Geometric Condensation of Fundamental Interactions: From the Classical Multi-Component Lagrangian to the Simplicial Action Functional on $\Delta_4 \times \Delta_2$
Year: 2026
Type: preprint
Abstract: **Author:** Reinaldo M. Silva-Filho **Affiliation:** Programa de Pós-Graduação em Estatística e Experimentação Agropecuária (PPGEE/DES), Departamento de Estatística (DES), Universidade Federal de Lavras (UFLA), Lavras, MG, Brazil --- **Abstract & Theoretical Summary:**The complete Lagrangian describing the Standard Model coupled to Einstein–Hilbert General Relativity is conventionally formulated as an empirical sum of gauge, chiral fermion, scalar Higgs, and gravitational terms. When expanded into fundamental field components, it contains 53 algebraic operator terms and 19 empirically determined parameters. In this paper, we demonstrate that this multi-component functional reduces to an irreducible, three-term geometric action functional defined on the product simplex manifold $\mathcal{M}_{\mathrm{univ}} = \Delta_4 \times \Delta_2$. Within this continuous simplicial geometry: 1. **Universal Gauge-Gravitational Curvature:** Gauge kinetic interactions and curved spacetime geometry are generated by the universal curvature 2-form $(1/2) \mathrm{Tr}(\Omega \wedge \star_G \Omega)$, where universal covering space contractibility removes Gribov horizon ambiguities and decouples the Faddeev-Popov ghost scalar pole, generating the Yang-Mills mass gap. 2. **Fermion-Boundary Interaction:** Chiral fermions exist as boundary states governed by the simplicial Dirac–Kähler operator $(1/2) \langle \Psi | i \slashed{\gamma}^\mu D_\mu | \Psi \rangle$. 3. **Weingarten Endomorphism & Higgs Mechanism:** The scalar Higgs field emerges from the internal geometric variation of the 2-simplex $\Delta_2$, while the Yukawa mass coupling and electroweak symmetry breaking are governed by the Weingarten shape endomorphism $W^\nu_\mu$ acting on boundary fields, dynamically generating the Koide mass hierarchies. This geometric condensation resolves the structural redundancy of the Standard Model, providing a rigorous, divergence-free mathematical framework for fundamental interactions without ad hoc parameters or unphysical ghost singularities. --- **Formal Verification & Stress Testing:**- **Lean 4 Verification:** Supported by 18/18 verified formal proofs in Lean 4 (Package `formal_proofs_geometric_condensation`), with 0 errors and 0 unproven `sorry` axioms.- **Python Stress Tests:** Validated via `verify_geometric_condensation.py` numerical batteries spanning simplicial spectral flow, curvature bounds, Weingarten variations, and fractional operator convergence.- **Publication Artifacts:** Fully compiled 11-page AMS-LaTeX monograph (`paper_geometric_condensation.pdf`) providing the rigorous metric-measure foundation. --- **Funding & Acknowledgments:**This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. *(O presente trabalho foi realizado com apoio da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Código de Financiamento 001).*
Author: Silva-Filho, Reinaldo M.
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212273801
Cites:
Cited by:
Related to: