The Kaleidoscopic Filter, Part I : A Structural Resolution of Restricted Integer Partitions

Abstract

The integer partition function $p(n)$ has historically been constrained by recursive series and asymptotic approximations, causing severe memory bottlenecks. We present a fundamental geometric resolution for the restricted partition function $p_k(n)$ via the Stratified Simplicial Decomposition of the Ehrhart partition polytope within the affine $A_{k-1}$ Weyl group. By establishing the Rational Structure Theorem, the Simplicial Resonance Algebra, and the cross-convolution operator $\mathcal{B}_k$, we prove that evaluating $p_k(n)$ collapses to a deterministic $\mathcal{O}_k(1)$ invariant. Rigorously bounding fractional boundary defects strictly below $0.5$, we bypass explosive recursions, establishing a global closed-form evaluation via generalized Bernoulli polynomials and a nearest-integer rounding operator $\lfloor \cdot \rceil$. This framework extends to unrestricted $p(n)$ via Sylvester-Ramanujan waves and a Durfee-Ehrhart formulation. Furthermore, it unifies additive number theory: trivializing Euler's distinct-odd identity, aligning with MacMahon's $Ω$-calculus, and modeling Dyson's Rank. Crucially, we introduce Kaleidoscopic Filter Theorem: applying Weyl reflection coefficients of dimension $k$ to $p(n)$ cancels all lower-dimensional geometry, yielding exactly the number of partitions of $n$ formed exclusively by parts strictly greater than $k$. Finally, we establish the Prime Resonance Theorem, proving the collapse of Ramanujan sums to the Möbius function for prime masses. We reveal the Mock Modular genesis within the cyclotomic tail via Indefinite Theta Toric Fibrations, healing rational vertex defects via generalized Dedekind-Rademacher sums. Mapping these exact polyhedral volumes into a Toeplitz-Hessenberg matrix, we derive a novel, non-recursive geometric closed form for the prime-counting function $π(x)$.