The HULYAS master equation

Most Zeq computes resolve to a closed-form solver. When they don't, the engine does not guess — it integrates a field equation. That equation is the HULYAS master equation, and this page teaches exactly what it is, what every term means, and how the engine solves it. Every number below is reproducible against a live node.

The equation

$$\Box\phi \;-\; \mu^2(r)\,\phi \;-\; \lambda\phi^3 \;-\; e^{-\phi/\phi_c} \;+\; \phi_{42}\sum_{k=1}^{42} C_k(\phi) \;=\; T^\mu{}_\mu \;+\; \beta\,F_{\mu\nu}F^{\mu\nu} \;+\; J_{ext}$$

It is a nonlinear scalar field equation. Read left to right:

Term	Name	What it does
$\Box\phi$	wave operator	how the field $\phi$ propagates in space and time — the $\partial^2/\partial t^2 - \nabla^2$ core
$-\mu^2(r)\,\phi$	mass / stiffness	a position-dependent restoring term; sets the local "stiffness" of the field
$-\lambda\phi^3$	self-interaction	the nonlinearity — what lets the system model real, coupled behaviour rather than a pure wave
$-e^{-\phi/\phi_c}$	decay	damps the field over distance/time; $\phi_c$ is the critical field value
$+\,\phi_{42}\sum_{k=1}^{42} C_k(\phi)$	operator coupling	direct coupling to the 42 kinematic operators
$T^\mu{}_\mu$	stress–energy trace	the matter source
$\beta F_{\mu\nu}F^{\mu\nu}$	electromagnetic	the field source
$J_{ext}$	external	anything you drive it with

What the engine actually integrates

The □ core is a well-posed φ⁴ Klein–Gordon field, □φ = ∇²φ − μ²φ − λφ³. For a compute the engine reduces the equation to a second-order ODE in time and integrates it. In first-order form the state is $y = [\,\phi,\ \dot\phi\,]$ and the acceleration is:

$$\ddot\phi \;=\; -\mu^2(r)\,\phi \;-\; \lambda\phi^3 \;-\; e^{-\phi/\phi_c} \;+\; \phi_c^2 \sum_{k} C_k(\phi)$$

(source: shared/api-core/src/lib/zeqSolver.ts).

The integrator: RK4

It is solved with fourth-order Runge–Kutta on a uniform time grid (default tMax = 5.0 s, dt = 0.01 s). Each step combines four slope estimates:

$$y_{n+1} = y_n + \tfrac{dt}{6}\,(k_1 + 2k_2 + 2k_3 + k_4)$$

RK4 is the same integrator used across scientific computing (it is what scipy.integrate defaults toward for non-stiff problems) — a deliberate, boring, verifiable choice. The KO42 metric rides the same clock: ds² = gμν dxμ dxν + α·sin(2π·1.287·t)·dt².

The result: functional energy

The full trajectory φ(t) is reduced to one characterising scalar, the functional energy:

$$E = P_\phi \cdot Z(M, R, \delta, C, X)$$

where $P_\phi$ is the RMS field momentum $\sqrt{\langle(\dot\phi)^2\rangle}$ and $Z$ folds in the mass $M$, length scale $R$, and the coupling factors. This is the value the ODE master-equation fallback returns when no closed form matched.

Worked example (reproducible)

Integrate the field for a 1 kg object on Earth:

curl -sS -X POST https://zeq.dev/api/solve \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{"prompt":"a 1 kg pendulum on Earth","mass":1.0}'

The response carries the integrated trajectory and the value of every term of the equation, so you can see the physics, not just a number. From a live run:

masterEquationTerms:
  muSquared        : 0
  nonlinear (−λφ³) : −173.565
  decayTerm        : 165659.848
  operatorCoupling : 0
  stressEnergy Tᵘᵤ : 4.956
  baseline         : 9.139

functionalEnergyTerms  (E = P_φ · Z):
  P_φ (RMS momentum): 20.428
  Z                 : 4.9617×10⁶
  M = 1   R = 6.371×10⁶ (Earth)   δ = 0.05

energy: 248.499     functionalEnergy: 1.0136×10⁸

Every field in masterEquationTerms maps one-to-one to a term in the equation above. tEval is the time grid (0 → 5 s), solution is φ(t) at each grid point — the raw trajectory the RK4 loop produced.

When you meet this equation

• as the ODE master-equation fallback — when an operator has no closed form, the engine integrates this equation on your inputs and returns its functional energy, transparently labelled so it is never mistaken for a closed form;
• directly, via POST /api/solve (single field) and POST /api/multibody (coupled fields);
• under the hood of the strict solver (/api/solve/strict), which iterates the KO42 settings to drive the fit error down.

Read next

• KO42 · the gauge mode — the modulation and clock that ride the integration.
• The functional equation — E = P_φ · Z in full.
• The solvers — closed forms first, this equation as the fallback.
• Precision & proof — how a result is checked and signed.