A Spectral Reduction of the Collatz Conjecture via Phantom Orbit Shadowing - v3

May 13, 2026Open Access

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Key Points

The study aims to reformulate the Collatz conjecture using a finite-dimensional spectral approach, introducing phantom orbits and shadowing elements.
Developed a phantom-set taxonomy and created an episode graph for phantom orbits.
Verified mathematical constructs using Lean 4 with Mathlib for rigorous formalization.
Constructed a finite matrix encoding first-return behaviors and applied a weighted Collatz–Wielandt bound.
Numerical results show bounds of 0.052 for T ≤ 16, j ≤ 32; cross-node operator spectral radius at 0.0366, lower than worst-case scenarios.
A deterministic finite-residue-cell certificate demonstrates exact rational transition matrices for various states.
Slope improvements in spectral bounds indicate a stabilization trend for large j values.

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Demonstrates a spectral reduction of the Collatz conjecture in mathematical contexts, suggesting novel pathways for understanding this problem.

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Piero Borgatta (2026) studied this question.

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