IEEE submission notice. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.
The focused pooled-feedback manuscript was submitted to IEEE Transactions on Communications on 15 October 2026. Submission does not imply acceptance or completed peer review.
Complete Research 7 release, including the pooled-feedback frontier and command-distribution cost extensions. Start with the updated complete guide, then read Pooled Phase Feedback for Simultaneous Binary Uplinks: Energy Bounds and Command Distribution Costs and its standalone proof supplement. The focused manuscript has eight pages; the supplement has fourteen. This record is a preprint and reproducibility archive, not peer-reviewed publication.
New centerpiece. Across M independently phased binary transmitter pairs, with R counted digital control bits per pair, the claimed sharp leading reserved signal-energy coefficient is the lower convex envelope of f(R)=3-2 cos(pi 2^(-R)) for 0<=R<=log2(3), followed by coefficient 2. Its supporting line is 5-2.009123154 R, tangent at R=1.107042776 and coefficient 2.775814726. Total reserved transmitted signal energy is [Cbar(R)+o(1)] N0 n ln n. The converse permits arbitrary nonlinear or shared receiver acquisition, randomized phase-independent dictionaries, differently corrupted recipients and correlated epoch selections within an explicit full-vector phase-independent digital-control kernel. It requires fresh uncoded fair binary payloads, normalized uplink gains, constant block phases, fixed block alphabets and an aggregate conditional-mean cap on every applied outcome. Each pair occupies its own simultaneous payload coordinate; pooling concerns control descriptions. M and coherence length L grow, while the epoch count may stay bounded or grow. Uncounted analog transmitter phase information, payload-history control and unequal physical gain weights require different theorems.
Command distribution is charged separately. The 20.75% limiting compression benefit concerns description bits. For 64 pairs, separate two-symbol ternary commands use 128 transmitted symbols and 128 recipient receptions; one global word uses 102 and 6,528. Under fixed linear reserved transmission/listening symbol prices in the stipulated fixed-length OOK family, some grouping saves cost exactly when eR/eT<1/9; triple groups attain the threshold. The global 64-pair word instead needs eR/eT<13/3200. A conservative all-recipient noise reservation makes the global word cost 12.82% more reserved waveform energy than separate commands. This is a sufficient union-bound calculation, not measured joules, a hardware experiment, or globally optimal finite-SNR performance. Growing groups retain the limiting description rate with subquadratic listening work; clock/LO idle cost, phase aging, setup time and realistic supply energy remain physical validation questions.
Broader theory and boundary audit. The release includes a precisely quantified cap converse, a phase-only context extension, explicit outside-class counterexamples, and a separate expected-signal-energy law [(1-epsilon)Cbar(R)+o(1)] N0 n ln n under a maximum word budget and intentional common outages. The cutoff factor is an established principle and is not claimed as new. Targeted primary-source comparisons credit prior shared feedback indices, quantized phase control, joint power/feedback allocation, distortion-volume covering and feedback circuit accounting. No exhaustive historical-priority clearance is claimed.
Everything important is included. Research_7_Complete_Release_Pooled_Feedback.zip contains the new paper and source, full proofs, both latest extension/audit packages, all accompanying graphs and numerical data, calculators and checks, v3-v6 manuscripts, earlier fixed-command extensions, complete noisy two-bit software and results, an explicitly unexecuted hardware-validation plan, supplied original archive and development history. The preceding complete 410-member archive is preserved byte for byte within the history. Earlier superseded claims are labeled history. The older Research_7_Complete_Release.zip remains an earlier snapshot. Public-copy transformations and regenerated hashes are recorded; original local sources are unchanged.
Earlier executable evidence. The two-bit protocol includes actual noisy uplink references, noisy OOK downlink observations, wrong command reception, phase correction, fresh payload generation and causal Bayesian decisions. At reserved signal energy 16 N0 per tuple and coherence L=64, the synthetic full-block failure estimate is 0.030315 versus 0.29471 for the tested restricted optimized open-loop candidate; at L=8 it is worse, 0.14061 versus 0.10028. Orthogonal signaling remains more reliable where its second payload dimension is feasible. Allocation searches are restricted, rare-error and quadrature limitations are documented, and current-pair Bayes decisions are not globally optimal block-tuple decoding. These simulations do not establish physical device savings.
Validation and AI disclosure. OpenAI Codex substantially assisted derivation, proof development/reconstruction, literature comparison, writing, software and verification. The author reported personally reading the underlying proofs and finding them satisfactory. All model-assisted mathematical audits and computational controls remain internal. No outside specialist proof certification, external replication, SDR hardware trial, measured supply-energy advantage, calibrated deadline performance or qualifying deployed one-coordinate requirement has been established. The hardware log is empty and calibration values are unmeasured. Private correspondence, portal account data, installed dependencies and third-party full-text caches are excluded. No additional reuse license is asserted; retained components keep their existing terms.