Strict Entropy Decrease of Clausius Entropy in an Isolated System with Energy-Form Conversion: Theoretical Proof, Numerical Illustration, and Critical Examination

This paper is accountable only to explicitly stated physical assumptions and strict logical inference. Its goal is to run a rigorous stress test of second-law claims within the Clausius framework. We work directly with \textbf{Clausius's entropy definition} for an isolated composite with energy-form conversion. Heat is withdrawn from a cold releasing subsystem with relatively small heat capacity, converted to electrical energy, and then delivered as heat to a hotter subsystem. In the ideal limit, the electrical leg contributes negligibly to Clausius entropy accounting, so the modeled reservoir Clausius sum is $$\Delta S_{\mathrm{Cl}} = Q\!\left(\frac{1}{T_B}-\frac{1}{T_A}\right) < 0.$$ The paper provides a derivation, numerical illustrations, and a scope analysis; any claimed contradiction should be interpreted as a compatibility issue between different axiom sets, not as an algebraic error in the Clausius bookkeeping above.