17 / C3-O: Adjoint Core Balance, the Cancellation Corridor, and Balance–Dynamics Separation

C3-N left open whether the gauge/advection/diffusion terms generated by the cutoff function itself can be stripped away entirely. This round uses a backward-parabolic adjoint cutoff (setting $\partial_t\chi+u\cdot\nabla\chi+\nu\Delta\chi=0$ and solving backward from a terminal condition) to absorb exactly these three terms, arriving at the Adjoint Core Balance Theorem (Theorem 4.1): $E_\chi'+D_\chi=A_\chi+B_\chi$, where $A_\chi=-2\int\chi\det S$ is the bulk self-amplification and $B_\chi=\int\nabla\chi\cdot(\frac13F_B+F_p)$ is a clean boundary correction flux. Defining the ratio $\rho_I=B_I/A_I$, the Hard Depletion Barrier (Theorem 10.1) states: if $\rho_I\le-1$, the window in question cannot be a window of positive local strain growth — this is a genuine hard exclusion zone. $\rho_I\to-1^+$ survives, but must pay a "Cancellation-Precision Debt" (§12 — large bulk generation plus a large opposing boundary flux plus a small residual, with an exact lower bound of $\Omega(A_I)$); $\rho_I\to+\infty$ survives, with the boundary/pressure flux becoming the dominant carrier. $\rho_I\to0$, however, is not a case of "survival" but a non-identifiability/no-go: the constant cutoff $\chi\equiv1$ on all of space gives $B_{\chi\equiv1}=0$ identically, hence $\rho_I=0$, but this carries no information capable of recovering the omitted operator $\mathcal P_{NS}$ — $\rho\to0$ by itself cannot imply $\mathcal P_{NS}\to0$; scalar balance information alone is not enough to decide whether the operator is truly small. The paper's most important result is the Balance–Dynamics Separation No-Go (Proposition 17.1): invoking Miller's decomposition of the full strain equation into a "strain self-amplification model plus an orthogonal perturbation term $\mathcal P_{NS}$," it proves that $\langle\mathcal P_{NS},S\rangle=0$ over all of space — the full N–S equations and this simplified model share exactly the same global strain–enstrophy growth identity. But Miller's own model does genuinely blow up in finite time for certain classes of initial data. So orthogonality ($\langle\mathcal P_{NS},S\rangle=0$) does not imply smallness ($\mathcal P_{NS}=0$) — an energy balance that looks like the simplified model does not mean the dynamics are actually close to that model. The document constructs a scale-invariant candidate diagnostic $\mathfrak P_I$ (an operator-level relative size), but explicitly flags it as, for now, only a candidate diagnostic, not a proven stability criterion. The scalar-ratio route has formally reached its limit; the next round must escalate to the operator level.