The SATOR Matrix - Branch-Exchange Tests for the Refined Decagonal Lift
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2026
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| _version_ | 1866901632582680576 |
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| author | Somma, Massimo Michele Edoardo |
| author_facet | Somma, Massimo Michele Edoardo |
| contents | <p>This note tests whether the branch-exchange symmetry required by the weak saturation route is already present in the refined decagonal pentagrammatic lift. The answer is negative for the lift as currently written, and the obstruction appears already at the root-return support level of the 70-state decagonal lift, not merely as a feature of the minimal non-oriented pentagonal quotient.</p> <p>The shell and arm return supports decompose into a different number of connected components — two disjoint cycles versus a single cycle — and since component count is invariant under permutation conjugacy, no relabelling of decagonal phases can exchange shell and arm. The same obstruction is visible at the level of the full directed type graph: its automorphism group acts independently on shell and arm types and never mixes them. The diagonal lifted parity operator implements a character shift but is not a permutation of shell and arm states.</p> <p>The branch-exchange symmetry capable of forcing a full-fibre top-exterior response is therefore not a hidden symmetry of the present decagonal lift; it would have to be added as a genuine extension or replaced by a stronger full-fibre covariance principle.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20261067 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | The SATOR Matrix - Branch-Exchange Tests for the Refined Decagonal Lift Somma, Massimo Michele Edoardo <p>This note tests whether the branch-exchange symmetry required by the weak saturation route is already present in the refined decagonal pentagrammatic lift. The answer is negative for the lift as currently written, and the obstruction appears already at the root-return support level of the 70-state decagonal lift, not merely as a feature of the minimal non-oriented pentagonal quotient.</p> <p>The shell and arm return supports decompose into a different number of connected components — two disjoint cycles versus a single cycle — and since component count is invariant under permutation conjugacy, no relabelling of decagonal phases can exchange shell and arm. The same obstruction is visible at the level of the full directed type graph: its automorphism group acts independently on shell and arm types and never mixes them. The diagonal lifted parity operator implements a character shift but is not a permutation of shell and arm states.</p> <p>The branch-exchange symmetry capable of forcing a full-fibre top-exterior response is therefore not a hidden symmetry of the present decagonal lift; it would have to be added as a genuine extension or replaced by a stronger full-fibre covariance principle.</p> |
| title | The SATOR Matrix - Branch-Exchange Tests for the Refined Decagonal Lift |
| url | https://doi.org/10.5281/zenodo.20261067 |