We present a large-scale computational study of the local structure of theprime numbers through two independent but complementary statistical probes. The first probe is information-theoretic: it measures the order-2 conditionalmutual information (CMI2) between consecutive residue triplets (p₍-₁, pₙ, p₍+₁) modulo q, over 50. 8 million primes up to x = 10⁹. The secondprobe is Fourier-based: it exploits the phase representation uₙ (alpha) =alpha pₙ and its increments Deltaₙ (alpha) = alpha gₙ for rationalvalues a/q and their neighbourhood. Version 8 incorporates a critical revision after expert peer review. The threesolid results are: (i) the G9 non-Markovianity of the prime residues, establishedagainst an order-1 Markov null on 16 prime moduli q spanning phi in 2, 388, with 10⁴ permutations per experiment, one-sided empirical p-value <= 10^-4, confirmed by three information estimators (plug-in, Miller-Madow, Grassberger) ; (ii) a compact analytic form for the spectral norm of the c₂ term of LemkeOliver and Soundararajan 2020, spec (q) ~ phi^0. 77 * (log phi) ^1. 63 on 167prime moduli up to phi = 996 with R² = 0. 9986, preferred to a pure powerlaw by a likelihood-ratio test (AIC gap = 361) ; (iii) the trimodality of theeffective slope b (phi), a robust empirical observation on the computed grid (62 moduli, 8 decades of x up to xmax = 10¹2). The claim of a predictive spectral decomposition of the trimodality, centralin earlier versions, is withdrawn: it does not survive a correct out-of-samplevalidation (detrended correlation r = +0. 22, p = 0. 23). The work is presentedas an experimental contribution. Target journal: Experimental Mathematics. No new theorem in the classical analytic sense is proved.
Sebastien Icard (Wed,) studied this question.