The Inequality That Became an Equality: Clausius's Postulate and the Entropy It Defined into Existence

Between 1850 and 1865 Rudolf Clausius founded thermodynamics as a quantitative science — and its founding quantity, entropy, enters the record in an unusual way: it is not found and measured but defined into existence. The second law is laid down as a postulate, in two equivalent wordings; the postulate, together with the reversibility idealization, guarantees that \oint \delta Q_{\mathrm{rev}}/T = 0; and a quantity whose cycle integral vanishes is a state function, so S exists, with dS = \delta Q_{\mathrm{rev}}/T. This paper evaluates the founding step on declared cycles: the defined entropy closes around every declared cycle at exactly 0.000000 J/K — closure being the definition auditing itself — while the postulate's entire empirical content, the uncompensated transformation N = -\oint \delta Q/T, returns 0.000000 on the reversible cycle and strictly positive values off it (0.640350 J/K at a declared finite-difference transfer, 3.371224 J/K on a declared free-expansion cycle): a sign, holding exactly where the idealization fails. The Carnot returns, the bookkeeping of heat, and entropy's working reign across three sciences are the papers' achievements and stand entire. What is counted here is the derivation status of their foundation: 1 law-postulate carrying 1 defined state function, 0 derivations in 1850/1865 of either. Census headline, unchanged: NOT SOLVED / PARTIAL · producing-count 0.