Torus Tilings, Crystallographic Restriction, and the π/8 Ising–Thales Correspondence
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2026
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| author | De Jesus, Elias |
| author_facet | De Jesus, Elias |
| contents | <p><strong>Scope and status clarification.</strong></p> <p>This paper establishes a <em>geometric unification</em> of the three exactly solved two-dimensional Ising models via the identity</p> <p>e^{2K_c}=\cot\!\left(\frac{\pi}{2f}\right),</p> <p>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).</p> <p><strong>Interpretive status of the coherence corridor.</strong></p> <p>The identification of this angular gap with the empirical “coherence corridor” [\sqrt{e},\,7/4] is <em>structural and interpretive</em>, not dynamical. The paper does <strong>not</strong> 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 <em>geometric admissibility window</em> within which many near-critical systems operate.</p> <p><strong>What is not claimed.</strong></p> <ul> <li> <p>No modification of Ising theory, statistical mechanics, crystallography, or information theory is proposed.</p> </li> <li> <p>No derivation of critical exponents, scaling laws, or dynamics is asserted.</p> </li> <li> <p>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.</p> </li> </ul> <p><strong>What remains robust.</strong></p> <ul> <li> <p>The cotangent identity and its geometric interpretation via Thales angles.</p> </li> <li> <p>The role of crystallographic restriction as a <em>hard geometric constraint</em> generating a forbidden region.</p> </li> <li> <p>The framing of the corridor as an admissible region shaped by symmetry obstruction, not by fine-tuned dynamics.</p> </li> </ul> <p><strong>Relation to subsequent work.</strong></p> <p>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 <em>constraint geometry</em>—an organizing principle for where coherence can emerge—rather than as a universal attractor or invariant.</p> <p><strong>Bottom line.</strong></p> <p>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 <em>admissibility and frustration</em>, not in prediction of fixed numerical outcomes.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18778863 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Torus Tilings, Crystallographic Restriction, and the π/8 Ising–Thales Correspondence De Jesus, Elias Ising model, critical coupling, flat torus tilings, crystallographic restriction theorem, Thales semicircle, five-fold symmetry, quasicrystals, geometric frustration, phase transitions, coherence corridor, Shannon optimality, statistical mechanics, lattice geometry, holographic correspondence <p><strong>Scope and status clarification.</strong></p> <p>This paper establishes a <em>geometric unification</em> of the three exactly solved two-dimensional Ising models via the identity</p> <p>e^{2K_c}=\cot\!\left(\frac{\pi}{2f}\right),</p> <p>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).</p> <p><strong>Interpretive status of the coherence corridor.</strong></p> <p>The identification of this angular gap with the empirical “coherence corridor” [\sqrt{e},\,7/4] is <em>structural and interpretive</em>, not dynamical. The paper does <strong>not</strong> 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 <em>geometric admissibility window</em> within which many near-critical systems operate.</p> <p><strong>What is not claimed.</strong></p> <ul> <li> <p>No modification of Ising theory, statistical mechanics, crystallography, or information theory is proposed.</p> </li> <li> <p>No derivation of critical exponents, scaling laws, or dynamics is asserted.</p> </li> <li> <p>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.</p> </li> </ul> <p><strong>What remains robust.</strong></p> <ul> <li> <p>The cotangent identity and its geometric interpretation via Thales angles.</p> </li> <li> <p>The role of crystallographic restriction as a <em>hard geometric constraint</em> generating a forbidden region.</p> </li> <li> <p>The framing of the corridor as an admissible region shaped by symmetry obstruction, not by fine-tuned dynamics.</p> </li> </ul> <p><strong>Relation to subsequent work.</strong></p> <p>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 <em>constraint geometry</em>—an organizing principle for where coherence can emerge—rather than as a universal attractor or invariant.</p> <p><strong>Bottom line.</strong></p> <p>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 <em>admissibility and frustration</em>, not in prediction of fixed numerical outcomes.</p> |
| title | Torus Tilings, Crystallographic Restriction, and the π/8 Ising–Thales Correspondence |
| topic | Ising model, critical coupling, flat torus tilings, crystallographic restriction theorem, Thales semicircle, five-fold symmetry, quasicrystals, geometric frustration, phase transitions, coherence corridor, Shannon optimality, statistical mechanics, lattice geometry, holographic correspondence |
| url | https://doi.org/10.5281/zenodo.18778863 |