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Thermodynamics as limiting behaviors of statistics is generalized toarbitrary system with probability {\it a priori} where thermodynamicinfinite-size limit is replaced by multiple-measurement limit. A dualitysymmetry between Massieu's and Gibbs' entropy arises in the limit of infinitelyrepeated observations, yielding the Gibbs equation and Hill-Gibbs-Duhemequation (HGDE) as dual pair. If a system has thermodynamic limit satisfyingCallen's postulate, entropy being an Eulerian function, the symmetry is lost:the HGDE reduces to the Gibbs-Duhem equation. This theory provides ade-mechanized foundation for classical and nanothermodynamics and offers aframework for distilling emergence from large data, free from underlyingdetails.

Emergence and Breaking of Duality Symmetry in Generalized Fundamental Thermodynamic Relations