A geometric reading of the Fermat finiteness conjecture via a deterministic prime machine

Description: This paper presents a geometric reading of the classical conjecture that only finitely many Fermat primes exist. The conjecture states that the sequence Fₙ = 2^ (2ⁿ) + 1 contains only the five known primes 3, 5, 17, 257, 65537, and that no further Fermat prime can be found beyond F₄. No classical proof of this finiteness is known. The approach is constructive and geometric. We first introduce a deterministic prime machine: for each primorial Mₖ = p₁ p₂. . . pₖ, the elementary abelian group Gₖ = Z₂^ (k-1) acts on the unit group of Mₖ by sign-flipping the residues modulo each odd prime. Orbit minima in the interval (pₖ, pₖ²] are the output of the machine. Two foundational results are proved in full classical rigour: 1. The machine is complete: every prime greater than 13 appears exactly once as an orbit minimum at the appropriate level. 2. The machine is fully symmetric: each orbit is a torsor under Gₖ, the mirror involution a -> Mₖ - a coincides with the diagonal group element and commutes with the whole Gₖ-action, and every orbit is graph-isomorphic to the hypercube Q_ (k-1). The paper then translates the Fermat finiteness conjecture into the language of the machine at the smallest Fermat-pure base ring M₃ = 30. The unit group U₃0 has eight elements; the orbit decomposition under G₃ = Z₂² matches the f-vector (8, 24, 32, 16) of the four-dimensional cross-polytope (the 16-cell), with stages vertices, edges, triangles, cells. A self-reference identity Fₙ = (F_ (n-1) - 1) ² + 1 provides a depth labelling on the Fermat chain. The known Fermat primes fill exactly the four dimensional stages of the 16-cell: F₀ at the generator level (base ring), F₁ on vertices, F₂ on edges (mirror pair 17 13), F₃ on triangles, F₄ on cells. Within this framework we give a structural argument that no further stage can be supported, so the Fermat chain is bounded by length five. Three structural pillars carry the argument, all stated and proved as strict theorems: Pillar 1 (Fermat-purity of the base ring): M₃ = 30 is the unique largest primorial whose non-trivial generators are all Fermat primes. Status: classical. Pillar 2 (Collapse onto class 17 modulo 30): For every n at least 2, Fₙ is congruent to 17 modulo 30. The proof is an explicit induction on 16ᵐ congruent 16 modulo 30. Status: proved. Pillar 3 (Dimensional uniqueness of the 16-cell as carrier): Across all dimensions d, only d = 4 satisfies vertexcount (betad) = phi (M_ (d-1) ), namely 8 = 8 at d = 4. For every other d the two quantities diverge, and from d = 5 onward phi (M_ (d-1) ) grows at least geometrically with factor 4 while 2d grows linearly. The proof is an explicit induction. The cross-polytope carrier exists as a faithful geometric realisation of the unit group of the base ring in exactly one dimension. Status: proved. The structural finiteness conclusion follows from these three pillars together with one named geometric reading (Reading A): each Fermat prime is localised on one f-vector component of the 16-cell in monotone k-stage order. Reading A decomposes into two sub-statements: Reading A. 1 (distinct Fermat primes live on distinct f-vector components): derivable from the BahnMin construction of the machine, since the k (n) values for distinct Fₙ form a strictly monotone sequence and each new k-stage adds a new incidence layer. Reading A. 2 (the monotone k-to-f correspondence): a named geometric modelling commitment. A direct derivation from the machine architecture is left as future work. Under these statements, the finiteness of Fermat primes follows by an exact count of f-vector components: four in d = 4, plus F₀ as base generator, total five. The author is aware that this is not a classical proof of the Fermat finiteness conjecture. It is a proof under one explicitly named geometric modelling commitment (Reading A. 2). The three pillars and Reading A. 1 are derivable; Reading A. 2 alone carries the remaining geometric assumption. The author leaves the final judgement on the status of the framework to the mathematical community: whether Reading A. 2 is acceptable as a structural modelling commitment, or whether it must be replaced by an independent classical statement before the finiteness conclusion can be considered established. A Gauss-Wantzel bridge is included as a remark: the five known Fermat primes are precisely the odd primes admitting constructible regular polygons. The finite list of Fermat primes that the machine can accommodate matches the finite list of Fermat primes that classical geometric construction can use. Keywords: Fermat primes, Fermat finiteness conjecture, primorial, unit group, orbit minima, prime machine, 16-cell, cross-polytope, hypercube, Cayley graph, Chinese Remainder Theorem, Gauss-Wantzel, geometric number theory, dimensional uniqueness