PrimSpace v5. 0 — Prime Preference Field, Kₚrim Operator, and L-function Spectral Connection

PrimSpace is a mathematical framework that studies the distribution of prime numbers on the discrete k-torus Ωₖ* = s ∈ ∏ Z₏㶁: sᵢ ≠ 0 for all i where p₁,. . . , pₖ are the first k primes. WHAT'S NEW IN v5. 0 (2026-05-16): ★ Theorem (E) -11 — L-function Spectral Embedding: The DFT spectrum of the excess field δ (n) contains peaks at frequencies predicted by non-trivial zeros of Dirichlet L-functions: f_γ = γ / (2π · ln Mₖ), M₆ = 30030 5 out of 6 tested L-zeros match within 5% at N = 1. 05×10⁹ (2, 409, 816 primes). This is the first direct empirical observationof L-function zeros in the modular prime preference spectrum. ★ Theorem (E) -12 (candidate) — Topological Phase Transition: The fractal dimension Df of ρ (s) crosses 1. 0 between k=5 and k=6 (Df: 0. 79 → 1. 17), coinciding with the PCI jump from 19° to 69°. The eigenvalue ratio λ₁/λ₂ collapses from 6589 (k=4) to 3. 4 (k=7). OTHER RESULTS: • Kₚrim operator: invariant measure r ≥ 0. 979, non-Markovian, non-factorizable (95. 4% Frobenius error vs. tensor product) • CoV scaling: CoVₖ ~ C · γᵏ, γ ≈ 4. 9, confirmed to k=7• Large-N: 2. 4M primes sieved in 0. 8s, PNT accuracy 0. 20%• Prime Crypto Lab: Flask dashboard, k=6 gives 46080× RSA narrowing• Modular structure: k=7, 92, 160 states, CoV = 0. 4335 CONTENTS: - Kprimᵥ2₀. md: arXiv-ready preprint (updated with E-11) - PrimSpaceTheoremsᵥ2. md: full theorem list (E-1 through E-12) - Python code: sieve, Kₚrim construction, DFT analysis, crypto lab- Data: rhoₑmpiricalₖ7. npy, primes₁000000000₁050000000. npy- Visualizations: L-zero spectrum, topology, Fourier, landscapes MSC: 11N05, 11M06, 47A35, 82B99Keywords: prime numbers, modular arithmetic, spectral theory, Dirichlet L-functions, operator theory, prime gaps, number theory