsq also A ∈ R^{m×n} with m,n ∈ [2,4]
`singularity_policy`	SINGULAR_MATRIX on inverse of a singular A; κ₁ → infinite (null) when A is singular; Cholesky NOT_SYMMETRIC / NOT_POSITIVE_DEFINITE; pivot/rank tests are scale-aware (ε·max(m,n)·‖A‖_max), not a fixed 1e-15 floor
`branch_policy`	not applicable — real matrices; eigen may report conjugate pairs
`tolerance`	note: scale-aware pivot / SVD rank vs ε·max(m,n)·‖A‖_max (or σ_max); LINEAR_ALGEBRA_SINGULARITY_THRESHOLD is the O(1) documented floor, not the runtime test
`convergence`	not applicable — dense ≤4×4 direct factorizations; Jacobi SVD is the bounded iterate
`conditioning`	κ₁ = ‖A‖₁‖A⁻¹‖₁; ill_conditioned warning when κ₁ ≥ engine threshold; unique solve publishes residual_2 = ‖Ax−b‖₂ and η = ‖Ax−b‖₂ / (‖A‖₁‖x‖₂ + ‖b‖₂) (denominator 0 → η = residual_2). REST exposes algorithm, numeric_backend, condition_number, warnings. Engine also returns singularity_threshold.
`rounding`	ieee754_round_ties_to_even
`overflow`	ieee754_nonfinite_or_coded_error
`reproducibility`	deterministic + Capability@Version
`independent_oracle`	cvp:math.linear_algebra O2 (linear-algebra-reference) / Evidence published cases
`golden_vectors`	Evidence golden + boundary + property (and published_reference_case when listed)

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

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

Example

One numerical check. The calculator page holds worked steps and FAQ.

2x+y=5, x+y=3 → x=2, y=1

Open this example in the calculator

Trust

CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes. Computational verification is defined by CVP. Supplemental domain review (internal) and named expert review are defined by the Review policy and do not by themselves determine CVP conformance.

CVP: CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Verified scope: all published modes
Assurance: Core
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)
Calculation version: 1.14.16
Verification: 58/58 calculation tests · 29/29 legacy golden · 6/6 legacy boundary · 23/23 legacy property
Production surface contract: 6/6
Versions: Calculation 1.14.16 · CVP protocol 1.0.0-proposed · Evidence 2026-10-30.cholesky-numeric-contract-v2
Verification revision: 2026-10-30.cholesky-numeric-contract-v2 · 10/10 property · digest 2f8af10fde34
Reference: O1 model · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP
ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 24/24 (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. Raw HTTP REST/JSON-RPC transport rejection 5/5; raw-number source and conflicting pins are transport-only checks, not engine/SSR input-source parity.
CVP suite: 478/478 golden · 8/8 CVP boundary · 78/78 invalid · 10/10 property · 2/2 metamorphic · 3/3 round-trip · 67/67 cross-interface · 2/2 cross-calculator · 6/6 CVP contract
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.14.16 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.14.16 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓

Call this capability

Human page, REST, and MCP share the same engine and calculation_version. Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Reference Calculation Standard — Production surface contract 6/6.

curl -sS -X POST 'https://www.calculatorx.com/api/v1/calc/linear-system' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"mode":"solve","order":2,"a11":2,"a12":1,"a21":1,"a22":1,"b1":5,"b2":3},"calculation_version":"1.14.16"}'

{
  "inputs": {
    "mode": "solve",
    "order": 2,
    "a11": 2,
    "a12": 1,
    "a21": 1,
    "a22": 1,
    "b1": 5,
    "b2": 3
  },
  "calculation_version": "1.14.16"
}

POST /api/v1/calc/linear-system. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

Tool id: linear-system
Engine: linear-algebra
REST: POST /api/v1/calc/linear-system
MCP: execute_calculation with capability_id: "math.linear_algebra"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.14.16)
Versioned output schema: Immutable pin (1.14.16)

Operations

Declared calculation modes for this capability.

chol

References

Human-readable sources plus CVP reference evidence classes.

Sources

Reference evidence classes: O1 model · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 numerical_behavior

Try with an agent

Inspect, then execute. Same engine as REST and the human calculator.

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: inspect_capability' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"math.linear_algebra"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: execute_calculation' \
  -d '{"jsonrpc":"2.0","id":2,"method":"tools/call","params":{"name":"execute_calculation","arguments":{"capability_id":"math.linear_algebra","inputs":{"mode":"solve","order":2,"a11":2,"a12":1,"a21":1,"a22":1,"b1":5,"b2":3},"calculation_version":"1.14.16"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.14.16) when the result must stay reproducible.

Inspect the trust tier and required assurance level before execution. Full protocol notes: MCP docs.