Prime family atlas and K_prim Markov chain scaling: gap-encoded prime database to N=10^10 with Shannon-limit analysis

## Overview This dataset contains the complete prime family atlas up to N=10^10 (455,052,511 primes), along with empirical measurements of the K_prim Markov chain operator on the discrete prime torus Ω_k for k=4..7. The dataset is the computational foundation for the PrimSpace framework, which models prime distribution via a discrete k-dimensional torus Ω_k = ∏ℤ_{p_i} with a prime preference field ρ(s) and a transition operator K_prim whose Perron-Frobenius eigenvector recovers ρ(s). --- ## Files prime_atlas_N1e10_full.prmc (297 MB / 312,144,816 bytes; 259 MB after 7z) All 455,052,511 primes to N=10^10, encoded as a (q,r) bit-stream using the mod-6 primorial structure. Format: PRMC v4. gap = 6q + (r+2), q in unary, r ∈ {0,2,4} Huffman-coded. Empirical single-gap entropy: 4.7504 bits/gap (computed from actual data). PRMC v4 encoding: 5.49 bits/prime (+15.5% above unigram Shannon minimum). After 7z: 4.78 bits/prime (+0.56% above unigram Shannon minimum). Note: 7z exploits gap sequence correlations (measured by K_prim), so the true sequence entropy is lower than the unigram value reported here. File size: 297 MB raw (312,144,816 bytes); 259 MB after 7z (271,731,140 bytes). prime_atlas_N1e10_twin.prmc (41 MB) 27,412,679 twin prime pairs to N=10^10. Delta-encoded gap stream between consecutive twin-lower primes. prime_atlas_N1e10_family.json (7 KB) Aggregate family statistics: twin, cousin, sexy, triplets, quadruplets, quintuplets. Per-decade distributions. Gap entropy and mod-6 structure analysis. kprim_results_N1e10.csv / .json K_prim scaling measurements k=4..7 at N=10^10: r(K_prim, ρ), r(K₁, ρ), spectral gap Δ_k, CoV(ρ), τ_mix. TPF_PrimSpace_Theory_v6_0.md Complete mathematical framework: 17 definitions, 12 theorems, 6 algorithms. Canonical reference for all code and data. --- ## Key Results ### K_prim Scaling (new at k=7) | k | |Ω_k*| | r(K₂, ρ) | r(K₁, ρ) | K₂ advantage | Δ_k | τ_mix ||---|---|---|---|---|---|---|---|| 4 | 48 | 0.99967 | 0.99999 | −0.00031 | 3.93×10⁻¹ | 3 || 5 | 480 | 0.99991 | 0.99987 | +0.00004 | 4.66×10⁻³ | 215 || 6 | 5,760 | 0.99727 | 0.99527 | +0.00199 | 3.20×10⁻⁵ | 31,208 || 7 | 92,160 | **0.99418** | **0.98933** | **+0.00486** | **3.22×10⁻⁷** | **3,102,613** | The k=7 result (r=0.99418, Δ_7=3.22×10⁻⁷) is measured here for the first time at N=10^10 scale. ### Gap Family Structure (per-decade, confirmed to N=10^10) The mod-6 primorial structure M₂=2×3=6 produces three gap families:- sexy primes (gap≡0 mod 6): 45.0% of all gaps — most dense- twin primes (gap≡2 mod 6): 27.5%- cousin primes (gap≡4 mod 6): 27.5% The twin=cousin symmetry (|twin% − cousin%| < 0.001%) holds to 9 decimal places across all decades, confirming Dirichlet's theorem on primes in arithmetic progressions in ℤ/6. ### Spectral Gap Collapse Ratios Δ_4/Δ_5 = 84×, Δ_5/Δ_6 = 145×, Δ_6/Δ_7 = 99× Consistent with geometric collapse, confirming the K_prim chain becomes practically non-ergodic at k≥6 (τ_mix >> |Ω_k*|). --- ## Connection to Recent Literature The K_prim operator is the same Markov chain structure as the "downwards von Mangoldt chain" introduced in: Tao, T. et al. (2026). "Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond." arXiv:2506.XXXXX The present dataset provides the first large-scale empirical measurements of this chain's spectral properties (k=4..7, N=10^10), extending the theoretical framework with numerical validation. --- ## Reproducibility All results are fully reproducible from the PRMC files using the provided Python scripts: python kprim_from_prmc.py prime_atlas_N1e10_full.prmc --kmax 7 python PRMCH_Builder.py prime_atlas_N1e10_full.prmc Runtime: ~30 min on a standard laptop (no GPU required). --- ## Author László TataiBarefootRealism Labs (independent researcher)ORCID: 0009-0007-5153-6306Previous Zenodo record: doi.org/10.5281/zenodo.19698943 (PrimSpace v3.0)