Exact Realizer Congruences and Rational-Shadow Bounds for Collatz Words
Year: 2026
Type: article
Abstract: This expository technical note assembles the exact congruence conditions characterizing finite valuation words of the accelerated Collatz map, with explicit attribution to the classical literature. It describes least positive realizers, nested residue classes, and the relationship between bounded least prefix realizers and positive-integer realization of an infinite word. For each finite block, its rational shadow is defined as the fixed point of the associated affine map. Terminal oddness supplies an additional binary digit in the divisibility condition, strengthening the resulting gap bound by a factor of two. An explicit example attains both the local lower bound and its transport to the initial seed. For blocks selected by returns to an integer layer, the note derives a weighted carry identity that retains the inherited phase and interior layer differences. It provides elementary carry bounds, identifies the hypotheses needed for a stronger estimate, and exhibits an infinite family showing why return length alone cannot force the normalized carry to grow. The discussion distinguishes cancellation from reduced rational height and separates three unresolved questions: existence of suitable blocks, their occurrence along a trajectory, and the strength of the bound after transport. Exact proofs and reproducible finite computations are distinguished throughout. No divergence exclusion, cycle exclusion, or resolution of the Collatz conjecture is claimed.
Author: De Jesus, Elias
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225001407