Processus-Algebra: Numbers as Dynamical Systems

We introduce a novel algebraic structure called processus-algebra in which each element simultaneously encodes a value, a dynamics, and an attractor. A processus is a triple \((v, f, a)\) where \(v\) is the current value, \(f = dv/dt\) is the flow, and \(a\) is the target attractor. Multiplication in this algebra incorporates the Leibniz product rule at the structural level: \(\text{flow}(p \cdot q) = \text{val}(p) \cdot \text{flow}(q) + \text{val}(q) \cdot \text{flow}(p)\). We prove that this structure admits a natural Lyapunov function and that the flow always contracts toward the attractor. The main results establish superposition of flows, equilibrium simplification of products, gap additivity, and monotonic decrease of the Lyapunov functional. All 10 theorems are formally verified in the Platonic proof system with 14 explicit hypotheses and 5 type axioms.

Keywords: processus-algebra, dynamical systems, Lyapunov stability, arithmetic structures, attractor convergence

MSC 2020: 37C75 (Stability theory), 46H99 (Topological algebras), 12H05 (Differential algebra)