Counterfactual Entropy Resistance as a Necessary Thermodynamic Condition for Active Living States

We propose a necessary thermodynamic condition for active living states: sustained counterfactual resistance to entropy growth on a pre-registered system boundary and observable manifold. Let H (t) be the coarse-grained Shannon entropy, in bits, of operationally specified observables. The corrected resistance rate is Rᵣes (t) = (dH/dt) ₚassive − (dH/dt) ₐctual. The earlier raw rate Rᵣaw (t) = −dH/dt is retained as a useful special case for growth, repair, recovery, ordering, and error-correction episodes, but it is not a general criterion for life: homeostatic non-equilibrium steady states normally have dH/dt ≈ 0 while remaining actively maintained. Under Markovian dynamics with local detailed balance, the entropy balance gives dH/dt = ΠH + JH, where ΠH ≥ 0 is internal entropy production and JH is entropy flux into the system, both in bits per unit time. For finite-time raw entropy reduction, the minimum work cost is ⟨W⟩ ≥ Δ⟨E⟩ + kB T ln2 ∫ Rᵣaw (t) dt = ΔFₙeq, with Fₙeq = ⟨E⟩ − kB T ln2 · H. Passive detailed-balance relaxation monotonically decreases the relative entropy D (pₜ‖pₑq) and the non-equilibrium free energy; Shannon entropy itself is not generally monotonic. This is why the life-relevant condition must be counterfactual rather than raw. Beneath this macroscopic criterion we supply a certified mechanism. In a bipartite split of an active system into a regulated subsystem and its controller, the stochastic thermodynamics of continuous information flow makes sustained negative apparent entropy production, σₐppX = ΠX + İX < 0, a necessary signature of active ordering — bounded below by the information flow the controller supplies and unattainable for a passive, uncontrolled reference. A minimal autonomous model realises this signature at a genuine non-equilibrium steady state, holding the regulated subsystem ordered while total entropy production remains strictly positive. The framework is description-relative, falsifiable, and intended as a Level-1 necessary condition rather than a complete definition of life. Computational examples are interpreted as schematic consistency checks and negative controls, not empirical validation. Direct measurement of Rᵣes (t) in biological systems remains an open experimental task.