Unified Mass Decomposition in ECM and Cosmology

Author: Soumendra Nath Thakur | Tagore's Electronic Lab, India | January 30, 2026

Abstract: 

This statement formalizes the equivalence between cosmological mass relations from Chernin et al. (2013) and Extended Classical Mechanics (ECM) formulations, demonstrating 

M_M = M_G - M_DE 

and 

M_M = M^eff + |M^app|, 

with M_M = M_ORD + M_DM. 

These unify local field effects and large-scale observations, decoupling intrinsic matter mass from gravitational effects.

The relations 

M_M = M_G - M_DE (cosmology) and 

M_M = M^eff + |M^app| (ECM) 

hold consistently, decomposing total matter mass M_M into ordinary M_ORD and dark matter M_DM components.

Cosmological Framework

Chernin et al. (2013) establish gravitational mass as 

M_G = M_M + M_DE, where 

M_DE ∼ −(8π/3)ρΛR^3 < 0

induces antigravity at large radii (e.g., Coma cluster beyond R_ZG. Rearranging isolates matter mass: 

M_M = M_G - M_DE = M_ORD + M_DM,

matching observations where dark energy separates from clustered matter.

ECM Local Derivation

ECM defines effective mass via field-energy interactions (NAM): 

M^eff = M_M - M^app 

with (M^app < 0) (negative apparent mass from -∆PE_ECM/c^2. 

Thus, 

M_M = M^eff + |M^app| = M_ORD + M_DM, 

recovering intrinsic mass (e.g., photons:

M^eff = -2M^app 

applicable locally in motion or gravitation.

Presentation

Component | Cosmology | ECM Formulation

Gravitational Mass | M_G = M_M + M_DE | M_G = M^eff     

Matter Mass | M_M = M_G - M_DE | M_M = M^eff + |M^app| 

Subcomponents | M_M = M_ORD + M_DM | M_M = M_ORD + M_DM

Repulsive Term | M_DE < 0 | M^app < 0 

Conclusion:

The mass decompositions 

M_M = M_G − M_DE 

derived from large-scale cosmological observations and 

M_M = M^eff + ∣M^app∣ 

obtained within Extended Classical Mechanics are formally and physically equivalent. Both arise from Newtonian, force-based gravitational analysis, not from curved-spacetime or relativistic constructs.

Chernin et al's observational formulation reflects a classical interpretation of anti-gravitating field contributions at large radii, while ECM derives the same separation locally through field-energy redistribution via negative apparent mass. In both cases, intrinsic matter mass remains conserved and decomposes naturally into ordinary and dark components without invoking relativistic postulates.

This establishes that gravitational mass and inertial mass are not universally identical, while preserving classical mechanics at all scales. ECM thus reproduces cosmological observations using extended Newtonian principles, employing a generalized force law rather than the unmodified classical form, thereby avoiding unnecessary theoretical inflation and preserving conceptual continuity from laboratory-scale physics to cosmic structure.

Phase Advance, Phase Lag, and Time Measurement in ECM

January 30, 2026

Within the Extended Classical Mechanics (ECM) framework, phase lag corresponds to an observed time delay, whereas phase advance corresponds to an observed time advance. In both cases, the directly measurable quantity is the accumulated time shift relative to a reference clock, while clock time itself remains strictly positive, cyclic, and normalized to the local oscillatory standard.

Although phase shifts in ECM may be positive (phase advance) or negative (phase lag), clock-based observations record only the resulting time offset. Consequently, both phase advance and phase lag manifest operationally as time delays, with the directionality of the underlying effect encoded in the inferred phase or frequency relationship rather than in the clock time itself.

This operational nuance highlights why ECM may initially appear to differ from conventional physical interpretations. The apparent contradiction is not a failure of consistency, but a consequence of ECM’s explicit separation of conceptual variables—phase, frequency, and clock time—that are typically conflated in classical and relativistic frameworks. When these distinctions are properly accounted for, ECM reproduces all known experimental results while providing a phase-based, observer-accessible description of gravitational phenomena. In this view, ECM is not an alternative to physics; it is a refined framework that reveals the hidden structure of phase, time, and frequency interactions in gravitational fields.