POST /solve

Integrate the master equation □φ − μ²(r)φ − λφ³ − e^(−φ/φ_c) + φ_c²·Σ C_k(φ) = T^μ_μ + βF² + J_ext for one body and return the trajectory plus a structured register-dump of derived quantities.

Method POST Path /api/framework/solve Group Framework Precision ≤0.1%

Request body

The math reads only numerical / categorical fields. The prompt field is metadata — never inspected by the solver.

Field	Type	Default	Notes
`prompt`	string	(required)	Label for the run. Threaded through the response but never read by the math. Max 500 chars.
`mass`	number	`1.0`	Object mass in kg. Pass explicit numerical value — no string parsing.
`location`	string	`"earth"`	Categorical: `"earth" \| "mars" \| "moon" \| "jupiter" \| "sun"`.
`medium`	string	`"air"`	Categorical: `"air" \| "water" \| "vacuum"`.
`object`	string	`"unknown"`	Free-form object label — not read by math.
`alpha`	number	`1.0`	KO42.1 automatic-tensioner amplitude.
`beta`	number	`0.0`	KO42.2 manual-tensioner amplitude. When `> 0`, enables drag + manual-β mode.
`koSettings`	`{string: number}`	`{}`	Operator weights, keyed by operator ID (e.g. `{ KO42: 1.0, QM01: 0.5, GR35: 0.3 }`).
`tMax`	number	`5.0`	Simulation horizon in seconds. Clamped to `[0.1, 30]`.
`dt`	number	`0.01`	Integrator timestep. Clamped to `[0.001, 0.1]`.
`useOperatorModules`	boolean	`false`	When `true`, the Σ C_k coupling uses operator-specific physics values from the SDK operator modules instead of the legacy `Σ w_k · φ`.
`normalizeOperators`	boolean	`true`	When `useOperatorModules` is on, divide each operator's raw value by its calibrated reference scale so KO42 (∼10⁻³) and GR35 (∼c²) are commensurate per unit weight.
`coreOnly`	boolean	`false`	When `useOperatorModules` is on, restrict the coupling to the 8 Core Operator Families (QM/NM/GR/CS/Awareness/HF/AGO/HRO/KO42).
`referenceMode`	string	`"model"`	Analytic comparison: `"free-fall"` compares to `y₀ + v₀t − ½gt²`, `"shm"` compares to `A·cos(ωt + φ₀)`, anything else compares to a no-operator self-reference trajectory.

Response body — `SolverResult`

{
  // Identity + labels
  "ckoId": "CKO_1715349438246",
  "prompt": "...",                              // echo of input — not read by math
  "object": "feather",                          // echo of input
  "location": "earth",                          // echo of input
  "mass": 1e-4,                                 // echo of input
  "mode": "auto-α",                             // "auto-α" | "manual-β"
  "koSettings": { "KO42": 1.0 },

  // Trajectory (downsampled to ≤512 points)
  "tEval": [0.0, 0.0098, ...],                  // simulation time in seconds
  "solution": [1.3725, 1.3726, ...],            // φ(t)

  // Error metric
  "errorPct": 0.0019,                           // calibrated mean |φ−ref|/|ref| × 100
  "energy": 1.3239,                             // mean(½·m·dφ² + ½·m·φ²) over trajectory

  // Term-by-term diagnostic of the master equation at t=tMax
  "masterEquationTerms": {
    "muSquared": 0.0, "nonlinear": ..., "decayTerm": ...,
    "operatorCoupling": ..., "stressEnergy": 4.96, "drag": 0.0, "baseline": ...
  },

  // HULYAS Functional Equation: E = P_φ · Z(M, R, δ, C, X)
  "functionalEnergy": 20895.4320,
  "functionalEnergyTerms": {
    "P_phi": 2.0828e1,                          // RMS field momentum √mean((dφ/dt)²)
    "Z":     1003.5,                            // M · R · (1+C) · (1+|X|) · exp(−δ·t_total)
    "M": 1e-4, "R": 6.371e6, "delta": 0.05,
    "C": 1.0, "X": 0.0
  },

  // Register dump — structured derived quantities (mirrors the spec's three-body output)
  "registerDump": {
    "phi_final":          -9.2295,
    "phi_min":            -15.4340,
    "phi_max":             9.1000,
    "phi_range":          24.5340,
    "phi_mean":           -1.5706,
    "phi_std":             6.8074,
    "phi_rms":             6.9863,
    "dphi_final":          5.8305,
    "dphi_mean":          -1.1350,
    "dphi_rms":           20.4851,
    "zeroCrossings":       4,
    "period_s":            2.5000,              // 2 × tMax / zeroCrossings (NaN if no crossings)
    "frequency_Hz":        0.4000,              // 1 / period_s
    "freq_ratio_fH":       0.3108,              // frequency / 1.287 Hz
    "energy_K_mean":   314728.36,               // mean ½·m·dφ²
    "energy_U_mean":    36605.80,               // mean ½·m·φ²
    "energy_total_mean":351334.16,
    "momentum_final":    8745.74,               // m · dφ_final
    "angular_proxy_mean":-10659.54              // scalar analogue of L: mean(m·φ·dφ)
  }
}

Call it

Minimal:

curl -X POST https://zeqapi.com/api/framework/solve \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{
    "prompt":  "feather drop",
    "mass":    1e-4,
    "location":"earth",
    "medium":  "air",
    "koSettings": { "KO42": 1.0 }
  }'

With operator-modules mode (SDK formula path) and core-only filtering:

curl -X POST https://zeqapi.com/api/framework/solve \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{
    "prompt": "QM + GR coupling",
    "mass": 9.109e-31,
    "location": "earth",
    "medium": "vacuum",
    "koSettings": { "KO42": 1.0, "QM05": 0.5, "GR34": 0.3 },
    "useOperatorModules": true,
    "coreOnly": true
  }'

For analytic-anchored error (free-fall calibration target):

curl -X POST https://zeqapi.com/api/framework/solve \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{
    "prompt": "free-fall calibration",
    "mass": 1.0,
    "koSettings": { "NM19": 1.0, "NM24": 0.3 },
    "tMax": 0.4, "dt": 0.001,
    "referenceMode": "free-fall"
  }'

Compose

Chain with POST /solve/strict for autotune toward ≤ 0.1 % error.
Chain with POST /multibody for N coupled bodies (vector master equation).
Every response carries a Zeqond phase stamp so downstream consumers can phase-align results to the 1.287 Hz HulyaPulse.

Reference

Source: app/artifacts/api-server/src/routes/framework.ts (route) → src/lib/zeqSolver.ts:runMasterExperiment (math).
Spec: master-equations.md §4 (master equation) + §5 (functional equation).
Operator catalogue: Kinematic Spectrum — 8 Core Families.
Papers: Zeq Paper · Zeq Framework

Middleware active. Kernel on the 1.287 Hz HulyaPulse. Awaiting next Zeqond.

Request body​

Response body — SolverResult​

Call it​

Compose​

Reference​

Request body

Response body — `SolverResult`

Call it

Compose

Reference