Statistical Probes of the Local Structure of the Prime Numbers: Order-2 Conditional Mutual Information and Detuned Rational Singularities of the Phases {alpha p_n}

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 (pₙ₋₁,p_n, pₙ₊₁) modulo q, over 50.8 million primes up to x = 10^9. The secondprobe is Fourier-based: it exploits the phase representation u_n(alpha) ={alpha p_n} and its increments Delta_n(alpha) = {alpha g_n} 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 10^4 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 c_2 term of LemkeOliver and Soundararajan 2020, spec(q) ~ phi0.77 * (log phi)1.63 on 167prime moduli up to phi = 996 with R^2 = 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 = 10^12). The claim of a predictive spectral decomposition of the trimodality, centralin earlier versions, is withdrawn: it does not survive a correct out-of-samplevalidation (detrended correlation r = +0.22, p = 0.23).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.