The Kaleidoscopic Filter on the Partition Manifold, Part I: The Simplicial Ehrhart, Barycentric Pascal Product, and Continuous Bernoulli Exact Closed Forms for $p(n)$

The partition function $p(n)$ has historically been constrained by recursive series and asymptotic approximations, coupling evaluation to sequential memory bottlenecks. Here we derive three exact, non-recursive closed-form identities for $p(n)$ and $p_k(n)$. Via the Stratified Simplicial Decompositi