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)

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).