Analytic Reduction of the Collatz Problem to a Finite Residual Frontier
Year: 2026
Type: article
Abstract: We present a standalone analytic reduction of the Collatz problem to a finite explicitresidual frontier. The argument proceeds through reduction to odd multiples of 3, analyticdescent in the regime e(u) = v2(9u + 1) ≥ 7, reduction of the residual regime e(u) ≤ 6to stable dyadic leaves, passage from depth-3 stable leaves to primitive families, andreorganization of the resulting primitive layer into 64 admissible affine families modulo192. The active branch of each admissible family is reduced to a distinguished terminalfamily associated with 104, whose remaining 34 base cases are finite and explicitly listed.The only configurations not closed analytically in the present paper are the rigid first-exitseeds arising from the admissible active families. These seeds form a finite explicit residualfrontier of cardinal at most 722. No claim is made in the present manuscript that thisresidual frontier is itself closed analytically within the manuscript.
Author: redero, julian
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160936283
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