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Mathematical Methods operators

16 operators in the mathematical_methods category of the live registry. Each is a named formula you can compose inside a state contract or call directly through POST /api/zeq/compute. KO42 is always on; add up to three more per call (total ≤ 4), per the 7-step protocol.

OperatorDescriptionEquation
ECHO0Resonant kinship operator: inverse-distance golden ratio coupling between Zeq entities.K_r = (1/N) · Σ ‖φ - φ_k‖⁻¹ · sin(2π·1.287·t + θ_k)
ECHO1Temporal harmony synchronizer: exponentially stabilized sinc-pulse at 1.287 Hz.τ_sync = τ₀ · exp(-β_T ∫|∇φ|² dt) · sinc(2π·1.287·t)
ECHO3Pulse-coherent memory matrix: time-sampled correlation function at HulyaPulse intervals.M_ij(t) = Σ φ(t-nT) · C_ij[n] · e^(-|n|/τ_c), T=1/1.287
ECHO4Cross-domain resonance bridge: an integral coupling three field channels (physical, computational, and a third) at the HulyaPulse tick.B_x = ∫ φ_phys · φ_comp · φ_3 · δ(2π·1.287·t) dt
IO7Gamma function evaluation: integral of x^6 * e^(-x) from 0 to infinity equals 6! = 720.I_7 = \int_0^\infty x^6 e^{-x}dx = 720
LYRA1Lyra pulse gain operator: signal amplification modulated by the 1.287 Hz HulyaPulse.LYRA01 = G_φ(t) = G_0 · (1 + ρ·sin(2π·1.287·t)) · φ_c^{-1}
Nyx1Zeq temporal unit (Zeqond): fundamental time quantum of 777,000,777 nanoseconds (0.777000777 s exact).ZEQOND = Duration = Duration::from₋nanos(777₋000₋777)
Nyx2Cosmic origin epoch: Big Bang timestamp as the zero reference for Zeq universal time.BIG₋BANG₋EPOCH = u64 = 0
Nyx3Universal Time Pulse counter: atomic counter tracking elapsed HulyaPulse cycles.UTP₋COUNTER = AtomicU64 = AtomicU64::new(0)
PS_F1313th Fibonacci number F(13) = 233 from the recursive sequence.F_{13} = 233
PS_F55th Fibonacci number F(5) = 5 from the recursive sequence.F_5 = 5
PS_H3Third Hermite polynomial H3(x) = 8x³ - 12x for quantum harmonic oscillator solutions.H_3(x) = 8x^3 - 12x
UCO10Multiscale correlation operator: power-law decay with scaling function for critical phenomena.C(r,t) = \langle O(\vec{x},t) O(\vec{x}+\vec{r},t) \rangle \sim r^{-(d-2+\eta)} f(r/\xi(t))
UCO11Universality class operator: set of critical exponents characterizing a phase transition.\mathcal{U} = {\alpha, \beta, \gamma, \delta, \eta, \nu} \quad \text{critical exponents}
UCO12Framework completion operator: product of Gaussian completion factors with cross-entropy bonus.F_{complete} = \prod_{i=1}^N \left[1 - e^{-(O_i - O_{i,min})^2/2\sigma_i^2}\right] \cdot \left[1 + \frac{H_{cross}}{H_{total}}\right]
UCO9Fisher information metric on the space of probability distributions for information geometry.ds^2 = \sum_{ij} g_{ij}(\theta) d\theta^i d\theta^j, \quad g_{ij}(\theta) = \mathbb{E}\left[\frac{\partial \ln p}{\partial \theta^i} \frac{\partial \ln p}{\partial \theta^j}\right]

Compute with one of these​

curl -sS -X POST https://zeqsdk.com/api/zeq/compute \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{"operators":["ECHO0"],"inputs":{}}'

The response carries the bare physics value, its unit and uncertainty, the generated master equation, and a signed envelope you can verify on any node.

See also​