Abstract:   We introduce an algebraic vector closure $\mathfrak M$ set within a complex vector space $\mathcal{S}\subset\mathbb{C}^3$ equipped with a non-Hermitian symmetric bilinear product: \[ \mathfrak M[\mathbf{v}, \mathbf{B}, \mathbf{E}] := \left\{ \mathbf{v}, \mathbf{B}, \mathbf{E} \in \mathcal{S} \;\middle|\; \frac{\mathbf{B} \cross \mathbf{E}}{\mathbf{B} \vdot \mathbf{B}} – \mathbf{v} = 0, \quad \frac{\mathbf{E} \cross \mathbf{v}}{\mathbf{v} \vdot \mathbf{v}} – \mathbf{B} = 0, \quad \mathbf{v} \cross \mathbf{B} – \mathbf{E} = 0 \right\}. \] Under a Maxwell-compatible transport law, this closure reconstructs the source-free Maxwell–Heaviside curl equations while leaving the transport speed $v$ as an initially free parameter. Consequently, $\mathfrak M$ acts as a structural constraint on the underlying d’Alembert wave operator $\Box = \nabla^2 – v^{-2}\partial_t^2$. While the standard d’Alembert equation permits arbitrary linear wave packets, the non-linear algebraic requirements of $\mathfrak M$ restrict allowable field configurations to continuous 3D spatial-phase rotations $R = R_{\rm x}R_{\rm y}R_{\rm z}$. This constraint unlocks non-trivial vortical and helical wave solutions across 1, 2, and 3 spatial dimensions—such as travelling vortices and helical centreline structures—that are unobtainable through d’Alembert’s equation alone.

The vacuum itself is classically quantised: an elementary voxel $l_o^3$ with axial length $l_o$ and an elementary time $t_o$ carries a quantised disturbance, whose electric displacement is not a response to an applied field but the primitive excitation agent generating the fields. Defining a quantised displacement as $D_{\rm E}:=e/l_o^2$ constrains a charge density and charge tensity, providing a propagation modulus $\mathcal T=l_o^2/t_o^2$ which autonomously establishes $c=\sqrt{\mathcal T}$ and thereby fixes the previously free transport speed in $\mathfrak M$ to $v=c$ before any constitutive parameter is derived. A parameterisation theorem then derives explicit expressions for the quantities $(\phi,\psi,\epsilon_0,\mu_0)$, and the elementary scales $l_o$ and $t_o$.

A second, structurally parallel construction introduces an action modulus $\mathcal{H}$ built from the same primitive charge variables to fix Planck’s constant $h$. Consequently, both fundamental constants of vacuum electrodynamics ($c$ and $h$) emerge not as independent empirical inputs, but as numerical calibrations of closure-native moduli within the algebraic framework.

Furthermore, the elementary time scale bounds the excitation spectrum: the upper bound eliminates the classical Rayleigh–Jeans divergence while aligning with ultra-high-energy photon limits from the Pierre Auger and Telescope Array observatories, and the lower bound yields an emergent, strictly positive electromagnetic mass gap $\Delta_{\mathrm{EM}}$, read as a vortex-nucleation threshold above an otherwise gapless excitation branch. These results establish $\mathfrak M$ as a fundamental framework for vacuum electrodynamics.

Key Words:   Vacuum, Quantising Space and Time,Vortical Solutions, EM Mass Gap,

Posted in:   1 Physical Vacuum, what is it?

Article Reference:   2128

Article Status: Preprint

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