Thank you, Prof. Thakur, for opening this discussion. From my own framework, the Theorem of Latent Conservation of Numerical Value (CLVN–LRR™), I find resonance with ECM yet with a different emphasis.

In ECM, the equivalence Mᴅᴇ = −Mᵃᵖᵖ explains the negative nature of dark energy as the counterpart of apparent mass effects. In my view, this corresponds to a latent compressed state: mass is not annihilated but transferred into a reversible mode of latency.

Thus, when the condition Mɢ = 0 is reached, I don’t interpret it as the “neutralization” of gravity, but rather as a pivot of non-manifest trajectories: the numerical value (mass) remains conserved in latency and may re-emerge under different structural conditions.

From this perspective, cosmic expansion is not driven by a growing negative force, but by a structural reversion process where matter progressively shifts into latent states. What ECM observes as dynamic dark energy, CLVN–LRR™ formalizes as reversible latent resonance.

Dear Lionel Gustavo Raggio,

Thank you for your thoughtful and engaging response. I appreciate your effort to connect the principles of your Theorem of Latent Conservation of Numerical Value with the Extended Classical Mechanics (ECM) framework. It is encouraging to see parallel explorations that seek to deepen our understanding of cosmic phenomena.

However, I would like to address some key distinctions. In ECM, the equivalence Mᴅᴇ = −Mᵃᵖᵖ is not merely an interpretation but a relationship grounded in both observational evidence and theoretical derivation. The negative nature of dark energy (Mᴅᴇ < 0) has been supported by observations such as those of A. D. Chernin and colleagues in the study of the Coma cluster of galaxies. ECM’s independent derivation of −Mᵃᵖᵖ—based on extensions of classical mechanics and explained in detail in Appendix A—aligns with these findings, offering a coherent explanation that integrates both local and cosmological scales.

Your interpretation that mass enters a latent compressed state rather than being annihilated resonates partially with ECM’s view that mass and energy can transform between states without loss. ECM likewise recognizes that such transformations were critical in the early universe, where potential energy transitioned into kinetic energy during the formation of the first existential events. Yet, the observational reality of dark energy’s negative effective mass goes beyond conceptual abstraction—it reflects measurable cosmic behaviour.

Regarding the condition Mɢ = 0, ECM interprets this not as the annihilation of gravity but as its neutralization. The inflationary expansion of the universe came to a halt when Mᴍ = −Mᵃᵖᵖ led to Mɢ = 0, effectively balancing gravitational attraction without erasing it.

Finally, while you propose that cosmic expansion arises from a structural reversion process rather than a growing negative force, ECM’s formulations point to the anti-gravitational effects generated by frequency-dependent mass deficits in particles like photons. Without addressing these mechanisms, it becomes challenging to account for phenomena such as gravitational escape and cosmic acceleration. Dark energy exhibits similar behaviour—differing in frequency and detectability—but equally relevant in its contribution to the universe’s expansion.

I value this opportunity for scientific dialogue and exchange. Such discussions are vital to refining our understanding and challenging assumptions. I hope this clarification helps distinguish ECM’s approach while recognizing the thoughtfulness behind your framework.

Best regards,

Soumendra Nath Thakur

Thank you for this detailed explanation of the ECM framework. The equivalence you highlight,

MDE=−Mapp,M_{DE} = -M_{app},MDE​=−Mapp​,

is indeed illuminating because it captures the idea that what we interpret as dark energy is not a separate fluid or constant, but the manifestation of an effective negative mass emerging from the dynamics of matter itself.

In our recent work, we have approached the problem from a complementary angle: rather than introducing dark energy as a distinct entity, we consider that gravitational memory, encoded in a scalar field ϕ(t)\phi(t)ϕ(t), provides the missing piece. This field records the past history of gravitational interactions and modifies the effective potential experienced by matter. At galactic and intergalactic scales, the scalar field contributes both positive (halo-like) and negative (expansion-driving) terms, depending on the geometry and persistence time τ\tauτ of the memory.

In this way, your ECM interpretation and our gravitational memory model are deeply connected:

• Both see dark energy not as constant, but as an emergent effect tied to matter’s interaction history.
• Both imply that as matter evolves (whether through frequency-dependent apparent mass in ECM or memory persistence in our model), the balance shifts toward an accelerating expansion.
• Both open the possibility of testable predictions: in our case, via correlations between supermassive black holes and halo structures (SPARC, SLACS, MaNGA datasets), and in yours through dynamical tracking of effective mass balance across scales.

This convergence suggests that the next step is to unify these approaches into a broader phenomenology of cosmic evolution, where matter, memory, and apparent mass are not independent concepts but manifestations of a deeper informational dynamics of spacetime.