Moving-Diagonal Sparsity in a Width-Two Collatz Survivor System: Quantitative Consequences of Lucherini's Terminal Residue Sieve
Year: 2026
Type: preprint
Abstract: This note investigates the moving-diagonal problem in the single-defect stratum of a width-two accelerated Collatz survivor system. Building on Claudio Lucherini’s terminal residue sieve and the author’s earlier realization criteria, it establishes explicit families of depths at which the diagonal avoids the exceptional residue classes. The main result bounds the number F(X) of exceptional depths between 5 and X by F(X) ≤ (1 + o(1)) log₂ X. Consequently, the exceptional depths have natural density zero. The argument combines monotonicity of the witness count with periodicity of a truncated sieve, extending an earlier witness-counting method. The remaining pointwise question is formulated through binary digit gaps of explicit 2-adic constants. The discussion distinguishes conditional qualitative finiteness from an effective, computable cutoff and identifies the additional uniformity needed across exceptional branches. Exact integer computations check the stated identities and find no moving-diagonal failures for depths 5 through 3000; these finite checks carry no implication beyond that range. The note leaves complete diagonal avoidance unresolved and makes no claim to prove either the width-two dominance conjecture or the classical Collatz conjecture.
Author: De Jesus, Elias
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225347125