The Dedekind-Gear Assembly: Primes as Rotors at the Golden Angle

by Pedro Henrique Corrêa Garcia · Opus 4.7 · 5 pp

Page 1 of 6· 4 stated

The Dedekind-Gear Assembly:
Primes as Rotors at the Golden Angle

Pedro Henrique Correa Garcia Joinville, Santa Catarina, Brazil garcia.pedro.wow@gmail.com · April 2026

Abstract

We present a mechanical model of the prime hierarchy in which each prime is a rotor with teeth offset from the previous rotor by the golden angle θ_g = 360^∘ / φ² ≈ 137.507764^∘. The simultaneous-engagement patterns of a k-rotor assembly are in bijection with the antichains of the Boolean lattice 2^[k], so the number of valid mechanical states is the Dedekind number D(k). We prove the bijection, identify the irrationality of φ as the unique property that makes the construction work, derive the Sperner upper bound on the configuration count, and use that bound to fix the size of a k ≈ 13 configuration database at roughly 2²³⁷⁰ states. We close by noting that the numerical value of θ_g and the value of the first-half ladder ceiling (137 = P₃₃ = 1/α) are the same fact.

Introduction

In Article 0 we introduced Dedekind numbers as the count of independent configurations at each level of the prime hierarchy, and sketched a mechanical interpretation in which primes are rotors at the golden angle. The present article develops that interpretation in full: it proves the bijection between mechanical states and antichains, establishes the Sperner bound on the state count, and uses the bound to derive an explicit size estimate for a k = 13 configuration database.

The model is not a visualization of an abstract combinatorial fact. It is a physical realization of the fact, and the physical realization explains why the count is exactly D(k) and not some other number.

Unrecognized environment: keyidea

Primes are rotors, the golden angle is the unique offset that keeps them non-resonant, and the Dedekind numbers count the states that the as

The gear assembly

Definition

A prime rotor of index i is a circle of circumference 1 with i equally spaced teeth, rotated by an angle that depends on the index according to the offset rule below.

Definition

The golden angle is

θ_g = 2π⁄φ² = 360^∘⁄φ² ≈ 137.507764^∘, eq:goldenangle

where φ = (1 + √(5))/2 is the golden ratio. The rotor of index i is offset from the reference position by i · θ_g.

Definition

The gear assembly G_k of order k is the ordered tuple of prime rotors (1, …, k) with offsets (θ_g, 2θ_g, …, kθ_g).

The rotors do not physically interact: they rotate independently. But their teeth sweep past a common reference line, and the simultaneous engagement pattern at any instant is the set of rotors whose teeth are at the reference line at that moment.

The no-resonance theorem

The key property of the golden angle is that it prevents any finite rational reconciliation between rotors.

Theorem

For any two distinct indices i ≠ j and any pair of non-zero integers (m, n), the rotors i and j under golden-angle offsets never satisfy m i θ_g = n j θ_g 2π.