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 $p_k(n)$, addressing the limitations of traditional recursive and asymptotic methods. By introducing the Simplicial Successive Decomposition (SSD) framework, we demonstrate that the partition polytope $\mathcal{P}<em>{n,k}$ is not an arbitrary geometric object, but admits a rigid minimal unimodular triangulation into exactly $N_k = \binom{k}{2}$ simplices. This cardinality is determined by the positive root system of the $A</em>{k-1}$ Weyl chamber.We decompose Euler's generating function into a finite sum of simplicial rational transforms. By applying Brion's localization theorem and the negative binomial expansion, we derive an exact closed-form formula with $O(1)$ computational complexity. The validity of the model is confirmed through Ehrhart-Macdonald reciprocity, ensuring accuracy in the "Core Collapse" regime where the polytope's interior is empty and continuous volume approximations are inapplicable.