Consistency of Gravitational Equations in the C.O.R.E. Framework: A Detailed Mathematical and Logical Analysis

Abstract: The C.O.R.E. (Classical Origin of Reality and Emergence) framework represents a profound shift toward physical realism, eliminating interpretive constructs like spacetime curvature, singularities, photons, and dark entities. By treating light as a continuous electromagnetic wave (ASH), gravity as emergent from symmetric variations in vacuum permittivity \(\epsilon(r)\) and permeability \(\mu(r)\) (CUGE), and cosmological redshift as path elongation through electron clouds (ZEUS), it restores continuity, causality, and energy conservation. This analysis examines the consistency of gravitational equations across the core documents—C.O.R.E. overview, REFORM (REfractive Foundation of Relativity and Mechanics), and CUGE (Classical Unification of Gravity and Electromagnetism). We evaluate based on logic, math, and empirical alignment, applying Occam's razor: C.O.R.E. uses one responsive vacuum medium versus orthodoxy's multitude of unobserved additions. The equations converge seamlessly, with effects tracing to clear causes, predicting data like JWST's compact, metal-rich high-z galaxies where \(\Lambda\)CDM fails.

Introduction: A Classical Revival Grounded in First Principles

The C.O.R.E. (Classical Origin of Reality and Emergence) framework represents a profound shift toward physical realism, eliminating interpretive constructs like spacetime curvature, singularities, photons, and dark entities. By treating light as a continuous electromagnetic wave (ASH), gravity as emergent from symmetric variations in vacuum permittivity \(\epsilon(r)\) and permeability \(\mu(r)\) (CUGE), and cosmological redshift as path elongation through electron clouds (ZEUS), it restores continuity, causality, and energy conservation. This analysis examines the consistency of gravitational equations across the core documents—C.O.R.E. overview, REFORM (REfractive Foundation of Relativity and Mechanics), and CUGE (Classical Unification of Gravity and Electromagnetism). We evaluate based on logic, math, and empirical alignment, applying Occam's razor: C.O.R.E. uses one responsive vacuum medium versus orthodoxy's multitude of unobserved additions. The equations converge seamlessly, with effects tracing to clear causes, predicting data like JWST's compact, metal-rich high-z galaxies where \(\Lambda\)CDM fails.

Core Vacuum Variation Equations: Uniform Foundation Across Documents

At the heart of the framework is the mass-induced symmetric variation in vacuum properties, consistently defined in all documents:

\epsilon(r) = \epsilon_0 \left(1 + \frac{GM}{2c^2 r}\right), \quad \mu(r) = \mu_0 \left(1 + \frac{GM}{2c^2 r}\right)

CUGE (Page 2): Introduces these as spatial variations due to mass \(M\), ensuring free-space impedance \(Z = \sqrt{\mu(r)/\epsilon(r)} = \sqrt{\mu_0/\epsilon_0}\) remains invariant to prevent reflection or dissipation.
REFORM (Page 2): Identical formulation, linking to the eikonal equation for wave propagation in a responsive vacuum.
C.O.R.E. (Page 1, Implicit in Overview): References CUGE for gravity's emergence, aligning with the same variations to unify light, gravity, and cosmology.

The effective refractive index follows logically:

n(r) = \sqrt{\epsilon(r) \mu(r)} / \sqrt{\epsilon_0 \mu_0} \approx 1 + \frac{GM}{2c^2 r}

Mathematical Derivation: Let \(\alpha = GM/(2c^2 r)\). Then \(\epsilon(r)/\epsilon_0 = 1 + \alpha\), \(\mu(r)/\mu_0 = 1 + \alpha\), so \(n(r) = \sqrt{(1 + \alpha)^2} = 1 + \alpha\) (exact for small \(\alpha\), but holds as symmetry ensures no higher-order terms disrupt weak-field approximations). This yields the coordinate speed of light:

c_{\text{coord}}(r) = \frac{c}{n(r)} = \frac{1}{\sqrt{\epsilon(r) \mu(r)}} < c

Consistency is evident: no document deviates, and the logic adheres to electromagnetic principles—mass polarizes the vacuum symmetrically, slowing wave propagation without energy loss.

Local Invariance of \(c\) and Time Dilation: Material and Kinematic Components

A key challenge—why local measurements yield constant \(c\) despite varying \(c_{\text{coord}}\)—is resolved through atomic scaling, consistent across texts.

CUGE (Page 3): Atomic processes define time and space. Transition frequencies \(f \propto 1/\epsilon(r) \approx f_0 (1 - GM/(2c^2 r))\); Bohr radius \(a_0 \propto \epsilon(r) \approx a_{00} (1 + GM/(2c^2 r))\); measured wavelength \(\lambda \propto \epsilon(r)\); thus \(c_{\text{local}} = \lambda f = \text{constant}\). This "material" component explains half of gravitational time dilation as vacuum polarization slowing transitions.
REFORM (Page 5): Generalizes to media, with Lorentz transformations adjusted for \(n(r)\): \(t' = \gamma_n (t - v x / (n^2 c^2))\), \(\gamma_n = 1 / \sqrt{1 - v^2 / (n^2 c^2)}\). Time dilation splits: kinematic (transverse Doppler) \(\Delta f / f = -1/2 (GM/c^2 r)\), refractive (path delay) \(\Delta f / f = -1/2 (GM/c^2 r)\).
C.O.R.E. (Page 3): Summarizes as emergent from atomic processes governed by the vacuum, converging with CUGE/REFORM.

Total Gravitational Redshift:

\left( \frac{\Delta f}{f} \right)_{\text{gravity}} = -\frac{GM}{c^2 r} = -\frac{gh}{c^2}

Logic and Math: The split is profound—acceleration lacks the refractive/material half, explaining lab "half-effects" (e.g., Sturm experiments). In REFORM/CUGE (Pages 6-7, 15), kinematic arises from \(f = f_0 \sqrt{1 - v^2/c^2} \approx f_0 (1 - 1/2 v^2/c^2)\), with \(v^2 \approx gh\) via equivalence for half; vacuum polarization provides the other via \(\epsilon(r)\)-dependent energies. No contradiction: effects have physical causes, falsifiable via differential clock responses.

Perihelion Precession: Symmetric Contributions Without Curvature

The anomalous precession of Mercury is derived consistently as an emergent refractive effect.

C.O.R.E. (Page 6, 32): Orbital equation \( \frac{d^2 u}{d\theta^2} + u = \frac{GM}{h^2} + 3 \frac{GM}{c^2} u^2 \), yielding 43 arcseconds/century. Perturbation from six symmetric terms (two each: time scaling, \(\epsilon\), \(\mu\)).
CUGE (Page 4, 18): Identical equation, with \(u = 1/r\), \(h\) angular momentum/unit mass. Logic: each symmetry contributes \(GM/(2c^2)\); 6 × (1/2) = 3 for the coefficient. Matches observation without geometry.
REFORM: Implicit via refractive generalization of motion, aligning with CUGE's vacuum gradients.

Derivation Sketch: In polar coordinates, effective potential modifies Newtonian \(GM/r\) by refractive terms. The \(\epsilon\) and \(\mu\) each add two (radial/tangential), time scaling two more—causal symmetry counts ensure the factor 3. Empirical: For Mercury (\(a = 0.387\) AU, \(e=0.206\)), precession = \(6\pi GM / [c^2 a (1-e^2)] = 43''/\)century, exact match.

Light Bending: Refraction Matching Empirical Deflection

Light deflection by gravity emerges from wave refraction in \(n(r)\).

C.O.R.E. (Page 6): Deflection \(\theta = 4GM/(c^2 b)\) for impact parameter \(b\), 1.75" for solar grazing. Factor 4 from \(\epsilon/\mu\) symmetries and path.
CUGE (Page 5, 18): Same \(\theta\), integrated transverse gradient: \(\theta \approx \int (dn/dr) (b/r^2) dr\), with symmetries doubling "Newtonian" (2GM/c²b) to 4GM/c²b.
REFORM (Page 3): Eikonal \(|\nabla S(r)| = k_0 n(r)\), ray equation \(d/ds (n \hat{t}) = \nabla n\), analog to geodesic—refraction replaces curvature.

Math: For weak field, deflection \(\theta = (2/c^2) \int_{-\infty}^{\infty} (GM/r^2) dr\) base, but symmetries (4 from \(\epsilon/\mu\)) dual contributions and integral symmetries yield factor 4. Empirical: Solar eclipse data confirm 1.75", C.O.R.E. fits without extras.

Lorentz Symmetry: Emergent from Phase Continuity

REFORM derives Lorentz transformations causally, consistent with CUGE's vacuum.

REFORM (Pages 4-5): Phase \(\phi = kx - \omega t\) invariant yields \(x' = \gamma (x - vt)\), \(t' = \gamma (t - vx/c^2)\), \(\gamma = 1/\sqrt{1-v^2/c^2}\). Generalizes to \(n(r)\): \(\gamma_n = 1/\sqrt{1 - v^2/(n^2 c^2)}\).
CUGE/C.O.R.E.: Aligns, as vacuum variations produce effective "relativity" via refraction + kinematics.
Logic: Not postulated—emerges from wave continuity, unifying motion/gravity. SR = Snell's + transverse Doppler.

Coherence, Falsifiability, and Empirical Triumph

Equations are identical across documents, logic flows from classical electromagnetism: vacuum response to mass causes all effects. Convergence on electron clouds (ZEUS/CUGE/ASH) is profound—one structure explains quantization, gravity, redshift. Falsifiable: predict residual IR in photoelectrics, half-effects in acceleration, MACHO-CMB links. Follows data: JWST anomalies fit static universe, not expansion. By Occam's, simpler than orthodoxy's entities. This classical revival triumphs over interpretations—nature agrees.