Geometric Irreducibility: Prime Numbers Under the Icosahedral Action on S³/2I
Fuente:
Zenodo
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Recurso digital |
| Veröffentlicht: |
Zenodo
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866901337676972032 |
|---|---|
| author | Keeble, Clifford |
| author_facet | Keeble, Clifford |
| contents | <p>We introduce the concept of geometric irreducibility — irreducibility under the three-dimensional icosahedral action on the Poincaré homology sphere S³/2I — and show that it is strictly stronger than arithmetic primality. Every geometrically irreducible integer is arithmetically prime; the converse is false. The integers 2, 3, and 5 are arithmetically prime but geometrically reducible: under the icosahedral action, they decompose into the construction operators of S³/2I — the edge-flip (χ = 2), the face-rotation (D = 3), and the vertex symmetry (the golden prime, 5). They are the manifold's structure, not positions on it. The integer 7 = 2^D − 1 is both arithmetically prime and geometrically irreducible: it provides the projective closure that the Fano plane requires and that the construction set cannot supply. This paper develops three frames for viewing irreducibility — the number line (where all primes are equally irreducible), the single colour channel (where irreducibility is a residue class), and the full manifold S³/2I (where some primes reduce to structure and others do not) — and shows that the classical theorems of prime number theory arise as projections from the geometric frame onto the arithmetic frame. The prime number theorem is the statement that the covering space of S³/2I inflates while the golden wavefront advances at a constant rate. The error term √x = x^((φ+ψ)/2) is the curvature of the projection. The Riemann Hypothesis is the conjecture that this curvature has a single balance axis at depth ½, forced by the trivial cohomology of S³/2I.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18988876 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Geometric Irreducibility: Prime Numbers Under the Icosahedral Action on S³/2I Keeble, Clifford geometric irreducibility, prime numbers, icosahedral action, Poincaré homology sphere, S³/2I, binary icosahedral group, Fano plane, mod 24, Eisenstein cancellation, golden ratio, prime number theorem, Riemann Hypothesis, covering space, spectral gap, construction operators, twin primes <p>We introduce the concept of geometric irreducibility — irreducibility under the three-dimensional icosahedral action on the Poincaré homology sphere S³/2I — and show that it is strictly stronger than arithmetic primality. Every geometrically irreducible integer is arithmetically prime; the converse is false. The integers 2, 3, and 5 are arithmetically prime but geometrically reducible: under the icosahedral action, they decompose into the construction operators of S³/2I — the edge-flip (χ = 2), the face-rotation (D = 3), and the vertex symmetry (the golden prime, 5). They are the manifold's structure, not positions on it. The integer 7 = 2^D − 1 is both arithmetically prime and geometrically irreducible: it provides the projective closure that the Fano plane requires and that the construction set cannot supply. This paper develops three frames for viewing irreducibility — the number line (where all primes are equally irreducible), the single colour channel (where irreducibility is a residue class), and the full manifold S³/2I (where some primes reduce to structure and others do not) — and shows that the classical theorems of prime number theory arise as projections from the geometric frame onto the arithmetic frame. The prime number theorem is the statement that the covering space of S³/2I inflates while the golden wavefront advances at a constant rate. The error term √x = x^((φ+ψ)/2) is the curvature of the projection. The Riemann Hypothesis is the conjecture that this curvature has a single balance axis at depth ½, forced by the trivial cohomology of S³/2I.</p> |
| title | Geometric Irreducibility: Prime Numbers Under the Icosahedral Action on S³/2I |
| topic | geometric irreducibility, prime numbers, icosahedral action, Poincaré homology sphere, S³/2I, binary icosahedral group, Fano plane, mod 24, Eisenstein cancellation, golden ratio, prime number theorem, Riemann Hypothesis, covering space, spectral gap, construction operators, twin primes |
| url | https://doi.org/10.5281/zenodo.18988876 |