Dynamical Universality from Critical Partition Functions

Critical Partition Function Theory (CPFT) treats a universal critical partition function not as a microscopic finite-scale identity but as a representative of a critical equivalence class after irrelevant corrections, regular backgrounds, and metric freedoms have been separated. In this paper we ask whether the differential dynamics extracted from such a representative possesses a further, nontrivial universality quotient. Starting from the weighted rank-one Hessian sector of CPFT, we deform the critical Hessian relation within the same weighted differential language. The undeformed face reproduces the normalized-Esscher finite-rank dynamics of CPFT III. A constant weighted Hessian deformation gives a variable-coefficient linear equation of Gauss-hypergeometric type, showing that finite differential-Hankel rank detects only constant-coefficient hidden linearity. The next affine weighted deformation reduces, after a hodograph transformation and a canonical Sundman time, to the autonomous third-order family $p'''+a p p''+b(p')^2+(c p^2+e)p'=0.$ Its scale quotient is described by two dimensionless parameters $(J_1,J_2)=(b/a,c/a^2)$ together with the nonzero lower-weight parameter $e$ up to time scaling. We derive three polynomial-integrability strata, including two regular CPFT divisors carrying degree-four and degree-six first integrals. Their distinguished single-valued points are the Chazy V and VI principal classes. The corresponding affine V and VI equations themselves belong to previously studied Chazy--Bureau/Painlev\'e-type families; this prior art is separated explicitly from the claims made here. For the constant-coefficient affine-VI specialization we give levelwise birational coordinates that reduce every spectrally regular first-integral surface, on a dense open set, to a diagonal linear flow on $(\C^*)^2$. The same spectral cubic canonically defines an auxiliary elliptic curve whose nonzero $2$-torsion abscissae are the three linear-flow eigenvalues; its discriminant is $16$ times the spectral discriminant, and for $e\neq0$ the selector ratio $\chi_{VI}=C/e^3$ is equivalent to its $j$-invariant. We then derive an all-order symmetric-power obstruction theorem: transverse homogeneous reconstruction is controlled by explicit resonance factors and one-coefficient compatibility functionals in eigen-coordinates. Finally, admissible quadratic phase curves of the affine family have Gauss-hypergeometric normal variational equations. Combining this with Kimura's solvability criterion and a direct local leading-transverse-term lemma yields a rigorous meromorphic non-integrability theorem whenever at least one admissible phase curve has non-solvable normal variational Galois group. An exact regular CPFT example with class coordinates $(J_1,J_2)=(3,1)$ has non-solvable normal variational Galois group and therefore admits no nonconstant meromorphic first integral near the selected phase curve. These results suggest that the differential output of a CPFT universality class is naturally organized by a dynamical universality signature containing relation ideals, extraction data, polynomial-integrability strata, spectral fibrations, singularity data, and variational-Galois obstructions. A stronger all-degree polynomial-rigidity statement is isolated as an open conjecture rather than used in the main theorems.