**Title** Zeta: A Deterministic Parameter-Free Architecture over the Thirteen-Adic Ring A deterministic, parameter-free alternative to statistical sequence models v. 7. 3 Author: Dávid Navrátil e-mail: david. navratil2016@gmail. com **Description** This preprint studies the algebra generated by a single integer matrix: the companion matrix of the Tribonacci cubic λ³ − λ² − λ − 1 in SL (3, Z). It is the only matrix that occurs; every further object is a power of it, a linear combination of its powers, or is built from it and its transpose. The work is carried out over the integers, the finite fields of thirteen and one hundred sixty-nine elements, the finite rings Z/13ᵏ, and the thirteen-adic integers. No real numbers, no complex numbers, and no metric notion appear: comparison of two states is by the thirteen-adic valuation alone, which counts agreeing digits. The working prime is derived rather than chosen. Thirteen is the smallest prime at which the cubic has splitting type (1, 2), the reduced matrix attains the maximal order q² − 1, and two is a quadratic nonresidue, so that the norm form of the residue extension is anisotropic. At that prime the generated algebra splits as a product of the residue field and its quadratic extension, and the state space of 13³ elements splits into pieces of rank one and rank two. Both idempotents are exhibited as outer products of eigenvectors with a dual basis, and all three eigenvectors follow from one formula. A second, independent splitting comes from the antisymmetric part of the matrix, and the two splittings share no common basis. The matrix has multiplicative order one hundred sixty-eight, a number reached from four independent definitions. Its cyclic group is given in full, with all sixteen subgroups listed by generator, order, and index. The same matrix arises a second time as the incidence matrix of a substitution on three letters, to which it is conjugate in SL (3, Z). Around this core the paper develops: the lattice of fields, with the decomposition of the working prime in each, up to a Galois extension of degree twenty-four; the ultrametric notions of valuation, depth, and geometric weight, where classes at a given depth are nested or disjoint and the smallest of three agreement counts always occurs at least twice; the combinatorial geometry of a tree of degree fourteen and an affine building of type Ã₂ whose spherical residue is thick of thickness fourteen and carries two thousand five hundred sixty-two chambers; and an exact transform with its admissible sizes and convolution rule. Groups usually described by a diagram are introduced by presentations instead, so that no angle is named anywhere. Every numerical statement was verified by exact computation over the integers and finite rings, with two consecutive runs producing identical output and no randomness anywhere. Figures are drawn as vector graphics directly from the computed data and contain no fitted curve or interpolation. Statements are separated throughout into those for which a map identifying the two sides is exhibited, and those that are equalities of counts only; readings that attach external names to the objects are marked as assignments rather than results. The preprint runs to 127 pages and contains 359 displayed equations, 234 numbered results each with a proof, and 45 figures. **Keywords** Tribonacci companion matrix; finite field; thirteen-adic integers; idempotent splitting; anisotropic torus; valuation depth; Bruhat-Tits tree; affine building; number theoretic transform; deterministic architecture **Mathematics Subject Classification** 15B33, 11T06, 11S15, 20G40, 11E88, 16K20, 20E42, 11R16
Dávid Navrátil (2026) studied this question.