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The aim is to derive critical parameters in a modified gravity framework using a heat-kernel expansion on a Schwarzschild background.

July 13, 2026Open Access

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RG-S Framework v25: Complete Lagrangian, Minimal MERA Berry Phase, Falsifiability, and Heat-Kernel Derivation of γ = 0.0931

JAJuan Arroyo

Key Points

The aim is to derive critical parameters in a modified gravity framework using a heat-kernel expansion on a Schwarzschild background.
Derivation of γ parameter from a 1-loop heat-kernel expansion on curvature-dependent mass
Employing scalar-vector interactions involving a Z2 scalar and Proca vector
Analysis of gravitational wave echo characteristics under modified field theories.
γ parameter derived as 0.0931 ± 0.0004, indicating profound implications for gravitational dynamics
Predicted GW echo amplitude Aecho/Ainit = 0.03 ± 0.01 at 45 ± 5 ms for a stable remnant
Defined topological hysteresis as a geometric Berry phase requiring a massive number of coherent quanta.

Abstract

Public version contains strictly astrophysical fundamental theory only. Full La-grangian with scalar-vector interactions and engineering mappings is reserved underU.S. provisional patent filings April-July 2026. We present the sanitized v26 of theRG-S framework. From the 1-loop heat-kernel expansion on Schwarzschild back-ground with a Z2 scalar φ1 and massive Proca vector φ2μ with curvature-dependentmass m2eff = m20 + ξK, K = RμνρσRμνρσ, we derive γ = 0.0931 ± 0.0004, ξ = 1/6,predicting a stable remnant M∗ = √γ/4πMP l = 0.0861MP l with S = 0, T = 0.The framework predicts a falsifiable GW echo Aecho/Ainit = 0.03 ± 0.01 at 45 ± 5 msfor ∼ 62M⊙ remnant, independent of Proca mass. Topological Hysteresis is definedas a geometric Berry phase in a MERA cortex requiring Nent ≥ 5 × 1053 coherentquanta for kpc displacements. Three falsifiers are defined.Note on versions: v25 full version with explicit gsp coupling and interaction terms is underrestricted access due to patent-pending matter. This v26 is the open astrophysical-onlyversion. Concept DOI: 10.5281/zenodo.21253878. This version DOI: v26 sanitized. Keywords Modified Gravity, General Relativity Extension, Symmetron Field, Proca Field, Fluid Geometry, Galactic Dynamics, Astrophysics, Theoretical Physics, Open Science

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RG-S Framework v25: Complete Lagrangian, Minimal MERA Berry Phase, Falsifiability, and Heat-Kernel Derivation of γ = 0.0931

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Abstract

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