ECM Unified Gravitational–Cosmological Equation: Phase Kernel Formalism, Effective Gravitational Mass and Clarification of Shapiro Delay and Gravitational Lensing

This study presents a unified gravitational–cosmological equation derived within the Extended Classical Mechanics (ECM) framework. Using the Phase Kernel Formalism, the model introduces an effective gravitational mass representation (Mᵉᶠᶠ) that links phase evolution, frequency dynamics, and cosmological energy redistribution. The framework provides reinterpretations of gravitational lensing and the Shapiro time delay while maintaining compatibility with Planck-scale physical constants, suggesting that gravitational phenomena may emerge from deterministic phase dynamics across cosmological scales.ECM provides a unified framework connecting the fundamental oscillatory origin of existence, phase evolution, frequency variation, energy redistribution, mass manifestation, and gravitational interaction with cosmic evolution. Time emerges as a consequence of the phase progression of primordial oscillations, while events arise from deviations from the primordial frequency. Central to ECM, the Phase Kernel Formalism represents cumulative phase delays along a path, governing the interaction of energy, frequency, and mass. Effective Gravitational Mass (Mᵉᶠᶠ), defined as the sum of matter mass (Mᴍ) and negative apparent mass (−Mᵃᵖᵖ), links energy redistribution to observable gravitational phenomena.Gravitational effects, including lensing and the Shapiro time delay, are interpreted as arising from local, symmetric modulations of a photon's instantaneous frequency and momentum, rather than any change in the intrinsic speed of light. The manuscript develops a unified ECM gravitational–cosmological equation that integrates these mechanisms, capturing phase, frequency, momentum, and effective gravitational mass in a single coherent framework.The derivation is presented gradually to allow comprehension at multiple levels— from conceptual understanding suitable for students to advanced mathematical formalism for researchers. A canonical set of ECM equations is provided as the minimal mathematical backbone of the theory, illustrating how local photon–gravity interactions and large-scale cosmic evolution emerge from the same fundamental phase–frequency–energy dynamics.