Kramers–Wannier Duality as Angular Complementarity: A Cotangent Law for the Regular-Tiling Ising Critical Couplings

Scope and status clarification. This paper establishes a geometric unification of the three exactly solved two-dimensional Ising models via the identity e^{2K_c}=\cot\!\left(\frac{\pi}{2f}\right), where f is the face number of the underlying regular tiling of the flat torus. This identity is an exact re-expression of known results (Onsager, Wannier, Houtappel) and remains fully valid. The crystallographic restriction theorem then implies that the formally defined f=5 (pentagonal) case is geometrically inaccessible to periodic lattices, creating a well-defined angular gap between \pi/10 (five-fold, forbidden) and \pi/8 (four-fold, allowed). Interpretive status of the coherence corridor. The identification of this angular gap with the empirical “coherence corridor” [\sqrt{e},\,7/4] is structural and interpretive, not dynamical. The paper does not claim that systems universally lock to a fixed ratio or that the corridor represents a conserved quantity. Rather, it proposes that five-fold frustration—arising from the incompatibility of locally preferred five-fold coordination with global periodic order—creates a geometric admissibility window within which many near-critical systems operate. What is not claimed. No modification of Ising theory, statistical mechanics, crystallography, or information theory is proposed. No derivation of critical exponents, scaling laws, or dynamics is asserted. No claim is made that the corridor enforces scale-invariant ratios across physical systems; later analyses show that such ratios are contextual and history-dependent. What remains robust. The cotangent identity and its geometric interpretation via Thales angles. The role of crystallographic restriction as a hard geometric constraint generating a forbidden region. The framing of the corridor as an admissible region shaped by symmetry obstruction, not by fine-tuned dynamics. Relation to subsequent work. Later investigations (including dark-matter–motivated tests) indicate that partition ratios vary with environment and evolution. These results do not contradict the present paper: they clarify that the corridor should be understood as a constraint geometry—an organizing principle for where coherence can emerge—rather than as a universal attractor or invariant. Bottom line. This paper should be read as a geometric synthesis: it identifies a precise mathematical correspondence, a genuine topological obstruction (five-fold symmetry on the torus), and a plausible structural interpretation for why many systems cluster near a narrow operating window. The explanatory power lies in admissibility and frustration, not in prediction of fixed numerical outcomes.