Critical Partition Function Theory VII

We develop an algebraic and operator formulation of finite-size dynamics in Critical Partition Function Theory (CPFT). The point of departure is a finite-size family of critical probability measures Pc(s;L) rather than a prescribed correction-to-scaling ansatz. In a regular finite-support sector, the moment Hankel data reconstruct the support polynomial Q and the finite quotient algebra Aₛₚ=[z]/(Q). We prove that every normalized first-order deformation of an n-point measure is represented uniquely by a pair (U,V)∈ Aₛₚ⊕φ, where U transports support and V transports weights. This yields a natural connection on the moving quotient algebra and, after dualization, an explicit rank-n flat connection for the Laplace section Ψ=e-aS^φ. Scalar elimination gives the annihilator equations Q(Dₐ)=0, [∂₋V(Dₐ)+aU(Dₐ)]=0, with Dₐ=-∂ₐ, and the polynomial transport identity is exactly their scalar compatibility condition. The previously appearing mixed hierarchy Q(Dₐ)ʳ⁺¹∂_ʳ=0 is proved and reinterpreted as a jet shadow of the first-order system rather than an independent hierarchy. We then place the finite-size flow in a filtered algebraic setting. For a finite-size derivation D and its fixed-point ideal I, the I-adic filtration and extended Rees algebra separate the full nonlinear flow from its first nonzero graded scaling symbol. In the hyperbolic case the conormal endomorphism produces the graded correction spectrum and its additive semigroup; under analytic linearization these become ordinary power corrections, while Jordan blocks give power--log terms. In one-dimensional nonhyperbolic sectors, a first nonzero homogeneous term of degree >0 gives inverse-logarithmic scaling (log L)-1/ and its observable-dependent powers. These asymptotic forms are therefore different local shadows of one filtered finite-size derivation, not independent ansatz classes. Finally, the Hankel moment pairing defines an effective spectral algebra Aeff=Aₛₚ/ B_φ. On a regular constant-rank stratum its radical is parallel, so Hankel rank is preserved while the regular transport remains defined; leaving that stratum separates algebraically into support collision and weight extinction. Classical ingredients used here include Prony/Hankel reconstruction, transport of finitely atomic measures, Rees and normal-cone geometry, D-module/Pfaffian ideas, normal-form analysis of scaling corrections, and Toda/Lax spectral methods. To our knowledge, however, these ingredients have not previously been assembled in the present CPFT form. The precise architecture is a common-base construction: the same physical-size derivation produces a filtered/Rees scaling shadow and, when a regular finite spectral family is present over the base, a quotient-algebra/flat-Laplace shadow whose scalar elimination recovers the annihilator system. We state explicitly which parts are classical and which CPFT-specific synthesis claims remain conditional on the existence of an appropriate physical finite-size relation space.