Frequency and Energy in Extended Classical Mechanics (ECM):

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

Email: postmasterenator@gmail.com

Date: July 21, 2025

In Extended Classical Mechanics (ECM), the concepts of frequency and energy are reinterpreted with frequency (f) taking precedence as the most fundamental descriptor of a physical state. This framework positions frequency as more intrinsic to a system's dynamical evolution, independent of external reference frames, and dimensionally foundational for constructing observable quantities.

Frequency as the Primary Descriptor of Oscillation

In ECM, the argument for frequency's primacy is robust:

Synonymous with Vibration: Frequency isn't just a measure of vibration; it's considered synonymous with it. To talk about something vibrating or oscillating is, by definition, to imply a recurring event, and frequency quantifies that recurrence.

Mathematical Representation: ECM posits that frequency is the "only valid mathematical representation" of repetitive motion. Without a defined frequency, the concept of a regular, repeating vibration or oscillation becomes ill-defined. Consider a simple pendulum: its swing is described by its frequency (or period), not just its displacement.

Definitional Precedence: The idea is that vibration cannot be defined without frequency. This establishes frequency as the primary descriptor for any oscillatory system. It's the intrinsic characteristic that defines the very essence of its repetitive movement.

This perspective elevates frequency from being merely a derived quantity (like energy in the E = hf relation) to being the foundational property from which other related concepts emerge.

Reinterpreting the Planck Relation

The classical Planck relation, E = hf, is fundamentally reinterpreted in ECM. Here's how:

Energy as an Emergent Quantity: Unlike traditional physics where energy is a primary physical quantity, ECM views energy as an emergent concept. It's considered a secondary representation derived from a system's intrinsic frequency of phase transition or oscillation.

Conversion Constant: Instead of Planck's constant (h), ECM introduces its own conversion constant, for example, k = 5.558 × 10^−34 Js, which scales the system's intrinsic frequency to yield energy. This suggests that energy is a consequence of frequency, rather than frequency being a characteristic of energy.

Primacy of Frequency

ECM emphasizes the primacy of frequency over both energy and time (Δt) for several reasons:

Inherent to Dynamical State: Frequency is considered inherent to a system's dynamical state. It directly characterizes the ongoing evolution of a system without requiring an external timekeeper. This contrasts with energy, which depends on factors like system configuration (mass, motion, potential), and time, which is seen as relational and observer-dependent.

Fundamental for Oscillation: Frequency is deemed synonymous with vibration or oscillation and is presented as the only valid mathematical representation of such repetitive motion. It is argued that vibration cannot be defined without frequency, thus establishing frequency as the primary descriptor for oscillatory systems.

Cosmological Implications: In ECM cosmology, phenomena like redshift and energetic shifts are framed in terms of Δf (change in frequency) rather than time-based expansion. This perspective suggests that apparent energetic imbalances (ΔE) or mass variations (ΔM) over a duration (Δt) are governed by the fundamental phase-frequency evolution, not by the passage of time itself.

In essence, ECM proposes a shift from an energy-centric view to one where frequency is the foundational element driving physical processes and defining the nature of reality.

Comparative Analysis Report: Terry McMahon's Reformulations vs. Extended Classical Mechanics (ECM) Framework.

Prepared by: Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803
Date: July 21, 2025

Scholarly Cross-Comparison Now Live: Read on ResearchGate

Introduction

This report offers a focused, point-by-point comparative analysis between the reformulated physics approach proposed by Terry McMahon in his 2025 paper, Quantum gravity, special relativity & unification QGSM, and the Extended Classical Mechanics (ECM) framework developed independently. Both approaches challenge the foundations of relativity and quantum field theory, offering energy-centric alternatives to spacetime geometry and abstract force-carrying entities. Despite different terminologies, the two share foundational similarities as well as critical divergences.

Comparative Summary Table

Key Theme	Terry McMahon's View	ECM View	Commentary
Lorentz Factor & SR	Lorentz transformation introduces non-physical frame effects. Replaced with an index-based formulation.	SR time dilation is rejected; frequency governs Δt. Frame effects arise from energy-frequency shift, not geometry.	Both reject the Lorentz factor and spacetime-based time dilation. ECM emphasizes frequency displacement instead of dynamic index.
Photon Mass	Photons possess momentum → they must have mass. Mass evolves from bound (<c) to liberated (at c).	Photon/gamma ray emission represents displaced apparent mass (−Mapp), hence carries effective mass component.	Both assert non-zero mass association for free photons. ECM frames this via mass-displacement (ΔMM).
Planck Constant	Δf varies with energy scale. Not truly constant.	k = 5.558 x 10−34 Js derived in ECM as fundamental, linked to f0.	McMahon sees h as dynamic; ECM anchors a similar constant as derived from f0 via energetic continuity.
Gravity Mechanism	G is dimensionally flawed; gravity is energy density dependent, not geometric.	Gravity is mass-binding confinement of Meff; anti-gravity is liberated Mapp. No curved space required.	Both reject GR's geometry. ECM formalizes the binding/release mechanism through energy-mass structuring.
Gravitons & Fields	Gravitons unnecessary; field theory flawed.	No graviton; no field quantization. All transitions are real, observable mass shifts (ΔMM).	Total agreement on non-necessity of virtual field carriers.
Time & Causality	SR's time is geometrical fiction; proper time must emerge from energy-frequency behaviour.	Time is derived as f0; real dynamics follow frequency shift, not relativistic time.	McMahon calls for time redefinition; ECM implements it via f → Δt transformation.
Quantum Gravity	Must emerge from internal energy distributions, not curved spacetime.	ECM sees gravity and anti-gravity as reversible via ΔMM-mediated transitions.	Agreement in rejecting geometrisation in favour of physical transitions.
Unification of Forces	Energy-scaled parameters (running parameters) unify EM and gravity.	Mass-frequency correspondence allows unification without particle mediation.	Parallel strategies: dynamic parameters vs. frequency-mass equivalence.
Resistance from Academia	Journals resist anti-relativistic reformulations.	ECM development has been independent due to similar institutional resistance.	Shared experience of marginalization for paradigm-challenging frameworks.

Conclusion

Terry McMahon's reformulations and the ECM framework arrive independently at converging conclusions: spacetime curvature is not fundamental, time is emergent from energy behaviour, and photons are not massless. Where McMahon uses "running parameters," ECM formalizes transitions using mass-displacement (ΔM_M), effective and apparent mass (M^eff, M^app), and frequency (f) as foundational. This comparative insight opens the door to potential synthesis or mutual reinforcement of both models under a shared principle of energy-centric realism.

Intellectual debate fosters a deeper and more accurate understanding for everything.

July 21, 2025

This statement underscores the transformative power of engaging with diverse perspectives through reasoned discourse. When individuals with differing viewpoints engage in intellectual debate, it's not merely about winning an argument, but about a collaborative process of inquiry.

Through this process, assumptions are challenged, logical inconsistencies are exposed, and ideas are rigorously tested against alternative interpretations and evidence. Participants are compelled to articulate their positions with greater clarity, to consider counter-arguments, and to integrate new information that might refine or even alter their original understanding. This dynamic exchange moves beyond superficial comprehension, leading to a more nuanced, comprehensive, and ultimately, more accurate grasp of the subject matter, whatever it may be. It's a continuous cycle of questioning, defending, learning, and evolving one's understanding.

- Soumendra Nath Thakur

Fundamental Role of Frequency in Physical Description: A Dimensional and Conceptual Justification

Author: Soumendra Nath Thakur | ORCiD: 0000-0003-1871-7803  Email: postmasterenator@gmail.com

Date: July 20, 2025

Scientific Statement:

In Extended Classical Mechanics (ECM), frequency is posited as the most fundamental descriptor of physical state, more primary than both energy (E) and time (Δt), based on its direct association with intrinsic dynamical evolution, independence from external reference systems, and dimensional precedence in the construction of observable quantities.

Description:

In the classical Planck relation E = hf, energy appears as the product of a constant h and frequency f, suggesting an equivalence. However, ECM reinterprets this relation: energy is not a primary physical quantity, but rather an emergent one - a secondary representation of the system's intrinsic frequency of phase transition or oscillation, scaled by a conversion constant (e.g., ECMs k = 5.558 x 10⁻34 Js).

Unlike energy, which depends on system configuration (mass, motion, potential), and time Δt, which is relational and observer-dependent, frequency is inherent to the systems dynamical state. Whether referring to the oscillation of a field, the cycling of a phase, or the emission profile of a particle, frequency directly characterizes the ongoing evolution of the system without requiring an external timekeeper.

This primacy is reinforced by the role of frequency in ECM cosmology, where redshift and energetic shifts are framed in terms of Δf rather than time-based expansion. The apparent energetic imbalance ΔE or mass variation ΔM over a duration Δt is governed by the fundamental phase-frequency evolution, not by the passage of time itself.

Footnote: Frequency as Fundamental

Frequency is more fundamental than energy. It is synonymous with vibration or oscillation, since frequency is the only valid way to mathematically represent such repetitive motion. Vibration cannot be defined without frequency; hence, frequency precedes energy as the primary descriptor of oscillatory systems.

Dimensional Analysis:

Quantity	Symbol	Dimensional Form
Frequency	f	[T⁻1]
Time	Δt	[T]
Energy	E	[ML2T⁻2]
Planck constant or ECM constant	h or k	[ML 2 T⁻1]

From the relation E = hf ⇒ [E] = [h][f]
Substituting dimensions: [ML2T⁻2] = [ML2T⁻1] x [T⁻1]

This confirms frequency is the more elementary term: both energy and Planck's constant depend dimensionally on it. Furthermore, frequency derives solely from inverse time, while energy couples mass, length, and squared inverse time.

Importantly, frequency does not depend on energy, but energy cannot be defined without frequency, if one accepts E = hf as fundamental. Similarly, Δt is just the inverse of frequency, and thus frequency subsumes the role of time as well:

f = 1 / Δt      ⇒      Δt = 1 / f

Therefore, both E and Δt are derived quantities, contingent upon f.

External Validation of Frequency s Primacy in Physical Definition

The ECM position-that frequency is more fundamental than both energy and time-finds clear support in both experimental standards and foundational quantum theory.

1. Time Defined via Frequency in SI Units
   The International System of Units defines the second via the fixed frequency of radiation from cesium-133 atoms:
   

   The second is defined by taking the fixed numerical value of the caesium frequency ∆ν(Cs) to be 9,192,631,770 when expressed in the unit Hz (s⁻¹). - NIST

   This makes time a derived quantity from frequency, not vice versa.
2. Quantum Energy as Frequency of Phase Evolution
   In quantum mechanics, energy is expressed through phase evolution:
   ψ(t) ∝ e−iEt/ħ = e−2πift
   Energy (E) is interpreted as the product of frequency (f) and Planck's constant. This supports the ECM position that frequency governs state evolution.
3. Dimensional Coherence Reinforced
   From E = hf:
   [E] = [h][f] = [ML2T⁻1] x [T⁻1] = [ML2T⁻2]
   Again showing energy is a frequency-scaled quantity.
4. Experimental Clocks Rely on Frequency
   All modern atomic clocks define time intervals by counting cycles of atomic transitions-i.e., frequency. This operational priority supports the ECM claim that time Δt is secondary to frequency.
5. Uncertainty Principle Prioritizes Frequency's Accuracy
   In the energy-time uncertainty relation, a precise energy value requires a long-duration observation of a stable frequency. Thus, frequency accuracy is logically prior to energy definition.

Conclusion:
These empirical and theoretical pillars collectively reinforce the ECM stance:

Frequency is not merely a useful parameter but the ontological basis of time, energy, and mass relations.

Post-Latent Energetic Dynamics and the Dual-State Evolution of the Universe in ECM

Soumendra Nath Thakur | ORCiD: 0000-0003-1871-7803 | Tagore's Electronic Lab, India | postmasterenator@gmail.com

July 18, 2025

Abstract

In Extended Classical Mechanics (ECM), the origin and evolution of the universe are fundamentally reinterpreted through a frequency-based cosmological framework that eliminates the need for an initial mass singularity or explosive Big Bang. Instead, ECM posits that the universe begins from a quiescent field of latent potential energy defined purely by frequency, void of mass, motion, or classical energy. The transition from this latent state into observable dynamism occurs via a frequency displacement—a post-latent transition—initiating the reversible emergence of apparent mass and kinetic energy. This leads to a primordial energetic imbalance where the magnitude of negative apparent mass exceeds gravitational mass, generating a repulsive dynamic that drives early cosmic expansion.

In ECM, mass is not intrinsic but emerges through the reversible extraction of potential energy. The early universe evolves in two distinct energetic states: an unobservable pre-Planckian regime where negative apparent mass manifests as dark energy, and an observable regime where parts of that energy settle into dark matter or stabilize into visible matter. The equivalence ΔMᴍ ≡ −Mᵃᵖᵖ governs these transformations, highlighting the cyclic and reversible energetic rhythm of the cosmos. ECM thus proposes a non-singular, wave-governed universe, where frequency, phase, and time govern mass emergence and cosmic structure through reversible energetic displacement rather than irreversible mass-based collapse or inflation.

Keywords:

Extended Classical Mechanics, latent frequency field, apparent mass, post-latent transition, dark energy, dark matter, frequency displacement, energetic asymmetry, cyclic cosmology, reversible dynamics, gravitational stabilization, mass-energy transformation, Planck threshold, kinetic extraction,

In the framework of Extended Classical Mechanics (ECM), the universe does not begin with a massive explosion or singularity, as is commonly depicted in traditional cosmological models. Instead, it emerges from a quiet, uneventful field of latent potential energy, a primordial state defined entirely by frequency. This field carries no mass, no motion, and no classical energy in the conventional sense—it is a state of pure potential, not yet disrupted by asymmetry or displacement.

This latent frequency condition represents the pre-dynamical phase, prior to any observable transformation. Once a displacement occurs within this field—triggered by frequency change over time or phase—dynamism begins. This moment is referred to in ECM as the post-latent transition. It marks the point where a portion of the potential energy field is reversibly extracted, leading to the manifestation of mass-equivalent dynamics.

Emergence of Apparent Mass and the Initial Energetic Imbalance:

According to ECM, the universe’s earliest energetic condition is defined by an imbalance between two forms of mass: gravitational mass and apparent mass. Gravitational mass corresponds to the familiar attractive force that shapes structure in the cosmos. In contrast, apparent mass—particularly in its negative form—acts as a repulsive kinetic driver, emerging from displaced potential energy and carrying the dynamical role of expansion.

At the origin, ECM proposes that gravitational mass is less than the magnitude of negative apparent mass. This asymmetry is critical, as it leads to a powerful phase of repulsive expansion that temporarily overwhelms gravitational attraction. In ECM interpretation, this is the true beginning of observable cosmic evolution—driven not by mass but by the displacement of frequency-defined energy into kinetic form.

From Frequency Displacement to Cosmic Expansion:

Once the latent field undergoes displacement, the frequency shift results in the appearance of mass-like behaviour, most notably in the form of negative apparent mass. This mass is not permanent but rather a transient energetic state, fuelled by the kinetic extraction of energy from the original potential. It serves as the key agent behind the initial high-velocity expansion of the universe—effectively functioning as the ECM analogue to inflation.

As this expansion proceeds, frequency gradually decays, and the kinetic energy associated with the apparent mass begins to transform again. Gravitational mass slowly increases, or the magnitude of apparent mass decreases, leading eventually to a phase of gravitational stabilization. This transition represents a cyclic energetic rhythm, as the universe oscillates between potential and kinetic dominance, between displacement and restoration.

Two Distinct States in ECM Cosmology:

ECM cosmology defines two distinct energetic states of the universe:

1. The Unobservable State:

This state exists prior to or beyond the Planck threshold. It is characterized by a frequency regime that lies outside the limits of measurement or observation. During this phase, frequency displacement initiates the emergence of mass-equivalent forms. A portion of the resulting apparent mass maintains sufficient dynamism to remain in motion, manifesting as what ECM identifies as dark energy—a kinetic component exhibiting anti-gravitational behaviour. Some of this dark energy continues into the observable state and may later partially transform into ordinary mass, associating itself with what is recognized as dark matter.

2. The Observable State:

This state occurs within the Planck threshold, where frequency values fall into measurable and interpretable domains. In this phase, the same dynamic principle continues. Portions of the earlier apparent mass, now with reduced dynamism, behave as dark matter—retaining some of the anti-gravitational characteristics of dark energy but with diminished motion. Simultaneously, other portions stabilize further and contribute to the formation of observable matter, such as massive particles and atomic structures.

Cyclic Energetic Interpretation:

Central to ECM is the idea that apparent mass is reversible. It arises from the subtraction of potential energy and can, under the right conditions, revert to its original form. This reversibility is not just a theoretical curiosity—it forms the backbone of a cyclical universe model where energy constantly oscillates between latent potential, kinetic dynamism, and gravitational structure.

In this view:

• Dark energy represents the active phase of apparent mass, driving accelerated expansion.

• Dark matter marks the residual, decelerated phase, still unobservable but less dynamic.

• Ordinary matter emerges as the stabilized result of kinetic energy transformations entering gravitational equilibrium.

ECM thus presents a cosmological model that is wave-based rather than mass-based, governed by transformations of frequency, phase, and time—not by initial mass concentration or singularities. The universe is born from energy displacement and evolves through reversible energetic phases, offering a fundamentally different lens through which to understand its origin, structure, and fate.

Energetic Imbalance at the Origin: The Role of Apparent Mass and Frequency Displacement in ECM

In Extended Classical Mechanics (ECM), the foundational relations among frequency f, phase φ, and time Δt—including f₀, f₁, and Δf—adhere to classical wave principles across all scales. Crucially, the quantum constants such as Planck’s constant h and quantized energy E = hf are not applicable at the universal origin, since these constants are defined only within or below the Planck threshold and lack physical meaning when extrapolated to pre-Planckian regimes. (See: Relevant Appendix/ces on ECM constants and Planck limitations.)

At the heart of the universe’s origin, ECM proposes a fundamental mass-energy asymmetry:

Mɢ < |−Mᵃᵖᵖ|

This inequality suggests that the negative apparent mass dominates over gravitational mass during the universe's earliest phase. The implications of this asymmetry are central to ECM cosmology:

• High-velocity expansion is initiated by −ΔMᴍ-mediated kinetic energy,

• A net repulsive dynamic temporarily overcomes gravitational attraction,

• Gravitational stability only emerges later, as Mɢ increases or |−Mᵃᵖᵖ| decreases.

In earlier ECM discussions, it was shown that the universe originates not from a singular mass event but from an uneventful field of latent potential energy, defined solely by frequency. This latent energy undergoes frequency displacement, initiating the appearance of kinetic mass-like phenomena, particularly negative apparent mass −Mᵃᵖᵖ, which acts as a dynamical driver of cosmic inflation and initial acceleration.

This energetic cycle gradually evolves: as frequency decays, gravitational mass Mɢ vanishes, and the system reverts to a frequency-defined potential state—restoring the pre-inflationary conditions in a rhythmic manner. This formulation provides a mathematically coherent connection between cosmological redshift, apparent mass displacement, and energetic reversion, highlighting a cyclic rhythm of expansion and contraction rooted in frequency dynamics rather than mass-based singularities.

The central equivalence governing this transformation is:

ΔMᴍ ≡ −Mᵃᵖᵖ

Here, −Mᵃᵖᵖ serves as the kinetic driver, translating reversible frequency changes into observable energetic behaviour. (See: Appendices on Energetic Reversion, Mass Displacement, and Cosmological Frequency Cycles for derivations and further discussion.)

Latent Frequency Field and the Dual-State Emergence of the Universe in ECM:

Post-Latent Transition:

In Extended Classical Mechanics (ECM), the universe does not originate from a singular mass-based event, but rather from an uneventful field of latent potential energy, defined solely by frequency. The governing energetic expression for this primordial state is:

PEᴇᴄᴍ,ᵤₙᵢᵥ = (PEᴇᴄᴍ,ᵤₙᵢᵥ − ΔPEᴇᴄᴍ,ᵤₙᵢᵥ) + ΔPEᴇᴄᴍ,ᵤₙᵢᵥ

Where:

• −ΔPEᴇᴄᴍ,ᵤₙᵢᵥ → −Mᵃᵖᵖ

• −ΔPEᴇᴄᴍ,ᵤₙᵢᵥ → −ΔMᴍ

This reflects that the observable energetic portion of the universe arises from a reversible extraction of potential energy, transforming into kinetic mass-equivalent dynamics.

Two Fundamental States of the Universe in ECM:

ECM distinguishes between two primary energetic states of the universe:

Condition 1: Unobservable State (Beyond the Planck Scale, Pre-Origin):

• Defined by f₀ → f₁, where frequency displacement initiates energetic asymmetry.

• Governing equivalence:

ΔPEᴇᴄᴍ,ᵤₙᵢᵥ ≡ KEᴇᴄᴍ,ᵤₙᵢᵥ → ΔMᴍ ≡ −ΔMᴍ

• In this pre-Planckian regime, a portion of ΔMᴍ sustains cosmic dynamism as KEᴇᴄᴍ,ᵤₙᵢᵥ and manifests as dark energy (DE) — exhibiting anti-gravitational behaviour.

• Some of this dark energy persists into Condition 2, partially converting into observable mass Mᴍ, binding with dark matter (DM).

Condition 2: Observable State (Within the Planck Scale):

• Transition defined by fᴘ → fᴏᵦₛₑᵣᵥₐᵦₗₑ, where the frequency enters the detectable domain.

• The same kinetic-potential equivalence applies:

KEᴇᴄᴍ,ᵤₙᵢᵥ → ΔMᴍ ≡ −ΔMᴍ

• A portion of ΔMᴍ that loses dynamism (↓KEᴇᴄᴍ,ᵤₙᵢᵥ) or regains gravitational coupling becomes dark matter (DM) — still exhibiting anti-gravitational behaviour but with reduced kinetic drive.

Energetic Interpretation in ECM

• The equivalence ΔMᴍ ≡ −ΔMᴍ expresses the reversible nature of energetic transformation: apparent mass emerges from the subtraction of potential energy and may later revert.

• Dark energy corresponds to the active kinetic component of apparent mass that fuels expansion.

• Dark matter emerges as the residual, decelerated kinetic component, gravitationally reactive but still non-luminous.

This dynamic view frames the entire evolution of the universe as an oscillatory frequency-driven transformation, where latent frequency displaces into mass-equivalent motion, then reverts through decay. Mass is thus not fundamental but a transient expression of energetic displacement across phase and time.

Alphabetical List of Terms in ECM (with Descriptions, Units, and Appendix References)

Term	Description	Unit	Relevant Appendix
Δf	Change in frequency; describes the shift during frequency displacement, central to redshift dynamics.	Hz	Appendix 6, Appendix 29
ΔMᴍ	Change in mass-equivalent dynamics, interpreted as reversible kinetic transformation of apparent mass.	kg	Appendix 6, Appendix 29, Appendix 30
ΔPEᴇᴄᴍ	Change in potential energy in ECM framework; represents extracted energy from a latent frequency field.	J	Appendix 6, Appendix 29
f₀, f₁	Initial and displaced frequencies before and after cosmic energetic transition.	Hz	Appendix 6, Appendix 29
fᴘ	Planck threshold frequency—defines boundary between unobservable and observable cosmological regimes.	Hz	Appendix 30
fₒᵦₛₑᵣᵥₐᵦₗₑ	Frequency range within measurable and gravitationally interactive state-space.	Hz	Appendix 6, Appendix 29
G (Gravitational Mass, Mɢ)	Mass component contributing to gravitational attraction. In ECM, G <	−Mᵃᵖᵖ	at origin.
KEᴇᴄᴍ	Kinetic energy in ECM, equivalent to extracted potential energy; initiates mass-like motion.	J	Appendix 6, Appendix 29
Mᵃᵖᵖ (Apparent Mass)	Emergent, reversible, often negative mass associated with frequency displacement; not intrinsic but dynamic.	kg	Appendix 6, Appendix 29, Appendix 30
Mᴍ	Observable mass formed through transformation and stabilization of ΔMᴍ.	kg	Appendix 29
PEᴇᴄᴍ	Potential energy defined in ECM by the frequency field prior to displacement.	J	Appendix 6, Appendix 29
φ (Phase)	Angular phase variable used in wave dynamics to interpret phase-time transformations.	radians	Appendix 6
t, Δt	Time and time interval associated with phase or frequency shift; crucial for dynamism in ECM.	s	Appendix 6, Appendix 29

 

End of statement.

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:

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