CCEGA Part IV: 5D Bulk Geometry and Dimensional Reduction to 4D Effective Gravity (v1.0)

Building on the geometric framework of Part I and thermodynamic justification in Part II, we present the complete 5D dimensional reduction that underlies the predictions of P28-P43. The maximum neutron star mass Mₘax = 4. 26 M_☉ derived via TOV integration in P34 is verified numerically here using the Israel junction formalism. Description/Abstract: We present the complete theoretical framework for dimensional reduction in the Curvature-Coupled Exponential Gravity (CCEGA) scenario from five-dimensional bulk geometry to four-dimensional effective gravity. The Israel junction conditions relating brane stress-energy to bulk geometry are derived rigorously and verified numerically via modified TOV integration. KEY RESULTS: • Complete 5D warped metric ansatz with scalar field coupling to brane matter • Israel junction boundary conditions: analytical derivation with numerical verification • Modified TOV integrator with density-dependent coupling Gₑ (ρ) = GN exp (-ρ/ρc) • Confirmation of maximum neutron star mass Mₘax = 4. 26 M_☉ (consistent with prior work) • Gauss-Codazzi decomposition formalism for 5D-to-4D projection • Weyl tensor projection and effective dark matter interpretation framework • Complete numerical algorithms and implementation details This work provides the complete mathematical and numerical foundation for CCEGA dimensional reduction. The functional form of the Weyl tensor projection E_μν (ρ) and its relation to the exponential coupling form remain open questions for Part IV v1. 1. Astrophysical predictions from Parts I-III and papers P28-P42 remain fully consistent with this framework. The geometric origin of the effective coupling and the role of Weyl projection in dark matter phenomenology constitute the next phase of investigation. TECHNICAL DETAILS: - 30 pages of theoretical derivations and numerical methods - 4 main sections + 4 detailed appendices - TOV integration verified for APR4, SLy, MPA1 equations of state - 5D bulk integration algorithms documented Israel junction conditi ons implemented and tested numerically