A Spectral Reduction of the Collatz Conjecture via Phantom Orbit Shadowing

This work proposes a structural reformulation of the Collatz conjecture in terms of a finite-dimensional spectral problem. Starting from the empirical observation that "rebel" orbits of the Syracuse map shadow, for finite stretches, periodic rational points of the 2-adic dynamics associated to negative rational fixed points, we construct an episode graph on the space of phantom orbits. The strongly connected component (SCC) of this graph that supports the slowest descent is encoded as a finite weighted transfer operator on a refined phase state at the critical node (k=12, c=2, b=1). We prove an exact congruential shadowing lemma reducing arbitrarily long shadowing episodes to a residue condition n ≡ qw (mod 2bA+1), where qw is the 2-adic representative of a phantom word w. For each truncation depth T and high-bit lift count j we build a finite matrix FULLT,j encoding first-return behavior at the critical node, decompose it as FULL = CORE + TAIL, and apply a weighted Collatz–Wielandt bound with respect to the right Perron eigenvector v of FULL. Numerical results show bound(T, j) ≤ 0.052 for T ≤ 16 and j ≤ 32, with monotone but geometrically decaying increments. The cross-node operator on the SCC has spectral radius 0.0366, lower than the single-node bound at the critical vertex. This is not a proof of the Collatz conjecture. The structural problem is reduced to two explicit a priori estimates on finite matrices: the asymptotic boundedness of the weighted bound under refinement in T, and the closure of the empirical signature support under refinement in j. We believe both can be approached via Lasota–Yorke + Hennion's theorem + Keller–Liverani spectral perturbation theory, but we have not carried this out. Research program note This paper is the first artifact of a personal research program by the author, an independent researcher, exploring whether iterative collaboration with large language model assistants can support non-standard attempts at open mathematical problems. The author led the research direction throughout: the conceptual framing (gravitational debt, 2-adic corridors, the shadowing intuition), the strategic choices, the decision to publish at this stage. The technical work — Python implementation, mathematical formalization of the shadowing lemma and the weighted bound, and the LaTeX manuscript — was developed in iterative collaboration with multiple LLM systems: OpenAI Codex and Google Gemini for the early empirical phase, and Anthropic's Claude (via the Claude Code interface) for the spectral program, the formalization, and the manuscript drafting. As of this version, the work has not been reviewed by a human mathematical expert. Numerical results are deterministic and reproducible from the supplementary scripts. Mathematical claims rest on AI-assisted formalization and have not been externally validated; the work is submitted explicitly to invite expert scrutiny. A planned next phase is to mechanically verify the shadowing lemma in Lean 4 with Mathlib; a revised version of this preprint will incorporate the formal verification when complete. Constructive feedback from researchers in symbolic dynamics, p-adic dynamical systems, and transfer operator theory is explicitly invited. Source code: github.com/PieroBorgatta/CollatzMethodology and contribution split: see METHODOLOGY.md in the supplementary archive.