FDCL Part X: FDCL Generation Operators — Spectral Counting and Controlled Dynamics

Abstract

We analyze an explicit unbounded two-component FDCL generation operator and a family sharing its leading block structure. A sum-of-squares form, logarithmic cutoff and graph-norm estimate identify its self-adjoint realization, weighted operator domain and compact resolvent. Unitary re-pairing compares it with a classical Meixner channel and a diagonal arithmetic ladder, with error at most 991/1000. The remainder consists of a persistent coupling of essential norm 1/2 and a Hilbert–Schmidt correction, yielding a limiting spectral displacement at most 1/2. A sufficient coefficient-family criterion makes the scope of this reduction explicit. We retain uniform count and rank envelopes for a declared response multiset and classify coincidences among its six exact sectors. Coordinate elimination gives memory through two boundary channels; finite observability determines whether cancellation persists and which Taylor order first leaks. Energy-tail and commutator estimates give propagation error O((1+T)R−1/2) on bounded form-energy classes and O((1+T)R−1) on bounded operator-moment classes. A finite-band extension states the corresponding growth assumptions. Classical spectral, projection and approximation principles are used with explicit coefficient and domain proofs. No physical Yang–Mills Hamiltonian, vacuum or continuum mass gap is identified here. Series and status. FDCL Part X of twelve, Version 1.0 (manuscript dated 7 October 2026, 41 pages); unsubmitted working paper. The FDCL series studies the Fractal Diagonal Cut Lattice, the three-dimensional self-similar set generated by six dyadic corner maps, and the graph, operator and gauge models associated with it. This part cites Part IX as a companion manuscript. Files: the manuscript as PDF and a source archive (25 files) with the LaTeX source and optional exact finite checks. The other parts are archived separately. AI use disclosure. Generative AI (GPT-6.0, OpenAI; Claude Opus 5.5, Anthropic) was used substantively in preparing this work, including literature comparison, the development and checking of proofs and counterexamples, exact computations and the writing and running of verification code, and drafting and editing. The research questions, framework and final claims were directed and reviewed by the author, who takes full responsibility for the content, including the accuracy of all references and reported numbers. Repository metadata were prepared with assistance from Claude (Anthropic).