Critical Partition Function Theory VI

Abstract

We study the rank-one scaling equation that arises in the two-variable sector of Critical Partition Function Theory (CPFT), $a\Phi\Phi''+bu\Phi'\Phi''-c(\Phi')^2=0,$ with normalized local data $\Phi(0)=\Phi'(0)=1$. The individual ingredients used below---nonlocal linearization of nonlinear ordinary differential equations, Riccati/projective dynamics, binary quadratic forms, and degenerate Monge--Amp\`ere geometry---all have substantial classical literatures. The result here is a CPFT-specific synthesis. We prove that the nonlinear equation admits an explicit local derivative-dependent reparametrization under which $X=u\Phi'$ and $Y=\Phi$ satisfy a constant-coefficient $2\times2$ linear system. Its projectivization is a Riccati flow generated by a canonical traceless matrix $H_{\rm II}\in\mathfrak{sl}_2$, and the associated binary quadratic determines an unordered divisor of two fixed points on $\PP^1$. The CPFT exponent duality $(\rho,\sigma)\mapsto(1-\rho,1-\sigma)$ becomes exchange of these two roots. The same generator is obtained algebraically from the CPFT III second-order operator after an explicit coefficient map, with a genuine second-order realization when $b\neq0$ and time rescaling $\tau=bs$. We show that this coincidence is structural rather than accidental: the common binary quadratic is sent to the common generator by the canonical symplectic identification $\Sym^2(V^*)\simeq\mathfrak{sp}(V)=\mathfrak{sl}_2$ in dimension two. The generator further satisfies $H_{\rm II}^2=(\Delta/4)I$, where $\Delta=(a+b+c)^2-4ac$. Hence the discriminant simultaneously classifies the projective fixed-point divisor and the semisimple/nilpotent type of the $\mathfrak{sl}_2$ generator; over the reals this is the hyperbolic/parabolic/elliptic classification of the induced M\"obius flow. On the canonical real CPFT lift, $\Delta=c=(\rho-\sigma)^2\ge0$, excluding the elliptic sector. In addition, after centering the exponents by $x=\rho-1/2$ and $y=\sigma-1/2$, we prove that the affine canonical coefficient surface is the algebraic quotient $\mathbb A^2/\{\pm1\}$, equivalently the $A_1$ quadric cone $UW=V^2$; its unique singular point is the self-dual exponent point $(\rho,\sigma)=(1/2,1/2)$. We also give finite-jet reconstruction of the exponent duality orbit, branch-free parametric solutions, explicit regular and degenerate examples, and a literature-audited comparison with critical scaling, thermodynamic geometry, Monge--Amp\`ere theory, and nonlinear-ODE linearization. To our knowledge, the combined CPFT identification of the rank-one scaling equation with this constant projective generator, root-divisor geometry, exponent duality, and CPFT II--III generator correspondence has not previously been stated in this form.