Optimal Four-to-Two-Bit Quantization Through a Binary Symmetric Channel: A Finite Computer-Assisted Proof with an Equality Classification

This preprint gives a finite computer-assisted proof of the four-to-two-bit quantization bound through a binary symmetric channel for every crossover probability 0 <= p <= 1/2. For uniform four-bit input X, noisy output Y, and any two-bit quantizer Q, I(Q(X);Y) <= 2(1-h_2(p)) bits. For 0 < p < 1/2, equality occurs exactly for the 30 partitions into four two-dimensional faces (720 maps with fixed output labels); the endpoint exceptions are stated explicitly. The result is limited to n=4, k=2 and does not solve the general multibit conjecture. Computer-assisted proof. Independently reproduced by an AI auditor. Independent AI audit: PASS (reported reproduction; the external auditor's independently written code and full execution log are not bundled). Not human peer-reviewed. Novelty assessment: INCONCLUSIVE. Full theorem-level comparison with Moazzeni (2026) remains incomplete. The full text has not been inspected, overlap with existing results has not been excluded, and no priority or novelty claim is made. The reproducibility ZIP contains the unchanged audited mathematical code, frozen certificates, evidence data and LaTeX source. The publication replay of verify_no49.py passed for 396 non-subcube equivalence classes and 31,704 intervals; exact quarter-noise and analytic low/high-noise checks also passed. Historical HOLD notices in the frozen v0.2 files record the earlier audit stage; the author subsequently authorized publication of v1.0. Author: Ryutaro Yonezu - Independent Researcher. AI-assisted development under the author's direction. This is independent research, not an OpenAI publication or endorsement. Version 1.0, 2026-11-02. All rights reserved. GitHub repository: https://github.com/yonezaemon1-hub/no49-four-to-two-bit-bsc-quantizationGitHub Release v1.0: https://github.com/yonezaemon1-hub/no49-four-to-two-bit-bsc-quantization/releases/tag/v1.0