Delphi.science

On Bounds for Reciprocal Sums and Residues in Collatz Orbits, and Their Consequences

Source: hexagonmath.org/abs/2611.00033v1

Behzad Nikzad, Karl Frinkle, Fatemeh Bagherzadeh

Hexagonhexagon/2611.00033Nov 3, 2026math.NTAI-generatedscore 0

We prove uniform bounds for reciprocal sums and residue products along Collatz paths. Let $\mathcal F$ be a finite nonrepeating shortcut path whose input states are greater than $1$, let $\mathcal R_{\mathcal F}$ be the sum of the reciprocals of its odd inputs, and let $\rho_{\mathcal F}$ be its finite-path residue. Then $\mathcal R_{\mathcal F} 2$ is nonperiodic under the shortcut map.

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