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Mathematical revision of Research #6: Binary Recovery from Sums with Independent Random Phases. The problem concerns independent labeled binary variables encoded separately as vectors, each multiplied by an independent uniform phase, with only their sum in n complex observations available to the receiver. No realized phase or encoder-seed side information is supplied.

The manuscript claims arbitrarily reliable full-message recovery if and only if K <= 2n, with unbounded signal energy permitted for achievability. Above threshold it claims a positive dimension-dependent average-error lower bound uniform over finite codebooks, independent private stochastic encoders and decoders.

VALIDATION STATUS: EXTERNALLY SUBMITTED, BUT MATHEMATICALLY UNVALIDATED. Substantial OpenAI ChatGPT/Codex assistance was used in derivation, proof development, drafting, literature research and verification code. Independent human proof validation and formal mathematical peer review have not yet occurred. Internal symbolic and numerical checks are consistency checks, not formal proof certificates or external validation. Historical priority is not asserted.

The previous engineering presentation was desk-rejected by IEEE Transactions on Information Theory on 6 October 2026 for fit, motivation, presentation and concerns about the absence of human proof validation. That decision does not constitute a mathematical referee verdict. This revision recasts the problem as recovery under a torus nuisance action and simplifies terminology.

This is not a practical wireless performance, battery-energy, coverage or deployment claim. The extremely conservative constants are not useful engineering error-rate estimates. No new physical trials or link experiments accompany this presentation revision. Version 1 remains available at DOI 10.5281/zenodo.23123476. Private reviewer correspondence and endorsement codes are excluded. No additional reuse license is asserted; retained materials retain their existing terms.

This version contains only the complete 13-page revised manuscript PDF, as requested by the author. Earlier source and verification archives remain in version 1 and are not reuploaded or represented as a newly verified supplement to this revision. The mathematical rewrite includes the proof and explicit-bound appendix.

MSC 2020: 94A15 (primary); 62C10, 60B15, 15A15 (secondary). Keywords: binary recovery; random phase; statistical decision problem; overlap of probability measures; independent transversals.