POST /multibody

N-particle extension of the master equation. Each body gets its own scalar field φ_i(t); bodies attract one another through a pairwise coupling term γ · m_j / ((φ_i − φ_j)² + ε) · sign(φ_j − φ_i). Same numerical safety as the scalar path (acceleration clamped at ±10⁶, decay exponent clipped to ±5).

Method POST Path /api/framework/multibody Group Framework Precision ≤0.1% per body

The math

For each body i ∈ [0, N):

d²φ_i/dt² = (scalar master-equation terms applied to φ_i with body i's m, M, R, medium)
          + Σ_{j ≠ i} γ · m_j / ((φ_i − φ_j)² + ε) · sign(φ_j − φ_i)

Reduces to the scalar /solve when there's only one body. Each body's m, location, medium, optional φ₀, dφ₀ are independent.

Request body

Field	Type	Default	Notes
prompt	string	(required)	Label for the run. Math never reads.
bodies	BodyInput[]	(required)	Array of 2–16 body specifications (see below).
koSettings	{string: number}	{}	Operator weights — shared across all bodies.
alpha, beta	number	1.0, 0.0	Tensioner amplitudes.
tMax, dt	number	5.0, 0.01	Integration horizon + step. Clamped.
pairwiseCoupling	number	0.05	The γ in the coupling term. Set 6.674e-11 for Newton's G.
pairwiseSoftening	number	1e-6	The ε² softening prevents singularity when bodies cross.
useOperatorModules	boolean	false	Same semantics as scalar /solve.
normalizeOperators	boolean	true	Same.
coreOnly	boolean	false	Same.

BodyInput

Field	Type	Default	Notes
mass	number	1.0	Body mass in kg.
location	string	"earth"	Categorical: "earth" | "mars" | "moon" | "jupiter" | "sun".
medium	string	"air"	Categorical: "air" | "water" | "vacuum".
object	string	"unknown"	Label only — not read by math.
phi0	number	(derived)	Optional explicit initial φ. If omitted, derived from m and operator weights.
dphi0	number	(derived)	Optional explicit initial dφ/dt.

Response body — MultibodyResult

{
  "ckoId": "CKO_MB_1715349438246",
  "prompt": "...",
  "mode": "auto-α",
  "tEval": [0.0, 0.0098, ...],

  "bodies": [
    {
      "index": 0,
      "object": "sun",
      "mass": 1.989e30,
      "location": "sun",
      "medium": "vacuum",
      "trajectory":  [...],             // φ_i(t), downsampled to ≤512 points
      "dtrajectory": [...],             // dφ_i/dt(t), downsampled
      "registerDump": {
        "phi_final": ..., "phi_min": ..., "phi_max": ..., "phi_range": ...,
        "phi_mean": ..., "phi_std": ..., "phi_rms": ...,
        "dphi_final": ..., "dphi_mean": ..., "dphi_rms": ...,
        "zeroCrossings": ..., "period_s": ..., "frequency_Hz": ..., "freq_ratio_fH": ...,
        "energy_K_mean": ..., "energy_U_mean": ..., "energy_total_mean": ...,
        "momentum_final": ..., "angular_proxy_mean": ...
      }
    },
    // ... one entry per body
  ],

  "interactions": [
    {
      "bodyA": 0, "bodyB": 1,
      "F_avg":     0.0060,              // mean γ·m_i·m_j / ((φ_i−φ_j)²+ε) over t
      "F_final":   0.0060,              // pairwise force at t = tMax
      "separation_min":  ...,
      "separation_max":  ...,
      "separation_mean": 9.945e27
    },
    // ... N·(N−1)/2 pairs
  ],

  "totalEnergy": 9.836e85,              // Σ over bodies of registerDump.energy_total_mean
  "pairwiseCoupling": 0.05,
  "pairwiseSoftening": 1e-6
}

Call it — Sun-Earth-Moon (the spec's three-body example)

curl -X POST https://zeqapi.com/api/framework/multibody \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{
    "prompt": "sun-earth-moon",
    "bodies": [
      { "mass": 1.989e30, "location": "sun",   "medium": "vacuum", "object": "sun"   },
      { "mass": 5.972e24, "location": "earth", "medium": "vacuum", "object": "earth" },
      { "mass": 7.342e22, "location": "moon",  "medium": "vacuum", "object": "moon"  }
    ],
    "koSettings": { "KO42": 1.0, "NM21": 0.5, "GR35": 0.3 },
    "tMax": 5,
    "dt": 0.01
  }'

Returns 3 per-body register dumps + 3 pairwise summaries (sun–earth, sun–moon, earth–moon).

Call it — quantum binary (two-body)

curl -X POST https://zeqapi.com/api/framework/multibody \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{
    "prompt": "electron pair",
    "bodies": [
      { "mass": 9.109e-31, "object": "e1" },
      { "mass": 9.109e-31, "object": "e2", "phi0": 2.0 }
    ],
    "koSettings": { "KO42": 1.0, "QM02": 1.0 },
    "useOperatorModules": true,
    "coreOnly": true
  }'

Notes

• Bodies start at slightly offset φ₀ unless you specify phi0 explicitly, so the pairwise term is well-defined from t = 0. The ε² softening guards against bodies crossing.
• For large mass ratios (Sun-Earth, ratio 3.3 × 10⁵) the lighter body's φ stays approximately bound to the heavier body's φ over the integration window. Visible orbital motion requires a tMax long enough to traverse one period; at default tMax = 5 s the trajectory is in the early-transient regime.
• pairwiseCoupling and pairwiseSoftening are exposed so callers can dial the interaction strength against the master-equation terms. Setting pairwiseCoupling = 0 disables interaction and the bodies evolve independently.

Compose

• Combine with /solve to validate that a single-body's registerDump matches the same body in a multi-body run with pairwiseCoupling = 0.
• The wizard / textbook-dispatch path (/api/zeq/compute) does NOT carry register-dump or pairwise summaries — it operates on a different code path designed for single-formula evaluation.

Reference

• Source: app/artifacts/api-server/src/routes/framework.ts (route) → src/lib/zeqSolver.ts:runMultibodyExperiment (vector ODE solver).
• Vector RK4 integrator: src/lib/zeqSolver.ts:rk4IntegrateVector.
• Spec: master-equations.md §4 (master equation, scalar form — generalised here to vector).

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