Extended Classical Mechanics' Pre-relativistic mass-energy equivalence Principle.

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

December 31, 2025

Extended Classical Mechanics' Pre-relativistic mass-energy equivalence Principle.

1. Classical Force Laws:

Newton’s Second Law

F = m a

Force is the cause of acceleration of inertial mass.

2. Newton’s Law of Gravitation:

Fɢ = G (M m/r²)

This gives the force acting on a mass (m) due to a source mass (M).

3. Dynamical Equivalence:

Set gravitational force equal to inertial force:

m a = G (M m/r²)

Cancel (m):

a = GM/r²

This shows that gravitational acceleration is independent of the test mass.

4. Gravitational Field:

g(r) = GM/r²

So the gravitational force becomes

Fɢ = mg(r)

5. Physical Causal Chain:

M → g(r) → a → F

This is exactly what Newtonian gravity means physically:

Mass produces a field, the field produces acceleration, and acceleration produces force.

6. Classical Total Mechanical Energy:

Eₜₒₜₐₗ = PE + KE

This states that the energy of a classical system is the sum of its stored (potential) and motion (kinetic) energy.

For motion in a gravitational field,

Eₜₒₜₐₗ = mgh + ½mv²

This provides the exact classical baseline, against which the ECM generalization is being built.

7. Equivalence of Inertial and Gravitational Mass — ECM Formulation:

In classical mechanics, inertial mass and gravitational mass are empirically identical:

mɪₙₑᵣₜᵢₐₗ = mɢᵣₐᵥᵢₜₐₜᵢₒₙₐₗ

mɪ = mɢ

This Weak Equivalence Principle states that the mass that resists acceleration is the same mass that produces and experiences gravity.

Classical physics treats this equality as fundamental, without explaining why it holds.

8. The ECM Conceptual Breakthrough:

Classical mechanics assumes mass is a single primitive quantity:

mɪ = mɢ = m

This works because classical theory does not resolve how energy, inertia, and gravity are stored or generated within mass.

Extended Classical Mechanics (ECM) rejects this assumption.

In ECM, mass has internal structure.

9. Mass Structure in ECM:

In ECM, gravitational and inertial behaviour arises from the effective mass:

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ) = Mɢ

where

• Mᴍ is matter mass (existence field)

• −Mᵃᵖᵖ is Negative Apparent Mass (NAM) generated by manifested potential energy

−Mᵃᵖᵖ ≡ −ΔPEᴇᴄᴍ

Thus, gravitational mass is not just matter mass — it is matter plus manifested energy field.

10. Classical Mechanics as a Special Case of ECM:

When no manifestation occurs,

−ΔPEᴇᴄᴍ = 0 → −Mᵃᵖᵖ = 0

so ECM collapses to

Mᵉᶠᶠ = Mᴍ

which gives

mɪ = mɢ

Therefore, the equivalence principle is not fundamental — it is the zero-manifestation limit of ECM.

11. What Classical Physics Cannot See:

Classical mechanics unknowingly treats

m = Mᴍ + (−Mᵃᵖᵖ)

as a single undifferentiated constant.

ECM reveals that:

• gravity

• inertia

• kinetic energy

• dark mass effects

all arise from the hidden NAM field, generated by −ΔPEᴇᴄᴍ.

Classical physics contains:

• no −Mᵃᵖᵖ

• no ΔMᴍ

• no mass–energy exchange

• no internal mass structure

So it reduces to:

Mɢ → m, Mᴍ → m, −Mᵃᵖᵖ → 0

which automatically yields

mɪ = mɢ

12. Physical Meaning:

Classical mechanics treats the entire ECM mass–energy structure as if it were already collapsed into a single constant (m).

That is why Newtonian gravity works — but it does not know why.

ECM provides the missing physics:

Classical mass = matter mass + hidden NAM field

but classical theory cannot separate them.

13. Final ECM Statement:

Mɢ = Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ), −Mᵃᵖᵖ ≡ −ΔPEᴇᴄᴍ

In classical mechanics:

−ΔPEᴇᴄᴍ → 0 → Mɢ = Mᴍ

hence

mɪ = mɢ

ECM shows that this equality hides a deeper mass-energy structure that becomes visible only when motion, gravity, and manifestation occur.

14. Physical interpretation:

Classical mechanics treats the entire mass–energy structure of ECM as if it were already “pre-collapsed” into a single constant (m).

That is why classical gravity works — but it does not know why.

ECM tells the missing physics:

Classical mass matter mass + hidden NAM field

but classical theory cannot separate them.

15. Conclusion:

In classical mechanics, mɪ = mɢ because classical physics assumes −Mᵃᵖᵖ.

ECM reveals that this equality hides a deeper structure:

m = Mᴍ + (−Mᵃᵖᵖ),

which becomes visible only when motion and gravity exist.

16. ECM Extension of the Classical Force Law:

Extended Classical Mechanics (ECM) is built on the principle that kinetic energy is not independent of mass. Instead, kinetic energy arises from the redistribution of mass through the entropic release of stored potential energy,

−ΔPEᴇᴄᴍ.

In ECM, motion is powered when a portion of the system’s stored potential energy PEᴇᴄᴍ is converted into a manifested mass component,

−ΔPEᴇᴄᴍ ≡ −Mᵃᵖᵖ

which generates effective mass

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)

and it is this effective mass that participates in inertia, gravity, and acceleration.

Thus, what appears in classical physics as “kinetic energy” is, in ECM, the dynamic manifestation of mass created by potential-energy release.

17. Classical Force Law (Newtonian Interface)

F = M a

This equation remains exactly valid in ECM — but ECM now reveals what the symbol (M) really contains.

In classical mechanics, (M) is treated as a single primitive quantity.

In ECM, that same (M) is the collapsed form of

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)

where

• Mᴍ is matter mass (existence field),

• −Mᵃᵖᵖ is Negative Apparent Mass (NAM) generated by −ΔPEᴇᴄᴍ.

Therefore, the classical force law becomes, in ECM meaning:

F = Mᵉᶠᶠ a = (Mᴍ + (−Mᵃᵖᵖ)a,

showing that force acts on the combined matter-mass plus manifested energy-mass.

18. The ECM Bridge Behind Newton’s Law:

The Newtonian equation

F = M a

corresponds in ECM to the causal chain

F ⟶ −ΔPEᴇᴄᴍ ⟶ −Mᵃᵖᵖ ⟶ Mᵉᶠᶠ ⟶ a.

Thus, force does not directly produce acceleration. It first releases potential energy, which generates NAM, which modifies effective mass, and only then produces motion.

This is the hidden physics behind Newton’s law.

19. Newtonian Limits (Recovered Exactly)

For classical fixed mass (M):

a ∝ F

For fixed force (F):

a ∝ 1/M

These relations are not altered in ECM.

They are simply the collapsed limit where

−Mᵃᵖᵖ ⟶ 0, Mᵉᶠᶠ ⟶ Mᴍ,

so that ECM reduces to

F = Mᴍ a,

which is ordinary Newtonian mechanics.

What ECM adds?

Newton gives the result.

ECM provides the engine.

Newton sees:

F ⟶ a.

ECM reveals:

F ⟶ −ΔPEᴇᴄᴍ ⟶ −Mᵃᵖᵖ ⟶ Mᵉᶠᶠ ⟶ a.

This is why ECM can generate gravity, inertia, dark mass, and cosmic dynamics from a single physical process — the manifestation of mass from potential energy.

20. ECM Insight: Mass is Not Static in Motion

In Extended Classical Mechanics (ECM), motion is powered by mass–energy conversion, governed by the manifestation identity

−ΔPEᴇᴄᴍ ↔ ΔKEᴇᴄᴍ ↔ ΔMᴍ

This means that when potential energy is released, it does not disappear into abstract “energy”; it manifests as matter-mass.

Therefore, a moving body possesses more mass than its rest matter mass:

Mᴍₘₒₜᵢₒₙ = Mᴍᵣₑₛₜ + ΔMᴍ

where ΔMᴍ is the kinetic mass created from the released potential energy −ΔPEᴇᴄᴍ.

21. Total Energy in ECM

Classically,

Eₜₒₜₐₗ = PE + KE = mgh + ½mv².

In ECM,

Eᴇᴄᴍ,ₜₒₜₐₗ = PEᴇᴄᴍ + KEᴇᴄᴍ

But ECM resolves how this sum is formed:

PEᴇᴄᴍ + KEᴇᴄᴍ = (PEᴇᴄᴍ - ΔPEᴇᴄᴍ) + KEᴇᴄᴍ

Since

KEᴇᴄᴍ ≡ −ΔPEᴇᴄᴍ ≡ −Mᵃᵖᵖ ≡ ΔMᴍ,

we obtain

Eᴇᴄᴍ,ₜₒₜₐₗ = PEᴇᴄᴍᵉᶠᶠ + KEᴇᴄᴍ,

where:

PEᴇᴄᴍᵉᶠᶠ = PEᴇᴄᴍ − ΔPEᴇᴄᴍ = Mᵉᶠᶠgᵉᶠᶠh

This expresses that motion reduces stored potential energy and creates kinetic mass.

22. Potential Energy of the Effective Mass

The released energy modifies the gravitationally active mass.

The system now carries effective mass

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ),

so the effective potential energy becomes

PEᴇᴄᴍᵉᶠᶠ = Mᵉᶠᶠgᵉᶠᶠh

Thus gravity in ECM acts on matter mass plus manifested NAM.

23. Kinetic Energy as Frequency-Generated Mass

In ECM, kinetic energy is not an abstract term — it is mass oscillating at frequency.

The inherent (de Broglie) component is

KEᴇᴄᴍ,ᵢₙₕₑᵣₑₙₜ = ½(−Mᵃᵖᵖ,ᵢₙₕₑᵣₑₙₜ)c² = ½(ΔMᴍ,ᵢₙₕₑᵣₑₙₜ)c² = ½(fᵈᴮ)c².

When this inherent oscillation interacts with the gravitational source field Mɢ, a Planck-type interactional component appears. The total kinetic energy becomes

KEᴇᴄᴍ↑ = ½(−Mᵃᵖᵖ,ᵢₙₕₑᵣₑₙₜ −Mᵃᵖᵖ,ᵢₙₜₑᵣₐᴄₜᵢₒₙₐₗ)c² = ½(fᵈᴮ + fᴾ)c² = ΔMᴍc² = hf

Thus,

KEᴇᴄᴍ = ΔMᴍc² = hf.

For photons (pure oscillatory systems),

KEᴇᴄᴍ = ΔMᴍc² = hf.

24. Frequency-Governed Kinetic Energy Law

KEᴇᴄᴍ = (ΔMᴍᵈᴮ + ΔMᴍᴾ)c² = (αΔMᴍᵈᴮ + (1 − α)ΔMᴍᴾ)c² = h(fᵈᴮ + fᴾ) = hf

This shows that

• de Broglie frequency governs inertial motion

• Planck frequency governs field-coupled motion

• their sum governs total kinetic mass and energy

Final ECM Meaning

Classical mechanics writes

KE = ½mv².

ECM reveals the hidden structure:

KEᴇᴄᴍ = ΔMᴍc² = hf,

where the mass ΔMᴍ is created by

−ΔPEᴇᴄᴍ

Motion is therefore the physical manifestation of mass released from potential energy.

This is the engine beneath Newton’s equations.

25. Alphabetical List of ECM Terms and Denotations

α (alpha)

Weighting coefficient that determines how much of the total manifested kinetic mass comes from de Broglie (inertial) versus Planck (field-coupled) frequency contributions.

aᵉᶠᶠ

Effective acceleration produced by the action of force on the effective mass Mᵉᶠᶠ.

c

Speed of light. In ECM it acts as the mass-to-frequency conversion constant via KEᴇᴄᴍ = ΔMᴍc² .

ΔKEᴇᴄᴍ

Change in kinetic energy in ECM, generated by the manifestation of mass from released potential energy.

ΔMᴍ

Manifested matter-mass produced from released potential energy.This is the physical carrier of kinetic energy.

ΔPEᴇᴄᴍ

Change in ECM potential energy. A negative change −ΔPEᴇᴄᴍ generates NAM and kinetic mass.

f

Total effective frequency associated with manifested mass:

f = fᵈᴮ + fᴾ

fᵈᴮ (de Broglie frequency). Frequency associated with inertial motion of manifested mass.

fᴾ (Planck frequency). Frequency associated with field-coupled (gravitational) interaction of manifested mass.

gᵉᶠᶠ

Effective gravitational field acting on the effective mass Mᵉᶠᶠ.

h

Planck constant, relating frequency to manifested energy-mass via

KEᴇᴄᴍ = hf = ΔMᴍc²,

KEᴇᴄᴍ

Kinetic energy in ECM. It is not abstract motion energy but mass created from released potential energy:

KEᴇᴄᴍ = hf = ΔMᴍc²

Mᵃᵖᵖ (Negative Apparent Mass, NAM). Mass equivalent of released potential energy:

−Mᵃᵖᵖ ≡ −ΔPEᴇᴄᴍ

It is the field-like mass that drives gravity, inertia, and kinetic energy.

Mᵉᶠᶠ (Effective Mass)

Total gravitational and inertial mass of a moving system:

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ).

Mɢ (Gravitational Mass)

In ECM: Mᵉᶠᶠ = Mɢ

Gravity responds to both matter mass and NAM.

Mᴍ (Matter Mass)

The rest-existence mass of matter — the static mass stored in the potential field PEᴇᴄᴍ.

Mᴍ,ₘₒₜᵢₒₙ

Total mass of a moving body:

Mᴍ,ₘₒₜᵢₒₙ = Mᴍ,ᵣₑₛₜ + ΔMᴍ

PEᴇᴄᴍ

Stored ECM potential energy — the existence field carried by matter mass Mᴍ.

PEᴇᴄᴍᵉᶠᶠ

Effective potential energy after manifestation:

PEᴇᴄᴍᵉᶠᶠ = PEᴇᴄᴍ − ΔPEᴇᴄᴍ = Mᵉᶠᶠgᵉᶠᶠh

−ΔPEᴇᴄᴍ

Released ECM potential energy that creates NAM and kinetic mass.

−Mᵃᵖᵖ

Negative Apparent Mass created by −ΔPEᴇᴄᴍ.

It is the engine of gravity, inertia, and kinetic energy.

ΔMᴍᵈᴮ

Portion of manifested mass generated by de Broglie (inertial) oscillations.

ΔMᴍᴾ

Portion of manifested mass generated by Planck (field-coupled) oscillations.

ΔMᴍc²

Energy equivalent of manifested mass:

ΔMᴍc² = hf

Phase-Kernel

The 0-D ECM source where frequency-driven mass manifestation occurs, generating NAM, gravity, and kinetic energy.

NAM (Negative Apparent Mass)

The mass-equivalent of released potential energy.

It produces gravitational attraction, inertial resistance, and kinetic energy.

Weak Equivalence Principle (Classical Limit)

In ECM collapse limit −Mᵃᵖᵖ → 0,

Mᵉᶠᶠ → Mᴍ

so inertial mass equals gravitational mass.

Gravity and Motion as Entropic Duals

Soumendra Nath Thakur

ORCID: 0000-0003-1871-7803

December 30, 2025

Gravity and motion are two complementary manifestations of entropy within Extended Classical Mechanics (ECM).

At the most fundamental level, both gravity and motion arise from a frequency-controlled redistribution of mass–energy. This single underlying process governs how matter, energy, and apparent mass continuously transform throughout the universe.

In ECM, Negative Apparent Mass (NAM) emerges dynamically from the redistribution of energy–mass within a gravitational field. It is not an independent substance, but a consequence of entropic mass–energy conversion.

An effective mass—which functions as gravitational mass—is formed by combining the matter mass with negative apparent mass (NAM). Both baryonic matter and dark matter contribute to the matter-mass component of this effective mass.

Under this framework, gravity is not a fundamental force nor a curvature of spacetime, but a mass-binding condition. It represents a system’s capacity to confine and organize mass–energy within a stable effective gravitational structure.

Motion, in contrast, is a mass-radiating and dispersive state. During motion, matter mass is transformed into negative apparent mass in proportion to entropic decay, giving rise to anti-gravitational behavior.

Thus, gravity and anti-gravity emerge naturally from frequency-governed mass transformation. Where mass condenses into effective mass, gravity dominates; where mass disperses into NAM, motion and expansion prevail.

The fundamental phenomena of the universe—kinetic energy, gravitation, and cosmic expansion—are therefore not independent. They are all emergent expressions of a single frequency-regulated mass–energy redistribution process.

Through this mechanism, ECM establishes the missing physical link between microscopic vacuum decay and the large-scale regeneration of the universe.

In conclusion, gravity and motion are entropic duals: as entropy increases, mass radiates into NAM and motion dominates; as entropy decreases, mass condenses into effective mass and gravity prevails.

New Review Published: ECM Bridge between Sen’s Conjecture and Penrose’s Conformal Cyclic Cosmology

December 29, 2025

I have published a new review report titled:

“Extended Classical Mechanics Bridge between Sen’s Conjecture and Penrose’s Conformal Cyclic Cosmology”

DOI: https://doi.org/10.13140/RG.2.2.23039.47521

This paper examines how Extended Classical Mechanics (ECM) provides a physical energy-based mechanism that links two of the most profound ideas in modern theoretical physics:

• Ashoke Sen’s conjecture, which explains how matter and branes dissolve into vacuum through tachyon condensation

• Roger Penrose’s Conformal Cyclic Cosmology (CCC), which proposes that the universe passes through endless cosmic aeons

The review shows that ECM’s frequency-governed mass and energy redistribution supplies the missing physical bridge between microscopic vacuum decay and the large-scale rebirth of the universe. It explains how matter, radiation, and even time itself can dissolve into a non-eventful energetic state, and later re-emerge as a new cosmic cycle — without violating energy conservation.

This work offers a post-relativistic, energy-driven interpretation of cosmic origin, cosmic expansion, dark energy, and the fate of the universe, unifying them within a single coherent framework.

Phase Shift, Energy Loss, and ECM Mapping

December 25, 20205

Soumendra Nath Thakur

A phase shift is not merely a geometric or angular quantity. Physically, a phase shift represents a fractional loss of completed oscillatory cycles in a propagating wave.

Because frequency is defined as the number of cycles completed per unit time, any loss of cycles immediately implies a reduction in effective frequency. This establishes the first physical link:

Phase shift → fractional cycle loss → frequency reduction.

Through Planck’s relation (E = hf), frequency directly determines the energy carried by an oscillatory quantum. Therefore, a fractional loss of cycles produces a proportional loss of Planck energy:

Δf/f₀ = Δt/T = x°/360 → ΔE = hf₀(x°/360).

This establishes the complete physical bridge:

phase → time distortion → frequency shift → energy loss→ redshift.

In Extended Classical Mechanics (ECM), this Planck-quantified energy loss is not an abstract bookkeeping change. It corresponds to a real physical conversion of stored potential structure into dynamical and mass-like manifestations:

-ΔPEᴇᴄᴍ ↔ ΔKEᴇᴄᴍ ↔ ΔMᴍ ⟶ observable Planck energy loss.

Thus, phase drift is the physical trigger by which oscillatory energy is removed from the wave, converted into Negative Apparent Mass (NAM) and associated kinetic and mass manifestations, and finally observed as redshifted radiation.

This provides a direct, causal, and Planck-consistent bridge between wave phase dynamics and ECM’s mass–energy conversion framework.

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Phase Shift Calculations and Example:

To illustrate the practical application of the phase–time relation T(deg), an example is presented.

Example 1 — 1° Phase Shift on a 5 MHz Wave

The time shift associated with a phase change is given by

T(deg) = x°/360f

For x = 1° and f = 5 MHz = 5 × 10⁶ Hz:

Now, plug in the frequency (f) into the equation for T(deg):

T(deg) = 1/(360 × 5 × 10⁶) = 5.556 × 10⁻¹⁰ s 

T(deg) ≈ 555 picoseconds (ps)

Thus, a 1° phase shift on a 5 MHz wave corresponds to a time shift of approximately 555 ps.

This calculation demonstrates how to determine the time shift caused by a 1° phase shift on a 5 MHz wave. It involves substituting the known frequency (f = 5 MHz) into the equation for T(deg).

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Phase Shift Equation 1.1 — General Form

For a x° phase shift on a f₀ Hz Wave:

T(deg) = x°/360f₀

By plugin the values of frequency (f₀) and phase shift (x°) into the equation, the calculated value of T(deg):

T(deg) = x°/360f₀ ≈ Δt

So, a x° phase shift on a f₀ Hz wave corresponds to a time shift of approximately Δt s.

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Infinitesimal Loss of Wave Energy Equations:

These equations relate to the infinitesimal loss of wave energy (ΔE) due to various factors, including phase shift:

The Planck energy-frequency equation:

• E = hf

So for a small change,

• ΔE = hΔf.

We write this in fractional form relative to the source frequency f₀:

• ΔE = hf₀(Δf/f₀) → hf₀(x°/360)

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Or, if phase–time coupling:

• ΔE = hf₀(Δt/T)

because only fraction of a cycle changes energy.

Derivation of hf₀(Δt/T):

A wave with period T has:

f₀ = 1/T

A phase shift means that the wave is no longer completing full cycles.

If the time shifts by Δt, the fractional cycle loss is:

Δt/T.

The fractional cycle loss is exactly fractional frequency loss:

Δf/f₀ = Δt/T = x°/360

This is the definition of frequency as cycles per unit time.

• [ΔE = hf₀(Δf/f₀) = hf₀(Δt/T) = hf₀(x°/360)]

This expression states:

Phase drift → fractional cycle loss → frequency reduction → Planck-quantified energy loss.

In Extended Classical Mechanics (ECM), this lost oscillatory energy is not abstract. It corresponds to a real conversion:

• -ΔPEᴇᴄᴍ → ΔKEᴇᴄᴍ → ΔMᴍ ⟶ Planck-quantified energy loss.

with the measurable manifestation appearing as the Planck energy deficit

• ΔE = hΔf.

Thus, phase drift directly generates Negative Apparent Mass (NAM) through the loss of oscillatory existence.

This is one of the key bridges between Planck physics and ECM’s NAM–phase–redshift mechanism.

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This equation determines the infinitesimal loss of wave energy (ΔE) from Planck’s constant (h) when the source frequency (f₀) and either the phase shift (x°) or the corresponding time shift (Δt) are known. It represents Planck energy scaled by the fractional loss of oscillatory phase.

When the phase shift in degrees (x°) is known, the infinitesimal energy loss is

• ΔE = hf₀(x°/360).

Since a phase shift corresponds to a fractional time shift (Δt) of one oscillation period (T), the energy loss may equivalently be written as

• ΔE = hf₀(Δt/T).

Dimensionally, (T) is the time duration of one oscillation cycle, whereas (360°) is the angular phase span of one cycle; the two are related by the fractional-cycle identity, not by numerical substitution.

These expressions form the foundation for analyzing phase shift, time distortion, frequency change, and the resulting infinitesimal loss of wave energy. They apply to both theoretical and practical wave analyses and align directly with the ECM interpretation of phase drift → energy loss → redshift → ΔMᴍ (NAM mapping).

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Loss of Wave Energy Calculations and Example:

Loss of Wave Energy Example 1: 

To illustrate the practical applications of the derived equations of loss of wave energy, example calculation is presented:

To determine the energy (E₀) and infinitesimal loss of energy (ΔE) of an oscillatory wave with a frequency (f₀) of 5 MHz and a phase shift x° = 0°, use the following equations:

Oscillation frequency 5 MHz, when 0° Phase shift in frequency. 

Calculate the energy (E₀) of the oscillatory wave:

E₀ = hf₀

Where:

h is Planck's constant ≈ 6.626 × 10⁻³⁴ Js .

f₀ is the frequency of the wave, which is 5 MHz (5 × 10⁶ Hz).

E₀ = (6.626 × 10⁻³⁴ Js) × (5 × 10⁶) = 3.313 × 10⁻²⁷ J.

So, the energy (E₀) of the oscillatory wave is approximately 3.313 × 10⁻²⁷ Joules.

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Calculate the time distortion T(deg) of the oscillatory wave when phase shift x°=0°:

For an oscillatory wave of frequency f₀ = 5 MHz with zero phase shift,

T(deg) = x°/360f₀ = Δt

Since x° = 0°,

T(deg)  = Δt = 0.

Thus, there is no time distortion because no phase shift has occurred. ECM-consistent chain: Phase → time distortion → energy change.

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Calculate the infinitesimal loss of wave energy (ΔE₀) when both Δf₀ and Δt are zero:

The infinitesimal energy change is given by

ΔE₀ = hΔf₀.

Since Δf₀ = 0,

ΔE₀ = 6.626 × 10⁻³⁴ × 0 = 0.

Therefore, the infinitesimal loss of wave energy (ΔE₀) is 0 joules because there is no time distortion (Δt = 0), no phase shift (x° = 0°), no frequency shift (Δf₀ = 0), meaning there is no infinitesimal loss of wave energy during this specific time interval.  

Conclusion for the zero-phase-shift case

These calculations demonstrate that for an oscillatory wave of frequency f₀ = 5 MHz with x° = 0°:

• the time distortion Δt = 0,

• the frequency change Δf₀ = 0,

• and the infinitesimal energy loss ΔE₀ = 0.

The wave therefore retains its full Planck energy

E₀ = hf₀ = 3.313 × 10⁻²⁷ J.

The energy (E₀) of the oscillating wave with a frequency 5 MHz and no phase shift (x° = 0°) is approximately 3.313 × 10⁻²⁷ joules. Due to the absence of a phase shift, there is no time distortion (Δt) and no infinitesimal energy loss (ΔE) of the wave during this specific time interval. 

This establishes the correct reference state against which phase-drift, redshift, and ECM-based energy conversion can be measured. 

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Loss of Wave Energy — Example 2

To illustrate the practical application of the derived equations for wave-energy loss, the following example is presented.

Consider an oscillatory wave with an original frequency

f₀ = 5 MHz

that undergoes a phase shift of

x° = 1°.

This x° phase shift produces a slightly reduced oscillation frequency f₁ and a corresponding infinitesimal loss of wave energy ΔE.

This example demonstrates how to determine:

• the new wave energy E₁,

• the infinitesimal energy loss ΔE, and

• the resulting shifted frequency f₁,

relative to the original frequency f₀, when the wave experiences a phase shift

x° = 1°.

To determine the energy E₁, the energy loss ΔE, and the resulting frequency f₁ for a wave with a (1°) phase shift from the original frequency f₀ = 5 MHz, proceed as follows:

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Calculate the energy E₁ of the oscillatory wave with the shifted frequency f₁:

Using Planck’s energy relation,

E₁ = hf₁

where

h is Planck’s constant ≈ 6.626 × 10⁻³⁴ J·s,

f₁ is the frequency after the phase shift.

Determine the frequency change Δf produced by a phase shift of x° = 1°:

A phase shift represents a fractional displacement of one oscillation cycle.

Therefore, the corresponding fractional change in frequency is:

Δf/f₀ = x°/360°

so,

Δf = (x°/360°)f₀

For x° = 1° and f₀ = 5 MHz = 5 × 10⁶ Hz,

Δf = (1/360) 5 × 10⁶ = 13,888.89 Hz

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The shifted frequency is therefore

f₁ = f₀ - Δf

for a red-shifting (energy-losing) phase drift in ECM. 

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Now that the frequency shift Δf has been determined, the shifted frequency f₁ is:

f₁ = f₀ - Δf

Substituting the values,

f₁ = f₀ - Δf

f₁ = (5.0 × 10⁶) - (13,888.89) = 4,986,111.11 Hz

Thus, the resulting frequency of the oscillatory wave after a 1° phase shift is approximately

f₁ = 4.98611111 × 10⁶ Hz

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This correctly implements the ECM rule:

Δf/f₀ = x°/360°

So a 1° phase drift produces a (1/360) fractional frequency reduction — and therefore a proportional energy and mass decrement, exactly as required by ECM-consistent chain: phase-drift → ΔE → ΔMᴍ mapping.

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Calculate the energy (E₁) using the new frequency (f₁):

E₁ = hf₁

E₁ ≈ (6.626 × 10⁻³⁴) × (4.98611111 × 10⁶) Hz.

E₁ ≈ 3.3048 × 10⁻²⁷ J

Thus, the energy of the oscillatory wave with a frequency of approximately 4,986,111.11 Hz and a x° = 1° phase shift is approximately E₁ ≈ 3.3048 × 10⁻²⁷ Joules.

This reflects the ECM relation

ΔE/E₀ = Δf/f₀ = x°/360°

so a 1° phase drift produces a real, proportional Planck-energy loss, exactly as ECM's phase-drift → energy-loss → redshift → ΔMᴍ chain requires.

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To determine the infinitesimal loss of energy (ΔE) due to the phase shift, use the formula:

ΔE = hΔf = hf₀(Δt/T) = h(f₀)²Δt

Where:

h = 6.626 × 10⁻³⁴ Js is Planck's constant. 

f₀ = 5 Mhz = 5 × 10⁶ Hz 

f₁ ≈ 4.98611111 × 10⁶ Hz

Δf =  f₀ - f₁ = 13,888.89 Hz = 0.01388889 × 10⁶ MHz

Δt ≈ 555 ps = 5.55 × 10⁻¹⁰ s, corresponding to a 1° phase shift on f₀.

ECM-consistent chain: Phase drift (x°) → Δt → Δf → ΔE = hΔf → ΔMᴍ   

So, 

ΔE = hΔf = 9.2036 × 10⁻³⁰ J

or,

ΔE = hf₀(Δt/T) = 9.2036 × 10⁻³⁰ J, where T = 1/f₀ = 2.0 × 10⁻⁷ s  

or,

ΔE = h(f₀)²Δt = 9.2036 × 10⁻³⁰ J

Thus, the infinitesimal loss of energy (ΔE) due to the 1° phase shift is approximately 9.2036 × 10⁻³⁰ Joules.

Resolved, the energy (E₁) of this oscillatory wave is ≈ 3.3048 × 10⁻²⁷ Joules.

Resolved, the infinitesimal loss of energy (ΔE) due to the 1° phase shift is approximately 9.2036 × 10⁻³⁰ Joules.

Resolved, the resulting frequency (f₁) of the oscillatory wave with a 1° phase shift is ≈ 4.98611111 × 10⁶ Hz.