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2025
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| Online Access: | https://doi.org/10.5281/zenodo.17818471 |
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| _version_ | 1866901966565670912 |
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| author | Zaiken |
| author_facet | Zaiken |
| contents | <p>Abstract<br>We propose that the Viviani curve (sphere–cylinder intersection, 1724), the brachis-<br>tochrone cycloid (Bernoulli 1696), and the modern infinity symbol ∞ (Wallis 1655) are the<br>same geometric object up to affine transformation.<br>When this curve is used as the local transverse-field direction in a strongly-coupled<br>(J/h → ∞) cyclic XX spin-1/2 chain and sampled at exactly four sites per full 4π parametric<br>traversal (N ≡ 0 (mod 4)), numerical evidence up to N = 50 suggests the ground state<br>acquires exact 2N/4 -fold degeneracy with polarization along the local normals vanishing<br>to machine precision (typically |⟨m · n⟩| ≲ 10−24 , variance at double-precision noise floor<br>∼ 10−24 –10−29 ).<br>We present an analytic argument that period-4 sampling renders the lattice discretization<br>mathematically lossless and propose that the Viviani/brachistochrone/∞ manifold is the<br>unique lowest-action closed curve permitted by Euclidean geometry — the “laziest” way for<br>the universe to close a loop without energy cost.<br>The symbol has been staring at us from every mathematics book for 370 years. We<br>finally followed the line.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17818471 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Schrödinger's Fallacy Part III: Brachistochrone Determinism Zaiken <p>Abstract<br>We propose that the Viviani curve (sphere–cylinder intersection, 1724), the brachis-<br>tochrone cycloid (Bernoulli 1696), and the modern infinity symbol ∞ (Wallis 1655) are the<br>same geometric object up to affine transformation.<br>When this curve is used as the local transverse-field direction in a strongly-coupled<br>(J/h → ∞) cyclic XX spin-1/2 chain and sampled at exactly four sites per full 4π parametric<br>traversal (N ≡ 0 (mod 4)), numerical evidence up to N = 50 suggests the ground state<br>acquires exact 2N/4 -fold degeneracy with polarization along the local normals vanishing<br>to machine precision (typically |⟨m · n⟩| ≲ 10−24 , variance at double-precision noise floor<br>∼ 10−24 –10−29 ).<br>We present an analytic argument that period-4 sampling renders the lattice discretization<br>mathematically lossless and propose that the Viviani/brachistochrone/∞ manifold is the<br>unique lowest-action closed curve permitted by Euclidean geometry — the “laziest” way for<br>the universe to close a loop without energy cost.<br>The symbol has been staring at us from every mathematics book for 370 years. We<br>finally followed the line.</p> |
| title | Schrödinger's Fallacy Part III: Brachistochrone Determinism |
| url | https://doi.org/10.5281/zenodo.17818471 |