Research #4: Exact Phase-Adversarial Finite-Alphabet Hierarchies — All-User 16-QAM, Global Certificates, and Algorithms

Abstract

Research #4 consolidates exact results for finite-alphabet superposition under independently adversarial phases. The main manuscript proves the global optimum for every finite number K of square 16-QAM users under unit squared-amplitude normalization: D_K* = delta / sqrt(sum from h=0 to K-1 of (1+3 sqrt(2))^(2h)), where delta is nearest-neighbor spacing. Every optimizer is a permutation of the normalized geometric hierarchy with base 1+3 sqrt(2), and its minimum labeled separation equals the optimum at every phase tuple. The computer-assisted induction uses two fixed 22-node rational certificates and includes a reusable sufficient finite-spectrum criterion. The complete release also preserves the exact three-user 16-QAM optimum and 47-node global certificate, separate four- and five-user certificates (186 and 719 nodes), the sharp universal endpoint-spread and all-user QPSK results, and the sorted radial predecessor algorithm with O(N log N) fixed-allocation evaluation and fixed-user rational optimization. Original archives are included unchanged, together with LaTeX sources, exact standard-library checkers, corruption tests, optional discovery code, pinned dependencies, audit records and SHA-256 manifests. The separate PDFs and original packages are provided for convenient access; Research_4_Complete_Release.zip contains the full consolidated release. RESEARCH_4_README.md explains reading order and superseded historical open-problem statements. Fresh exact checks reproduced the recorded certificates and rejected all 24 tested certificate corruptions. The result concerns worst-phase labeled geometric separation, not a practical coded-service optimum, phase identifiability, channel capacity, RF performance or measured energy savings. Historical priority remains unresolved; the work has not undergone external peer review or proof-assistant formalization. AI assistance was used in derivation, writing and code preparation. Paper 2 service-feasibility and Paper 3 deployment studies remain separate projects. No additional reuse license is asserted for the author's work; any third-party material retains its existing terms. Research_4_Working_History_Supplement.zip preserves substantive source-folder snapshots, including six early exploratory scripts and the three-user certificate-regeneration records. Those historical numerical explorations do not supersede the final exact proofs. RESEARCH_4_COMPLETENESS_AUDIT.md maps every requested deliverable to its archive, and RESEARCH_4_SUPPLEMENT_SHA256SUMS.json covers the added supplement and audit.