The support wave of Rule 30: exact anatomy of an irreducible pattern (with companion papers on local grammars, arrows of time, and the odometer structure)
Year: 2026
Type: preprint
Abstract: Three companion papers on the regular flank of elementary cellular automaton Rule 30, with all verification scripts and data. Flagship (flagship.pdf): the support-wave calculus in the Lucas basis — OEIS A094604 characterised exactly as odometer carries of the support crest; front speed 1/2 generated by an explicit finite automaton; an exact centre-column formula (with refutation of a recently published identity); pure-transient law of windows. Grammar (grammar.pdf): classification of all sixteen family rules by locality of the diagonal grammar; soundness precondition for joint-induction certification; complement-conjugacy lemma; certified reversal asymmetry (arrow of time); the clocks of the family — four distinct irregular period-doubling sequences, three of them new; the seed-moduli space of clocks, the tail-front bijection, and the absorption lemma characterising the A094604 basin. Odometer (odometer.pdf): dyadic eigenfunctions; coin digits as literal digits of absolute time; a measurable skew product over the dyadic odometer with exact window entropy 0.680082 bits per symbol; the all-ones vacuum as the unique obstruction to a topological odometer factor. Code (Python/C), minimal forbidden word lists, and reproduction instructions included. Licenses: CC-BY-4.0 (papers), MIT (code). Version 3 adds: the anti-shift conjugacy theorem (the natural NAND pattern is the complemented time-shift of the Rule 30 pattern, and likewise for the implication pair, with all crest rigidities as corollaries); the certified ten arrow-of-time rules of the 256-rule three-argument family with seven new clock sequences (OEIS A399493-A399495 cover the two-argument family); the reclassification of the non-SFT hypothesis as a square-root discrepancy statement, with exact integer counts over full periods; the reduction of unconditional non-SFT to uniform second moments (new section of the odometer paper); and the boundary-of-the-rigid-regime discussion in the flagship, characterising the diagonal polynomials as near-maximal-degree with random-scale weight deviations vanishing exactly at antiperiodic levels. Version 4 adds: a corrected proof of the consecutive-doublings theorem (the earlier argument relied on an inequality between maxima of OR-convolutions that is false in general; the new proof uses the unique-pair argument), with the corollary that two consecutive gaps of one never occur in A094604; the deficit-ladder theorem (a gap of two lowers the crest's plateau deficit by exactly one) and a complete stochastic model of the gaps of A094604 - an exact ladder, the plateau criterion, fair coins and a geometric reset - replicated on the seed-moduli space (56,602 carries over the four rules OR, NOR, implication, reverse implication), giving P(gap=1)=1/4, P(gap=2)=P(gap>=3)=3/8 and mean gap 5/2; the theorem that the carry copies the Boolean derivative of the rule (valid for all 256 three-argument rules, verified on the ten certified ones), which explains why the two-argument class shares one gap mechanism and the genuinely three-argument rules do not; and a withdrawal of the claim that the NOR support maximum jumps across powers of two (the exact odometer holds for the whole class). New scripts: verify_crest_thms, crest_copy, gap_struct, second_step, deficit_ladder, plateau_bits, gap_model, gap_orbits, gap_orbits2, family_crest, family3_crest, family3_copy. Version 5 (correction of files only): the PDFs included in version 4 were stale builds that predated the version-4 additions (the LaTeX sources in that archive were current). This version replaces them with freshly compiled PDFs of all three papers (flagship 9 pages, grammar 11 pages, odometer 6 pages); no other change. Version 6 adds: in the grammar paper, the classification of all 256 three-argument rules by the cofactor of the rule in its newest argument (64 linear, 160 nonlinear, 32 degenerate), which decides the gap mechanism of the clock and is verified on all live rules; in the odometer paper, the flip lemma, the coin lemma (on doubling levels the phase bit is a provably fair coin independent of everything below), the peeling theorem (the density of any word on a window of levels is at most 2^-N with N the number of doubling levels in the window), its corollaries (the double-zero density equals 1/4 exactly after two consecutive doublings; all diagonals are uniformly non-degenerate conditional on bounded first differences of A094604), and the exact structure of the zero runs of the driver; plus the memos of two multi-agent proof assaults on the balancedness question, recording what survived refereeing and what was refuted.
Author: Fomin, Daniil
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204654199
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