Substantial Space. Part III: Quantitative Verification, Soliton Mass Spectrum, Elastic Nature of Constants, and Cosmic Intersections

Abstract This paper presents a decisive stage in the mathematical formalization of the pointless space ontological basis previously proposed by the author in Part I. Within the framework of space density geometrodynamics, both gauge fields and gravitation are derived directly from the dynamics of a real scalar field of substance density Φ in a strictly 4D continuum. The long-standing conceptual conflict of scales is resolved: the Planck scale \ell_P is defined not as a discrete spatial quantum, but as a dimensional conversion constant ([L^{-2}]) that maps field values onto the metric tensor components, fully preserving the absolute continuity of the integration measure over Heyting locales. 

Based on the model's gauge relationships, exact quantitative estimates of the vacuum medium parameters are obtained for the first time: the dimensionless vacuum expectation value is topologically fixed at Φ₀ = 1/16π ≈ 0.01989, and the effective field theory (EFT) cutoff scale is mapped precisely onto the Great Unification scale (Λ ~ 10¹⁶ GeV). A non-linear anisotropic fourth-order self-interaction potential V(Φ, H₀) is formulated, translating the background primary helicity parameter H₀ to dynamically break the Φ → -Φ symmetry and provide a calculable mechanism for the baryon asymmetry of the Universe. 

By constructing the angular momentum operator via shift generators of the mereotopological boundaries of open domains, we strictly demonstrate a homomorphism between the lattice transformation group \mathbb{Z}_4 and the double covering of the SO(3) rotation group, establishing a topological origin for fermion spin 1/2. Finally, a dynamic wave equation for shear deformations is derived, quantitatively matching the transverse tensor modes registered by LIGO due to scattering on soliton density gradients, while the macroscopic stability of the Maxwellian electromagnetic equations is mathematically proven by the existence of a massive plasmon gap (M_p \ge 200 GeV) in the spectrum of longitudinal compression modes.
Keywords: geometrodynamics, mereotopology, Heyting algebra, scalar solitons, Lorentz gauge, no-ghost condition, LIGO tensor modes, non-singular cosmology, pointless space, space density, Victor Muravyov