Sharp Fluctuation Bounds and the Unique Optimal Code in a Spin-3/2 Quantum Tetrahedron

AUTHOR-SUBMITTED PREPRINT • v2.5 A binary measurement can hide logical-state dependence on its first read and reveal it when repeated. We study rank-two logical codes in singlet sectors of four equal positive spins, requiring all three opposite-pair correlations to have state-independent means. Spin 3/2 is the smallest spin supporting such a code. In its four-dimensional singlet sector, every fixed-pair-mean code obeys sum_O g_O >= 64/75 and max_O g_O >= 64/225, where g_O is the width of the compressed second moment. Both bounds are attained by a unique code plane. The optimizer is the span of the unique totally antisymmetric and totally symmetric singlets and, equivalently, the maximal-absolute-eigenvalue subspace of the oriented triple product Q = J_1 dot (J_2 cross J_3). Its two sharp-Q states have opposite handedness while sharing all individual pair-measurement distributions. Under sequential binary readout with classical records, inaccessible measurement environments, and no quantum bypass memory, a self-contained Bell-recovery bound yields an explicit response inequality. Exact recovery at detector strength eta > 0 forces each conditional-response width to satisfy d_i = eta g_i/2, and known-code conditional restoration attains this requirement. Uniform pair selection therefore has the attained minimum D = 32 eta/225. Accessible word privacy alone remains insufficient when unread noise can erase the logical state. PUBLICATION STATUS This is the author's original submitted v2.5 manuscript and technical supplement. It was submitted to the Journal of Mathematical Physics on October 22, 2026 (manuscript JMP26-AR-03103), and is not the journal's published version of record. No acceptance or peer-review approval is claimed here. If a journal version is eventually published, cite the final version separately. PUBLIC FILES QM_Phase1_v2_5_JMP.pdf — complete submitted manuscript, 19 pages. QM_Phase1_v2_5_SM.pdf — associated technical supplement, 3 pages. license.txt — CC BY-NC 4.0 licence notice for the two PDFs. LICENCE Copyright 2026 Kent Nimmo. The manuscript and supplement are shared under Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0), subject to its full legal terms: https://creativecommons.org/licenses/by-nc/4.0/. The licence applies to the two publicly deposited PDFs and does not release private code, other materials or patent rights. REPRODUCIBILITY This is theoretical work supported by analytical proofs and exact finite-dimensional computations. The verification scripts, certificates and replay materials are available from the corresponding author upon reasonable request; they have not been publicly deposited here. AUTHOR Kent Nimmo — Independent Researcher, Vancouver, British Columbia, Canada. ORCID: https://orcid.org/0009-0003-8503-8891. Contact: [email protected]. No external research funding received.