Critical Partition Function Theory VI

We study the rank-one scaling equation that arises in the two-variable sector of Critical Partition Function Theory (CPFT), aΦΦ''+buΦ'Φ''-c(Φ')²=0, with normalized local data Φ(0)=Φ'(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=uΦ' and Y=Φ satisfy a constant-coefficient 2×2 linear system. Its projectivization is a Riccati flow generated by a canonical traceless matrix HII₂, and the associated binary quadratic determines an unordered divisor of two fixed points on ¹. The CPFT exponent duality (ρ,σ)↦(1-ρ,1-σ) 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≠0 and time rescaling τ=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 ²(V^*)(V)=sl₂ in dimension two. The generator further satisfies HII²=(Δ/4)I, where Δ=(a+b+c)²-4ac. Hence the discriminant simultaneously classifies the projective fixed-point divisor and the semisimple/nilpotent type of the sl₂ generator; over the reals this is the hyperbolic/parabolic/elliptic classification of the induced M\"obius flow. On the canonical real CPFT lift, Δ=c=(ρ-σ)²≥0, excluding the elliptic sector. In addition, after centering the exponents by x=ρ-1/2 and y=σ-1/2, we prove that the affine canonical coefficient surface is the algebraic quotient A²/\±1\, equivalently the A₁ quadric cone UW=V²; its unique singular point is the self-dual exponent point (ρ,σ)=(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.