A Free-Energy Upper-Bound Criterion for Entropy-Active Element Counts in High-Entropy Materials

High-entropy alloys and compounds are usually introduced through a lower configurational-entropy threshold, but their largest stable element count is not a universal integer. We formulate the upper-bound problem as a free-energy comparison between the configurational entropy of a declared mixed sublattice and the nonconfigurational penalty required to maintain a disordered target phase against the lowest composition-conserving competitor. The relevant count is the entropy-active integer support on the mixed sublattice, not simply the number of element names in the formula. Local chemistry is represented by a bonding-compatibility manifold, where mismatch is measured as a Christoffel-corrected covariant displacement and projected through phase-preserving constraints into retained soft channels. In the single-sublattice metallic limit, the covariant penalty reduces to a scalar metallic stiffness multiplied by the actual variance of the selected elements along a metallic compatibility coordinate. Under the controlled uniform-path envelope, the ideal entropy gain grows as lnN, whereas the mismatch variance grows as N2, allowing an upper crossing to emerge without imposing a universal chemical integer. The reported upper count is nevertheless obtained only by direct integer back-checking of the actual selected subset, residual penalty, and composition-conserving competitor envelope. The same variational principle extends to constrained compounds after mixed-sublattice normalization and explicit bonding, charge, defect, and competitor constraints. The framework therefore predicts how the admissible entropy-active count changes with temperature, bonding stiffness, selected-subset spread, mixed-sublattice fraction, and constraint rank.