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

PrimSpace is a mathematical framework that studies the distribution of prime numbers on the discrete k-torus Ω_k* = { s ∈ ∏ Z_{p_i} : s_i ≠ 0 for all i } where p_1, ..., p_k 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 M_k), M_6 = 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 D_f of ρ(s) crosses 1.0 between k=5 and k=6 (D_f: 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:• K_prim operator: invariant measure r ≥ 0.979, non-Markovian, non-factorizable (95.4% Frobenius error vs. tensor product)• CoV scaling: CoV_k ~ C · γ^k, 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:- Kprim_v2_0.md: arXiv-ready preprint (updated with E-11)- PrimSpace_Theorems_v2.md: full theorem list (E-1 through E-12)- Python code: sieve, K_prim construction, DFT analysis, crypto lab- Data: rho_empirical_k7.npy, primes_1000000000_1050000000.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