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

This is a supplementary section appended to the Zenodo record of version v2.

Version v3 of the manuscript closes the single remaining open status item of v2.

In v2, the finiteness argument rested on three strictly proved pillars together with one named geometric reading (Reading A), of which Reading A.2 (the monotone k-to-f correspondence) was declared a yellow geometric modelling commitment. The author left the question whether Reading A.2 could be derived from the machine architecture as a separate future task.

Version v3 settles this question in the affirmative, internal to the framework. What was Reading A.2 in v2 is replaced in v3 by Theorem A: under Pillars 1, 2, 3 and the self-reference identity F_n = (F_(n-1) - 1)^2 + 1, the assignment F_n -> f_(n-1) is the unique order-preserving injection of the Fermat chain into the f-vector of the 16-cell. The proof is given in five steps in Section 4.4 of v3:

Step 1 (Localisation): by Pillar 3, the carrier is uniquely the 16-cell, so each F_n with n at least 1 lives on one of the four f-components f_0, f_1, f_2, f_3.

Step 2 (Disjointness): the BahnMin construction assigns each prime to its smallest k-stage. The Fermat primes occupy strictly monotone k-stages (k(F_1) = 2, k(F_2) = 3, k(F_3) = 7, k(F_4) = 55), and distinct k-stages correspond to distinct f-vector components.

Step 3 (Count): the assignment is an injection into a four-element set, so at most four Fermat primes beyond F_0 can be accommodated.

Step 4 (Order): the self-reference identity makes the depth index n an intrinsic invariant of the Fermat tower, and depth n is in one-to-one monotone correspondence with the k-stage at which F_n enters the machine.

Step 5 (Assignment): the f-vector carries a canonical incidence hierarchy f_0 less f_1 less f_2 less f_3 (vertices contained in edges, edges in faces, faces in cells). An order-preserving injection from a four-element ordered set into a four-element ordered set is uniquely the identity. The unique assignment is F_1 -> f_0, F_2 -> f_1, F_3 -> f_2, F_4 -> f_3.

The updated status table contains four green entries:

Pillar 1 (Fermat-purity of M_3) - green (classical)
Pillar 2 (collapse onto class 17 mod 30) - green (proved)
Pillar 3 (16-cell as unique carrier in d = 4) - green (proved)
Theorem A (unique ordering F_n -> f_(n-1)) - green (consequence of Pillars 1 through 3 plus self-reference)

No yellow modelling commitment remains.

A methodological note is added before Theorem A. The prime machine of this paper is not an algebraic abstraction and not an analogy to an existing structure. It originates from a purely geometric construction, the matchstick game, in which prime numbers emerge from incidence rules between matchsticks on a finite grid. The algebraic description in terms of G_k = Z_2^(k-1) acting on U_(M_k) is a faithful encoding of this geometric construction, not the construction itself. Geometry is primary; algebra is consequence.

What remains open after v3, and is honestly named in the manuscript, is the question whether the dimensional uniqueness of d = 4 (Pillar 3) admits an independent classical interpretation outside the machine framework.

The author leaves the final judgement on the framework, and on the validity of the chain Pillars 1 plus 2 plus 3 plus Theorem A implies finiteness of Fermat primes, to the mathematical community.

Description V2:

This paper presents a geometric reading of the classical conjecture that only finitely many Fermat primes exist. The conjecture states that the sequence F_n = 2^(2^n) + 1 contains only the five known primes 3, 5, 17, 257, 65537, and that no further Fermat prime can be found beyond F_4. No classical proof of this finiteness is known.

The approach is constructive and geometric. We first introduce a deterministic prime machine: for each primorial M_k = p_1 p_2 ... p_k, the elementary abelian group G_k = Z_2^(k-1) acts on the unit group of M_k by sign-flipping the residues modulo each odd prime. Orbit minima in the interval (p_k, p_k^2] 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 G_k, the mirror involution a -> M_k - a coincides with the diagonal group element and commutes with the whole G_k-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 M_3 = 30. The unit group U_30 has eight elements; the orbit decomposition under G_3 = Z_2^2 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 F_n = (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: F_0 at the generator level (base ring), F_1 on vertices, F_2 on edges (mirror pair 17 <-> 13), F_3 on triangles, F_4 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): M_3 = 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, F_n is congruent to 17 modulo 30. The proof is an explicit induction on 16^m congruent 16 modulo 30. Status: proved.

Pillar 3 (Dimensional uniqueness of the 16-cell as carrier): Across all dimensions d, only d = 4 satisfies vertex_count(beta_d) = 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 F_n 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 F_0 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