Toward an Emergent Electrodynamic-to-Gravitational Hierarchy (〖10〗^39) from Trefoil-Knot Anapole Casimir Interaction in the Zero-Point Field

The multi-order-of-magnitude discrepancy between the electrostatic force and the Newtonian gravitational force in fundamental particle pairs ( for proton-electron; for proton-proton) remains an unexplained empirical boundary condition in modern physics [4, 5]. Historical stochastic electrodynamic (SED) models—most notably by Sakharov [1], Puthoff [2], and Haisch-Rueda [8, 9]—successfully derived an attractive inverse-square () Casimir radiation-pressure deficit from the Zero-Point Field (ZPF) [6, 7], but relied on a circular Planck-frequency cutoff () that embeds Newton's into its own derivation. This paper presents a non-circular framework for Newton's gravitational constant and the fundamental force hierarchies from first principles by analyzing an extended electromagnetic soliton configured as a self-confined trefoil torus knot () of circulating displacement current [12, 14]. The paper establishes a rigorous distinction between analytically derived theorems and well-motivated candidate hypotheses. Analytically Derived Results: 1. The primary topological linking number () yields the quantized Coulomb monopole force (). 2. Fourier harmonic orthogonality of the knot trajectory causes the electric dipole, magnetic dipole, and off-diagonal quadrupole moments to vanish identically, leaving vacuum radiation coupling dominated by the toroidal anapole moment [15, 16, 17]. 3. A dual-cycle topological scale cascade across the toroidal and poloidal windings fixes the flux-tube aspect ratio to , analytically suppressing vacuum energy coupling by the anapole cavity filtering factor per soliton. 4. Because the extended soliton maintains an active, persistent displacement current immersed in an omnidirectional ZPF energy flux, steady-state energy-momentum conservation () guarantees an exact geometric inverse-square () force law across macroscopic distances, evading the retardation decay of dead-vacuum dispersion models. 5. In an optically thin medium (mean free path , governed by the elastic scattering cross-section ), extensive summation over constituent nucleons naturally yields Newton's mass proportionality () and preserves the Weak Equivalence Principle [18]. 6. An incoherent ballistic ray-tracing calculation provides an exact analytical lower bound of , which underestimates empirical by a factor of due to plane-wave phase-space dilution and the breakdown of geometric optics in sub-wavelength regimes. Candidate Physical Hypotheses: 1. A two-stage momentum transfer combining shadow production at Soliton 1 () with shadow interception at Soliton 2 () establishes the candidate gauge coupling , which yields the universal coupling power . 2. In the coherent topological limit, the torus knot winding density () and transverse field polarization projection () establish the candidate prefactor . Evaluating with these candidate values yields , matching CODATA within , and establishes the two-proton ratio . For the proton-electron system, bandwidth truncation by the electron Compton frequency scales the interaction by , yielding . The neutron is resolved as an unknotted twisted electron loop coaxially nested within the central aperture of the trefoil knot [13], neutralizing the external electric monopole while preserving the bulk Casimir interaction. We identify direct numerical evaluation via 3D finite-difference time-domain (FDTD) simulations as the definitive test to compute the non-linear coupling factor and coherent enhancement factor from first principles.