The Geometric and Interference Theory of Coulomb Forces: Phase Coupling and Gauging the Discrete Vacuum Bubble Substrate

Introduction

In the preceding papers of this cycle, we demonstrated that the vector reductionism introduced by Heaviside and Gibbs effectively blinded classical field dynamics to the volumetric compressibility of the vacuum medium [1,2]. To restore spatial continuity and complete the field equations without adopting the artificial kinematic constraints of Minkowski space, we reintroduced the Maxwellian scalar potential S = Real (DA) ≠ 0, based on Maxwell’s original formulations [5-7]. This successfully yielded a regular, singularity-free model of the electron acting as a localized, non-linear toroidal vortex governed by Verhulst–Van der Pol logistic dynamics.

However, the historical description of this sub-microscopic substrate has been heavily contaminated by the quantum-mechanical term “spacetime foam”, which introduces an unnecessary probabilistic and stochastic ambiguity. In this paper, we strictly abandon the concept of “foam”.

The Hierarchical Bubble Substrate: Macro-Cavities and Connected Micro-Bubbles

To eradicate the dualistic paradox of “particle vs. field” that plagues vector physics, we formulate a strict hierarchical, fractal model of the vacuum substrate, returning field dynamics to the rigorous domain of continuous media mechanics [3,8].

The Macro-Bubble (The Charge Core): An elementary charge (such as an electron or proton) is modeled as a localized macroscopic bubble—a cavitational cavity of rarefaction within the bubble field. This macro-cavity possesses no rigid physical boundary, but acts as a dynamic spatial interface. when a macro-bubble rests, it mechanically tensions the surrounding matrix. The classical electric field E is thus revealed as chains of radially elongated and bound micro-bubbles. Near the core, the micro-bubbles experience maximum elongation, mapping the high intensity of the near-field. As the radial distance increases, the micro-bubbles relax back into their spherical, non-tensioned state, causing the field intensity to naturally decrease according to the inverse-square law without generating infinite energy divergences at r → 0, matching the smooth solutions verified in quaternionic field analysis [9].

The Interference Theory of Coulomb Forces: The “Lock-and-Key” Mechanism of Vacuum Bubble Complements

Quaternionic Superposition of Pulsating Fields

Let two macro-bubbles (charges), located at coordinates x₁ = −d/2 and x₂ = +d/2, continuously pulsate at the characteristic frequency ω driven by their internal attractors. Each core emits a longitudinal wave of scalar pressure into the surrounding bound micro-bubble matrix. The individual scalar pressure profiles are expressed as:

where r₁ = |x − x₁| and r₂ = |x − x₂|. The net scalar tension field along the interaction axis is formed via direct quaternionic wave interference, applying the non-commutative sum and alignment rules established by Hamilton [4]: S_total = S₁ + S₂. Averaging the superposition over time, the spatial distribution of the field reduces to:

The Topologies of Interaction: “Spike-to-Spike” vs. “Spiketo-Hole”

The phase difference breaks the geometric symmetry of deformation, yielding two strict regimes that map the bio-chemical “lock-and-key” mechanism onto continuous media mechanics:

Experimental Proof via the Compton Scattering Framework

The validity of this phase-interference theory is verified by the Compton effect and localized wave mechanics within sub-planckian medium models [14,15]. Rather than a particle collision in empty space, Compton scattering is reinterpreted as a mechanical impact of a moving high-frequency micro-bubble autowave packet against the rotating boundary of a macro-bubble core. The shifting wavelength expressionλ = Λc(1−cosθ) mirrors our derived interference phase coefficient. The Compton wavelength Λc = 2πc/ωc = 2πRe acquires a strict geometric definition: it is the literal spatial perimeter of the toroidal electronic macro-vortex core.

Mathematical Evaluation of the Interference Force and Alignment with Coulombs Law

The local force F acting on a macro-bubble core of volume V_e is driven by the spatial gradient of the scalar pressure field: F = −V_e · ∇_Stotal(r). Differentiating our regular sinc(r) profile relative to the distance r yields:

As ωR → ∞ in the macro-range, the time-averaged flux converges to a strict invariant: Φ (R) = Const = Q . Gauss’s Law is thus proven E ε0 to be a direct hydrodynamic law of volumetric conservation of the compressible bubble matrix, tracking the constant volume of medium displaced by the macro-cavity core.

Vector Visualization of Interference Fields: The Topological Structure of Spikes and Holes

To visually map the non-linear “lock-and-key” mechanics of the bubble field, we construct the spatial profile of the modified scalar potential Stotal(x) across the collision axis of the macro-cores. Figure 1 displays the numerical distribution of the interaction fields. The upper curve maps the constructive phase alignment (+1), forming a localized regular compression “Spike” that prevents further proximity of single-sign defects. The lower curve maps the inverted destructive phase alignment (-1), carving an intensive geometric “Hole” that accommodates the incoming wave front and governs the mechanics of mutual attraction.

Conclusion

The physical essence of electric field interaction is freed from point-like abstractions: like charges conflict via identical compression spikes, whereas opposite charges experience an immediate “lock-and-key” phase locking, allowing a pressure spike to cleanly drop into a corresponding vacuum hole. This completely unifies the parameters of static fields and the Compton scattering boundaries under a single, non-singular quaternionic framework.

Figure 1: The Holographic Topology of Coulomb interactions: The Red Curve Illustrates the Compression Pressure Spike sinc(x) Generated During Like-Charge Conflicts, While the Blue Curve Displays the Inverse Rarefaction Hole −sinc(x) Enabling Opposite-Charge Phase Locking

References

1. Ostasevicius, A. (2026). The Unified Field Paradox: Restoring Maxwell’s Scalar Potential and Decoupling the Heaviside Vector Reduction via Quaternion Field Formulations. viXra Preprints, 2608.0023.
2. Ostasevicius, A. (2026). The Non-Linear Dynamic Architecture of the Hydrogen Atom: Overcoming the Paradoxes of Linear Quantum Mechanics via Quaternion Attractors. Journal of Electrical Electronics Engineering, 2026, Vol. 5, Issue 4, pp. 01–08.
3. Euler, L. (1765). Theory of the motion of solid or rigid bodies. Rostochii.
4. Hamilton, W. R. (1866). Elements of Quaternions. Longmans, London.
5. Maxwell, J. C. A. (1873). Treatise on Electricity and Magnetism. Oxford.
6. Heaviside, O. (1893). Electromagnetic Theory. Volume 1. London.
7. Gibbs, J. W. (1901). Vector Analysis. Scribner’s Sons, New York.
8. Kowalevski, S. (1889). On the problem of the rotation of a rigid body about a fixed point. Acta Mathematica, vol. 12, pp. 177–232.
9. Kravchenko, V. V. (2003). Applied Quaternionic Analysis. Heldermann Verlag, Lemgo.
10. Tesla, N. (1978). Colorado Springs Notes: 1899–1900. Nolit, Belgrade.
11. Tesla, N. (1990). The Problem of Increasing Human Energy. The Century Magazine.
12. Meyl, K. (2000). Scalar Waves. INDEL GmbH, Villingen-Schwenningen.
13. Meyl, K. (2003). Scalar Wave Technology. 1st edition, INDEL GmbH, Publishing Department, Villingen-Schwenningen.
14. McKenna, K. M. (2025). Tesla’s Longitudinal Experiments. The Institute of Substrate Physics.
15. Borisov, A. M. (2026). Transversality or Longitudonality. Academic Journal of Substrate Physics.