The Folding Game: Protein Structure as Nash Equilibrium in the Latent Algebra

We establish an exact correspondence between protein folding dynamics and strategic game theory through the Latent algebra's grade decomposition. Under the Nash-Boltzmann duality, amino acid residues become players, dihedral angles become strategies, and the conformational energy function becomes the (negative) payoff. The Latent Number \(\rho\) — measuring the exponential decay rate of interaction grades — determines both the computational complexity of the resulting game and the thermodynamic stability of the native fold. We prove five structural results: (i) Levinthal's paradox resolves because proteins play grade-2 dominated games (\(\rho \gg 1\)), which admit polynomial-time approximate Nash equilibria; (ii) thermal denaturation corresponds to \(\rho\) crossing 1, a Nash equilibrium bifurcation from a unique pure equilibrium to a mixed ensemble; (iii) protein sequence design maps exactly to mechanism design, with stability \(\Delta G_{\text{fold}}\) playing the role of auction revenue; (iv) misfolding diseases correspond to \(\rho \to 1\), placing the folding game at the PPAD-hardness boundary; (v) molecular chaperones act as game moderators that increase \(\rho\), with quantifiable efficiency bounds. All results are formalized in the proof kernel (20 theorems, 0 sorry) and unify two previously independent kernel modules across game theory and structural biology.