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The equations on every envelope

Every computation seals a CKO envelope. Beyond the value, proof and ZEQOND receipt, the envelope carries the framework's core equations genuinely evaluated on the result that was just produced — not recorded as strings. Here is exactly what rides on it and under which JSON key.

Always present​

KO42 — the metric tensioner. operatorChain[0] is always KO42. It is the ϕ_c²·ΣC_k coupling term of the master equation carrying the 1.287 Hz HulyaPulse, and it is what makes every sealed state breathe by ±α.

R(t) — the Zeq Equation (HulyaPulse modulation). The universal proper-time modulation R(t) = S(t)·[1 + α·sin(2π·f_H·t)] is evaluated at the seal's Zeqond phase, with S(t) = |value| and α = 1.29×10⁻³. Fields: S_t, R_t, sin_term, pulse_hz. Because it is taken at the live Zeqond phase, two seals of the same value carry different R_t.

The 7-step protocol. protocol_steps — SELECT · BIND · VALIDATE · COMPUTE · VERIFY · PULSE · RETURN, where VERIFY carries the honest precision verdict (verified when precisionActual ≤ 0.001, else observable-differential).

The ZEQOND receipt. zeqond_receipt — pulse_frequency, time_unit, and the continuum-state fields the compute genuinely produced (temperature, displacement, stress, pressure, potential, …). See the receipt.

Present when the solver produced a field​

A solver that produces a trajectory, a spectrum or a profile also carries the two runtime equations and the master-equation term breakdown, evaluated on that field. A solver whose result is a single scalar with no field leaves these null — honestly, never fabricated.

HULYAS Master Equation. master_equation_terms — the six terms of □ϕ − μ²ϕ − λϕ³ − e^{−ϕ/ϕ_c} + ϕ_c²·ΣC_k = T^μ_μ + βF² + J_ext evaluated on the field: wave (□ϕ), mass, nonlinear, decay, coupling (KO42), stressEnergy (T), and a relative residual. See the master equation.

Functional Equation. functional_equation — E = P_ϕ · Z(M,R,δ,C,X) with P_ϕ = √⟨(dϕ/dt)²⟩ and Z = M·R·(1+C)·(1+X)·e^{−δt}. Fields: E, P_phi, Z, terms. See the functional equation.

Spectral-Topological. spectral_topological — the propagator Ψ(x,t) = ∭ K(x,x′,t,t′) ϕ dx′dt′, K = K_spectral·K_temporal·K_chaos, computed by a real DFT + statistics of the field: psi (Ψ), spectralEntropy (the K_spectral entropy H), dominantMode, coherence (K_temporal, pulse coherence), chaosMixing (K_chaos). A clean oscillation → low entropy, high coherence, low chaos, large Ψ; noise/chaos → the opposite.

Verified, not asserted​

These are real evaluations, deterministic and reproducible. A worked example: the same solver value sealed at two different Zeqonds carries two different R_t; a smooth diffusion field seals chaosMixing ≈ 0.07 while a turbulent one seals spectralEntropy ≈ 1.0; a static field seals E = 0 (no field momentum). Verify any envelope: POST /api/zeq/verify.