From Faraday and Maxwell to Neural Cycle Topology: The Operations of a Coupled Conductive-Loop Lattice

Abstract

We derive the governing equation of neural cycle topology (NCT) computing from Maxwell’s equations and set what follows from it against measurements of a 362-cavity lattice read by a pickup on every cavity. In the magnetoquasistatic limit, charge conservation confines the conductor currents to the cycle space of the network graph, and Faraday’s law integrated around each cycle gives L di/dt + R i = −dΦ/dt, with R and L the Gram matrices of a cycle basis under the resistive and the Neumann inductive inner products and Φ the flux the drive threads through each cavity. The lattice’s rates run from 0.223 to 4.812 s⁻¹, so at every driven line, from 1.618 to 60 Hz, it operates in the inductive regime, where the equation reduces to î = −L⁻¹Φ̂ − (j/ω)L⁻¹RL⁻¹Φ̂. Every cavity moves at once: 84.0% of cavity responses begin within 5 ms of the strike. The operator is real to within λ/ω, so a phase relation fixed across inputs can sit only at 0° or 180°; the five steadiest pairs sit at −180.0°, +0.3°, 179.8°, +0.2° and −0.5°. Superposition is exact: a type’s ringing with a 204-type key superposed equals its ringing alone at 1.0000 in pattern and amplitude for all 204 types. Measured on the lattice: scaling, filtering, holding, forgetting, phase lock, direction into order, convergence, divergence, fast decision and selection. A binary processor reproduces one 2 s presentation in about 5 × 10⁸ multiply-adds; the lattice does it by ringing for 2 s.