CVP verified calculation capability

Polynomial roots Calculation Capability

All roots of a real polynomial of degree 1–8, or Horner evaluation at a real x. Closed form for linear, quadratic, and cubic (Cardano); Durand–Kerner otherwise. Cubic discovery /cubic-equation. Evaluate discovery /polynomial-evaluate. Not the quadratic formula page, not Ferrari, not a single Newton root, not CAS.

Math · Polynomial Capability ID math.polynomial.roots version 1.2.2 CVP VERIFIED CURRENT Core

Model

All n roots

All n roots: c0 + c1 x + … + cn x^n = 0
Linear: x = −c0 / c1
Quadratic: x = (−b ± √(b² − 4ac)) / (2a)
Cubic: Cardano / trigonometric closed form
Degree 4–8: Durand–Kerner simultaneous iteration

Inputs / Outputs

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

Symbol	Quantity	Unit	Role
`coeffs`	—	—	input
`coefficients`	—	—	input
`a`	—	—	input
`b`	—	—	input
`c`	—	—	input
`d`	—	—	input
`x`	—	—	input / output
`method`	all_roots/all/roots/durand_kerner/durand/weierstrass/dk;cardano/cubic/cubic_equation/cubic_formula;evaluate/eval/horner/polynomial_evaluate/polynomial_evaluation	—	input
`review`	—	—	output
`api_version`	—	—	output
`api_contract_version`	—	—	output
`spec_version`	—	—	output
`page`	Canonical calculator page URL	—	output
`links`	—	—	output
`degree`	—	—	output
`n`	—	—	output
`solver`	—	—	output
`roots`	—	—	output
`residual_max`	—	—	output
`cubic_case`	—	—	output
`delta`	—	—	output
`value`	—	—	output
`n_mul`	—	—	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`	Exact rational square-free GCD multiplicities; stable quadratic / Cardano, bounded scaled Durand–Kerner; exact rational Horner projected once.
`valid_domain`	Finite real constant-first binary64 coefficients, degree1–8, nonzero leading term. Strict decimal/scientific tokens≤512chars; total text≤4608; exact arithmetic≤16384bits. Raw nonzero→zero rejects NUMERIC_UNDERFLOW.
`singularity_policy`	Unknown/conflicting aliases reject; root/value range errors explicit. Unresolved root disks or real isolation reject NON_CONVERGENCE. No universal finite-input success guarantee.
`branch_policy`	not applicable — real iterate
`tolerance`	note: Exact Weierstrass/Gershgorin disjoint root disks; relative L1 inclusion budget2e-10, normalized L1 residual≤2e-12; independent fixed-vector roots4e-10 relative L2 plus2 subnormal quanta; Horner1ULP.
`convergence`	max_iter: 2×2048 DK; Sturm512levels/4096steps; criterion: Complete-set inclusion certificate plus residual; small residual alone is insufficient
`conditioning`	Clusters are ill-conditioned under coefficient perturbations; multiplicities refer to the exact accepted binary64 polynomial. Nullable absolute residual/delta indicate projection range.
`rounding`	ieee754_round_ties_to_even
`overflow`	ieee754_nonfinite_or_coded_error
`reproducibility`	deterministic algorithm + Capability@Version
`independent_oracle`	CPython Fraction factor construction and Decimal250/750;67 complete packs and independent diagnostic recomputation
`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

The Result Card includes an engine-linked figure of p (markers are published real roots on y=0; complex roots stay off the real plane). Same Horner nesting as REST/MCP. Curve samples are not a REST field.
Real coefficients
Leading coefficient nonzero
Degree ≤ 8
Not a second quadratic page
Not Ferrari quartic
Not CAS factoring

Example

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

coeffs=-6,11,-6,1 → 1, 2, 3

Open this example in the calculator

Trust

CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods. 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 methods· View Manifest · CVP overview · Specification
Verified scope: all published methods
Assurance: Core
Declared partition coverage: PASS · 4/4 declared partitions (all-roots, cardano, evaluate, invalid-domain)
Calculation version: 1.2.2
Verification: 21/21 calculation tests · 7/7 legacy golden · 6/6 legacy boundary · 8/8 legacy property
Production surface contract: 6/6
Versions: Calculation 1.2.2 · CVP protocol 1.0.0-proposed · Evidence 2026-10-28.metric-scope-v3
Verification revision: 2026-10-28.metric-scope-v3 · 3/3 property · digest eca488fa2ce7
Reference: O1 model · O2 expected_values · O2 numerical_behavior
Numerical scope: 67 listed independent full-pack vectors only
Interfaces: PASS · UI (SSR) / REST / MCP
ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 28/28 (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.
CVP suite: 67/67 golden · 3/3 CVP boundary · 34/34 invalid · 3/3 property · 2/2 metamorphic · 95/95 cross-interface · 1/1 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 · 1.2.2 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.2.2 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/polynomial-roots' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"coeffs":[-2,2]},"calculation_version":"1.2.2"}'

{
  "inputs": {
    "coeffs": [
      -2,
      2
    ]
  },
  "calculation_version": "1.2.2"
}

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

Full REST contract: REST API docs.

Tool id: polynomial-roots
Engine: polynomial-roots
REST: POST /api/v1/calc/polynomial-roots
MCP: execute_calculation with capability_id: "math.polynomial.roots"
Agent input schema: Canonical (MCP / Agent)
REST input schema: Compatibility (tol[] / tol1…)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.2.2)
Versioned output schema: Immutable pin (1.2.2)

Operations

Declared calculation modes for this capability.

all_roots
cardano
evaluate

References

Human-readable sources plus CVP reference evidence classes.

Sources

ISO 80000-2:2019 Quantities and units — Mathematics
NIST DLMF / Cardano cubic and Durand–Kerner simultaneous iteration
NIST DLMF §3.8 — Nonlinear Equations — §3.8
NumPy — numpy.polynomial.polynomial.polyval

Reference evidence classes: O1 model · 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.polynomial.roots"},"_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.polynomial.roots","inputs":{"coeffs":[-2,2]},"calculation_version":"1.2.2"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.2.2) when the result must stay reproducible.

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