Mathematical Foundations of Structured Space Theory (SST) — Supplement 2: General-Relativistic Recovery via Regge Calculus
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2026
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| _version_ | 1866901633621819392 |
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| author | Marson, Mauro |
| author_facet | Marson, Mauro |
| contents | <p><strong>Closing the missing gravitational field equation for the SST lattice</strong></p> <p>This supplement shows that the SST gravitational lattice admits a Regge-calculus realization, supplying the missing discrete gravitational field equation and extending the framework across weak-field, strong-field, rotating, and homogeneous cosmological regimes. It positions the <strong>gravitational </strong>sector of <strong>SST </strong>as <strong>formally complete</strong>, at the framework and continuum-limit level, within its Regge realization.</p> <p>We show that the SST gravitational lattice class admits a Regge-calculus realization. For the working FCC/tetrahedral–octahedral model, the construction is explicit, and SST lattice equilibrium is identified with Regge stationarity. We construct the simplicial decomposition and four-dimensional spacetime extension, derive the discrete Laplacian from the z = 12 cuboctahedral coordination shell, and prove that the linearized Regge equations reduce to the Poisson equation ∇²Φ = 4πGρ. In the weak-field regime, the SST response laws (E7–E8) are shown to parameterize the corresponding class of edge-length variations that extremize the linearized Regge action, establishing variational equivalence between SST equilibrium and the linearized Einstein field equations.</p> <p>This closes the missing discrete gravitational field equation for the SST lattice (Open Problem 6 of the Unified Pattern Framework), with the matter source term — including angular-momentum current for rotating sources — derived in the continuum limit from the UPF mass and centroid-circulation identifications. The vector Laplacian on the FCC lattice correctly discretizes all metric components, including the off-diagonal terms that encode frame dragging, thereby establishing the weak-field rotating sector. For homogeneous expansion, the Friedmann equations follow by applying the Regge cosmology result to the SST simplicial lattice, with flat spatial curvature (k = 0) fixed by the FCC topology rather than imposed as a boundary condition. In the strong-field regime, the full nonlinear Regge equations converge to the Schwarzschild solution, identifying the exponential form of E7–E8 as the weak-field approximation. Taken together, these results make the gravitational sector of SST formally complete, at the framework and continuum-limit level, within its Regge realization.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19376061 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Mathematical Foundations of Structured Space Theory (SST) — Supplement 2: General-Relativistic Recovery via Regge Calculus Marson, Mauro Structured Space Theory Regge calculus simplicial lattice discrete Einstein equations FCC honeycomb weak-field equivalence strong-field metric Kerr metric Friedmann equations cosmological redshift gravitational waves <p><strong>Closing the missing gravitational field equation for the SST lattice</strong></p> <p>This supplement shows that the SST gravitational lattice admits a Regge-calculus realization, supplying the missing discrete gravitational field equation and extending the framework across weak-field, strong-field, rotating, and homogeneous cosmological regimes. It positions the <strong>gravitational </strong>sector of <strong>SST </strong>as <strong>formally complete</strong>, at the framework and continuum-limit level, within its Regge realization.</p> <p>We show that the SST gravitational lattice class admits a Regge-calculus realization. For the working FCC/tetrahedral–octahedral model, the construction is explicit, and SST lattice equilibrium is identified with Regge stationarity. We construct the simplicial decomposition and four-dimensional spacetime extension, derive the discrete Laplacian from the z = 12 cuboctahedral coordination shell, and prove that the linearized Regge equations reduce to the Poisson equation ∇²Φ = 4πGρ. In the weak-field regime, the SST response laws (E7–E8) are shown to parameterize the corresponding class of edge-length variations that extremize the linearized Regge action, establishing variational equivalence between SST equilibrium and the linearized Einstein field equations.</p> <p>This closes the missing discrete gravitational field equation for the SST lattice (Open Problem 6 of the Unified Pattern Framework), with the matter source term — including angular-momentum current for rotating sources — derived in the continuum limit from the UPF mass and centroid-circulation identifications. The vector Laplacian on the FCC lattice correctly discretizes all metric components, including the off-diagonal terms that encode frame dragging, thereby establishing the weak-field rotating sector. For homogeneous expansion, the Friedmann equations follow by applying the Regge cosmology result to the SST simplicial lattice, with flat spatial curvature (k = 0) fixed by the FCC topology rather than imposed as a boundary condition. In the strong-field regime, the full nonlinear Regge equations converge to the Schwarzschild solution, identifying the exponential form of E7–E8 as the weak-field approximation. Taken together, these results make the gravitational sector of SST formally complete, at the framework and continuum-limit level, within its Regge realization.</p> |
| title | Mathematical Foundations of Structured Space Theory (SST) — Supplement 2: General-Relativistic Recovery via Regge Calculus |
| topic | Structured Space Theory Regge calculus simplicial lattice discrete Einstein equations FCC honeycomb weak-field equivalence strong-field metric Kerr metric Friedmann equations cosmological redshift gravitational waves |
| url | https://doi.org/10.5281/zenodo.19376061 |