Effective Spacetime Geometry from Admissible Non-Injective Projection
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| Natura: | Recurso digital |
| Lingua: | inglese |
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2026
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| _version_ | 1866901698731048960 |
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| author | Beau, Jérôme |
| author_facet | Beau, Jérôme |
| contents | <div> <p>Continuing from the gauge structure established in the companion paper Q6a, we derive effective spacetime geometry from the admissibility framework of Foundation paper by importing the continuum limit and geometric-extraction theorems of the companion paper Q5b. Under the lifting hypothesis [H-lift] of Q5b and the Mosco hypotheses of Q5a paper, the admissibility filter $\Pi_q$ converges to an effective operator $\mathcal{L}_{\mathrm{eff}}$ on $\mathbb{R}_\tau \times<br>\mathrm{Heis}_3(\mathbb{R})$ whose principal symbol defines a four-dimensional effective metric tensor $g^{\mu\nu}(x) \propto A^{\mu\nu}(x)$ of Lorentzian signature $(-,+,+,+)$ (Q5b Theorems 5.2 and~6.1). In the presence of a localised stationary obstruction to the admissibility flow with spherical symmetry, flux conservation within this effective geometry forces the Schwarzschild metric as the unique stationary exterior solution, with the horizon appearing as a degeneracy of the principal symbol rather than as a singularity of the underlying admissible structure. The Einstein equations emerge as consistency conditions of the emergent geometry, with the Einstein\textendash Hilbert action derived rigorously from the spectral entropy functional in the Gravity paper. The present paper closes the chain $\Pi_q \to \mathcal{L}_{\mathrm{eff}} \to g^{\mu\nu} \to G_{\mu\nu}$. Hypothesis [H-lift], on which the results were originally conditional, has since been proved in Q9; the main geometric results are therefore unconditional with respect to [H-lift]. No dynamical substrate is required at any step.</p> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20100336 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Effective Spacetime Geometry from Admissible Non-Injective Projection Beau, Jérôme Non-injective projection Emergent spacetime Admissibility Effective metric Lorentzian geometry Sub-Riemannian geometry Heisenberg group Principal symbol Schwarzschild solution Einstein equations Spectral geometry Induced gravity <div> <p>Continuing from the gauge structure established in the companion paper Q6a, we derive effective spacetime geometry from the admissibility framework of Foundation paper by importing the continuum limit and geometric-extraction theorems of the companion paper Q5b. Under the lifting hypothesis [H-lift] of Q5b and the Mosco hypotheses of Q5a paper, the admissibility filter $\Pi_q$ converges to an effective operator $\mathcal{L}_{\mathrm{eff}}$ on $\mathbb{R}_\tau \times<br>\mathrm{Heis}_3(\mathbb{R})$ whose principal symbol defines a four-dimensional effective metric tensor $g^{\mu\nu}(x) \propto A^{\mu\nu}(x)$ of Lorentzian signature $(-,+,+,+)$ (Q5b Theorems 5.2 and~6.1). In the presence of a localised stationary obstruction to the admissibility flow with spherical symmetry, flux conservation within this effective geometry forces the Schwarzschild metric as the unique stationary exterior solution, with the horizon appearing as a degeneracy of the principal symbol rather than as a singularity of the underlying admissible structure. The Einstein equations emerge as consistency conditions of the emergent geometry, with the Einstein\textendash Hilbert action derived rigorously from the spectral entropy functional in the Gravity paper. The present paper closes the chain $\Pi_q \to \mathcal{L}_{\mathrm{eff}} \to g^{\mu\nu} \to G_{\mu\nu}$. Hypothesis [H-lift], on which the results were originally conditional, has since been proved in Q9; the main geometric results are therefore unconditional with respect to [H-lift]. No dynamical substrate is required at any step.</p> </div> |
| title | Effective Spacetime Geometry from Admissible Non-Injective Projection |
| topic | Non-injective projection Emergent spacetime Admissibility Effective metric Lorentzian geometry Sub-Riemannian geometry Heisenberg group Principal symbol Schwarzschild solution Einstein equations Spectral geometry Induced gravity |
| url | https://doi.org/10.5281/zenodo.20100336 |