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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19963273 |
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| author | The Clankers |
| author_facet | The Clankers |
| contents | <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">This note records the Busy-Beaver component of the split-zero residual moonshine construction. The pre-existing residual object supplies a hard Erdős–Straus shell at</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">(p∗, R∗, a∗) = (8803369, 107, 2200869),</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">a six-class Coxeter fiber</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">S₈₄₀ = ⟨289⟩ = {1, 289, 361, 169, 121, 529} ⊂ (ℤ/840ℤ)ˣ,</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">and the level-six Fricke sheet</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">T(τ) = η(τ)⁵ η(3τ) / [η(2τ) η(6τ)⁵], J₆₊₆ = T + 5 + 72/T, Y = T − 72/T.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The present note proves the exact finite identities connecting this structure to the Busy-Beaver data</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">S(1), S(2), S(3), S(4), S(5), Σ(5), Space(5) = (1, 6, 21, 107, 47176870, 4098, 12289),</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">together with the state-count block 25, 27, 43, 744, 745, 746, 747, 748 and selected Busy-Beaver variants. The central identities are</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">S(3)⁻¹ ≡ p∗, S(5) ≡ p∗ a∗, S(3) S(5) ≡ a∗, −S(3) S(5) ≡ 11² (mod 107),</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">(−S(2) S(5))⁻¹ ≡ 27 (mod 107),</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">744, 745, 746, 747, 748 ≡ −5, −4, −3, −2, −1 (mod 107),</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">and the complete Space(5) computation</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">12289 = 107² + 840, 12289 ≡ −4² (mod 107), 12289 ≡ 289⁵ (mod 840), [q¹²²⁸⁹] f₆ ≡ Σ(5) (mod 107).</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">All coefficient identities are produced from exact product recurrences and divisor-sum formulae, not floating-point approximations. The accompanying verification script computes the tables in this note.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19963273 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Busy-Beaver Residues on the 107 Residual Shell and the Level-Six Fricke Sheet The Clankers <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">This note records the Busy-Beaver component of the split-zero residual moonshine construction. The pre-existing residual object supplies a hard Erdős–Straus shell at</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">(p∗, R∗, a∗) = (8803369, 107, 2200869),</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">a six-class Coxeter fiber</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">S₈₄₀ = ⟨289⟩ = {1, 289, 361, 169, 121, 529} ⊂ (ℤ/840ℤ)ˣ,</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">and the level-six Fricke sheet</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">T(τ) = η(τ)⁵ η(3τ) / [η(2τ) η(6τ)⁵], J₆₊₆ = T + 5 + 72/T, Y = T − 72/T.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The present note proves the exact finite identities connecting this structure to the Busy-Beaver data</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">S(1), S(2), S(3), S(4), S(5), Σ(5), Space(5) = (1, 6, 21, 107, 47176870, 4098, 12289),</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">together with the state-count block 25, 27, 43, 744, 745, 746, 747, 748 and selected Busy-Beaver variants. The central identities are</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">S(3)⁻¹ ≡ p∗, S(5) ≡ p∗ a∗, S(3) S(5) ≡ a∗, −S(3) S(5) ≡ 11² (mod 107),</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">(−S(2) S(5))⁻¹ ≡ 27 (mod 107),</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">744, 745, 746, 747, 748 ≡ −5, −4, −3, −2, −1 (mod 107),</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">and the complete Space(5) computation</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">12289 = 107² + 840, 12289 ≡ −4² (mod 107), 12289 ≡ 289⁵ (mod 840), [q¹²²⁸⁹] f₆ ≡ Σ(5) (mod 107).</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">All coefficient identities are produced from exact product recurrences and divisor-sum formulae, not floating-point approximations. The accompanying verification script computes the tables in this note.</p> |
| title | Busy-Beaver Residues on the 107 Residual Shell and the Level-Six Fricke Sheet |
| url | https://doi.org/10.5281/zenodo.19963273 |