The Icosahedral Digital Boundary: Why Prime Structure Breaks at e^142 and Creation Stops at e^184
Abstract
We show that icosahedral geometry creates two fundamental boundaries that bracket physical reality. At the digital boundary (e^142), mod 3 collapses and the Ramanujan tension transforms from −1/12 to −1/4. At gravitational closure (e^184), mod 5 locks out and tension transforms from −1/12 to −5/12. The operating physics lives between these boundaries where both mod 3 and mod 5 function, with tension −1/12 = −5/|A₅|. The prime product digit gaps discovered in our companion paper occur at these scales: staggered gaps requiring factor 3 at e^142, coincident gaps requiring factor 15 = 3 × 5 at e^184. We derive that the gap 55 − 53 = 2 equals the Euler characteristic χ because the icosahedron's triangular faces have f = 3 = D edges, making |v − f| = |5 − 3| = χ. The dimensional constant D = 3 plays dual roles: enabling structure (mod 3) and ensuring finiteness (mod 5). The boundary progression 137 → 142 → 184 follows icosahedral constants: span v = 5 (vertex degree), then V + E = 42 (vertices plus edges).
Fields of Study & Subjects
Quasicrystal Structures and Properties
Digital Image Processing Techniques
Mathematical Approximation and Integration
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 29 January 2026
- Document type
- Preprint