Life or Death
Every pipeline on this page integrates real physics and seals the result as a verifiable ZEQ envelope + ZEQOND receipt. They are independent — each is its own endpoint, MCP tool and CLI verb — but they share the 7-step wizard and one honest rule: a result only seals if it passes its own physical reality check.
Why "life or death"? Because for real dynamics the integrator decides whether a simulation stays physical or quietly turns to nonsense. Run an orbit with the wrong method and the energy drifts, the orbit spirals, and the satellite you were tracking hits the ground — in the numbers, not the sky. The framework refuses to seal a run that has drifted off its manifold. That refusal is the product.
One door — POST /api/zeq/auto (ZeqAuto)
You do not have to know which pipeline you want or which operators it computes with. Send the query and the constants, and the solver registry routes it:
curl -sX POST https://www.zeq.dev/api/zeq/auto \
-H 'Authorization: Bearer YOUR_KEY' -H 'Content-Type: application/json' \
-d '{ "query": "price a european call option",
"constants": { "S0":100, "K":100, "sigma":0.2, "r":0.05, "T":1 } }'
# → solver "quant", operators ["KO42","BLACK_SCHOLES"], value 10.43 $,
# matchedOn ["option","price","call"], + the same envelope & ZEQOND receipt
The reply names the solver it chose, the matchedOn keywords, the real
operators it computed with (heat→CHE17 Fick, flow→FL3 Navier–Stokes,
sat→CS81, gauge→HEP4, …), the value, and the sealed zeqProof +
zeqond_receipt — identical to calling the specific endpoint directly. Pass an
explicit "solver" to skip classification. Every solver below is reachable this
way, and so is zeq_auto over MCP and zeq auto on the CLI.
Verified live across pipelines: "lennard-jones argon liquid" → evolve
(NM19), "heat diffusion along a rod" → heat (CHE17), "price a call"
→ quant (BLACK_SCHOLES) — each classified, run, and sealed.
The pipelines
| Pipeline | Integrates | Method | Reality check | Receipt fields |
|---|---|---|---|---|
/solve | HULYAS master field ϕ (2nd-order ODE) | RK4 | ≤0.1% vs the reference | internal energy |
/multibody | N coupled fields / Newtonian N-body | RK4 | NVE energy drift ≤0.1% | internal energy, momentum |
/evolve — ZeqEvolve | molecular dynamics (Lennard-Jones + bonds) | velocity-Verlet (2nd-order symplectic) | NVE energy drift ≤0.1% | temperature, pressure, internal energy, momentum |
/action — ZeqAction | Hamiltonian trajectory (orbit / oscillator / pendulum) | Yoshida symplectic, order 4/6/8 | energy drift bounded (+ RK4 comparison) | internal energy |
/heat — ZeqHeat | heat / diffusion ∂T/∂t=α∇²T | explicit FTCS vs Crank–Nicolson | stability r=α·dt/dx²≤½ (explicit blows up) | temperature, heat flux |
/lorenz — ZeqLorenz | the Lorenz attractor (chaos) | RK4 | exact contraction −(σ+1+β); Lyapunov + boundedness | — (none, honestly) |
/wave — ZeqWave | wave equation / Maxwell | leapfrog / Yee FDTD | Courant C=c·dt/dx≤1; energy conserved | displacement, velocity / E, B |
/flow — ZeqFlow | incompressible Navier–Stokes | stable fluids (project) | ∇·u→0 after projection | velocity, pressure, viscosity |
/fracture — ZeqFracture | cohesive damage / fracture | peridynamic + Gauss–Seidel | quasi-static equilibrium residual | stress, strain, damage, elasticity, traction, contact |
/react — ZeqReact | reaction–diffusion (morphogenesis / excitable media) | explicit 2-D (Gray–Scott / FitzHugh–Nagumo) | stability D·dt/dx²≤¼; pattern bounded | concentration, reaction rate |
Between them, the ten pipelines fill every tier of the ZEQOND receipt with real solver output — ZeqFracture fills the material + boundary tier, and ZeqReact the chemical/biological tier, that nothing else touches.
All four are deterministic — identical inputs reproduce the trajectory and
therefore the zeqProof. All four fill the receipt only with fields they
genuinely computed; nothing is fabricated.
The proof of the point — symplectic vs RK4
ZeqAction integrates the same Kepler orbit two ways and returns
both. RK4 has smaller local error but no structure preservation, so its energy
drifts secularly; the symplectic composition's energy stays bounded by
construction. Measured, ~30 orbits, dt = 0.01:
| method | energy drift | angular momentum drift |
|---|---|---|
| RK4 (non-symplectic) | 7.5×10⁻⁵ % and growing | — |
| symplectic order 4 | 4.1×10⁻⁵ % (bounded) | ~10⁻¹⁶ (machine-exact) |
| symplectic order 6 | 2.1×10⁻⁷ % (bounded) | ~10⁻¹⁶ |
| symplectic order 8 | 1.6×10⁻⁹ % — ~48,000× better than RK4 | ~10⁻¹⁶ |
The RK4 number is the one that keeps growing with every orbit; the symplectic numbers oscillate and never accumulate. Over a long enough run the symplectic method wins at any order — that is the whole reason it exists.
Nothing was replaced
These pipelines were added, not swapped in. /api/zeq/compute, /solve and
/multibody are exactly as they were. ZeqEvolve and ZeqAction are new, independent
compute pipelines on the same seal — tap them from the API, the zeq_evolve /
zeq_action MCP tools, the evolve / action CLI verbs, contracts, or any state
machine.
Read next
- ZeqAction — the least-action symplectic integrator (this section's proof).
- ZeqEvolve — molecular dynamics as a sealed endpoint.
- The ZEQOND receipt — the fields these pipelines fill.
- The nano-zeqond — the integer-time grid ZeqAction integrates on.