A Staged-Null Anatomy of the Prime-Gap Signature: The B4 Symmetry Decomposition Attributes It, Block by Block, to the Short-Range Residue-Transition Grammar

The Watabe-Claude Method embeds a scalar series into 4D with a Takens window, reads the 24-cell occupancy,and decomposes 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) and the residual is smallest at second order there; by Takens delay,it lives entirely at adjacent gaps and is absent at every longer delay. The prime-gap B4 signature is, to theresolution of these tests, the short-range second-order residue-transition grammar modulo 30 -- entirely knownnumber theory, read out block by block. We make no claim about any prime conjecture; the decomposition is anexploratory measuring instrument. Text CC-BY-4.0; code AGPL-3.0-or-later.