Critical Partition Function Theory II

Abstract

We formulate a reconstruction theory for universal critical partition-function representatives in the presence of irrelevant corrections. The central object is the leading critical component of the logarithmic partition function, $K_c=\log Z_c$, defined through a weighted scaling filtration rather than through equality of microscopic finite-scale partition functions. We introduce a jet-stable filtration for which differentiation has a controlled scaling degree and prove a principal-relation descent theorem: a weighted-homogeneous differential-polynomial relation that holds to subleading order for every representative of a critical class descends to an exact differential relation for $K_c$. This provides a rigorous route from finite-scale critical relations to class-defining equations. We distinguish observed, reconstructive, and canonical universality signatures. A reconstructive signature consists of a finite presentation of linear and nonlinear differential relations together with branch and metric-normalization data; a canonical signature is instead extracted from a known representative inside a relation language fixed independently of that representative. For a parameterized normalized system $\cA(K,\theta)=0$, we define the fixed-class reconstruction defect $\delta=\dim\ker D_K\cA$. Under standard Banach-space regularity assumptions, the local solution manifold has dimension $\dim\Theta+\delta$. Critical observables reduce this dimension by their tangent rank, yielding a finite critical-data identification theorem. The theory is completed by a two-variable rank-one Hessian sector. The equations $(W-\rho)K_c=0$ and $\det\nabla^2K_c=0$ reduce to a second-order nonlinear equation for a scaling function $\Phi$. On the regular normalized branch, the fixed-class defect is zero. We derive a self-contained parametric solution, an all-orders formula for metric-invariant jet coordinates $I_n$, and show that $(I_2,I_3)$ locally identify the two-parameter universality family. Globally, the normalized shape possesses a discrete duality $(\rho,\sigma)\mapsto(1-\rho,1-\sigma)$. Thus a finite set of low-order critical invariants determines an entire normalized analytic scaling-function germ on the regular rank-one stratum. To the best of our knowledge, the universality-specific descent through irrelevant corrections, the resulting two-invariant finite reconstruction of this weighted rank-one critical family, and the associated exponent/parameter-inversion duality have not been reported previously in this form.