Critical Partition Function Theory V

Abstract

We develop a local physical-size scale-flow theory for critical phenomena within critical partition-function theory (CPFT), without assuming a renormalization-group transformation, a Hamiltonian, disorder, randomness, or a nonlinear sigma model as primitive input. The starting point is the finite-size generator introduced in CPFT IV. When a critical equilibrium has one simple center mode and a hyperbolic complement, a parameter-dependent center manifold produces a one-dimensional physical-time germ even when no global scalar projection exists. We identify a finite marginal dynamic signature whose contractions determine the generic fold germ and its conormal in critical-parameter space. On the fixed-point side the small eigenvalue vanishes as the square root of the signed distance to the fold; on the zero-free side the same local normal form produces a scale-passage time proportional to the inverse square root. Whenever the relevant small eigenvalue is physically identified with $1/\nu$, the paired amplitudes obey $A_T/A_\nu=2\pi$. These statements concern the local CPFT scale flow of any critical realization satisfying the stated hypotheses; they are not specific to Anderson localization. A second result concerns the rank-one scaling sector reconstructed in CPFT II. After fixing the normalized CPFT extraction coordinate $x=\log[\Phi(u)/\Phi(0)]$ and the physical-size scaling $d\log|u|/d\log L=1/\nu$, the canonical local scale flow $B(x)=d x/d\log L$ is determined by finite normalized critical data. On the regular chart $I_2\neq0$, writing $q=\nu B$, $a_\beta=I_2-1$, and $c_\beta=1-2I_2+I_3/I_2$, we obtain the closed rational equation $\frac{d q}{d x}=\frac{1+(a_\beta-c_\beta)q+c_\beta q^2}{1-c_\beta q},\qquad q(0)=0.$ The chart degeneracy is explicit: $\det[\partial(I_2,I_3)/\partial(\rho,\sigma)]=(\rho-\sigma)^4/[\rho^4(\rho-1)^4]$. The exceptional locus $I_2=0$ is not singular dynamically: the normalized rank-one equation forces $\Phi(u)=1+u$ and hence $B(x)=(1-e^{-x})/\nu$ exactly. The regular equation yields an elementary integral representation and a recursion for every Taylor coefficient of the canonical flow germ. We distinguish this normalized-shape statement from broad dynamical equivalence: a one-dimensional hyperbolic flow can be locally linearized by a regular observable redefinition, so only the fixed point and its linear eigenvalue are unrestricted local invariants. The general theory therefore assigns local scale-flow structure to CPFT critical phenomena before any model-specific beta-function interpretation is imposed. In a physical realization where the relevant scale-flow function is conventionally called a beta function, the same results become beta-function statements. Conversely, the construction does not require a beta function to have been supplied in advance: for a center-detecting observable, zeros of the constructed local flow are precisely local fixed points on the center manifold, its fixed-point slope is the reduced center eigenvalue, and simultaneous vanishing of the flow and its slope detects a local loss of hyperbolicity. This gives a conditional CPFT diagnostic for exploring critical states and critical-flow boundaries in realizations where a conventional beta-function description has not been identified or adopted. Anderson transitions provide an unusually stringent and data-rich realization rather than the defining setting: orthogonal-class data supply hyperbolic slope benchmarks through $B'(x_c)=1/\nu$, while a resummed symplectic beta function provides a concrete approximate fixed-point-annihilation example. The results clarify how local critical flow data may be finitely reconstructible and useful for criticality diagnostics even when a global flow graph is coordinate-, model-, or completion-dependent.