The Modified Second Law of Thermodynamics: Entropy Circulation and Memory Flux at Infinity

This paper introduces the Modified Second Law of Thermodynamics, an extension of the classical law that incorporates entropy flux through infinity. While local irreversibility (σ≥0\sigma \geq 0σ≥0) is preserved, a new term Ψ∞\Psi_\inftyΨ∞ accounts for entropy circulation across the infinite boundary, interpreted as a memory flux. This framework complements the previously proposed Modified First Law (Cromwell, 2025), where energy flux through infinity was recognized. Concrete examples demonstrate how the Modified Second Law applies to real physical systems: blackbody radiation (S˙=43L/T\dot S = \tfrac{4}{3}L/TS˙=34L/T), solar and terrestrial entropy export, and a gravitational-wave burst (GW150914). These cases show that entropy is not destroyed but continuously redistributed and recorded at infinity. The paper also anticipates critiques from thermodynamic purists, information theorists, cosmologists, experimentalists, and philosophers of physics, providing counterarguments that ground the new formulation in established physics while highlighting its novel insights. Together, the Modified First and Second Laws restore symmetry to conservation principles. Energy and entropy are never annihilated; they circulate through infinity, preserved locally and remembered globally.