"The Primorial Lattice ℤ₂₁₀ as the Minimal Spectral Foundation of the Standard Model: A Uniqueness Theorem with Physical Consequences

We present the Spectral Completeness Theorem: among all positive integers N, the primorial N = 210 = 2×3×5×7 is the unique minimal N that is squarefree, has φ(N) = 48 coprime residues admitting a full positive–negative spectral duality, and whose prime factorisation contains exactly the primes needed to generate the gauge structure of the Standard Model. We prove this theorem by exhaustive characterisation and show that no smaller N satisfies all four axioms simultaneously. From the primorial lattice ℤ₂₁₀ = ℤ₂×ℤ₃×ℤ₅×ℤ₇ we derive, without free parameters: the atomic orbital shell capacities 2, 6, 10, 14; the period lengths 2, 8, 18, 32 of the periodic table; spectral shell energy weights σ_l = sin²(π/p_l); and the cosmological dark energy fraction Ω_Λ = 24/35 ≈ 0.6857, matching Planck 2018 data to 0.1%. Submitted to Foundations of Physics, May 2026.Change 4 — Keywords. Add: primorial lattice, Standard Model, spectral completeness, unified theory, atomic structure, cosmological constant, ℤ₂₁₀, DUST