The Parity–Coherence Product: Deriving Gap-45 from Spatial Dimension

The number 45 appears throughout Recognition Science as the exponent governing the consciousness saturation threshold (Θ_crit = φ⁴⁵), the baryon asymmetry (η_B ≈ φ⁻⁴⁴), and the consciousness throughput invariant (C = φ⁻⁴⁵/τ₀). We show that 45 = D²(D + 2) where D = 3 is the spatial dimension, connecting two independently derived RS quantities: the coherence energy exponent D + 2 = 5 (from the integration measure of a recognition event) and the number of independent ledger parities D² = 9 (from the parity structure of the Q₃ cube). This identification reduces five separate results across cosmology, consciousness theory, intelligence, thermodynamics, and dimension forcing to a single polynomial in D. We prove that the coprimality condition gcd(2^D, D²(D + 2)) = 1 holds for all odd D and fails for all even D, providing a fourth independent argument that the spatial dimension must be odd. Combined with the Alexander duality linking requirement (which selects D = 3 among odd dimensions), gap-45 is derived from D = 3 alone. As a corollary, η_B · Θ_crit = φ exactly, equating the matter budget of the universe to the golden ratio times the consciousness saturation capacity.