Critical Partition Function Theory IV

We develop a finite-size sector of critical partition-function theory directly from a size-parametrized partition family, without assuming a renormalization-group ansatz, disorder, a random Hamiltonian, or a random-matrix ensemble. Exact size differentiation yields an all-order cumulant hierarchy and a weak transport--reweighting equation for normalized spacing measures; at fixed spacing weights, affine energy motion is an exact null direction of the normalized spacing-support sector. Static information is then shown to be insufficient for finite-size dynamics: neither a static critical state nor, in general, the complete one-variable partition function determines the transverse Hessian geometry or the size tangent. We therefore adjoin a projective Hessian orientation and the normalized rank-one defect ρH= H/( H)². For smooth positive-semidefinite Hessian families defined on a two-sided local parameter neighborhood, the pullback of ρH has vanishing first jet at a rank-one point. Minimization over relevant controls produces a reduced defect whose quadratic jet is a Schur complement, unifying exact rank-one roots with minimum-defect pseudocritical points. If the extended state is dynamically sufficient and its tangent assignment is regular, physical size composition induces an autonomous generator. The first nonconstant jet of an observable carries the induced symmetric-power Koopman action, so scaling exponents add on jet monomials. An independent Hessian-normal decay exponent ω_⊥ then competes with a tangential forcing exponent Ω: fast normal relaxation gives L-Ω, resonance gives L-Ωlog L, and slow normal memory gives L-ω_⊥. The accumulated defect is an exact positive weighted finite-time Gramian, whose Schur reduction describes dynamic pseudocritical optimization and distinguishes instantaneous from dynamically maintained rank one. The classical ingredients used here---finite-size scaling, Koopman algebras, normal-form resonance, positive invariance, Schur complements, and Gramians---are not claimed as new individually. The contribution is their derivation and organization within the critical-partition-function finite-size problem, together with the static-to-dynamic no-go results, the Hessian extension, and the resulting normal-mode and history-dependent finite-size laws.