CVP verified calculation capability

Alternate depth Calculation Capability

Rectangular both y from E, Q, b with E=y+Q²/(2gA²). Not specific energy E from y+v, not specific force M, not hydraulic jump y2, not critical depth yc as the product, not Manning n, not Q=Av.

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

Model

Specific energy

Specific energy: E = y + Q² / (2g A²)
Section: A = b y
Critical: Ec = (3/2) yc

Modes: depths · energy

Inputs / Outputs

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

Symbol	Quantity	Unit	Role
`b`	Rectangular surface width in m	—	input / output
`Q`	Volume flow in m³/s	—	input / output
`E`	Specific energy in m	—	input / output
`y`	One flow depth in m	—	input / output
`g`	Gravity in m/s²	—	input / output
`y_sub`	—	—	output
`y_super`	—	—	output
`y_alt`	—	—	output
`yc`	—	—	output
`Ec`	—	—	output
`v_sub`	—	—	output
`v_super`	—	—	output
`Fr_sub`	—	—	output
`Fr_super`	—	—	output
`model`	—	—	output

Assumptions / Validity

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

Assumptions

Rectangular alternate depths: E = y + Q²/(2g A²) with A=b y; both y_sub>yc and y_super<yc at the same E, Q, b; SI; default g=9.80665
Unknown is the pair (depths mode) or E (energy invert from one y); E>Ec=1.5 yc; Fr is reported not an input
Not specific energy E from y+v; not specific force M; not hydraulic jump y2 from Fr₁>1; not critical-depth yc as the product; not Manning n; not Q=Av

Not supported

specific energy from y+v
specific force M
hydraulic jump y2 from Fr₁>1
critical depth yc as the product
Manning n
Q=Av
trapezoid z (that is /calc/physics/trapezoidal-alternate-depth)

Example

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

E=1.05099 Q=2 b=2 → y_sub≈1.0000

Open this example in the calculator

Trust

CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Rectangular alternate depths from E = y + Q²/(2gA²) with energy invert; + O3 mpmath tabulated depths.. 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 · Rectangular alternate depths from E = y + Q²/(2gA²) with energy invert; + O3 mpmath tabulated depths.· View Manifest · CVP overview · Specification
Verified scope: Rectangular alternate depths from E = y + Q²/(2gA²) with energy invert; + O3 mpmath tabulated depths.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (depths, solve, alias, awkward, invalid-domain)
Deferred: Not jump y2, not specific force M, not trapezoid alternate depths, 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 b851963fc9ba
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Numerical scope: O2: rectangular alternate depths from E=y+Q²/(2gA²) vs a separate-module identity (≤2 ULP). Not jump y2, not specific force M, not trapezoid, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated alternate-depth vectors. Not jump y2, not M, and not trapezoid.
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/alternate-depth' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"E":1.05099,"Q":2,"b":2},"calculation_version":"1.0.0"}'

{
  "inputs": {
    "E": 1.05099,
    "Q": 2,
    "b": 2
  },
  "calculation_version": "1.0.0"
}

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

Full REST contract: REST API docs.

Tool id: alternate-depth
Engine: alternate-depth
REST: POST /api/v1/calc/alternate-depth
MCP: execute_calculation with capability_id: "physics.fluid.alternate_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 — Energy and volume flow
Chow, V.T. Open-Channel Hydraulics — Specific energy and alternate depths

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.alternate_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.alternate_depth","inputs":{"E":1.05099,"Q":2,"b":2},"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.