The Cascade's Powers
Year: 2026
Type: preprint
Abstract: All-Fields programme paper (All-Fields Run, Batch 18 (AF)), November 2026. Written November 2026; publicly deposited 9 November 2026. AF-41 graded the cascade — the composition of the two bounded readings — at depth one, and its falsifier F2 carried the grading to the composition iterated: the composed operator's own powers, the fourfold and sixfold compositions the family reaches by composing the cascade with itself. This paper grades the powers' kernels at the two grains the readings' ladders share, in AF-41's form entire and by AF-40's own grading function. Three facts are derived before any iterated number is read. The powers' kernels are the cascade kernel's circular self-convolutions, the fourfold its square and the sixfold its cube. Each power's symbol stands at modulus one exactly on the comb of grid frequencies where the stencil's period falls, and strictly below one off it — for the symmetric factor because its difference vanishes only there, for the one-sided factor by an algebraic identity in its denominator, which holds because τ₀ exceeds the grain at both shared grains. And the powers therefore concentrate on the comb: the limiting kernel is the equal spike train on the grain's lattice, 330 spikes of
Author: Aldred, William Alexander
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225621643
Cites:
Cited by:
Related to: