POST /api/zeq/closedform

Closed-form master-equation solve — the analytic counterpart to /api/zeq/solve.

Compiles polynomial couplings into effective (mu2_eff, lambda_eff, F_eff) and evaluates the EXACT branch of d2phi/dt2 = mu2_effphi + lambda_effphi^3 + F_eff (all / KO42 tensioner): free, uniform-accel, harmonic, hyperbolic, driven, and Duffing phi0cn(Omegat,k) via AGM. KO42 first-order on the linear branches; phi^2 / sqrt / exp shapes refused by name. Ships an independent RK4 closed-vs-open cross-check.

Method	POST
Path	/api/zeq/closedform
Feature area	Zeq core & solvers
Tag	Solvers
Auth	machine key — Authorization: Bearer $ZEQ_KEY
Tier	authenticated (machine-control)

Solver

Summary	Closed-form master-equation solve — the analytic counterpart to /api/zeq/solve. Compiles polynomial couplings into effective (mu2_eff, lambda_eff, F_eff) and evaluates the EXACT branch of d2phi/dt2 = mu2_effphi + lambda_effphi^3 + F_eff (all / KO42 tensioner): free, uniform-accel, harmonic, hyperbolic, driven, and Duffing phi0cn(Omegat,k) via AGM. KO42 first-order on the linear branches; phi^2 / sqrt / exp shapes refused by name. Ships an independent RK4 closed-vs-open cross-check.
Domain	Closed-Form Master Field
Numerical Method	Analytic branch (compile + cn via AGM) + KO42 perturbation
Operator Chain	KO42-tensioned
Unit	phi

See all sealed solvers.

Parameters

This operation publishes no path, query or header parameters.

Request body

Optional. Content type: application/json.

Field	Type	Required	Description
mu2_eff	number	no	Effective mu^2 (skip compile). mu2_eff<0 oscillates (omega0=sqrt(-mu2_eff)); >0 hyperbolic.
lambda_eff	number	no	Effective lambda (cubic term -> Duffing).
F_eff	number	no	Effective constant forcing.
couplings	array<object>	no	Polynomial couplings to fold: {a,b,c,d} cubic (c!=0 -> refused QUADRATIC) or {samples:[[x,y],...]} (>=4 pts; extra pts checked -> NONPOLY).
operators	array<any>	no	Operator IDs (or {id,equation,unit,drive}) compiled AS couplings: one input is driven base*(1+scale*phi), the real formula is sampled over the phi grid, fit to a cubic and folded. KO42 sets the frame. Non-polynomial operators (Kepler a^1.5, de-Broglie 1/v) refuse NONPOLY by name.
inputs	object<string, number>	no	Anchor values the operator formulas consume (mass, a, r, v, ...).
compile	object	no	—
compile.scale	number	no	Drive amplitude in base*(1+scale*phi). Default 0.25.
phi0	number	no	Initial field value. Default: equilibrium launch -F_eff/mu2_eff + 0.5.
phidot0	number	no	Initial field velocity. Default 0.
t	number	no	Evaluation time. Default 1.
ko42	boolean	no	Apply the KO42 tensioner first-order on the linear branches. Default true.
alpha	number	no	KO42.1 tensioner amplitude. Default 1e-3.
mode	string — one of "auto", "manual"	no	manual uses beta as the tensioner amplitude.

Responses

Status	Meaning
200	On solve: {ok:true, value:phi(t), branch, mu2_eff/lambda_eff/F_eff, phi_t, phidot_t, closedVsOpenPct, compile:{mode,ruleset,scale,operators,anchors,effective}, zeqProof, envelope}. On a shape the master equation cannot carry: {ok:true, refused:true, refusal:{code,term,message}} with code in QUADRATIC | NONPOLY | DRIVEN_DUFFING | SEPARATRIX | NO_BOUNDED_OSCILLATION | DUFFING_NEEDS_REST | PARAMETRIC_RESONANCE | UNKNOWN_OPERATOR | MISSING_INPUT | COMPILE_DEGENERATE (the refusal IS the answer).
401	Missing or invalid key
402	Insufficient compute credit

No response body schema is published for this operation — the node returns JSON; inspect a live call to see its shape.

Call it

curl -sS -X POST https://zeq.me/api/zeq/closedform \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{
         "mu2_eff": 1,
         "lambda_eff": 1,
         "F_eff": 1,
         "couplings": [
           {}
         ],
         "operators": [
           {}
         ],
         "inputs": {
           "mass": 70,
           "r": 1
         },
         "compile": {
           "scale": 1
         },
         "phi0": 1
       }'

Needs a machine key. Mint one: curl -sS -X POST https://zeq.me/api/demo-key/mint, then spin up your machine.

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Generated from the live OpenAPI description (https://zeq.me/openapi.json, spec version 1.287.0) by scripts/generate-reference.mjs. Do not edit — this file is rewritten on every run.