Invariant Rings of Adjacent-Quadratic Triangular Derivations

Let $k$ be a field of characteristic zero and $D_5$ the triangular locally nilpotent derivation $D_5(a_n)=a_{n+1}{a}_{n+2}$ on $A_5=k[a_0,\dots ,a_5]$. We prove $A_5^{D_5}=C_5[F_{46}^{\mathrm{i}\mathrm{n}\mathrm{t}}],$ hence finitely generated, where $C_5=k[p,q,J_1,J_2,F_0,H]$ is the normal hypersurface $q^{4}H-11025F_0^2-105q^2{J}_2^5{J}_1-9J_2^7=0$. Deleting $a_0$ gives the polynomial kernel $B^{D_5}=k[p,q,J_1,J_2]$, whose localization at $pq$ is the polynomial line $K_{pq}[W]$. Global descent fails twice, each time for an explicit reason. First, the weight-$24$, $a_0$-degree-$2$ invariant $H$ lies outside $k[p,q,J_1,J_2,F_0]$, and weight $24$ is the first such failure. Second, the weight-$46$, $a_0$-degree-$4$ invariant $F_{46}^{\mathrm{i}\mathrm{n}\mathrm{t}}$ lies outside $C_5$, and weight $46$ is the first such failure, with equality of graded pieces through weight $45$. Both failures come from algebraic residue initial forms at the primes dividing the pole-clearing denominator; this mechanism is isolated in a general multfloor valuation bound, sharp in both instances. A second slice chart from the $q$-primitive plinth element $f_{13}=p{J}_2^3\mathrm{/}35+p{q}^2{J}_1{J}_2\mathrm{/}4$ covers the locus where the first chart degenerates, and exact $q$- and $p$-saturation contract the resulting $p$-localized kernel to the closed ring. The closed ring is a codimension-two complete intersection with exact Hilbert series, torsion filtration, singular locus, and normality; the sharp $p$-adic top envelope is $\lceil 3m\mathrm{/}2\rceil$. The pole-cancellation mechanism is compared with the Daigle–Freudenburg example. At $N=6$ the $B$-level ring is exact by transport while the $A_6$ ring remains open.