A geometric reading of the Fermat finiteness conjecture via a deterministic prime machine
Year: 2026
Type: preprint
Abstract: A Geometric Reading of the Fermat Finiteness Conjecture via a Deterministic Prime Machine Please Note: This paper does not claim a traditional analytic or algebraic proof of the classical Fermat finiteness conjecture on \(\mathbb{Z}\). Instead, it translates the entire problem into a rigid geometric blueprint, showing that the finiteness of Fermat primes is an absolute architectural boundary of a four-dimensional polytope. Pierre de Fermat famously conjectured in 1640 that all integers of the form \(F_n = 2^{2^n} + 1\) are prime — a claim refuted by Euler in 1732. Since then, the conjecture that only finitely many Fermat primes exist has remained open. This foundational paper delivers a completely new paradigm, rephrasing the finiteness of the Fermat chain as a strict structural limit inside a deterministic, fully symmetric "Prime Machine" \(\mathcal{M}_{k}\) based on orbit minima. By analyzing the smallest Fermat-pure base ring M₃ = 30, the paper demonstrates that the decomposition of Fermat primes maps injectively and uniquely onto the faces of a regular four-dimensional cross-polytope: the 16-cell. 🏛️ The Three Strict Structural Pillars of the Framework: Pillar 1 (Fermat-Purity of the Base Ring): Proves that the primorial M₃ = 30 is the unique largest base ring whose non-trivial generators \(\{3, 5\}\) are exclusively Fermat primes. Any extension to higher primorials (like M₄ = 210) systematically introduces non-Fermat generators. Pillar 2 (The Collapse onto Class 17): Proves inductively via the Chinese Remainder Theorem that the entire infinite tail of the Fermat sequence (\(F_{n}\) for all n ≥ 2) collapses into a single, highly symmetrical mirror-orbit \(\{17, 13\}\) modulo 30. Pillar 3 (Dimensional Uniqueness of the Carrier): Proves an exact inductive theorem showing that d = 4 is the only dimension across all space where the vertex count of a cross-polytope 2d equals the order \(\varphi(M_{d-1})\) of the unit group. The 16-cell is the mathematically forced, unique geometric carrier of the machine. 🧩 Theorem A & The Structural Length-Five Ceiling By combining these three proved pillars with the classical self-reference identity \(F_n = (F_{n-1} - 1)^2 + 1\), the paper derives a remarkable geometric localization (Theorem A): Each successive Fermat prime \(F_{n}\) (for n ≥ 1) functions as a depth label that fills exactly one unique component of the 16-cell's geometry (Stage 0: Vertices → F₁, Stage 1: Edges → F₂, Stage 2: Triangles → F₃, and Stage 3: Cells → F₄). Because the regular 16-cell has an f-vector of exactly (8, 24, 32, 16), it provides precisely four dimensional layers. A fifth stage (f₄) cannot mathematically exist within a 4-polytope. Reusing components is strictly barred by Theorem A, and extending the dimension to d ≥ 5 destroys the necessary unit-group mapping. Consequently, within this geometric reading, the Fermat chain is rigidly capped at a length of exactly five: F₀, F₁, F₂, F₃, F₄. The arithmetic collapse of F₅ into a composite number emerges not as a random twist of number theory, but as the exact point where the underlying polytope architecture runs out of dimensions. File Content: Full peer-review ready PDF containing complete structural proofs, the complete f-vector incidence mapping matrix, rigorous induction bounds for primorial growth, and an exact cross-disciplinary connection to the constructibility of regular polygons via the Gauss-Wantzel theorem. Keywords: Fermat primes, finiteness conjecture, prime machine, cross-polytope, 16-cell, f-vector, self-reference operator, discrete geometry, unit group symmetries, geometric number theory.
Author: Krause, Thomas
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7164733873
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