Exact Small-Alphabet Distortion and Strict Alphabet Dependence for Length-Two Edit Distance

No.48 preprint v1.0. Ryutaro Yonezu - Independent Researcher. Let D_q(2) be the least distortion into real l1 of unit-cost insertion/deletion/substitution edit distance on all words of length at most two over a q-letter alphabet, including the empty word. Exact rational cut-cone certificates establish D_1(2)=1, D_2(2)=6/5, D_3(2)=17/14, and D_4(2)=123/101. An exhaustive integer cut inequality establishes D_q(2)>=785/643 for every q>=5, hence D_5(2)>D_4(2). Also supplied are the constructive upper bound 9/7 and a quantitative finite-alphabet limit comparison. The exact optimum for q>=5 and D_infinity(2) is not determined. Proof: Computer-assisted proof. Existing exact verifiers, all 240 symmetry actions, 1,048,576 q4 cuts, and 58,720,256 q5 canonical cases were rerun successfully; an independently constructed Python/NumPy q4 enumeration also PASS. Mathematical sections 1-6 and 35 protected code, evidence, and proof-note files are byte-for-byte unchanged from audited v0.2. The reported independent AI audit PASS is limited to the stated mathematics and finite computation. Not human peer-reviewed. Not a fully formalized Lean/Coq proof. Novelty: INCONCLUSIVE. Prior work: Andoni, Deza, Gupta, Indyk, and Raskhodnikova (SODA 2003, pp. 523-526; DOI 10.1145/644108.644196) already use finite-string cut-LP computations and report binary length<=3 distortion 4/3. The method is established prior work. OpenAI Math family 099 concerns asymptotic growth in maximum word length; this manuscript fixes length two and varies alphabet size. Its proof does not depend on correctness of OpenAI's result. No priority or general-optimum claim is made. License: All rights reserved. Copyright (c) 2026 Ryutaro Yonezu. GitHub repository: https://github.com/yonezaemon1-hub/no48-length-two-edit-distanceRelease: https://github.com/yonezaemon1-hub/no48-length-two-edit-distance/releases/tag/v1.0 SHA256 PDF: 0150365732196a784558f850ca866cb5dba7f9628359afa97cbb18c85565d498SHA256 ZIP: 1a6e3535c829abdf8a37020ab97d0471d32cf2670e4854a7403d50ad7379f173