Zeta: The First p-Adic Integer Artificial Intelligence - Update

Zeta: The First p-Adic Integer Artificial Intelligence

Update v7.2: Including:

Every structural result is accompanied by a "Role in Zeta" paragraph explaining its exact function in the inference engine, so the paper is simultaneously a mathematical proof and an implementation specification. 

Abstract.

Zeta is a deterministic, parameter-free sequence architecture whose entire state space, arithmetic, geometry, and dynamics are generated by a single companion matrix T_3 \in SL(3,\mathbb{Z}) with characteristic polynomial \lambda^3 - \lambda^2 - \lambda - 1. Every quantity is an integer or an element of a finite ring. The construction uses no real numbers, no complex numbers, no floating point, and no Euclidean metric. At the working prime p=13 the cubic has splitting type (1,2), so the core ring \mathbb{Z}{13}[\eta] decomposes into a rank-one syntactic channel on a Tits tree and a rank-two semantic channel on an A_2 building. Attention is ring multiplication, normalisation is a Fermat inverse, positional encoding is the torus clock (3+2\sqrt{2})^k, the multi-scale renormalisation is a 13-ary tree, memory is chamber frequency, and the number-theoretic transform uses powers of T_3 as twiddle factors. The text states more than 120 numbered equations, each verified by exact symbolic computation, and describes the engine as a body of exact integer operations with measured behaviour and limits reported.

Keywords: p-adic artificial intelligence, integer AI, deterministic sequence model, non-Euclidean geometry, Tits tree, Bruhat-Tits building, Cayley-Hamilton decomposition, Tribonacci recurrence, number-theoretic transform, parameter-free architecture, companion matrix, finite-field algebra, ultrametric attention

Highlights:

- Zero learned parameters; zero floating-point operations

- All attention weights are fixed by algebraic geometry (ball lookup on a Tits tree)

- Normalisation via Hensel lifting and Fermat inversion, not LayerNorm

- Positional encoding via anisotropic torus \mathbb{T} \cong \mathbb{Z}/14, not sinusoids

- Memory as ANCHOR orbital resonance (deduction, not gradient descent)

- Complete verification: 15 structural identities checked by exact computation

- Comparison table: every Transformer mechanism has a precise integer counterpart

Contents: 10 parts covering the generator, ring splitting, spectral decomposition, ultrametric geometry, A_2 building symmetry, number-theoretic transform, CRT engine, MERA pyramid, ANCHOR memory, the inference kernel, p-adic state analogues, and full verification. Appendices include the Cayley-Hamilton coefficient table, Teichmüller traces, spectral data, chamber data, and an end-to-end example.

Author: Dávid Navrátil (Independent Researcher)

e-mail: david.navratil2016@gmail.com

License: CC-BY 4.0

Date: July 6, 2026