Physics often describes matter in terms of fields. In this picture, particles are not tiny solid objects but stable ripples or disturbances in something spread throughout space. When physicists solve the equations describing these fields, a particular family of mathematical curves keeps appearing again and again: Bessel functions. These curves describe shapes that peak near a centre and fade smoothly outward, much like a ripple spreading across water.
But why should nature favour that particular shape?
The paper explores a possible answer. Instead of starting with fields, it starts one step deeper, with a framework in which the most fundamental structures are not particles or waves, but distinctions — the simplest boundaries that separate one state from another. In the VERSF framework these boundaries are called folds. A fold represents a minimal committed difference: the point at which something becomes permanently distinguishable from something else.
When many folds organize together, they can form stable closed patterns. These patterns behave like persistent structures moving through the underlying substrate — in other words, something very much like a particle. Importantly, the particle is not a tiny object sitting inside the system. It is a stable pattern formed by the relationships between folds, just as a whirlpool is a stable pattern formed by the motion of water.
The paper then asks what happens if you zoom out and describe this network of folds at larger scales. When the discrete structure is smoothed into a continuum description, the collective behaviour can be described by a wave equation. When that equation is solved for rotationally symmetric structures, the radial shape of the solution turns out to be exactly the same family of curves that appear in many particle models: modified Bessel functions.
In other words, the familiar Bessel-shaped field profiles may not be fundamental objects themselves. They may be the large-scale shadow of a deeper informational structure.
This does not yet prove that particles are made of fold-closure patterns. But it shows that if such a substrate exists, it would naturally produce the same mathematical structures that already appear in several field-theory models of particle structure. The work therefore builds a conceptual bridge between two very different ways of thinking about physics: one based on fields, and one based on information and topology.
If that bridge holds, the shapes that appear in particle equations may be telling us something profound: not just how particles behave, but how deep structure organizes itself when viewed from far away.
Interestingly, a separate paper in the VERSF research program explores this same mathematical structure from the opposite direction. In 3D String Theory: Electromagnetic Structure Theory of Fundamental Particles, particle structure is modeled as a twisted electromagnetic field configuration whose radial profile is described by the same family of Bessel functions that appear in the analysis above. In that framework the winding number of the field corresponds to the orbital angular momentum carried by the configuration.
Because light beams can also carry orbital angular momentum — so-called “twisted light” — this opens the possibility of experimental tests. If particles really possess internal structures related to these Bessel-mode field configurations, then scattering experiments using twisted light with matching angular momentum could produce distinctive resonance signatures. The EST paper develops this idea in detail, proposing concrete experiments that could probe whether such twisted internal structures exist.
From this perspective the two papers approach the same mathematical phenomenon from opposite directions. The EST work studies the continuum field solutions and their experimental consequences, while the VERSF framework investigates whether a deeper informational substrate could naturally produce those same structures when viewed at large scales.