"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 = (K₈) = 12), the characteristic classes (Todd, A, L, ᵧ), 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 (, z) whose central Fourier coefficients equal the odd divisor function d₎₃₃ (n), with Dirichlet series (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 (₀ (36), ) with the unique odd primitive Dirichlet character mod 3 (§5) —that we identify with the dark sector partition function. The level lift from the base ₀ (4) (forced by the Z₂ involution on T²/Z₂) to the twisted ₀ (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: ₃₌b = 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 ₂ = 1. 86. Five arithmetic selection rules governing the Kaluza–Klein excitation spectrum are derived from the multiplicative structure of d₎₃₃ (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, 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 (K₈) = 12 and zero continuous free parameters.