Finite-Length Switching and Limitations of Recursive Constructions in Binary Deletion Channels

For binary inputs of equal length transmitted through an independent deletion channel, this preprint studies the minimum total-variation distance between one-trace output laws as a function of deletion probability. Exact computer-assisted lower envelopes are certified for input lengths four through eight. The numbers of interior switches between distinct optimizing distance polynomials are respectively 1, 1, 2, 3, and 3. The certificates use integer subsequence counts, coefficientwise domination, exact rational Bernstein inequalities, and isolation of unique algebraic crossing points. An elementary fixed-length interval-stability result is included without a novelty claim. An exhaustive counterexample shows that, at deletion probability 4/5, none of the 256 common-bit insertion operations from the two optimal length-seven pairs reaches a length-eight optimum at the same parameter. Thus a proposed universal same-parameter optimality-preserving recurrence fails. A specific inserted pair is optimal at a different deletion probability, illustrating migration of optimal intervals; this observation is not a general theorem. The failed proof strategy and research chronology are preserved. Failure of this strategy does not refute the unbounded-switching conjecture. Independent AI audit: PASS for finite-length results (author-supplied outcome). Windows PowerShell reproduction of v0.2.1: PASS (author-reported 2026-11-02). Release verification reruns: all three Python verifiers PASS, their three output JSON files byte-identical to frozen evidence, and the unchanged PowerShell script PASS on Linux with PowerShell 7.6.9. Mathematical sections 1–7, verification algorithms, and frozen evidence are unchanged. Complete external-auditor and Windows execution transcripts were not part of the supplied source ZIP. Novelty has not been conclusively established. This preprint has not undergone human peer review. The conjectured unbounded growth in switching complexity remains unproved. Ryutaro Yonezu — Independent Researcher. No.50, v1.0, 2026-11-02. All rights reserved.