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 1.0.3

Open calculator REST API MCP Evidence Evidence JSON CVP Manifest Review policy

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`	Radicand. Share-URL alias: xInput	—	input
`xInput`	Share-URL alias for x	—	input
`real`	—	—	output
`imag`	—	—	output
`principal`	—	—	output

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: 1.0.3
Verification: 18/18 calculation tests · 11/11 legacy golden · 7/7 legacy boundary
Production surface contract: 3/3
Versions: Calculation 1.0.3 · CVP protocol 1.0.0-proposed · Evidence 2026-09-09.o3-boundary
Verification revision: 2026-09-09.o3-boundary · 4/4 property · digest aa716638904c
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Numerical scope: ≤2 ULP vs O3 applies to the published tabulated Square Root vectors, including non-negative named and seeded values, principal imaginary of -9 and -2, IEEE-754 min-subnormal, and max-finite. It is not a guarantee over the entire input domain.
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 not Core CVP. They are covered by npm run test:math-graph-e2e (MATH_GRAPH_E2E_ONLY=square_root).
CVP suite: 7/7 golden · 7/7 CVP boundary · 7/7 invalid · 4/4 property · 2/2 metamorphic · 3/3 round-trip · 21/21 O3 · 7/7 cross-interface · 5/5 URL→result→graph · 1/1 CVP contract
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.0.3 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.0.3 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 3/3.

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)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.0.3)
Versioned output schema: Immutable pin (1.0.3)

References

Human-readable sources plus CVP reference evidence classes.

Sources

NIST Digital Library of Mathematical Functions, Chapter 4 — Logarithm, Exponential, Powers
NIST DLMF §4.2(iv) — Powers: principal value z^a = exp(a ln z)
CalculatorX mathematical conventions — Principal real root; principal imaginary for x < 0

Reference evidence classes: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior

Try with an agent

Inspect this capability via same-origin MCP, then call execute_calculation.

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"}}}'

Inspect the trust tier and required assurance level before execution. Pin calculation_version (1.0.3) when reproducibility matters. Full protocol notes: MCP docs.