43 / C5-G: Pressure-Signature Defects, Vorticity Constraint Complements, and Fixed-Order Derivative-Gate Closure

C5-F compressed the residual network into four items: pressure-signature defects, vorticity-constraint-complement defects, fixed-order derivative-gate defects, and asymptotic-critical-order escape. C5-G prioritizes killing the fixed-order direct gate first, making a cleaner move: instead of routing through strain/shell transforms, it puts a global volume upper bound directly on the super-level sets of all components/signs of the raw $D^ku$. **C5-G.1 Direct Component-Volume Bound** (Chebyshev) proves $|V_{\lambda,k}^{\zeta,i,\pm}|\le\lambda^{-2}L_k^2/A_k^2$ holds uniformly for all component/sign combinations, with no strain/rotation decomposition, no shell-to-full-field conversion, and no need to guess which component first. Combined with C3-W's volume-to-line geometry lemma, this yields a theorem-ready 1D sparseness scale $r_{vol,k}\sim L_k^{2/3}A_k^{-2/3}$, which is compared directly against the true direct scale $r_{GX,k}$ of the published Grujić–Xu (2024, J. Math. Fluid Mech.) Theorem 3.5, defining the ratio $\mathfrak G_k^{dir}=r_{vol,k}/r_{GX,k}$. **The round's headline result, C5-G.3 Fixed-Order Direct Gate Closure Theorem**: if $\mathfrak G_k^{dir}(s)\le1$ at the theorem's admissible later time $s(t)$, then Theorem 3.5's spatial hypothesis genuinely holds, and $T_\ast$ is not a blow-up time — this is not a pre-gate; it is the first genuinely theorem-ready closure interface in the entire NS series. The old COMPSIGN (component/sign guessing) and SHELLFULL (shell-to-full-field) defect labels are thus formally bypassed along this direct-volume route, leaving only two items in the fixed-order residual: effective-volume (multiplicity) diffuseness and later-time misalignment. The exact $k=1$ form gives a gate between strain enstrophy and the raw gradient peak, $\|S\|_2^2\le C\|Du\|_\infty^{1/5}$, but the exponent $1/15$ is small, showing that although $k=1$ is already theorem-ready, genuinely closing it still needs very strong peak concentration; the general fixed-$k$ amplitude exponent $(4k-3)/(3(2k+3))$ approaches $2/3$ as $k$ grows, showing that higher derivative orders carry stronger leverage, but $L_k$ may grow in step as well, so this is not an automatic order-escalation closure. On the pressure-signature side: **C5-G.4/5** prove that if the joint far-field pressure-matrix signature (one-negative/two-negative) keeps switching while strong heredity holds (the relative change between adjacent matrices tends to zero), it must force the zero-determinant boundary $d_{\rm sig}\to0$ — giving a clean pressure-signature trichotomy (signature fixed ∨ signature-boundary defect ∨ pressure flip/source splitting). On the vorticity-dominant-leakage side: using the exact pressure-Poisson identity $\Delta p=-|S|^2+\tfrac12|\omega|^2$, **C5-G.6** proves that on the vorticity-dominant leakage set, pointwise $\Delta p>0$ — a genuine "pressure-Poisson return certificate," but it explicitly warns this is not yet $L^{3/2}$ pressure concentration (which needs additional control of the negative-value region, oscillation scale, and sign-consistent cutoff geometry). Separately, the exact constraint-complement pressure ledger $\nabla^2p=-(C_A+C_S+C_\omega)$ gives **C5-G.7 Vorticity-Complement Re-entry Trichotomy**: a large vorticity constraint complement must force one of a genuine pressure Hessian, an advection complement, or a strain-squared complement — the vorticity-constraint-complement defect cannot exist in isolation. **The most important honest boundary (NG-G7)**: repeated fixed-order direct-defect failure does not by itself imply that escape to $k\to\infty$ is necessary, unless every fixed-order recurrent defect has otherwise been excluded. The round closes by stating the hardest new question with full clarity: can a hypothetical singular survivor keep the $L^2$ mass of $D^ku$ sufficiently diffuse relative to its peak, at every fixed derivative order and at every theorem-admissible later time, so that $\mathfrak G_k^{dir}>1$? Formally hands off to C5-H — All-Order Effective-Volume Defects, Derivative Concentration Ladders, and Asymptotic-Critical Compatibility, with 8 targets left to prove.