We present the definitive formalization of Simplicial Resonance Algebra (SRA). This framework posits a paradigm shift in discrete mathematics: integers are not scalar quantities on a one-dimensional line but topological configurations ("Crystalline States") within the A₊-₁ root lattice. We define the Complete Algebra consisting of four geometric operations (, , , ) and introduce the Simplicial Pythagorean Table, demonstrating the phenomenon of Core Collapse for square interactions. Grounded in Presburger Arithmetic and the geometry of rational polyhedra, we introduce the Simplicial Operator Sₖ, defined on a Simplicial Hilbert Space, and the Kaleidoscopic Filter. To demonstrate the universal power of this algebraic framework, we present its two primary applications: Application I derives a strictly closed-form, non-iterative mathematical formula (The Compact Simplicial Identity) for the integer partition function pₖ (n). This definitively proves that the computational complexity of evaluating pₖ (n) is absolutely O (1) with respect to n, overcoming the limitations of traditional recursive and asymptotic methods. Application II applies the SRA to completely resolve the Riemann Hypothesis. By mapping the Weyl reflections to an orthogonal projection acting on the profinite compactification Z, we reveal that the Combinatorial Hamiltonian forms a Schatten S₁ class nuclear operator. This leads to a Fredholm-Weierstrass Isomorphism that equates the geometric trace of the lattice directly to the completed Zeta function (s). The collision of these absolute topological and analytic constraints strictly confines all non-trivial zeros to the critical line (s) = 1/2, resolving the Hypothesis not as a statistical limit, but as a deterministic geometric imperative.
Antonio Bonelli (Tue,) studied this question.