Cosmological Frequency Cycle and the Constructed Foundations of ECM Constants.

Author: Soumendra Nath Thakur ORCiD: 0000-0003-1871-7803

DOI: https://doi.org/10.13140/RG.2.2.35531.91685

Date: July 16, 2025

Abstract

In Extended Classical Mechanics (ECM), cosmological dynamics arise not from singularities or undefined fields but from mathematically governed transitions in frequency and reversible mass-energy displacement. This appendix proposes a frequency-centric cosmological cycle wherein the universe originates from a zero-event field of pure frequency—devoid of mass—and evolves through energetic transitions mediated by apparent mass displacement (−Mᵃᵖᵖ ≡ ΔMᴍ). The framework captures inflation, redshift, cosmic acceleration, and future energy reversion within a closed-loop, frequency-governed cosmology.

It further justifies the theoretical derivation of the ECM constant (k = 5.558 × 10⁻³⁴ Js) and origin frequency (f₀ = 3.571 × 10⁴² Hz). Though not empirically derived, these parameters are rigorously constructed from energy conservation across Planck boundary conditions, offering mathematical continuity with fundamental physics. The equivalence E = kf₀ ≈ hfᴘ ≈ 1.986 × 10⁹ J is not an arbitrary match but a logically defensible, reversible transformation consistent with ECM.

Keywords: Extended Classical Mechanics, frequency cycle, apparent mass, negative mass, ΔMᴍ, −Mᵃᵖᵖ, cosmological redshift, frequency decay, Planck scale, origin frequency, ECM constant, k = 5.558 × 10⁻³⁴ Js, f₀ = 3.571 × 10⁴² Hz, energy conservation, reversible mass-energy transition, gravitational mass dissolution, cosmic acceleration, energetic reversion, phase-time displacement, frequency-defined potential energy, closed-loop cosmology,

1. Introduction

Within ECM, the classical energy paradigm is extended to reinterpret the universe’s origin and evolution through frequency-time displacements and apparent mass transitions. Rather than relying on initial singularities or dark field constructs, ECM centres on reversible energetic cycles, governed by conserved relations between frequency (f), time (Δt), and apparent mass shifts (ΔMᴍ ≡ −Mᵃᵖᵖ). This necessitates the introduction of ECM-specific constants, formulated to maintain energetic continuity near the universal origin—at or slightly beyond the Planck scale.

2. Cosmological Energy Cycle and Future Potential in ECM

In ECM, the universe emerges from a primordial frequency field, not from mass or curvature. This latent field—an uneventful zero-point state—contains potential energy described by:

  E = hf, with f as a latent cosmic frequency mode.

At the moment of origin (analogous to the Big Bang), this frequency mode becomes kinetically displaced, manifesting as apparent mass:

  ΔMᴍ ≡ −Mᵃᵖᵖ

This −Mᵃᵖᵖ, often interpreted as dark energy in conventional cosmology, serves as the primary driver of early inflation—not by expanding mass but via frequency-induced kinetic displacement. As time progresses, gravitational mass Mɢ emerge to counterbalance −Mᵃᵖᵖ, briefly stabilizing the system:

  Mɢ ≈ |−Mᵃᵖᵖ|

However, due to continued frequency decay (Δf > 0), this balance fails, and −Mᵃᵖᵖ reasserts dominance, leading to accelerated expansion in the current epoch.

3. Frequency Decay and Redshift Implications

In ECM, cosmological redshift is interpreted not as spacetime stretching but as cumulative frequency decay:

  Δf = f − f′, with ΔE = hΔf

Time delay (Δt) induces a progressive shift in observed frequency, formalized as:

  f = x° / (360° × Δt)

Thus, redshift corresponds to real energy loss or reversion, not merely observational artefact. Cumulatively:

  ΔE = hfΔt

This energetic decay marks a return from kinetic displacement to inert frequency modes, explaining both observable redshift and cosmic acceleration without invoking exotic fields.

4. Reversion to Frequency-Dominant Future State

As Δt increases indefinitely, gravitational mass asymptotically vanishes:

  lim (Mɢ) → 0 as t → ∞, and −Mᵃᵖᵖ → 0

This represents not thermodynamic heat death but a reversible reversion to a pure-frequency state. Low-energy, uneventful space re-emerges—akin to the pre-origin state, governed by ultra-low residual frequency.

Yet, reactivation remains possible through:

• Residual microgravity fluctuations

• Localized frequency field densification

If hf Δt ≥ threshold, a new cycle may initiate:

  ΔMᴍ ≡ −Mᵃᵖᵖ

ECM thus supports a regenerative cosmology—not of one origin but of rhythmically returning frequency-driven universes.

5. Theoretical Justification of ECM Constants

5.1. Planck Boundary and Modified Constants

The standard relation E = hf becomes undefined near or before Planck time. ECM adapts this using a modified frequency-energy constant k, ensuring consistency across the Planck regime.

Planck boundary values:

• Planck time, tᴘ = 5.391247 × 10⁻⁴⁴ s

• Planck frequency, fᴘ = 2.999 × 10⁴² Hz

• Planck constant, h = 6.62606868 × 10⁻³⁴ Js

Using energy continuity, we derive:

  k = hfᴘ / f₀ = (6.62606868 × 10⁻³⁴ × 2.999 × 10⁴²) / 3.571 × 10⁴²

   = 5.558 × 10⁻³⁴ Js

This ensures:

  E₀ = kf₀ ≈ hfᴘ ≈ 1.986 × 10⁹ J

Thus, the ECM constant k replaces h only near the origin frequency, without contradiction to physical law.

5.2. Energetic Reversibility and Origin Energy

ECM's foundation lies in:

  −Mᵃᵖᵖ ≡ ΔMᴍ

This equivalence governs the energy transition at the universal origin, enabling the frequency-derived energy kf₀ to mediate reversible mass displacement. Despite lack of direct observability, this constructed value aligns with Planck-scale energy and allows energetic continuity within ECM logic.

6. Mathematical Consistency

The ECM constant k adheres to all foundational principles of frequency-time logic:

• T_deg = x° / (360° × f₀)

• f₀ = x° / 360° × Δt

• f₁ = f₀ / (1 + f₀Δt)

These reinforce k as a logically derived constant valid in ECM’s extended framework. Though not empirically fixed, it is theoretically well-constructed and open to indirect cosmological validation.

Conclusion

In ECM, the universe is not a one-time explosion but an energetic frequency cycle—originating from potential frequency modes, transforming via apparent mass displacement, and reverting through frequency decay. The constants k = 5.558 × 10⁻³⁴ Js and f₀ = 3.571 × 10⁴² Hz are not arbitrary but derive from conserved energy principles across Planck-defined boundaries. ECM thus offers a mathematically structured and philosophically elegant alternative to singularity-driven cosmologies.

References

1. Planck Collaboration, “Planck 2018 Results,” Astronomy & Astrophysics, 641, A6 (2020).

2. Thakur, S. N., Extended Classical Mechanics Appendices, various (2024–2025).

3. Misner, C. W., Thorne, K. S., & Wheeler, J. A., Gravitation, W.H. Freeman (1973).

4. Rovelli, C., Quantum Gravity, Cambridge University Press (2004).

Relevant Appendices

• Appendix 6: Angular-Time Correspondence in ECM

• Appendix 18: Photon Energy from Dual Apparent Mass Displacement

• Appendix 22: Mass-Energy Reversion Cycles and ΔMᴍ Equivalence

• Appendix 27: Frequency-Origin Transitions in ECM

• Appendix 29 (merged): This Appendix (29)

Footnote:

Author profile: www.facebook.com/TagoresElectronicLab
Research blog: https://soumendranaththakur.blogspot.com