Exact complex coherence in codimension two

We determine the minimum coherence of N unit vectors in C^(N-2) for every integer N >= 180. The optimum is given by an explicit four-block construction with balanced integer multiplicities. The same optimum holds among all complex N by N matrices with diagonal entries one and rank at most N-2. Every extremizer is equivalent to the balanced four-block Gram matrix. The proof uses kernel duality to reduce the matrix problem to a geometric comparison in C^2. Any competing matrix whose coherence is no larger than that of the construction admits a normalized two-dimensional left-kernel representation with every column near one of four directions. A strict local comparison then determines all equality cases. Rational arithmetic certificates establish the finite inequalities needed for the range N >= 180. When N is not divisible by 4, the exact value improves the classical codimension-two bound by a positive term of order N^(-3). This record contains the 47-page preprint, its LaTeX source archive and the manuscript rights notice. The source archive includes main.bbl and compiles with three pdfLaTeX passes. The accompanying certificates and verification code are archived at https://doi.org/10.5281/zenodo.23477764. That software release corresponds to GitHub commit 7fa9cdc7472c726c0db2dad8efd92f5f5ccabe57; its certificates and verifier match the mathematical snapshot b7a51dbc324ba260a92c2fafd3642e686cefe91a cited in the paper. For current reproduction instructions and software citation metadata, see the README and CITATION.cff on GitHub. Copyright (c) 2026 Xuan Li. Rights to the manuscript are retained by the author. See MANUSCRIPT_RIGHTS.txt for the terms applicable to the PDF and source files. OpenAI's Codex was used extensively in the research and preparation of the manuscript, including use of GPT-6 Astra with the Ultra reasoning setting. Its use included literature searches and comparisons with earlier results (Section 2); exploration and development of the four-block construction and its algebraic formulas (Sections 3 and 6); and formulation, refinement, and critical examination of arguments concerning the matrix reduction, global localization, local comparison, and equality cases (Sections 4, 7, and 8). Section numbers refer to the accompanying manuscript. Codex was also used in developing the computational work associated with Section 9 and the accompanying reproduction package. This included implementing and revising code for numerical exploration, certificate generation, rational-arithmetic verification, and regression tests. AI-generated drafts of mathematical arguments and explanatory prose were used and revised during the preparation of the manuscript. Codex was also used to organize the presentation, revise notation and cross-references, and edit the English. The author assumes responsibility for all content, including the mathematical claims, references, code, and final text. Paper and source at fixed GitHub commit:https://github.com/motives3/coherence-codimension-two-reproduction/tree/b471ef7f445cbeb8a411f1798f9cc7dba49021df/paper