FDCL Part XII: Haar-Preserving Assembly in FDCL SU(2) Gauge Models — State Preparation, Shared-Link Responses, and Delayed Transfer
Abstract
We study which information block assembly must retain to preserve responses in finite-graph SU(2) gauge theory. Scalar, pairwise and oriented triple invariants classify simultaneous conjugation orbits; we give exact consistency conditions including singular Gram ranks. Vertex multiplicities and physical path lengths determine Haar-unitary recoupling, and a concrete flower requires an odd child state in an even parent. Finite spectral interpolation prepares a specified native state without identifying its preparation word with multiplication. Two flowers sharing a path have an isometric product assembly with compressed electric energy 42I, but variance 3I. Their exact return extends to chains through an explicit Jacobi matrix, separating cyclic response from junction-location data. For a fixed two-square Hamiltonian, electric branch weights control the full interacting exterior resolvent. Center parity gives a positive fourth-order centered error and a certified compressed inverse for every real z < 36 on the responding sector, independently of other sectors' spectra. A separate two-detour Hamiltonian exhibits fourth-order rank loss, fifth-order reaction in the old output line and sixth-order recovery. The time-transfer determinant gives full rank near zero for every nonzero allowed coupling pair, and bounded holomorphic evolution implies full rank at almost every real time. Structural proofs and selected exact calculations retain the stated graph, physical metric, operator and input contracts. No uniform interacting gap or continuum construction is inferred. Series and status. FDCL Part XII of twelve, Version 1.0 (manuscript dated 7 October 2026, 50 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 Parts IX and XI as companion manuscripts. Files: the manuscript as PDF and a source archive (72 files) with the LaTeX source and exact check suites with their inputs and recorded outcomes. 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).
