Wheeler-DeWitt equation for holographic vacuum dynamics: Minisuperspace reduction and structural results

This paper derives a minisuperspace Wheeler–DeWitt equation for a two-field vacuum system — a radial Kähler modulus ρ and an angular axion phase θ — evolving in a multi-valley landscape potential, and shows that the system reduces from a three-variable partial differential equation to a one-variable ordinary differential equation plus an integer quantum number n. The reduction proceeds through two exact steps: U(1) factorisation of the angular phase, valid to corrections of order q ~ Λ/M_Pl^4 ~ 10⁻¹²², and Friedmann constraint elimination of the scale factor. Nine structural results follow from the reduction, independent of which specific field realises θ. The Mathieu parameter q linking the cosmological constant to the U(1) symmetry quality is identified as the best symmetry in known physics. The angular quantum number n provides a sharp selection rule for transient features in the cosmic expansion rate: n ≠ 0 if and only if a Hubble "mountain" exists. The correspondence principle identifies the classical Phase 2 dynamics as a coherent state with n ≫ 1, validating the semiclassical treatment of the DESI mountain feature near z = 1.32. The breathing mode σ = ln(V_K/V_K0) entering the reduced equation is identified with the field that produces the MOND equation through compact-dimension dynamics, establishing that the Wheeler–DeWitt equation quantises the MOND field. The angular field θ is identified with the conifold complex structure axion y_c of a flux-stabilised Calabi–Yau throat, with mass m_c ≈ 2.3 H₀ arising from flux integers (K, M) = (2, 1), W₀ = 0.009, V = 10⁵, and g_s = 0.1033. This identification resolves the Kreuzer–Skarke obstruction that prevents standard Kähler axion candidates from achieving the required instanton action. The Kähler–complex structure factorisation of the moduli space kinetic metric provides exponentially strong sector isolation (cross-coupling ratio ~10⁻⁷¹ in the warped-throat regime), ensuring that only G and ℏ_grav vary under compact-dimension compression, while c, α_EM, ℏ_EM, and particle masses are protected by Large Volume Scenario brane localisation. A UV completion candidate is identified as a double Swiss cheese Calabi–Yau with Hodge numbers (h¹¹, h²¹, χ) = (3, 165, −324): three Kähler moduli for the valley structure, a K3 fibre for inflation, and 165 complex structure moduli for the conifold axion. The thawing constraint m_c ≈ 2.3 H₀ implies a single DESI peak: Phase 2 is in its first half-oscillation and a second peak does not exist.