The Simplicial Geometry of Integer Partitions: An Exact $O(1)$ Formula via $A_{k-1}$ Root Systems

Abstract

We present a structural resolution to the exact evaluation of the partition function $p_k(n)$, systematically overcoming the limitations of traditional recursive and asymptotic methods. By framing the partition polytope $\mathcal{P}_{n,k}$ within the theory of rational polytopes and Ehrhart foliation, we prove that its discrete volume is exactly captured by a geometric Simplicial Spectral Decomposition. We establish the Rational Structure Theorem, demonstrating that the generating function of the spectral weights is a proper rational function defined rigorously over cyclotomic fields. Through partial fraction decomposition over complex roots of unity, we derive a strictly closed-form, non-iterative mathematical formula (The Compact Bonelli Identity). This rigorously proves that the strict arithmetic complexity of evaluating $p_k(n)$ is identically $O(1)$ with respect to $n$. Furthermore, we formally address the spatial complexity bottleneck for astronomically large $k$ by introducing a spatial memory reduction theory via Sylvester-Ramanujan waves, reducing the memory footprint to strictly $O(k)$. We additionally extend this structural framework to the unrestricted partition function $p(n)$, deriving an exact $O(\sqrt{n})$ closed form via Durfee squares, and establish its asymptotic limit as the geometric foundation of Euler's Pentagonal Number Theorem. Finally, we explicitly translate this additive framework into multiplicative number theory, establishing a geometric extraction of the divisor function $\sigma(n)$ and an exact non-recursive polyhedral closed form for the prime-counting function $\pi(x)$. In doing so, we formally identify a unique geometric basis of Ehrhart quasi-polynomials bridging continuous polyhedra and discrete arithmetic.