Foundation for the Mass Scaling Upgrade and Ramanujan Inspired p series Deformation

This paper is the first of two protocols that collectively build a complete theory of modified gravity based on Ramanujan's p-series. Protocol 1 focuses exclusively on the field-theoretic foundation of the deformation kernel. Key Results: 1. 5D Warped Geometry: The theory is formulated in a 5D anti-de Sitter (AdS_5) bulk with a warped extra dimension, bounded by a UV brane at z=0 and an IR brane at z=R_0.2. Action Principle: A bulk scalar field Psi(x,z) is coupled to a 4D gravitational potential phi(x) on the UV brane. The action is explicitly written and varied.3. KK Decomposition: Separation of variables leads to a Bessel equation for the extra-dimensional wavefunctions. Boundary conditions (Neumann at UV, Dirichlet at IR) quantize the KK spectrum to mu_n = n divided by R_0.4. Effective Potential: The static gravitational potential is proportional to (1/R) times the sum over n of (n/R_0) to the power (p-2) times e to the power (-nR/R_0).5. Poisson Resummation: The discrete sum is exactly evaluated to obtain the normalized deformation kernel: K_p(R) = zeta(p, 1 + R/R_0) divided by zeta(p).6. UV Regularization: The R=0 singularity is resolved by introducing a UV cutoff r_0. The regularized potential is Phi_reg(R) = -GM divided by the square root of (R squared plus r_0 squared) times K_p of (square root of (R squared plus r_0 squared) divided by R_0). The potential becomes finite at the origin.7. Regge-Wheeler Perturbations: Axial gravitational perturbations with angular momentum l=2 are analyzed. The effective potential is derived, and the shift in the quasinormal mode frequency is computed using third-order WKB: delta omega over omega is approximately equal to negative one-ninth times (r_0 divided by R_s) squared.8. LIGO-Virgo Constraint: From GW150914, the bound |delta omega / omega| is less than or equal to 0.015 gives: r_0 is less than or equal to 0.367 times R_s.9. Dynamical Scaling of r_0: To avoid low-mass black hole conflicts (such as GW230529), r_0 scales with the Planck length: r_0(M) = (Planck length squared times R_s) to the one-third power, which is proportional to M to the one-third power. This yields r_0 approximately 10 to the power -21 meters for stellar-mass black holes, satisfying all experimental bounds. Included Material: · Full derivations of all equations (no steps skipped)· Python code for WKB frequency shift calculation· Complete bibliography with 7 references· Summary tables of boundary conditions, quantization, and resolved critiques