🌀 LEDO-Field Force Model — Resonant Interactions between Rotons

This document defines the five fundamental force components governing pairwise and field-mediated interactions between Rotons in the LEDO-Field model.
Each Roton is a dynamic wave-source characterized by translational, rotational, and energetic properties.

⚛️ Roton Entity Definition

Each Roton i is defined by the following canonical parameters:

Derived quantities:

• Wavenumber $k_i = \omega_i / c_L$
• Amplitude $A_i = |C_i|$
• Phase $\phi_i = \arg(C_i)$

🔩 Force Overview

🧭 Common Definitions

Let two Rotons i and j have:

• Separation vector
  $\mathbf{r}{ij} = \mathbf{x}i - \mathbf{x}j,\quad r{ij} = |\mathbf{r}{ij}|,\quad \hat{\mathbf{r}}{ji} = \frac{\mathbf{x}_j - \mathbf{x}i}{r{ij}}$
• Axis parallelism $c_{ij} = \mathbf{t}_i \cdot \mathbf{t}_j$
• Planarity factor
  $\Pi_{ij} = (1-(\mathbf{t}i!\cdot!\hat{\mathbf{r}}{ij})^2),(1-(\mathbf{t}j!\cdot!\hat{\mathbf{r}}{ij})^2)$
• Chirality match
  $\chi_{ij} = \operatorname{sign}(\boldsymbol{\Omega}_i!\cdot!\mathbf{t}_i),\operatorname{sign}(\boldsymbol{\Omega}_j!\cdot!\mathbf{t}_j)$
• Spectral similarity
  $Q_{ij} = \exp!\Big[-\frac{(a_i-a_j)^2}{2\sigma_a^2}-\frac{(\omega_i-\omega_j)^2}{2\sigma_\omega^2}\Big]$
• Mean size $\bar a_{ij} = (a_i + a_j)/2$ and soft core $r_0$

1. F_AR — Axial Resonance Force

Description:
Acts along the rotation axis when two Rotons share nearly the same axis and spectral parameters (radius, angular velocity, frequency).
Independent of spatial distance — this is a non-local resonance coupling through the LEDO-field itself.

Formula: $$ \mathbf{F}^{\mathrm{AR}}{j\leftarrow i} = \kappa{!AR},Q_{ij},c_{ij}^{,p},\chi_{ij},\mathbf{t}_i $$

• $p$ sharpens axial alignment preference.
• Same chirality ($\chi_{ij}=+1$) → attraction; opposite → repulsion.
• Acts along the shared axis $\mathbf{t}_i$ or $\mathbf{t}_j$.

2. F_PR — Planar Resonance Force

Description:
Occurs when two Rotons rotate in parallel planes (axes roughly parallel) and their separation lies within that plane.
Decreases approximately with $1/d$.

Formula: $$ \mathbf{F}^{\mathrm{PR}}{j\leftarrow i} =\kappa{!PR},Q_{ij},c_{ij}^{,p},\Pi_{ij}, \frac{1}{\max(r_{ij},\bar a_{ij})},\hat{\mathbf{r}}_{ji} $$

• Optional $\chi_{ij}$ factor to include spin-sense dependence.
• Acts radially within the common rotation plane.

3. F_GR — General Resonant Attraction

Description:
Baseline isotropic attraction between similar Rotons, representing residual resonant coupling even if they are not axially or planar-aligned.
Decreases as $1/d^2$.

Formula: $$ \mathbf{F}^{\mathrm{GR}}{j\leftarrow i} =\kappa{!GR},\tilde Q_{ij}, \frac{1}{(r_{ij}+r_0)^2},\hat{\mathbf{r}}_{ji} $$

• $\tilde Q_{ij}$ can ignore spin and orientation for simplicity.
• Provides a weak global coherence background.

4. F_CL — Centric Lock-In Force

Description:
A resonant synchronization between coaxial Rotons whose angular velocities are integer multiples of each other.
When resonance occurs, both Rotons experience an additional inward axial attraction toward their shared axis center, temporarily shrinking their radius — a Sub-Roton oscillation.

Resonance condition: $$ \mathrm{Res}{ij} = \min_n \Big|\frac{\omega_i}{\omega_j} - n\Big| \le \delta\omega $$

Force form: $$ \mathbf{F}^{\mathrm{CL}}{j\leftarrow i} = -,\kappa{!CL}, \exp!\Big[-\Big(\frac{\mathrm{Res}{ij}}{\delta\omega}\Big)^2\Big], \Big( \Delta a_{ij},\hat{\mathbf{r}}_{ji}

• \Delta\omega_{ij},\mathbf{t}i \Big) $$ where
  $\Delta a{ij}=a_i-a_j,\quad \Delta\omega_{ij}=\omega_i-\omega_j.$

Effect:
Drives both radius and angular velocity toward harmonic multiples — forming locked, coaxial shells that pulsate through temporary radius contractions (Sub-Roton motions).

5. F_EDP — Energy-Density Pressure Force

Description:
A repulsive field-gradient force arising from local energy-density pressure.
Independent of orientation and chirality; pushes Rotons out of high-energy regions.

Define total field energy density: $$ \epsilon(\mathbf{x})=\sum_k \alpha_k,A_k^2,G(|\mathbf{x}-\mathbf{x}_k|;a_k) $$ with kernel
$$ G(r;a)=\frac{e^{-r^2/(2a^2)}}{r^2+r_0^2}. $$

Then $$ \mathbf{F}^{\mathrm{EDP}}j =-,\gamma,\nabla \epsilon(\mathbf{x}j) =-,\gamma\sum{i\ne j}\alpha_iA_i^2,\nabla G(r{ij};a_i) $$

Gradient term: $$ \nabla G(r;a) =\Big(\frac{-r}{a^2(r^2+r_0^2)} -\frac{2r}{(r^2+r_0^2)^2}\Big)e^{-r^2/(2a^2)},\hat{\mathbf{r}}. $$

⚙️ Combined Total Force

$$ \mathbf{F}j =\sum{i\ne j} \big( \mathbf{F}^{\mathrm{AR}}{j\leftarrow i} +\mathbf{F}^{\mathrm{PR}}{j\leftarrow i} +\mathbf{F}^{\mathrm{GR}}{j\leftarrow i} +\mathbf{F}^{\mathrm{CL}}{j\leftarrow i} \big) +\mathbf{F}^{\mathrm{EDP}}_j $$

⚖️ Implementation Notes

🧩 Modeling Notes

• F_AR + F_CL act mainly along the axial degree of freedom → rotational synchronization, axial resonance, and sub-Roton pulsation.
• F_PR + F_GR act in radial/planar space → clustering and field coherence.
• F_EDP ensures spatial separation and prevents collapse.
• Keeping these as separate analytical components is clearer and more controllable than merging into a single magnetic-dipole-like field; the angular kernels and decay exponents differ and represent distinct physical couplings.

LEDO-Field Simulation Framework – Force Definitions v3.1
© 2025, Olav Le Doigt