A categorical sketch for viability: Formalising the structural invariants of self-organising systems

We work in Set and define a P-system as a state object S with a self-production endomorphism φ:S→S, a functional membrane ∂S↪S, and a flux action ρ:R×S→S. The endomorphism is evaluation-based: φ reads the currently installed metabolism h through ev(h,a)=h(a), so the active metabolism is part of the state. Endogeneity is stated as an E-relative non-factorisation condition through a class K(S,E) of authorised mediations, with a separation lemma as the verification tool. The framework is worked out on an explicit finite Rosen-type model: 16 states, φ tabulated, membrane stability and permeability checked element by element, E-endogeneity verified by exhaustion, and a multi-level counterexample T=S×{0,1} in which the same subsystem admits one admissible embedding and one not. The formal object is a restricted generalised sketch with an avoidance clause, not an ordinary finite-limit sketch. Section 5 places the formalisation in dialogue with recent works on molecular autopoiesis, causation and codes, and empirical correlates of (M,R)-systems.