CONSECUTIVE PRIMES SEE CONGRUENCES, AND ONLY CONGRUENCES: PRE-REGISTERED TESTS AT 2 128 AND 2 256 AND OF FROBENIUS INDEPENDENCE BEYOND THE ABELIAN PART

Lemke Oliver and Soundararajan showed that consecutive primes avoid repeating their residue modulo q, and derived the bias from the Hardy–Littlewood (HL) prime k-tuple conjecture. HL sees primes only through congruences, which by Kronecker–Weber are exactly the abelian-over-ℚ part of a prime's Frobenius data. HL predicts that consecutive primes are correlated through their congruences, with the HL size (P+). Its natural extension, that Frobenius data of consecutive primes are independent beyond their abelian part, adds a second prediction: they are correlated through nothing else (P−). We test both in seven pre-registered entries; every test, band and piece of analysis code was frozen and hashed before its data existed. Part I (the congruences). On 72 million consecutive pairs from uniformly random starts at 2²⁵⁶ and 16 million at 2¹²⁸, 86 of 88 decisive tests came out as predicted (the two misses had planned powers 0.17 and 0.48). The two-term LO–S formula fails from q = 13 up, while their fuller equation (2.16) fits every prime q from 5 to 31. For consecutive residue triples we derive the triple analogue of (2.16); its residuals come from singular-series terms on three or more points that (2.16) drops. The untruncated HL density, computed by absorbing small primes into the modulus, reproduces them at both heights and predicted their growth from 2²⁵⁶ to 2¹²⁸ (z = +5.4); the individual 128/256 ratios fall about 2σ below the sharper prediction of about 3.5. Part II (nothing else). We test Frobenius data beyond congruences with two elliptic curves: non-abelian mod-7 classes of the non-CM curve 37a1, ray class data of ℚ(i) from the CM curve 32a1, and the Sato–Tate and Hecke angles of both curves. All 19 decisive tests came out as predicted, on up to 4.7·10⁷ consecutive pairs per test, from 10³ to random 128-bit starts. Where a statistic does carry congruence data, the bias appears at z between −71 and −1,380, at the size the residue-pair shares imply. Two pieces of new numerical evidence: the two-angle case of Mok and Zheng's independence conjecture through 16 symmetric-power moments (lag-1 correlation r₁₁ = +0.000155 ± 0.000143 for 37a1), and independence of the Hecke angles of consecutive split Gaussian primes beyond p mod 56. An exploratory computation, made after the data were seen but with no free parameters, shows that untruncated HL also accounts for the shortfall of (2.16) in the mod-4 pair deficit at every height tested (χ² = 5.9 on 5 df). Data and code:• Entries 017–021 (Part I): https://doi.org/10.5281/zenodo.23843120• Entries 022–024 (Part II): https://doi.org/10.5281/zenodo.23843174• Exploratory mod-4 computation: https://doi.org/10.5281/zenodo.23843225 The freeze manifests were recorded in append-only run logs but not externally timestamped; the predictions are produced by archived code that does not read the data and can be recomputed by anyone. This work was carried out with AI assistance (Claude, Anthropic), as described in the paper; the author approved every pre-registration and takes responsibility for the content.