The Cascade's Front
Year: 2026
Type: preprint
Abstract: All-Fields programme paper (All-Fields Run, Batch 17 (AF)), November 2026. Written November 2026; publicly deposited 9 November 2026. AF-40 graded the two bounded readings singly, as operators, and found no kernel on the ladder with a front-exact zero. Its falsifier F2 carried the grading to the one operator the family reaches without a new stencil: the composition of the two readings — the cascade, in either order — and its kernel's front. This paper grades the composed kernels. The cascade exists exactly at the grains the two readings' ladders share, the matched grain and the coarse; the one-sided reading has no fine rung, so no cascade stands there. Two facts are derived before any number is read: the two orders of composition are the same operator, because both symbols are functions of the grid's frequency alone and composition is their product; and the composed symbol's value at zero frequency is the product of the factors', 1. Against those, the composed kernel's front share is not a corollary of its factors' shares — the ahead/behind split of a convolution is set by the symbols' phases as well as their magnitudes — so the shares are graded on the bench, in AF-40's form entire, by AF-40's own grading function
Author: Aldred, William Alexander
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225647973
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