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The full catalogue

Twenty-six domain solvers, each an independent endpoint that integrates real physics or mathematics and seals the result as a verifiable ZEQ envelope + ZEQOND receipt through the same 7-step pipeline. Every one is reachable three ways — REST POST /api/zeq/<key>, the MCP tool zeq_<key>, and the CLI verb <key> — and every one is routed automatically by ZeqAuto / Zeq Math when you just describe the problem.

The honest part. A solver only seals verification: "verified" when its own reality-check — a genuinely checkable invariant, an exact solution, or a conservation law — comes in under the precision bound (precisionActual ≤ 0.001). Miss it and the same number seals as observable-differential instead: a real computed value, honestly labelled as not independently verified. Blow the scheme up entirely and nothing mints at all. The reality-check named in each entry below is exactly what that solver stakes its seal on.

Mechanics & dynamics​

  • ZeqEvolve — evolve · molecular dynamics. Velocity-Verlet NVE integration of a Lennard-Jones + bonded system. Check: total energy is conserved (the symplectic drift is the gate).
  • ZeqAction — action · symplectic mechanics. A least-action Yoshida composition map. Check: the symplectic invariant keeps a closed orbit where a same-order RK4 spirals in and crashes — the pipeline that proves the point.
  • ZeqLorenz — lorenz · chaotic dynamics. The Lorenz attractor by RK4. Check: exact phase-volume contraction ∇·f = −(σ+1+β) and boundedness; Lyapunov λ ≈ 0.905 measured live — deterministic and sensitive.

Heat, waves & fluids​

  • ZeqHeat — heat · thermal diffusion. 2-D ∂T/∂t = α∇²T, explicit FTCS vs implicit ADI. Check: the CFL stability limit, and agreement with the exact Fourier-mode decay.
  • ZeqWave — wave · wave mechanics. A 2-D membrane by leapfrog (plus 1-D Maxwell FDTD). Check: the discrete Hamiltonian is conserved to ~10⁻¹³, and the 2-D Courant number bounds it.
  • ZeqFlow — flow · fluid dynamics. 2-D incompressible Navier–Stokes by stable fluids. Check: the pressure projection drives ∇·u → 0 (the divergence residual is the gate).
  • ZeqClimate — climate · geophysical fluid dynamics. The 2-D shallow-water equations, seeded by a droplet. Check: total mass is conserved; wave speed is bounded by the CFL limit.
  • ZeqAcoustic — acoustic · acoustics. The resonant frequencies of a 1-D resonator (Helmholtz −p″=k²p) with open/closed ends — organ pipes, wind instruments, exhaust and HVAC ducts, muffler design. Check: the modal frequencies vs the exact closed forms (open–open n·c/2L, open–closed (2n−1)·c/4L).

Solids & structures​

  • ZeqFEA — fea · structural finite elements. 2-D plane-stress FEM (Q4, conjugate gradient). Check: the CG residual, and the tip deflection vs Euler–Bernoulli beam theory.
  • ZeqFracture — fracture · fracture mechanics. A 2-D bond-based peridynamic crack in a mode-I notched specimen, with the crack path traced. Check: below a critical strain the notch holds; above it the crack runs across the whole specimen (the relaxed-equilibrium residual seals each step).
  • ZeqBuckling — buckling · structural stability. The Euler critical load at which a compressed column/strut buckles, a generalised eigenproblem K φ=P·K_g φ from beam finite elements — columns, struts, landing gear, interstages. Check: P_cr vs Euler π²·EI/(K·L)² for the end condition (pinned K=1, fixed–fixed ½, cantilever 2, fixed–pinned ≈0.699).

Devices & power​

  • ZeqDevice — device · semiconductor devices. 2-D device electrostatics (∇²φ=0, matrix-free Jacobi-PCG) — the peak field that sets breakdown in a MOSFET/HEMT/diode — or the 1-D PN junction under dim:1. Check: vs the exact sinh Fourier mode + CG residual (2-D); vs the analytic built-in voltage (1-D).
  • ZeqGrid — grid · power systems. DC power flow B·θ = P on a bus network. Check: power balance at every bus (the KCL residual is the gate).

Life sciences​

  • ZeqEpi — epi · epidemiology. SIR/SEIR outbreak dynamics. Check: the total population S+I+R is conserved.
  • ZeqTumor — tumor · growth-front dynamics. The Fisher–KPP invasion front — oncology, invasion biology, a spreading gene. Check: the density stays in [0,1], and the front advances at the exact Ablowitz–Zeppetella speed c = 5√(rD/6).
  • ZeqLotka — lotka · population ecology. Lotka–Volterra predator–prey. Check: the conserved first integral H — RK4 holds it, a lazy integrator drifts to a false extinction and does not seal.
  • ZeqReact — react · reaction–diffusion. Gray–Scott / FitzHugh–Nagumo morphogenesis and excitable media. Check: the diffusion CFL D·dt/dx² ≤ ¼; the pattern forms and stays bounded.
  • ZeqDock — dock · molecular docking. Rigid-body minimisation of a Lennard-Jones + Coulomb binding energy. Check: the gradient norm converges to a genuine energy minimum.

Finance, optimization & computer science​

  • ZeqQuant — quant · quantitative finance. Geometric-Brownian-motion Monte-Carlo option pricing. Check: the MC price converges to the Black–Scholes closed form.
  • ZeqOpt — opt · optimization. Nelder–Mead / gradient descent on standard test functions. Check: the result vs the known global minimum — and it is honest when a multimodal surface traps a local optimiser.
  • ZeqKalman — kalman · state estimation. The Kalman filter (Joseph-form covariance) for tracking, sensor fusion and navigation. Check: the error covariance converges to the DARE fixed point; RMSE → √P∞.
  • ZeqSAT — sat · computational complexity. Random 3-SAT by DPLL, with certified results and NP-hard scaling. Check: a certified satisfying assignment or resolution — it verifies, it does not claim to settle P vs NP.
  • ZeqZeta — zeta · analytic number theory. Riemann–Siegel Z(t) zeros. Check: the zeros sit on the ½-line and match the counting formula N(T) — it verifies zeros, it does not claim to prove the Riemann Hypothesis.

Quantum & physics frontiers​

  • ZeqSchrodinger — schrodinger · quantum mechanics. The time-independent Schrödinger eigenvalue problem — the bound-state energy levels behind atoms, quantum wells/dots, molecular vibration and the qubit, by a Sturm-bisection of the tridiagonal Hamiltonian. Check: the computed spectrum vs the exact analytic levels (harmonic oscillator ℏω(n+½), square well n²π²ℏ²/2mL²).
  • ZeqGauge — gauge · lattice gauge theory. A compact U(1) lattice. Check: the mean plaquette ⟨cos θ_P⟩ vs the exact strong/weak-coupling Bessel ratio I₁(β)/I₀(β).
  • ZeqBlackhole — blackhole · semiclassical gravity. A radial Schwarzschild geodesic — a particle falling toward the horizon. Check: the proper-time dilation dτ/dt = √(1 − r_s/r) along a deterministic infall.