Critical Partition Function Theory II

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, Kc=log Zc, 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 Kc. 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 (K,θ)=0, we define the fixed-class reconstruction defect δ= DK. Under standard Banach-space regularity assumptions, the local solution manifold has dimension Θ+δ. 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-ρ)Kc=0 and ∇²Kc=0 reduce to a second-order nonlinear equation for a scaling function Φ. 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 Iₙ, and show that (I₂,I₃) locally identify the two-parameter universality family. Globally, the normalized shape possesses a discrete duality (ρ,σ)↦(1-ρ,1-σ). 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.