Geometric Theory of Atomic Structure and the Periodic Table of Elements in Discrete Geometric Physics: Derivation of the Mendeleev Table from the 26-Vertex Cubic Lattice

Modern atomic physics and chemistry describe the structure of atoms using quantum mechanics, but many fundamental questions remain unanswered: why are electron shells filled exactly according to the pattern 2, 8, 18, 32? Why do ionization energies, atomic radii, and electronegativities vary with nuclear charge in precisely the way they do? The Standard Model postulates the periodic law but does not derive it from deeper principles. In Discrete Geometric Physics (DGP), space is a 26-vertex cubic lattice with the topological invariant Σw = 14. All physical objects are solitons of this lattice. In this work, we show that the periodic system of elements and all atomic properties are direct consequences of the lattice geometry. All results are obtained from geometric invariants (Σw = 14, θ = arcsin(1/√3), φ = (1+√5)/2), and the numerical values of parameters such as F_top, η_crit, γ, ℓ_Pl, as well as the normalization constants for atomic properties (I₀, r₀, χ₀) are determined by solving variational minimization equations on the lattice and are not adjusted to fit experimental data. We present a complete table for all 118 elements, calculated using formulas derived from geometry, and compare them with experimental data. The theory uses the variational principle to derive dependencies, rather than empirical fitting coefficients. The only deviation for hydrogen (4.4% in the analytical formula) is eliminated by numerically solving the discrete Schrödinger equation on the lattice, confirming the correctness of the approach.