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 paper established that this assignment factorizes exactly into a magnitude channel (the axis-pair blocks, which read which 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 known number-theoretic correlations without saying which correlations, or how much. Here we take the full decomposition to the prime gaps and pin the attribution down. The consecutive-gap sequence (primes to 30 million, 1.86 million gaps) shows a consistent B4 block signature: every block separates the real gaps from an order-shuffle null at z between 5 and 8, in the same direction across all chunks. A discreteness control -- an iid series with the same marginal and the same heavy ties -- shows no such signature, so the block pattern is genuine consecutive-gap order structure, not an artifact of the 83 distinct gap values. We then walk a staged null ladder that removes progressively more structure: matching the marginal, then the full power spectrum (an IAAFT surrogate), then the mod-6 position classes, leaves the signature essentially intact -- it is not linear autocorrelation and not the mod-6 wheel. A residue-transition surrogate that preserves the transition bias between consecutive gaps' residues modulo q collapses it. Refining that null along three axes localizes the mechanism precisely: by Markov order, the signature dies stepwise, with the dominant explanatory order being two while third order removes the last of the blockwise sign consistency; by primorial modulus, it bottoms at q=30 (the 2-3-5 sieve); by Takens delay, it lives entirely at adjacent gaps and is absent at every longer delay. Freezing these parameters (q=30, order two, delay one) and applying them unchanged to an independent, non-overlapping prime range [30M, 100M) reproduces both the null-ladder collapse and the delay-one locality, with no re-tuning. The prime-gap B4 signature is, to the resolution of these tests, the short-range residue-transition grammar modulo 30, dominated by second-order transitions -- entirely known number theory, read out block by block. We make no claim about any prime conjecture; the decomposition is an exploratory measuring instrument. Text CC-BY-4.0; code AGPL-3.0-or-later.