CVP verified calculation capability

Square root Calculation Capability

Principal square root of x; non-negative returns real principal, negative returns principal imaginary √|x| i.

Math Capability ID math.square_root version 1.0.4 CVP VERIFIED CURRENT Core

Model

Definition

Definition: √x = y ⟺ y² = x, y ≥ 0 (principal)
Two real solutions: y = ±√x (when x > 0)
Exponent: √x = x^(1/2)
Babylonian step: b ← (b + x/b) / 2

Inputs / Outputs

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

Symbol	Quantity	Unit	Role
`x`	Finite binary64 number or strict decimal/scientific text ≤512 characters; nonzero-to-zero conversion rejects NUMERIC_UNDERFLOW	—	input
`xInput`	Finite binary64 number or strict decimal/scientific text ≤512 characters; nonzero-to-zero conversion rejects NUMERIC_UNDERFLOW	—	input
`real`	—	—	input / output
`imag`	—	—	input / output
`principal`	—	—	input / output

Trusted Math contract

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

Field	Declaration
`numeric_domain`	complex
`exactness`	float64
`algorithm`	Math.sqrt of the absolute radicand; binary64; full real/imag/principal result
`valid_domain`	Finite real JSON numbers or strict decimal/scientific strings up to512 characters. Matching x/xInput aliases; unknown fields reject.
`singularity_policy`	Zero maps to zero; nonzero text rounded to zero rejects NUMERIC_UNDERFLOW; nonfinite values reject INVALID_NUMBER.
`branch_policy`	Nonnegative real principal for x≥0; x<0 returns positive imaginary √\|x\|i with real=0 and principal=null.
`tolerance`	not applicable — closed form; capability Evidence owns comparison, no global default
`convergence`	not applicable — non-iterative
`conditioning`	Relative condition number1/2 for nonzero x; derivative singular at zero. Input rounding to binary64 is distinct from sqrt rounding.
`rounding`	ieee754_round_ties_to_even
`overflow`	ieee754_nonfinite_or_coded_error
`reproducibility`	deterministic algorithm + Capability@Version
`independent_oracle`	CPython Decimal2000 sqrt on exact normalized binary64 inputs; complete three-field comparison within2 ULP on declared vectors.
`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

Principal branch: nonnegative real root when x ≥ 0.
Unlike Log / Ln, this engine does not reject x < 0. It returns the principal imaginary √|x| i, not a real error.
The other real root for x > 0 is −√x; the other imaginary root for x < 0 is −√|x| i. The published value is the principal one.

Example

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

√144 = 12; √2 ≈ 1.41421356237; √(−9) = 3i

Open this example in the calculator

Trust

CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance. 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· View Manifest · CVP overview · Specification
Assurance: Core
Declared partition coverage: PASS · 6/6 declared partitions (positive-real, zero, negative-imaginary, tiny-subnormal, large-finite, invalid-domain)
Calculation version: 1.0.4
Verification: 18/18 calculation tests · 11/11 legacy golden · 7/7 legacy boundary
Production surface contract: 6/6
Versions: Calculation 1.0.4 · CVP protocol 1.0.0-proposed · Evidence 2026-10-26.square-root-complete-contract-v1
Verification revision: 2026-10-26.square-root-complete-contract-v1 · 4/4 property · digest 767ce88755a9
Reference: O1 model · O2 expected_values · O2 numerical_behavior
Numerical scope: 80 fixed and seeded complete results including both signs, zero, subnormal, max finite and rounding neighbors; not exhaustive
Interfaces: PASS · UI (SSR) / REST / MCP / URL→result→graph
Success 4/4. Error-path engine·REST·MCP 3/3 (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. Integration: URL → SSR result → Live graph current point (5/5). Hydration/slider/history are URL-canonical contracts, not a live browser session. Live-browser hydration, slider, Copy JSON, and Overview/Focus are covered in the current full review; Core ui-ssr is not a browser session.
CVP suite: 80/80 golden · 7/7 CVP boundary · 11/11 invalid · 4/4 property · 2/2 metamorphic · 3/3 round-trip · 7/7 cross-interface · 5/5 URL→result→graph · 6/6 CVP contract
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.0.4 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.0.4 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/square-root' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"x":144},"calculation_version":"1.0.4"}'

{
  "inputs": {
    "x": 144
  },
  "calculation_version": "1.0.4"
}

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

Full REST contract: REST API docs.

Tool id: square-root
Engine: square-root
REST: POST /api/v1/calc/square-root
MCP: execute_calculation with capability_id: "math.square_root"
Agent input schema: Canonical (MCP / Agent)
REST input schema: Compatibility (tol[] / tol1…)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.0.4)
Versioned output schema: Immutable pin (1.0.4)

Operations

Declared calculation modes for this capability.

sqrt

References

Human-readable sources plus CVP reference evidence classes.

Sources

Derived identities

CalculatorX mathematical conventions — Principal real root; principal imaginary for x < 0

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.square_root"},"_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.square_root","inputs":{"x":144},"calculation_version":"1.0.4"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.0.4) when the result must stay reproducible.

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