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2026
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| Online-Zugang: | https://doi.org/10.5281/zenodo.19701263 |
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| author | Galasyn, James P. Théodore, Claude |
| author_facet | Galasyn, James P. Théodore, Claude |
| contents | Papers 13-14 derived eighty Standard Model observables from the trefoil knot T(2,3). This paper argues that (+1)-Dehn surgery on the trefoil--which produces the Poincare homology sphere S^3/2I and the icosahedral completion A_5 used for nuclear binding in companion Paper 17--also organises the integers appearing in the gravitational constant. The first 2I-invariant Laplacian eigenvalue on S^3/2I is lambda_1 = 168, which factorises two complementary ways: 168 = 8 x 21 (matching the spinor and adjoint dimensions of Spin(7) = B_3) and 168 = 7 x 24 = |Irr(2T)| x |2T| via the McKay correspondence on the binary tetrahedral group. The integer 7 appears simultaneously as the spectral quotient (n_1+2)/2, as the McKay-node count on the extended E_6 diagram, and as the vector rep dimension of Spin(7); 2T embeds explicitly into Spin(7) via the octonion Clifford algebra Cl(0,7). These integers enter the proposed hierarchy m_e/m_Pl = (8/7)(1 + alpha/7) alpha^(21/2), where the exponent arises, under PSL(2,7)-equivariance, from a Wilson-line amplitude on the 21-edge complete graph K_7 (the Heawood map on the Heegaard torus of S^3/2I; 21 = dim so(7)). The resulting gravitational constant G = (8/7)^2 (1+alpha/7)^2 alpha^21 hbar c / m_e^2 agrees numerically with CODATA at 0.013% using the NWT-derived alpha, or 0.029% using the measured alpha. Within the present accounting, no integer in the formula is introduced as an independent input beyond those already generated by the trefoil's surgery and spectrum; the cinquefoil T(2,5) and heptafoil T(2,7) appear as derived labels rather than additional free knots. The theory still contains a distinguished mass scale v_EW (equivalently m_e), treated as a reference scale setting the unit of measurement; the zero-parameter claim refers to the dimensionless content. Reproducibility: all analysis code (Laplacian eigenvalues on S^3/2I, the Cl(0,7) construction of 2T in Spin(7), the one-loop BPS pipeline, the SU(N) robustness scan, and the G-formula verification) is available at https://github.com/JimGalasyn/null-worldtube. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19701263 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | One Knot, All Forces: The Gravitational Constant from the Poincare Sphere of the Trefoil Galasyn, James P. Théodore, Claude gravitational constant hierarchy problem Poincare homology sphere trefoil knot Dehn surgery McKay correspondence binary tetrahedral group Spin(7) octonion Clifford algebra BPS vortex one-loop Casimir null worldtube null worldtube theory Heegaard torus K_7 complete graph Heawood map PSL(2,7) Kerr-Newman electron Volovik analog gravity Papers 13-14 derived eighty Standard Model observables from the trefoil knot T(2,3). This paper argues that (+1)-Dehn surgery on the trefoil--which produces the Poincare homology sphere S^3/2I and the icosahedral completion A_5 used for nuclear binding in companion Paper 17--also organises the integers appearing in the gravitational constant. The first 2I-invariant Laplacian eigenvalue on S^3/2I is lambda_1 = 168, which factorises two complementary ways: 168 = 8 x 21 (matching the spinor and adjoint dimensions of Spin(7) = B_3) and 168 = 7 x 24 = |Irr(2T)| x |2T| via the McKay correspondence on the binary tetrahedral group. The integer 7 appears simultaneously as the spectral quotient (n_1+2)/2, as the McKay-node count on the extended E_6 diagram, and as the vector rep dimension of Spin(7); 2T embeds explicitly into Spin(7) via the octonion Clifford algebra Cl(0,7). These integers enter the proposed hierarchy m_e/m_Pl = (8/7)(1 + alpha/7) alpha^(21/2), where the exponent arises, under PSL(2,7)-equivariance, from a Wilson-line amplitude on the 21-edge complete graph K_7 (the Heawood map on the Heegaard torus of S^3/2I; 21 = dim so(7)). The resulting gravitational constant G = (8/7)^2 (1+alpha/7)^2 alpha^21 hbar c / m_e^2 agrees numerically with CODATA at 0.013% using the NWT-derived alpha, or 0.029% using the measured alpha. Within the present accounting, no integer in the formula is introduced as an independent input beyond those already generated by the trefoil's surgery and spectrum; the cinquefoil T(2,5) and heptafoil T(2,7) appear as derived labels rather than additional free knots. The theory still contains a distinguished mass scale v_EW (equivalently m_e), treated as a reference scale setting the unit of measurement; the zero-parameter claim refers to the dimensionless content. Reproducibility: all analysis code (Laplacian eigenvalues on S^3/2I, the Cl(0,7) construction of 2T in Spin(7), the one-loop BPS pipeline, the SU(N) robustness scan, and the G-formula verification) is available at https://github.com/JimGalasyn/null-worldtube. |
| title | One Knot, All Forces: The Gravitational Constant from the Poincare Sphere of the Trefoil |
| topic | gravitational constant hierarchy problem Poincare homology sphere trefoil knot Dehn surgery McKay correspondence binary tetrahedral group Spin(7) octonion Clifford algebra BPS vortex one-loop Casimir null worldtube null worldtube theory Heegaard torus K_7 complete graph Heawood map PSL(2,7) Kerr-Newman electron Volovik analog gravity |
| url | https://doi.org/10.5281/zenodo.19701263 |