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

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

Authors:

Navrátil, Dávid 

e-mail:

david.navratil2016@gmail.com 

X:

https://x.com/Dado50449061

Publication Date:

2026-06-18 

Resource Type:

Preprint 

Language:

English 

License:

Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) 

Funding: Independent research. No external funding.

I've present Zeta, an artificial intelligence architecture operating exclusively over the ring of 13-adic integers Z_13[η]. The architecture derives all computation from a single 3×3 integer matrix T_3 ∈ SL(3,Z) with characteristic polynomial χ_T3(λ) = λ³−λ²−λ−1. No real numbers, no complex numbers, no floating-point operations, no gradient descent, no softmax, and no dot-products exist anywhere in the system.

The single attention kernel is G(i,j) = Trib(|i−j|−1) · η^(−v_13(|i−j|)) ∈ Z_13[η], requiring two int64 lookup-table reads. Three algebraic facts determine the architecture without free parameters: (i) the Kummer condition 3|(p−1)=12 forces the cube root of unity ω=3 into F_13 natively; (ii) the discriminant disc(χ_T3)=−2²×11 identifies p=11 as the partially ramified prime used as a built-in anomaly detector; (iii) the Galois group S_3 = Z_2 ⋉ Z_3 of χ_T3 at p=13 yields six algebraically distinct attention heads without training.

We compute and show that the strong triangle inequality by two lines of integer divisibility, derive exact ultrametric ball structure (nested or disjoint, never partially overlapping), and construct the full Bruhat-Tits building B(GL(3,Q_13)) with 2562 chambers. The Tits tree T_13 of degree 14 replaces learned positional encodings. The Cayley-Hamilton table replaces learned embeddings. Hensel lifting replaces layer normalisation. The building retraction replaces the feed-forward network.

The architecture is planned to be implemented in C++ using six precomputed integer lookup tables totalling 462 KB. All eight self-tests pass in personal testing. Token reconstruction achieves 100% accuracy without training. The paper includes complete algebraic proofs, experimental validation, and topological memory extensions (braid group B_3, Burau representation, skyrmions, and the POCA automaton).

Mathematical guarantees: exact arithmetic (no rounding errors), deterministic inference (no random seeds), algebraic verifiability (every component is a theorem). All numerical claims are independently verified by finite arithmetic in F_13 and F_169. No machine learning training was performed. The 100% reconstruction result holds specifically for T3-generated sequences.

Planned further continuations of the documentation and expansion and addition of certain areas.

For any questions and future collaborations, contact me at X or email.

Communities:

computer-science

mathematics

artificial-intelligence

number-theory

Creative Commons Attribution Non Commercial 4.0 International (CC BY 4.0)