Extended Classical Mechanics Numerical Calculation of Phase Kernel & Weak-Field Formalism

This paper presents a numerical and analytical formulation of the Extended Classical Mechanics (ECM) phase kernel for gravitational propagation, weak-field lensing, phase delay, and related frequency transformations. The formulation begins with an operational frequency variation Δf, from which ECM defines an energetic phase measure x° and its associated phase-resolved temporal interval through x° = 360°fΔt. The gravitational propagation kernel is developed first in a GR-equivalent weak-field form, Φₖₑᵣₙ = −2Φₙ/c³, and is integrated along a photon or radio-signal trajectory to recover the standard leading-order Shapiro-delay structure. A numerical solar-grazing calculation gives a one-way delay of approximately 119.46 μs for the adopted geometry, providing a direct computational benchmark for the formalism. A phenomenological extension, designated Model B, introduces higher-order corrections through dimensionless coefficients α₁, α₂, … and the Schwarzschild-radius expansion rₛ/r. The resulting residual phase or time-delay contribution provides a parameterized route for testing departures from the GR-equivalent base kernel. The paper formulates a weighted least-squares procedure for estimating these coefficients from reduced radio-science observations and explicitly distinguishes fitted ECM parameters from published relativistic parameters such as γ. Cassini and Viking solar-conjunction observations, together with VLBI gravitational-delay measurements, are identified as independent observational benchmarks for such tests. The ECM treatment is further extended to gravitational deflection through an effective refractive-index or energetic-phase-gradient formulation. Retaining the proposed pre-Planck phase-threshold factor 360°/359.991934° yields an ECM phenomenological prediction corresponding to an effective PPN-equivalent deviation of approximately γECM − 1 ≈ 4.48 × 10⁻⁵. This value is presented as a model prediction rather than an established experimental result and is proposed as a direct target for precision light-deflection measurements. A possible Gaia astrometric or parallax signature is consequently formulated as a secondary, scanning-law- and covariance-dependent prediction requiring an explicit observation-level or AGIS-level simulation. Finally, the paper applies the ECM phase–frequency interpretation to cosmological redshift, describing the observed frequency reduction as an accumulated transformation of photon manifested mass-energy associated with transfer to a vacuum phase field represented by negative apparent mass. Within this interpretation, cosmological acceleration is associated with the dynamical negative apparent-mass field rather than an independently introduced cosmological constant. The resulting framework therefore provides a unified computational pathway connecting frequency variation, energetic phase, propagation delay, gravitational deflection, higher-order weak-field corrections, and cosmological phase–frequency transformation, while leaving the proposed deviations subject to quantitative observational validation.