General RQT Resonance-Calculation Framework

Starting from experimentally measured coupling strengths and inertial properties, RQT derives characteristic resonance distances, separation energies, phase-closure conditions, and confined resonance-mode energies. Applied to the electron response scale, these relations lead naturally to quon-scale energy levels of approximately $300,\mathrm{MeV}$, providing a possible energetic bridge between electron-scale and nucleon-scale structures within a unified resonance description.

Related Detailed Chapters

For detailed derivations and examples see:

• Electron Internal Structure
• Electron Interaction Modes
• Quon Resonance Structures
• Nucleus lattice structure

The present chapter serves as a condensed reference framework collecting the characteristic relations used throughout the RQT model.

(A) Residual Coupling Relations

The RQT describes resonance structures through two characteristic quantities:

• a distance-independent coupling strength $K$,
• and a characteristic resonance length $\ell$.

Residual interactions observed at distance $r$ are derived from these quantities and decrease with increasing distance from the resonance structure.

The residual coupling relations describe:

• externally visible residual interactions,
• field-like behavior,
• inverse-square dilution,
• long-range residual effects.

General residual form:

$$F(r)=\frac{K}{r^2}$$

Examples:

Electromagnetic:

$$K_E=k_e e^2$$

$$F_E(r)=\frac{K_E}{r^2}$$

Gravitational (dominantly):

$$K_G=Gm_1m_2$$

$$F_G(r)=\frac{K_G}{r^2}$$

RQT interpretation:

• $K$ is intrinsic coupling strength,
• $1/r^2$ is residual spherical dilution of open resonance channels.

(B) Resonance Locking and Redistribution Length

Characteristic resonance-locking distance:

$$d_R \sim \frac{K}{mc^2}$$

Experimantally derived electron (self-)response length:

$$\ell_e = \frac23\frac{K_E}{m_ec^2} \approx1.88,\mathrm{fm}$$

where the factor $\frac23$ comes as a geometric constant.

RQT interpretation:

• inertial response footprint,
• characteristic rephasing length,
• resonance redistribution distance.
• short range coupling distance
• the 2/3 factor might appear via the number of redistribution resonance channels
• or as $l_e = \boxed{\text{redistribution fraction}\times\text{coupling length}}$

RQT working hypothesis:

• $\boxed{d_R^{(e)}\approx\ell_e}$

(C) Residual Separation Energy

Weak externally visible separation work:

$$E_{\mathrm{sep}} = \int_d^\infty \frac{K}{r^2}dr = \frac{K}{d}$$

Electron scale:

$$E_{\mathrm{sep}} = \frac{K_E}{\ell_e} = \frac32 m_ec^2$$

RQT interpretation:

• residual observable coupling energy,
• not confinement energy.
• work required to separate two statically distance-locked resonance structures

(D) Confined Resonance Mode Energy

Localized internally confined resonance mode:

$$E_R(d) \sim \frac{\hbar c}{d}$$

derived from: $E=\hbar\omega$ and: $\omega\sim\frac{c}{d}$

RQT interpretation:

• coherent localized phase-mode energy,
• confinement energy,
• internally closed resonance redistribution.

(E) Multi-Mode / Network Energy

For coupled resonance structures:

$$E_{\mathrm{network}} \sim \eta N_{\mathrm{modes}} \frac{\hbar c}{d_R}$$

where:

• $N_{\mathrm{modes}}$ = confined resonance components,
• $\eta$ = geometric redistribution efficiency/symmetry factor.

RQT interpretation:

• inertia emerges as distributed resonance-network redistribution energy.

(F) Orbital Resonance Closure

For orbital resonance confinement:

$$\Delta\phi=2\pi n$$

with: $\phi=\frac{S}{\hbar}$ and rotational resonance action: $S\sim mvr$

leading to:

$$mvr=n\hbar$$

RQT interpretation:

• stable orbital resonance closure,
• coherent phase return after one cycle.

Example:

• An H-Atom would have a shared inertial redistribution term of $\mu_{ep}=\frac{m_e m_p}{m_e+m_p}$ with $\mu_{ep}\approx m_e$
• this gives a resonance closure at $\mu_{ep} v r = n\hbar$ with $r_n=\frac{n^2\hbar^2}{\mu_{ep}K_E}$
• and a ground state first orbital radius with $r_1=\frac{\hbar^2}{\mu_{ep}K_E}$ or $r_H \approx a_0\left(1+\frac{m_e}{m_p}\right)$

(G) Resonance Propagation Speed

Externally open systems: $v<c$ Internally confined systems: $v_{\mathrm{internal}}\sim c$

RQT interpretation:

• internally closed resonance channels permit maximal internal coherent redistribution propagation.

This allows for inertia calculations of electrons and quons. $m_R \sim \frac{E_R}{c^2}$

Example: For an electron response length of $d=\ell_e\approx1.88,\mathrm{fm}$ we get:

$$E_R\approx\frac{197.3,\mathrm{MeV,fm}}{1.88,\mathrm{fm}}\approx105,\mathrm{MeV}$$

So: $3E_R\approx315,\mathrm{MeV}$ indeed lands near one third of the experimentally confirmed proton rest energy.

Mapping to standard physics

Standard physic comparison:

For electromagnetism: $k_e=\frac{1}{4\pi\varepsilon_0} = 8.9876\times10^9;\mathrm{\frac{N,m^2}{C^2}}$ and $K_E=k_e e^2=\alpha,\hbar c$

Here the fundamental constant is $\varepsilon_0=8.8541878128\times10^{-12};\mathrm{\frac{F}{m}}$ as the vacuum permittivity.

And the formula $c^2=\frac1{\mu_0\varepsilon_0}$ links to $\mu_0$ as the vacuum permeability.

Modern particle phsyics prefers $\alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c}$ which gives $\alpha \approx \frac1{137.036}$

For the main gravity contributor: $G=6.67430\times10^{-11};\mathrm{\frac{m^3}{kg,s^2}}$

For quantum gravity discussions: $G=\frac{1}{m_P^2} * \hbar c$ where $m_p^2$ is the Planck mass $m_P = \sqrt{\frac{\hbar c}{G}}$

In RQT we keep running into this standard physics constant: $\boxed{\hbar c}$ denoting $\mathrm{Energy}\times\mathrm{Length}$ in $[\hbar c] = \mathrm{J,m}$

With this RQT interpretation: $\boxed{\hbar c = \text{resonance action propagated over one characteristic length}}$. carying phase, action and closure.

Where $\hbar$ carries: phase, action, closure and $c$ carries: propagation, redistribution speed.

The very interesing part is :

$$\boxed{ \frac{\hbar c}{K_E} = \frac1{\alpha} \approx137 }$$

So the residual electromagnetic energy is just: $\frac1{137}$ of the full confinement-mode scale.

This may turn out to be one of the most important observations in the entire derivation.

Summary

This is probably the minimal “master formula layer” before adding further: geometry and lattice structures

RQT interprets physical structures as coherent resonance systems which may exist in different coupling regimes. These range from weak residual interactions between externally open resonance channels to strongly confined resonance networks capable of storing substantial localized energy.

The purpose of this framework is to establish a common set of characteristic quantities and relations that can be applied consistently across different scales, including electrons, quons, nucleons, nuclei, and larger resonance structures.

Rather than introducing separate rules for each scale, RQT assumes that the same fundamental resonance principles remain valid throughout:

• residual coupling strength,
• resonance-locking distance,
• inertial response,
• phase closure,
• confined resonance propagation,
• and resonance-mode energy.

The following relations therefore serve as a scale-independent resonance framework from which both externally observable interactions and internally confined energy structures may be estimated.

A central result of the framework is the distinction between:

• residual coupling energy, describing the energy required to separate phase-locked resonance structures,
• and confined resonance-mode energy, describing the energy that can be accommodated within a coherent resonance structure itself.

Applying these relations to experimentally known electron properties leads to a characteristic resonance redistribution length in the femtometer range. When interpreted as the confinement scale of internally closed resonance modes, the resulting energy scale naturally enters the regime associated with quons (quarks), nucleons, and nuclear structures.

In this way, the framework provides a possible bridge between experimentally measured electron properties and the characteristic energy scales observed within atomic nuclei.