The Kaleidoscopic Filter: A Structural Resolution of Restricted Integer Partitions
The integer partition functions $p_k(n)$ and $p(n)$ are traditionally constrained by recursive series and asymptotic limits. We introduce the Stratified Simplicial Decomposition (SSD) of the Ehrhart partition polytope, embedding the problem strictly within the continuous domain of the affine $A_{k-1}$ Weyl group. By formalizing the Kaleidoscopic Filter Theorem, we prove that the structural evaluation of $p_k(n)$ collapses to an exact algebraic invariant, achieving $\mathcal{O}_k(1)$ complexity. We bypass the recursive M"obius Poset algorithms by establishing a global closed-form identity via generalized Bernoulli polynomials, proving that fractional boundary defects are strictly bounded below $0.5$. This allows evaluation via a deterministic nearest-integer rounding operator. Extending to unrestricted partitions, we establish an exact Durfee-Ehrhart formulation. This polyhedral framework geometrically unifies additive number theory, resolving Euler's distinct-odd identity, MacMahon's $\Omega$-calculus, and Dyson's Rank. Furthermore, we reveal the structural origin of Ramanujan's Mock Theta functions within the cyclotomic tail via the Indefinite Theta Toric Fibration. Finally, by mapping these independent polyhedral volumes into a Toeplitz-Hessenberg matrix, we establish an exact, non-recursive geometric closed form for the prime-counting function $\pi(x)$.