CVP verified calculation capability

Trapezoidal critical depth Calculation Capability

Trapezoid Fr=1 from Q, b, z with Q² T = g A³, A=(b+z y)y, T=b+2 z y. Not rectangular yc=(q²/g)^{1/3}, not yn, not E, not Manning Q, not Q=Av.

Physics · Fluid Capability ID physics.fluid.trapezoid_critical_depth version 1.0.0 CVP VERIFIED CURRENT Engineering

Model

Critical condition

Critical condition: Q² T = g A³
Area: A = (b + z y) y
Top width: T = b + 2 z y

Modes: depth · discharge

Inputs / Outputs

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

Symbol	Quantity	Unit	Role
`b`	Bottom width in m	—	input / output
`z`	Side slope run/rise	—	input / output
`Q`	Volume flow in m³/s	—	input / output
`yc`	Critical depth in m	—	input / output
`g`	Gravity in m/s²	—	input / output
`A`	—	—	output
`T`	—	—	output
`D`	—	—	output
`P`	—	—	output
`R`	—	—	output
`vc`	—	—	output
`Fr`	—	—	output
`model`	—	—	output

Assumptions / Validity

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

Assumptions

Trapezoid critical depth: Fr=1 with A=(b+z yc) yc and T=b+2 z yc so Q² T = g A³; SI; default g=9.80665; z>0 is run/rise
Critical speed vc = Q/A = √(g A/T) is reported, not an input; Fr=1 is the definition, not a Froude wrap
Not rectangular yc=(q²/g)^{1/3}; not normal-depth yn; not specific energy E; not Manning n; not Q=Av

Not supported

rectangular yc=(q²/g)^{1/3}
normal-depth yn
specific energy E
Manning n
Q=Av
z=0

Example

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

Q=2 b=2 z=1 → yc≈0.43298 m

Open this example in the calculator

Trust

CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Trapezoid Fr=1 Q²T = g A³ with A=(b+z y)y and T=b+2 z y; + O3 mpmath tabulated yc.. 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 · Engineering assurance · Trapezoid Fr=1 Q²T = g A³ with A=(b+z y)y and T=b+2 z y; + O3 mpmath tabulated yc.· View Manifest · CVP overview · Specification
Verified scope: Trapezoid Fr=1 Q²T = g A³ with A=(b+z y)y and T=b+2 z y; + O3 mpmath tabulated yc.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (depth, solve, alias, awkward, invalid-domain)
Deferred: Not rectangular yc=(q²/g)^{1/3}, not yn, not E, not jump, and not Q=Av.
Calculation version: 1.0.0
Verification: 6/6 calculation tests · 1/1 legacy golden · 0/0 legacy boundary · 5/5 legacy property
Production surface contract: 6/6
Versions: Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-26.o2-o3
Verification revision: 2026-09-26.o2-o3 · 1/1 property · digest c363ef73903f
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Numerical scope: O2: trapezoid Fr=1 Q²T = g A³ with A=(b+z y)y, T=b+2 z y vs a separate-module identity (≤2 ULP). Not rectangular yc=(q²/g)^{1/3}, not yn, not E, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated trapezoid-critical-depth vectors. Not rectangular yc, not yn, and not E.
Interfaces: PASS · UI (SSR) / REST / MCP
ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (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: 4/4 golden · 1/1 CVP boundary · 2/2 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 6/6 CVP contract
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.0.0 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.0.0 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/trapezoidal-critical-depth' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"Q":2,"b":2,"z":1},"calculation_version":"1.0.0"}'

{
  "inputs": {
    "Q": 2,
    "b": 2,
    "z": 1
  },
  "calculation_version": "1.0.0"
}

POST /api/v1/calc/trapezoidal-critical-depth. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

Tool id: trapezoidal-critical-depth
Engine: trapezoidal-critical-depth
REST: POST /api/v1/calc/trapezoidal-critical-depth
MCP: execute_calculation with capability_id: "physics.fluid.trapezoid_critical_depth"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.0.0)
Versioned output schema: Immutable pin (1.0.0)

References

Human-readable sources plus CVP reference evidence classes.

Sources

ISO 80000-4 — Mechanics — Length, speed, acceleration
Chow, V.T. Open-Channel Hydraulics — Critical flow in a trapezoidal section

Reference evidence classes: O1 model · O3 expected_values · O3 numerical_behavior · 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":"physics.fluid.trapezoid_critical_depth"},"_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":"physics.fluid.trapezoid_critical_depth","inputs":{"Q":2,"b":2,"z":1},"calculation_version":"1.0.0"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.0.0) when the result must stay reproducible.

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