35 / C4-I: Middle–Operator Gate Overlap, Angle Depletion, and Local Pressure Re-entry

This paper opens by picking up from C4-H, which already synchronized UV, middle strain, and the growth-aligned operator onto the same sequence of contracting record windows $J_j=(\tau_j,\tau_{j+1})$, each window carrying a middle-load integral lower bound $A_j$ and an operator-load integral lower bound $B_j$ — but had not yet proved same-instant overlap. C4-I attacks only two questions: what exactly is missing from genuine middle/operator same-window overlap, and when must pressure re-enter within the local adjoint core. The core is the Middle–Operator Capacity-to-Overlap Theorem (Theorem 5.1, pure measure inclusion–exclusion, $|E\cap F|\ge|E|+|F|-|J|$): it converts two independent integral lower bounds into a genuine overlap lower bound, provided the peak capacity (essential suprema $M$, $O$) is controlled. The document explicitly points out (§9, §11) that what is truly missing is not another marginally-divergent condition, but rather control of peak capacity and persistence — without an independent upper bound on the peak-to-average capacity ratio, integral information alone cannot force same-instant overlap; this is the Middle–Operator Peak-Capacity Desynchronization Debt. On the operator-angle side, defining $g=\zeta r_\nu$ as the operator's growth-aligned component, the Large-Ratio Non-Growth Routing theorem (Theorem C4-I.3) proves that a large Miller ratio without genuine growth ($g\le1$) must be either strongly anti-aligned ($g<-1$) or must carry a large orthogonal operator component $\|Q_\perp\|\ge\sqrt{R^2-1}$ — a far finer angular-dissipation classification than the plain difference $1-\zeta$. Citing Miller's own exact orthogonality result from 2026, $\langle P_{st}(\omega\otimes\omega),-\Delta S\rangle=0$, the document proves that the vorticity-quadratic term falls exactly within the growth-orthogonal subspace, so orthogonal congestion can only be borne by the vorticity-quadratic term or by orthogonal advection/strain-squared components (Theorem 18.1); while positive $\dot H^1$ growth itself can only be driven by advection alignment or strain-squared alignment (Theorem C4-I.4) — because of Miller orthogonality, the vorticity-quadratic term can never directly drive growth. The document also supplies a pure matrix-algebra counterexample (§22): taking the diagonal matrix $S=\operatorname{diag}(-2,-1,3)$, even though $\lambda_2=-1<0$, the local integrand can still be positive, proving that strain-squared-aligned $\dot H^1$ growth does not imply a pointwise positive middle eigenvalue $\lambda_2^+>0$. On pressure: exact whole-space orthogonality ($\langle\nabla^2p,-\Delta S\rangle=0$) means a growth-aligned operator cannot directly lower-bound pressure; but within the local adjoint core, C3-U's exact identity $M_\chi'=-B_\chi-P_\chi$ yields the Adjoint Mean-Rotation / Pressure Dichotomy (Theorem C4-I.5, a simple triangle inequality): a non-degenerate local quadratic mean driving force must be borne by either mean rotation or pressure; chaining this to C3-X's Hessian-sensitive pressure-oscillation bound gives Mean-Stability Forces Pressure Re-entry (Theorem C4-I.6). The round's core result is the Quadratic Forcing Three-Way Re-entry Theorem (Theorem C4-I.7): a large-magnitude local quadratic intensity must lead to one of matrix/spatial cancellation, mean rotation, or pressure concentration — pressure is not a universal consequence of the operator; pressure must re-enter only when the local quadratic driving force is coherent enough (not cancelled) and mean rotation has been sufficiently depleted. The document honestly lists a complete no-go catalogue (NG-I1 through NG-I5), confirming that C4's remaining genuine asynchronous degrees of freedom are no longer new physical channels, but two compensation mechanisms: temporal pulse separation, and pressure evasion via mean rotation or quadratic cancellation.