Geometric Completion of Thermodynamic Response: From the (T)-(V)-(S)-(P) Compass to Recursive $\lambda$-Jacobian Geometry

Abstract

We develop a rigorous geometric framework for thermodynamic response in a simple compressible system and connect it to a target-faithful Recognition Kernel construction. Starting from a smooth fundamental relation U(S,V), we derive the Maxwell relations, six constraint-dependent response coordinates, their product identities, and the closure C_P/C_V = K_S/K_T. For any four nondimensional response channels, the pairwise area brackets form an intrinsic skew 4 x 4 matrix of rank at most two satisfying an exact Plucker relation, with a noise-stable falsification bound. We then construct the Recognition bundle E_RK = ker A/(ker A intersect ker Pi) from pulled-back thermodynamic response jets, define the response-generated field Phi_th, and prove that compatible transport descends to a connection whose curvature is the infinitesimal obstruction to path-independent target reconstruction. A sharp decoder theorem gives source domination and excludes target-relevant blind modes. A finite Cut-Flow-Jet theorem, together with an analytic infinite-jet extension, controls recursive response transport and cut grading. Within a declared local, parity-even, diffeomorphism- and gauge-invariant low-derivative class, we classify the minimal action and derive the corresponding metric, gauge, and Recognition-field equations. We also establish exact finite-regulator quantum identities, a regulator-independent Gaussian continuum theory for positive physical Hessians, reflection positivity, unitary reconstruction, bosonic anomaly freedom, and stable metric-connection identifiability from propagation, scale, and transport data. The full nonlinear interacting continuum theory and empirical validation remain open.