The Octahedral Proton Mass Partition, Charge Cavity, and the √2 Invariant in VEQF

The Vacuum Energy Quanta Field (VEQF) framework models the vacuum as a physical, structured, elastic medium of bonded quanta. In this ontology, mass is not a substance but a measure of a topological soliton's resistance to agitation (m = Kf, where K is the vacuum impedance), and electric charge is not an intrinsic point property but a dynamically maintained strain feature of the vacuum medium. This paper presents a closed, parameter-free mechanical model of the proton, anchored by the empirical observation (Wang, 2025) that the bound-state down-to-up quark mass ratio is exactly √2. The proton is modeled as a tetrahedral flux network—comprising a Central Flux Junction and three quark poloidal pumps—embedded in an octahedral VEQ coordination shell. We establish three unified mechanical definitions for the nucleon's primary properties: MASS (Scalar Tension): Because the proton assembly tumbles, the effective coupling of each quark's pump to the ambient energy-density gradient is a time-averaged projection. The exact 45° geometry of the octahedral shell forces the tumbling-averaged coupling ratio between the down and up quarks to be √2. Partitioning the proton's total agitation budget by these geometric weights yields absolute bound-state quark masses governed by the silver ratio (δ_S = 1 + √2). These parameter-free masses sum identically to the macroscopic proton mass, reproducing Wang's √2 ratio exactly. CHARGE (Pseudoscalar Vorticity): Electric charge is defined as the vorticity moment of the continuous flow field generated by the quark pumps. The fast-moving flux creates a central, VEQ-hungry strain cavity via the aspirator effect. Charge is the pseudoscalar contraction of this cavity's vorticity with the junction's axis. The UUD geometry of the proton forces a non-zero net vorticity (yielding a +1 spin-gated charge that attracts electrons). Conversely, the UDD geometry of the neutron, constrained by the exact same √2 octahedral radial ratio, forces the vorticity vectors to cancel identically (yielding 0 charge). MAGNETIC MOMENT (Quadratic Circulation): The magnetic dipole moment is derived as a distinct, quadratic functional of the exact same flow field (weighting by cross-sectional area rather than channel length). This mathematical separation explains how the neutron possesses "circulation without chirality"—retaining a net magnetic moment (μ_n ≠ 0) despite having exactly zero net charge (Q_n = 0). Ultimately, the √2 invariant unifies the nucleon sectors, emerging simultaneously as an angular projection ratio (mass), a radial path-length condition (charge neutrality), and a native cavity eigenvalue (resonance). The document includes complete analytical proofs and numerical verification scripts.