Mathematical Derivation Supporting
Phase Dependent Formation To Fate of the Universe Through ECM Consistent Formalism
Abstract
This supplementary mathematical note develops a formal ECM derivation supporting the
phase-dependent cosmological framework proposed in
Phase Dependent Formation To Fate of the Universe Through ECM Consistent Formalism
(DOI:
10.5281/zenodo.20075747
).
The derivation establishes how manifest matter reduction,
apparent-mass growth, effective acceleration increase,
and wavelength contraction jointly imply a terminal cosmological condition
characterized by dimensional extinction and complete energetic unmanifestation.
The result provides a mathematically consistent ECM route from cosmological manifestation
to ultimate cosmic dissolution.
1. Fundamental Manifestation Law
ECM begins with the core manifestation identity:
ΔPEᴇᴄᴍ ↔ ΔKEᴇᴄᴍ ↔ ΔMᴍ
meaning:
- potential redistribution produces kinetic realization,
- kinetic realization produces manifested matter,
- all three are phase-linked expressions of the same transformation.
2. Apparent Mass Definition
Mᵃᵖᵖ ≡ −ΔPEᴇᴄᴍ
This represents latent energetic deficit:
[Mᵃᵖᵖ] = J
and functions as an unmanifest energetic reservoir.
3. Effective Mass Structure
Mᵉᶠᶠ = Mᴍ + (-Mᵃᵖᵖ)
Thus:
- if Mᴍ dominates, the universe remains manifest;
- if (−Mᵃᵖᵖ) dominates, dissolution begins.
4. Force Law
Mᵉᶠᶠ aᵉᶠᶠ = Fᴇᴄᴍ
or equivalently
F = {Mᴍ + (−Mᵃᵖᵖ)} aᵉᶠᶠ
so
aᵉᶠᶠ = Fᴇᴄᴍ / Mᵉᶠᶠ
therefore
aᵉᶠᶠ ∝ 1 / Mᵉᶠᶠ
Since:
Mᵉᶠᶠ = Mɢ
then
aᵉᶠᶠ ≡ gᵉᶠᶠ ∝ 1 / Mɢ
5. Apparent-Mass Growth Relation
As
Mᵃᵖᵖ ↑
effective mass decreases:
Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ ↓
therefore
aᵉᶠᶠ ↑
hence:
aᵉᶠᶠ ≡ gᵉᶠᶠ ∝ Mᵃᵖᵖ ∝ 1/Mɢ
Mass dissolution and gravitational mass reduction are directly proportional
to effective acceleration increase.
6. Mass Dissolution Chain
ΔMᴍ ↓ → Mᴍ ↓ → Mɢ ↓ → aᵉᶠᶠ ↑
This defines the ECM cosmic dilution pathway.
7. Wavelength Collapse
Initially:
λ ≥ ℓᴘ
(fully manifest domain)
Then:
λ < ℓᴘ
(transitional sub-Planck domain)
Then:
λ ≪ ℓᴘ
(unmanifest regime)
Finally:
λ → 0
which implies:
x, y, z → ∅
meaning spatial dimensionality loses physical meaning.
8. Terminal Condition at rₘₐₓ_ᵤₙᵢᵥ
At asymptotic cosmic expansion:
r → rₘₐₓ_ᵤₙᵢᵥ
with
(−Mᵃᵖᵖ) ≫ Mᴍ
and
Mᴍ → 0
and
λ → 0
therefore:
dimension → 0
Thus:
Universe → Complete Unmanifest State
9. Final ECM Fate Equation
rₘₐₓ_ᵤₙᵢᵥ :
(−Mᵃᵖᵖ) ≫ Mᴍ,
Mᴍ → 0,
λ ≪ ℓᴘ → 0
therefore
PEᴇᴄᴍ → Fully Latent
and
Cosmic Dissolution = Dimensional Extinction
This derivation supports the ECM proposition that the universe does not end in simple thermal decay,
but through progressive unmanifestation driven by apparent-mass dominance,
mass dissolution,
sub-Planck wavelength collapse,
and extinction of manifest dimensionality.
References
-
Planck, M. (1901).
On the Law of Distribution of Energy in the Normal Spectrum.
Annalen der Physik, 4(553), 1–11.
(Foundational formulation of the energy–frequency relation, E = hf.)
-
Planck, M. (1899).
Über irreversible Strahlungsvorgänge.
Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin,
5, 440–480.
-
Bureau International des Poids et Mesures (BIPM).
SI Brochure: The International System of Units (SI).
-
Planck Collaboration. (2018).
Planck 2018 Results: Cosmological Parameters.
-
Chernin, A. D., Bisnovatyi-Kogan, G. S., Teerikorpi, P.,
Valtonen, M. J., Byrd, G. G., & Merafina, M. (2013).
Dark energy and the structure of the Coma cluster of galaxies.
Astronomy & Astrophysics, 553, A101.
https://doi.org/10.1051/0004-6361/201220781
Related Works by the Author
-
Thakur, S. N. (2025).
Phase Dependent Formation To Fate of the Universe Through ECM Consistent Formalism.
https://doi.org/10.5281/zenodo.20075747
-
Thakur, S. N. (2025).
ECM Confidence: Planck-Scale Frequency-Length Invariance.
https://doi.org/10.5281/zenodo.20115415
-
Thakur, S. N. (2025).
Foundational Formulation of Extended Classical Mechanics: From Classical Force Laws to Relativistic Dynamics.
https://prereview.org/preprints/doi-10.20944-preprints202504.1501.v1
-
Thakur, S. N. (2026).
Clarification on the Units and Dimensions of Mᵃᵖᵖ ≡ −ΔPEᴇᴄᴍ, ΔMᴍ, and ΔKEᴇᴄᴍ in Extended Classical Mechanics (ECM).
https://doi.org/10.13140/RG.2.2.33366.20805
-
Thakur, S. N. (2026).
Extended Classical Mechanics (ECM) in Relation to Richard Lewis’s Fundamental Physics and The Conscious Brain: A Scholarly Alignment Analysis.
https://doi.org/10.13140/RG.2.2.24971.04649
-
Thakur, S. N. (2025).
Extended Classical Mechanics: Redefining Force, Mass, and Light.
https://www.preprints.org/reading-list/30
-
Thakur, S. N. (2025).
Extended Classical Mechanics (ECM): Consistent Fundamental Energy Principle - Planck-Scale Frequency Origins vs. Relativistic Mass-Energy.
http://dx.doi.org/10.2139/ssrn.6221099
-
Thakur, S. N. (2024).
Extended Classical Mechanics: Vol-1 - Equivalence Principle, Mass and Gravitational Dynamics.
Preprints.
https://doi.org/10.20944/preprints202409.1190.v3
-
Thakur, S. N. (2025).
Extended Classical Mechanics (ECM): Fundamental Terms, Field Dynamics, and Empirical Applications.
https://doi.org/10.13140/RG.2.2.20563.44327
-
Thakur, S. N. (2025).
A Dual Framework for Rest and Motion States: Energy and Mass Dynamics in Extended Classical Mechanics.
