Critical Partition Function Theory

We formulate a critical-partition-function theory in which a generalized partition function and the differential geometry of its logarithm are primary, while spatial dimension, symmetry, and interaction structure are treated, when identifiable, as quantities to be reconstructed from critical data. In a homogeneous two-variable critical sector, a rank-one condition on the leading Hessian yields a nonlinear differential equation for the scaling function. On the stated nondegenerate rank-one branch, removal of one vertical amplitude and one horizontal scale leaves a complete normalized scaling curve fixed by two scaling exponents. In the thermal/magnetic specialization these exponents reduce to conventional critical exponents; the same chart defines a critical-derived dimension coordinate that coincides with spatial dimension under ordinary hyperscaling and returns $4$ for mean-field Ising data. Separately, we encode a normalized critical spacing measure by its Laplace transform $L_c(a)$ and prove an exact reconstruction theorem. Writing $V=(\log L_c)''$, the running shape $\vartheta=(V/V')'$, together with two low-order initial data, determines the full transform and hence the measure whenever the Laplace representation is unique. The constant-shape sector is solved explicitly and, under exponential tilting, coincides with the classical natural-exponential-family power-variance classification. More generally, every regular lower edge $P_c(s)\sim Cs^\gamma$ forces $V(a)\sim(\gamma+1)a^{-2}$ and $\vartheta(a)\to-1/2$, providing an exact edge-to-curvature relation. We compare these results with scaling theory, RG, CFT, nonlinear-$\sigma$ models, Anderson critical statistics, random-matrix theory, fluctuation geometry, and arithmetic statistical mechanics. Finally, we prove a fixed-section non-identifiability result: critical differential data at each spacing-deformation section do not determine the inter-section normalization required for the exact spacing curvature. A microscopic spacing deformation supplies that normalization exactly, isolating the remaining critical-first closure problem as an inter-section transport problem rather than a fitting ambiguity.