Clarifying the Role of Mathematical Rigor and Experimental Expectation in ECM Interpretation.

Soumendra Nath Thakur
July 04, 2025

The term “rigorous mathematical derivation” is often misapplied when it is used to imply an objectively necessary standard for conceptual legitimacy, regardless of context. In reality, what is considered “rigorous” must be appropriate to the domain and purpose of the framework in question. In the case of Extended Classical Mechanics (ECM), the mathematical formulations are internally consistent and serve their interpretive purpose. The insistence on a particular form of "rigor" or demand for new experimental data as a gatekeeping criterion overlooks that ECM builds upon already validated phenomena—such as thermionic emission and the photoelectric effect—by reinterpreting them through a novel lens of apparent mass displacement (−Mᵃᵖᵖ), motion-energy dynamics, and gravitational scaling.

It is intellectually dishonest to dismiss such a framework simply because it does not conform to traditional formalism or peer-reviewed expectations, especially when those expectations were already fulfilled by the very classical and quantum experiments ECM draws upon. Expecting new data or traditional derivations from an interpretive theory—whose role is to explain, unify, or clarify existing data and models—is an unrealistic standard that serves more as an expression of entrenched bias than scientific openness.

For instance, it is unnecessary to use calculus to prove that 1 + 1 = 2. Likewise, ECM uses the mathematical structures appropriate to its framework—rooted in energy-mass transformations and apparent mass dynamics—without mimicking the exact derivational pathways of other frameworks. Simplicity, clarity, and honest consistency matter more than performative mathematical complexity.

In short, ECM presents a novel synthesis that does not require validation by arbitrary and externally imposed standards of mathematical formalism or redundant experimental repetition. Its value lies in the clarity of interpretation it brings to already understood but incompletely explained phenomena.

This statement reflects my considered position and serves as a direct response to prior critique. 

Energy-Mass States of Bound and Free Electrons: ECM Interpretation of Atomic Transitions, Thermionic Emission, and Photon Emission.

An Extended Classical Mechanics Interpretation: Energy-Mass States of Bound and Free Electrons. 

Soumendra Nath Thakur | ORCiD: 0000-0003-1871-7803 | Tagore's Electronic Lab, India | July 04, 2025

The total energy-mass of a free electron is equivalent to its rest mass energy, expressed as:

Eₜₒₜₐₗ = Mₑc² ≈ 0.511 MeV,

where Mₑ denotes the electron's rest mass.

Within an atom, an electron bound in the lowest energy orbital—the ground state (n = 1)—exhibits a significantly lower Eₜₒₜₐₗ due to the presence of negative electrostatic potential energy resulting from Coulomb interaction with the atomic nucleus. For a hydrogen atom, this energy level is quantized and is given by:

Eₙ = −13.6 eV for n = 1,

E₂ = −3.4 eV, E₃ = −1.51 eV, etc.

These values represent net bound-state energies, which are markedly lower than the energy-mass condition of a free, unbound electron at rest (0.511 MeV), emphasizing that atomic electrons possess lower Eₜₒₜₐₗ due to confinement.

In outer shells, the valence electron, often residing at the highest occupied energy level, possesses the greatest total energy relative to other bound electrons. Under thermal excitation, when sufficient energy is supplied (typically via heat), the electron may acquire enough kinetic energy to overcome the −Mᵃᵖᵖc² binding potential (where −Mᵃᵖᵖ denotes the negative apparent mass associated with electrostatic confinement), thereby escaping the atomic structure in a process known as thermionic emission.

During thermionic emission, the electron transitions from a bound state (Mᴍ < Mₑ) to a nearly free state, achieving:

ΔMᴍ = Mₑ − Mᴍ > 0,

accompanied by the displacement of −Mᵃᵖᵖ from the atomic system to the metallic boundary surface. This released electron becomes quasi-free and localized near the outer metallic surface, though not yet a completely free particle in vacuum.

In contrast, photoelectric emission occurs when incident photons of sufficient frequency (f) interact with valence electrons. If the photon energy hf ≥ |−Mᵃᵖᵖ|c², the electron overcomes its binding condition and is emitted from the material. Here, the interaction satisfies:

hf = ΔMᴍc² = −Mᵃᵖᵖc²,

highlighting the mass-energy equivalence of photon interaction with electron confinement energy.

Within atoms, when an electron transitions from a higher energy level (nᵢ) to a lower one (n𝒻), the energy difference is released as a photon:

ΔE = hf = Eₙᵢ − Eₙ𝒻,

signifying the conversion of potential and kinetic energy loss (−ΔPEᴇᴄᴍ and −ΔKEᴇᴄᴍ) into radiative output. This emission occurs only for bound electrons undergoing quantized transitions. In contrast, a truly free electron (Mᴍ = Mₑ) does not emit photons under motion in free space, as it lacks quantized energy states or orbital confinement.

Thus, under conservative dynamics, such as an electron moving within an electric potential, any gain in potential energy is reciprocated by a corresponding loss in kinetic energy, and vice versa:

−ΔPEᴇᴄᴍ = +ΔKEᴇᴄᴍ,

preserving total internal energy. During quantum transitions, the decrease in bound-state potential energy is manifest externally as photon emission—corresponding to the released hf, now separable from the atom as a radiative quantum of energy and mass.

Complementarity of Dynamic and Apparent Mass in ECM: (ΔMᴍ ↔ Mᵃᵖᵖ)

Soumendra Nath Thakur 

A core interpretive principle within ECM is the complementarity between dynamic mass displacement (ΔMᴍ) and apparent mass (Mᵃᵖᵖ). These quantities are not merely opposites in algebraic sign, but mutually defining constructs that gain physical significance only in relation to one another.

For example:

ΔMᴍ represents the emergent or emitted mass-equivalent energy due to frequency scaling, as in:

hf = ΔMᴍ c²

Mᵃᵖᵖ = −ΔMᴍ captures the corresponding loss or reduction in apparent mass from the source system.

This mutual dependence mirrors other foundational complements in nature:

Black and white as absence and presence of light

Potential and kinetic energy in transition

Finite and infinite as relational constructs

In ECM, neither ΔMᴍ nor Mᵃᵖᵖ has causal validity in isolation. It is their interaction—seen in transformations like:

KEᴇᴄᴍ = −Mᵃᵖᵖ c ² or ΔMᴍ = hf / c²

—that defines real physical outcomes such as radiation, gravitational weakening, and cosmic expansion.

This principle of complementarity reinforces ECM's broader stance: that energy and mass, emergence and loss, are not independent absolutes, but relational constructs whose meaning arises through causal symmetry.

Summary

ECM restores physical continuity and causality by linking frequency to mass-energy emergence, rejecting singularities and probabilistic quantum behavior. Its structural pillars are:

Frequency-scaling of force, energy, and displacement

Nonlinear collapse at Planck thresholds

Energetic boundary formation instead of metric expansion

Deterministic time onset defined by 

This unified interpretation enables ECM to model dynamics across photon, collapse, and cosmological scales with logical continuity and dimensional precision

Pre-relativistic framework and ECM:

Soumendra Nath Thakur

Relativity is not necessary for the very phenomena it is often praised for explaining. In truth, it diverted science away from rational foundations by introducing dilatable time and curved, blended space — abstraction that complicate rather than clarifying physical reality.

The pre-relativistic framework was already sufficient to support a more consistent and physically intuitive understanding of the universe. What was needed was not a leap into spacetime distortion, but a deeper refinement of classical principles.

This is where Extended Classical Mechanics (ECM) comes in — a framework with the potential to restore coherence and rational causality to physics. Once fully explored, ECM may well demonstrate that relativity’s perceived necessity was a historical detour, not a scientific inevitability.

🚀 ANNOUNCEMENT: Publication of ECM Appendices 20–22

🔬 Extending Causal Mass-Energy Theory Across Frequency, Collapse, and Cosmological Boundaries

By Soumendra Nath Thakur | Tagore's Electronic Lab, India

ORCiD: 0000-0003-1871-7803

I'm pleased to announce the release of a critical three-part extension to the ECM series, now published and available on ResearchGate:

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📘 Appendix 20: Frequency Scaling and Energy Redistribution in Extended Classical Mechanics

🔗 DOI: 10.13140/RG.2.2.11662.06725

This appendix introduces frequency as the core driver of energy and force scaling in ECM. Replacing wave-particle duality with dynamic mass displacement logic, it establishes causal relationships such as:

Redshift, blueshift, and radiation are modelled as frequency-governed mass-energy transitions.

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📘 Appendix 21: Planck Thresholds, Energy Quantization Limits, and Nonlinear Collapse in ECM

🔗 DOI: 10.13140/RG.2.2.15017.51042

This work recasts Planck-scale transitions as causal saturation points of reversible energy exchange. At

$$f \to f_{planck} \approx 2.999 \times 10^{42} \text{ Hz},\quad \Delta Mᴍ \to Mᴍ,ʀₑₛₜ$$ $$KEᴇᴄᴍ = -\Delta PEᴇᴄᴍ = \Delta Mᴍc²,\quad v > c$$

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📘 Appendix 22: Cosmological Boundary Formation and Mass-Energy Reconfiguration in ECM Expansion

🔗 DOI: 10.13140/RG.2.2.26761.56166

Redefining cosmic expansion through energetic redistribution, this appendix introduces a non-metric cosmological boundary where

$$\Delta Mᴍ \to 0,\quad Fᴇᴄᴍ \to 0,\quad \text{at } t = t₀$$

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🧭 These three appendices form a cohesive pillar in the ECM framework—bridging photon-scale quantization, trans-Planckian collapse, and universal boundary emergence.

Feedback, discussion, and collaboration are warmly welcomed.

🔗 View the full ECM series here: researchgate.net/profile/Soumendra-Nath-Thakur

— Soumendra Nath Thakur