March 3, 2026Open Access

The Unified Field of Singularities: Geometric Formalization of Riemann, Goldbach, and Twin Primes via Pencil Projections

Key Points

Goldbach's Conjecture is presented as a reflexive symmetry, highlighting its geometric underpinning.
The observed structural proximity invariance of Twin Primes reveals connections between prime distributions and geometrical constructs.
Analysis employs the Pencil Projection technique, enabling a novel deterministic geometric proof regarding singularities.
The findings suggest that the Riemann hypothesis should be viewed as a structural necessity rather than a probabilistic outcome.

Abstract

This work extends the Information Lattice Field Theory (ILFT) by introducing the ”Pencil Projection” methodology to formalize a unified field of prime singularities. The research focuses on the Modulo-12 Manifold, demonstrating thatGoldbach’s Conjecture emerges as a reflexive symmetry (Theorem 3.3) and TwinPrimes as a structural proximity invariance (Theorem 3.2). By mapping the criticalaxis through resonance and symmetry, the Pencil method provides a deterministic geometric proof for the distribution of singularities, identifying the Riemannhypothesis not as a probabilistic occurrence, but as a necessity of the lattice’sstructural rigidity

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Cite This Study

Silvio Gabbianelli (Wed,) studied this question.

synapsesocial.com/papers/69a75ce3c6e9836116a26229 https://doi.org/https://doi.org/10.5281/zenodo.18399848

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