"The Mock Modular Partition Function of K₈: Ramanujan's Mock Theta Functions and the Standard Model." Paper XI in MEF Series.

We compute the Spinᶜ-twisted partition function of the manifold K₈ = [(CP² × S²) ×w (T²/Z₂)]Spinᶜ central to the Master Equation Framework (MEF), and demonstrate that it is a mock Jacobi form of weight 0, level 2, and mock theta order 2. The computation proceeds through the complete Chern data of K₈ (five Chern numbers with c₄ = χ(K₈) = 12), the characteristic classes (Todd, Â, L, χy), and the elliptic genus, in which the Spinᶜ twist by L = O(3) breaks Liu rigidity by accessing the lower cohomological components of the CP² integrand. The resulting q-expansion is governed by an Appell–Lerch sum μ(τ, z) whose central Fourier coefficients equal the odd divisor function dodd(n), with Dirichlet series ζ(s)²(1 - 2⁻ˢ). The mock modular structure produces a Zwegers shadow—a weight-3/2 unary theta function whose Spinᶜ-induced modular residence is S3/2(Γ₀(36), χ) with χ the unique odd primitive Dirichlet character mod 3 (§5)—that we identify with the dark sector partition function. The level lift from the base Γ₀(4) (forced by the Z₂ involution on T²/Z₂) to the twisted Γ₀(36) is a direct consequence of the c₁(L) = 3h twist via the Serre–Stark level formula [6]. The squared amplitude ratio of dark to visible phase sums yields the dark matter ratio as an exact algebraic number: ΩDMΩb = 3 + √6 = 5.4495, in agreement with the Planck 2018 value [5] 5.36 ± 0.05 to 1.67%. The Standard Model parameters reside in the q⁰ (topological) sector of the partition function, with spectral corrections entering at O(q) ~ O(10⁻⁵) at the physical modular parameter τ₂ = 1.86. Five arithmetic selection rules governing the Kaluza–Klein excitation spectrum are derived from the multiplicative structure of dodd(n), replacing classical Ramanujan-type congruences with an orbifold tensor product factorisation. The number-theoretic hierarchy of MEF physical constants—topological integers, algebraic numbers in Q(√6), transcendental products of integers and π, multiplicative arithmetic functions, and Rademacher-type series—is established in full. Thirteen new predictions are catalogued, including the normal neutrino mass hierarchy, absolute proton stability, a null prediction for LISA electroweak-era gravitational waves, and the \2,3\ arithmetic constraint excluding all BSM models whose topological data require primes p ≥ 5. All results derive from the topology of K₈ with χ(K₈) = 12 and zero continuous free parameters.