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. 

Photon Trajectory Modulation in Convergent Gravitational Fields

Soumendra Nath Thakur | 
ORCiD: 0000-0003-1871-7803
January 29, 2026

Gravity itself does not “bend”; rather, gravitational field lines converge toward a massive body. The gravitational field of a massive object is a spherically symmetric potential field whose strength decreases with the inverse square of the distance from the source.

Photons, as carriers of electromagnetic energy and momentum, propagate through this convergent gravitational field. As a photon passes near a massive body, the spatial gradient of the gravitational potential alters the photon’s momentum direction through continuous interaction with the gravitational field. This interaction produces a gradual change in the photon’s trajectory, which is observationally interpreted as the bending of light near massive objects.

In Extended Classical Mechanics (ECM) terms, as a photon traverses a gravitational field, the spatial variation of gravitational potential produces a position-dependent phase and time delay. This cumulative phase modulation alters the effective propagation direction of the photon, resulting in an apparent deflection of its trajectory when passing near a massive body.

A Unified Paradigm of Cosmic Manifestation: Bridging Extended Classical Mechanics, Sen’s Conjecture, and Conformal Cyclic Cosmology

The fundamental architecture of modern theoretical physics is currently undergoing a period of profound re-evaluation. While the dual pillars of General Relativity and Quantum Field Theory have provided remarkably accurate descriptions of the macroscopic and microscopic worlds respectively, the persistent inability to reconcile these frameworks—particularly at the singularity of the Big Bang and the event horizons of black holes—suggests a missing ontological layer. The standard cosmological model, ΛCDM, relies on the existence of dark energy and dark matter, components that constitute the vast majority of the universe's mass-energy budget yet lack a definitive physical carrier or structural explanation. Simultaneously, string theory, in its quest for a theory of everything, has uncovered deep truths about the nature of vacuum stability and the decay of material structures, most notably through the conjectures of Ashoke Sen. Roger Penrose’s Conformal Cyclic Cosmology (CCC) offers a complementary geometric perspective, positing that the universe does not begin or end but iterates through infinite aeons. Extended Classical Mechanics (ECM), as formulated by Soumendra Nath Thakur, provides the necessary physical bridge between these diverse perspectives. By reinterpreting mass not as a static scalar but as a dynamic, redistributable energy reservoir governed by frequency and phase, ECM offers a mechanistic explanation for the transition from non-eventful potential to manifested existence, thereby unifying Sen’s microscopic dissolution with Penrose’s macroscopic cycles.

Extended Classical Mechanics (ECM) Insight: Why Vacuum Isn’t Empty

Soumendra Nath Thakur 

January 05, 2026

In Extended Classical Mechanics (ECM), mass is not a fixed substance. It is a dynamic quantity that changes when energy is released from gravitational or motional potential. Existence in the universe is distributed across different frequency bands rather than being confined to visible matter alone.

When a system converts stored potential energy into manifested energy, part of its material mass is reduced by what ECM calls Negative Apparent Mass (NAM). When this reduction becomes equal to the matter mass itself, the effective mass of the system becomes zero. At this point, matter does not continue moving as a particle — it transitions into a vacuum-mode state of the Phase Kernel.

This vacuum-mode is called f0.

This state does not mean “nothing exists.” It means the system no longer exists as matter. Instead, it exists as a pure energy-frequency state within the Phase Kernel. In this form it has no inertia, no gravitational mass, and no classical visibility — yet it still exists physically as a field excitation.

In ECM, ordinary matter occupies the lower, perceptible frequency band of existence. Dark matter and dark energy occupy a higher frequency band that lies above ordinary matter but below the Planck limit. They are not separate substances — they are the same underlying existence expressed in different frequency states.

Dark matter arises when matter is partially shifted into this higher-frequency vacuum-mode. Its mass is reduced and it becomes invisible, but it still interacts gravitationally with visible matter. Dark energy arises when this shift is complete, producing a fully vacuum-mode state that no longer attracts matter but instead generates a repulsive, anti-gravitational background.

Crucially, f0 is not just the beginning of the universe. It is not a historical moment. It is a physical state that can occur anywhere in space and at any time whenever matter is fully converted into Phase-Kernel excitation. The universe is continuously transforming matter into vacuum-mode energy and, under suitable conditions, back again.

This means vacuum is not empty.

Vacuum is the deepest energetic form of existence — the hidden frequency band of the universe.

ECM therefore connects gravity, motion, dark matter, and dark energy through a single physical process: matter transforming into higher-frequency vacuum-mode energy within the Phase Kernel.

This is not an addition to classical physics — it is what classical physics was missing.