No New Structure in the Prime Gaps: An Exact Reconstruction of the B4 Signature from the mod-30 Residue Grammar

Abstract

The title says no new structure in a specific, bounded sense: not that the prime gaps are without structure -- theyare richly structured by known number theory -- but that the 24-cell B4 signature we measure contains nothingbeyond that known structure, as we show by reconstructing it exactly from the residue grammar. TheWatabe-Claude Method embeds a scalar series into 4D with a Takens window, reads the 24-cell occupancy, anddecomposes it with the signed-permutation group B4 into orthogonal isotypic blocks. A companion paperestablished that this assignment factorizes exactly into a magnitude channel (the axis-pair blocks, which readwhich coordinate pair is largest) and a sign channel (the sign and resid4 blocks, which read the signs of that pair).An earlier study of real prime gaps found a large rank-space signal but could only classify it as a readout of knownnumber-theoretic correlations without saying which correlations, or how much. Here we take the full decompositionto the prime gaps and pin the attribution down. The consecutive-gap sequence (primes to 30 million, 1.86 milliongaps) shows a consistent B4 block signature: every block separates the real gaps from an order-shuffle null at zbetween 5 and 8, in the same direction across all chunks. A discreteness control -- an iid series with the samemarginal and the same heavy ties -- shows no such signature, so the block pattern is genuine consecutive-gaporder structure, not an artifact of the 83 distinct gap values. We then walk a staged null ladder that removesprogressively more structure: matching the marginal, then the full power spectrum (an IAAFT surrogate), then themod-6 position classes, leaves the signature essentially intact -- it is not linear autocorrelation and not the mod-6wheel. A residue-transition surrogate that preserves the transition bias between consecutive gaps' residuesmodulo q collapses it. Refining that null along three axes localizes the mechanism precisely: by Markov order, thesignature dies stepwise as first-, second-, and third-order residue transitions are conditioned on (higher-ordergrammar, not a first-order shadow, and not a pool-depletion artifact -- the iid control never collapses); by primorialmodulus, it bottoms at q=30 (the 2-3-5 sieve), where the aggregate residual is smallest at Markov order two whileorder three removes the last of the blockwise sign consistency; by Takens delay, it lives entirely at adjacent gapsand is absent at every longer delay. Freezing the resulting parameters (q=30, order two, delay one) and applyingthem unchanged to an independent, non-overlapping prime range [30M, 100M) reproduces both the null-laddercollapse and the delay-one locality, with no re-tuning. We then harden the small-residual claims with atwo-hundred-surrogate ensemble and two-sided empirical permutation p-values: the number of blocks that remaindistinguishable from the null falls from all five (through the marginal, spectrum, and mod-6 nulls) to four at thefirst-order transition null, two at second order, and none at third order, where every block lies inside the null's 95percent band (smallest p about 0.08). The same two-hundred-surrogate collapse reproduces on the independentrange with the modulus frozen, on the blocks that carry the signature. An exact residue-class null -- one thatpreserves the transition matrix of consecutive primes' reduced residues modulo 30 -- reconstructs the two blocksholding 99.3 percent of the signature, the dominant block already at first order; and the same grammar, fitted onone prime range and generated forward, predicts the B4 fingerprint of a disjoint range about four times better thana marginal baseline. The prime-gap B4 signature is, to the resolution of these tests, the short-rangeresidue-transition grammar modulo 30, dominated by second-order transitions and statistically indistinguishablefrom that known-arithmetic null once third-order transitions are conditioned on -- entirely known number theory,read out block by block, confirmed both by elimination and by forward prediction. We make no claim about anyprime conjecture; the decomposition is an exploratory measuring instrument. Text CC-BY-4.0; codeAGPL-3.0-or-later.