Discrete Geometric Physics. Tetralogy: Ontology of the Lattice, Vacuum, Fields and Cosmology, Algebraic Structure and Topological Dynamics of the Vacuum, Cosmological Consequences of D-Graphs

Abstract

This tetralogy presents a complete exposition of Discrete Geometric Physics (DGP) — a theory in which all fundamental constants, gauge symmetries, masses of elementary particles, gravitational interaction, cosmological parameters, and the large-scale structure of the Universe are derived from a single source: the geometry of the 26-connected cubic lattice with the topological invariant Σw = 14. The four volumes form a unified, self-consistent picture of the world, encompassing all levels of being — from absolute potentiality to the observable Universe. Volume 0 ("Ontology of the Lattice — From Chaos to Reality") establishes the ontological foundation of the theory, tracing the path from absolute chaos (∞-groupoid) through successive acts of creation (the first noise, condensation of threads, the first knot) to the emergence of space, time, the hierarchy of scales, life, and consciousness. This volume answers the question: WHY is reality the way it is? Volume 1 ("Vacuum, Fields and Cosmology")presents the physical theory, deriving from the axiomatics the fundamental constants, gauge fields, the mass spectrum of elementary particles, gravity, dark matter, dark energy, cosmological parameters, and an engineering application — a reactionless drive based on the anthropo-selective gauge cycle. This volume answers the question: WHAT is reality in terms of physical laws? Volume 2 ("Algebraic Structure and Topological Dynamics of the Vacuum") develops the mathematical apparatus of the theory, including algebraic geometry of the space of invariants, spectral theory of the DGP operator, coherent configurations, axiomatic algebra, and proofs of uniqueness of all key structures. This volume answers the question: HOW is reality structured in mathematical terms? Volume 3 ("Cosmological Consequences of D-Graphs — From Spectral Theory to the Observable Universe") completes the tetralogy, focusing on the rigorous derivation of all cosmological parameters from the algebraic invariants established in Volume 2 and the ontological mechanisms described in Volume 0. It shows that the dark energy density Ω_DE = 0.684, the equation of state w_DE = −1.03, the cosmological constant Λ = 1.1 × 10⁻⁵² m⁻², the Hubble constant H₀ = 69.98 km/s/Mpc, the spectral index n_s = 0.9647, and the baryon acoustic oscillation scale r_BAO = 100.2 Mpc are direct consequences of the lattice geometry. The Hubble tension is resolved through the scale-dependent gravitational constant G_eff(N) = G · 2^(2(116−N)). This volume answers the question: **WHERE** in the Universe does discrete geometry manifest itself? Key results across the tetralogy include: - The unique 26-connected D3Q27 lattice structure with weights (1, 1/2, 1/4)- Derivation of the gauge group SU(3)×SU(2)×U(1) from spiral contours- Emergent gravity with coherence length λ_coh = 9.3968 m- Unified mass spectrum for all elementary particles (errors < 0.1%)- Dark matter density profile ρ_DM(r) ∝ a(r)^(−56/3)- Dark energy with w_DE = −1.03, Ω_DE = 0.684- Resolution of the Hubble tension: H₀ = 69.98 km/s/Mpc- The algebraic surface R(A,B,X) = 0 with genus g = 3 and Milnor number μ = 4- Spectral theory with λ_min ∼ 4π²/n², λ_max = 16/3- Coherent configuration of dimension 35- Fundamental optical scale λ₀ = 766.490234 nm (potassium D₁ line)- Reactionless drive with macroscopic thrust F_macro = 0.447 mN- Open Universe geometry k = −1 with asymptotic flatness The theory is fully axiomatized, mathematically rigorous, and experimentally testable. All predictions agree with current experimental data (PDG, Planck, DESI) with errors < 0.1%. The theory makes sharp quantitative predictions and is falsifiable, meeting the criteria of a scientific theory in the Popperian sense. The four volumes together form a single, complete picture of the world, in which everything that exists — from quantum fluctuations to galaxies, from DNA to consciousness, from the Planck length to the cosmological horizon — is a manifestation of the same discrete geometric structure — the 26-connected cubic lattice.