PrimSpace Research Lab v4.0 — Primorial Field Dynamics: Phase Weight, Gap Geometry, and a New Language for Prime Numbers

PrimSpace is a mathematical framework that treats prime numbers not as isolated objects to be found, but as boundary states in a dynamical field defined over the natural numbers. Every natural number is embedded into a discrete torus via modular coordinates, and the resulting field structure is studied through the tools of operator theory, differential geometry, and Fourier analysis. This release (v4. 0) introduces a new vocabulary for prime number theory — one borrowed from physics and geometry, but made mathematically precise: - Phase weight Psiₖ (n, L) — accumulated field tension; the area under the phase curve- Phase pressure Pₖ (n) — local gradient of Psiₖ; predicts prime/gap transitions with 100% accuracy before n=131- Elasticity Eₖ (n, L) — how "stretched" the space is relative to its mean- Representational saturation Rₖ (n, L) — how close the torus is to full capacity- Gaussian curvature K (n) — saddle surface geometry of the phase space manifold KEY RESULTS: Theorem 4 (Gap Minimum Determination): The minimum possible prime gap following p is determined exclusively by p mod Mₖ — the torus coordinate of p. Exactly 15 residue classes mod 210 admit twin primes; all others are provably excluded. The twin prime condition p+1 = 0 (mod 6) is a direct consequence of the CRT embedding. Natural-to-pressure transition at n* = 131: Phase pressure correctly predicts prime positions in 100% of cases for n = 131 ("pressure-driven zone"). Conjecture 7: n* ~ p₊+₁^k/2. The 137 observation: 137 is the first prime in the pressure-driven zone with a strong correct phase pressure signal (+0. 02055). Its torus coordinate both permits and predicts this — a new geometric perspective on a number long considered mysterious. Conjecture 6 (Saddle Surface): K (n) ~ -C/ (ln n) ² * n — hyperbolic curvature in n-direction, parabolic in Psi-direction. The Psiₖ spiral is a torus geodesic projection; wave crests before primes are hyperbolic divergences. Cryptographic application: alphaₖ pre-sieve reduces Miller-Rabin candidates by ~77% with 0% false-negative rate. Cohen's d = 2. 59 (k=6) separation between primes and Carmichael numbers. The interactive Research Lab (single HTML file, runs in any browser, no installation) provides 9 visualization panels: alpha-Field, phi-Phase, Delta-Gap, Crypto, QFT, Psi-Weight, rho-Field, Theory, Export. This release is part of a series: - DVFM v1. 1: https: //doi. org/10. 5281/zenodo. 17675025- GPM: https: //doi. org/10. 5281/zenodo. 19692647- PrimSpace v1. 0: https: //doi. org/10. 5281/zenodo. 19698943- PrimSpace v3. 0 (direct predecessor): https: //doi. org/10. 5281/zenodo. 19794151 ## LICENSES (dual licensing — Zenodo supports multiple) | File | License ||------|---------|| `PrimSpaceResearchLabᵥ4₀. html` | Apache License 2. 0 (software) || `PrimSpacePressureTransition. html` | Apache License 2. 0 (software) || `READMEPrimSpaceᵥ4₀EN. md` | CC BY 4. 0 (document) || `PrimSpacePhaseWeightFrameworkᵥ1₀EN. md` | CC BY 4. 0 (document) || `PrimSpaceTheorem4GapMinimumᵥ1₀EN. md` | CC BY 4. 0 (document) |