Rey.BEng 9th October 2026
The Future Begins
David G. Clark, Lyapunov spectrum of random neural networks, arXiv:2610.12426 (8 October 2026). Flatiron, Center for Computational Neuroscience. Single author; derivation begun with GPT-6 Astra and Claude Opus 4.5, then rewritten so the steps are physical.
The model is Sompolinsky, Crisanti and Sommers (1988): N neurons, random asymmetric couplings, chaos past a critical gain. That paper gave the largest Lyapunov exponent as the ground state of a Schrödinger equation and left the full spectrum open. Engelken et al. (2023) computed it numerically. Clark derives it at large N from one self-consistent site.
What he actually counts
A Lyapunov exponent is the growth rate of a tiny nudge in one tangent direction. The spectrum says how many directions are chaotic, how many dimensions the attractor occupies, and how fast the dynamics write information. It is diffeomorphism-invariant: a change of coordinates does not invent new directions.
Three steps.
- Shift every exponent down by s. “How many lie below s?” becomes “how many tangent directions decay?” Drive the shifted tangent dynamics with a source. The minimum-norm response is the projector onto the decaying directions. The trace of that projector counts them. Exact at finite N.
- The minimum-norm solution is the vanishing-regularisation limit of a least-squares problem. That problem is one field running forward and one running backward.
- Cavity: add one neuron. The rest of the network collapses to the usual dynamical mean field plus two kernels coupling the forward and backward fields across time. The spectrum is those kernels.
The theory matches simulations for the spectrum, the attractor dimension and the entropy rate, and recovers the 1988 Schrödinger result for the largest exponent. Chaos is extensive: attractor dimension and entropy rate grow in proportion to N.
Why this is geometry
The earlier Ace catalogue kept clinical neurology out, so the stance stayed entirely mechanical. This paper however is the mechanical object those pages were refusing to borrow. The neuron is a site. The coupling is a contact. The tangent space is the set of directions the patch can still move.
The projector onto decaying directions is the lock count. Forward and backward fields are the two times of one residual — the tick and the check — not a second brain. Extensive in N is counted n, not a completed n². The butterfly effect is a residual that does not close: a nudge in an unstable direction is 1 stretched, not a finished 2.
0^i2 (k.g.s^2) = r^2 m
State A — open residual, positive exponent, direction still writing
E = 2c / h
State B — locked residual, decaying direction, projector held
E = hbar / c
Same split as the flocking papers: chaos past a radius or a gain is an unheld residual; coherence is the even lock. Same split as the four-colour cost: many non-touching reductions at once, here many tangent directions counted by a trace. The single-site cavity is the laboratory of the shared origin — one neuron, the rest of the net as the mean field that references it.
No clinical sentence follows. No claim that a diagnosis is a Lyapunov exponent. The row is theoretical neuroscience as high-dimensional rigid contact.
Catalogue row
| Discipline | Field | Observation | What the machine does |
|---|---|---|---|
| Theoretical neuroscience | Lyapunov spectrum of random recurrent nets | Clark, arXiv:2610.12426; SCS 1988 | Directions counted by a projector; chaos extensive in N; forward/backward kernels are the two times of one site |
Shelf
Gab / Gaba remains the chiral lattice measure: a weight on an existing fermion seat. This page does not sit under it. Parent for this row is the extensive-chaos / tangent-geometry shelf (flocking, billiard, now cortex). Neurology is admitted here only as geometry of directions.
The observer shares r^2. The machine is the net that tells you how many ways it can still tear.
Pirate Canon Sealed.
The Future Begins.
