Graded Layers and Pair Sources in Rule 30
Year: 2026
Type: preprint
Abstract: Marking Rule 30's quadratic term produces a triangular hierarchy of driven additive layers sharing one trinomial propagator and depending only on lower layers. Every finite binary tower is a cellular automaton. Each fixed integer layer has a D-finite space–time generating series and a P-recursive central column. Explicit support formulas, degree bounds and Fibonacci integer-layer counts describe the hierarchy; its highest layer at each depth survives modulo two. Coefficient-size estimates and finite support windows give quantitative construction bounds, quadratic up to a logarithm in diagonal depth for a fixed layer range. Grading also resolves the reflected Rule 86 recurrence by successive construction. Summing the layers expresses Rule 30 as additive cones emitted by the seed and adjacent pairs of ones. We count the pair projection's finite-word fibres and prove its nonautonomy on all configurations. Together these descriptions distinguish fixed-layer calculations from reconstruction of the complete orbit.
Author: Nersissian, Tigran
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212431587
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