18 / C3-P: Operator Escape, the Far-Pressure Harmonic Matrix, and the Finite-Dimensionalization No-Go

C3-O already proved that an energy balance which looks like the simplified model does not mean the dynamics are actually close to that model. This round formally escalates to the operator level. The core is the Exact Two-Model Gap (Theorem 5.1): reorganizing the full strain equation relative to both the "strain self-amplification model" ($\mathcal N_{SSA}$) and the "strain–vorticity interaction model" ($\mathcal N_{SV}$, for which Miller 2026 proves global smooth solutions for arbitrary $L^2$ initial data), the gap between the two models is an exactly writable operator $\mathcal G$. Regular-Model Operator Escape (citing Miller's 2026 theorem, §9): a hypothetical finite-time blow-up forces $\limsup_{t\uparrow T_\ast}\|\mathcal Q_{SV}(t)\|_2/\|-\Delta S(t)\|_2\ge1$ — full N–S cannot remain forever inside the perturbation tube of this known, globally regular model; this is a genuine operator-level necessary condition, not a scalar energy inference. But the document also notes that the SSA model itself can blow up in finite time (in a separate paper by Miller), so "being close to the SSA model" cannot be taken as a direction toward regularity — it is even compatible with blow-up under certain initial-data/perturbation assumptions. On the pressure side, using the pressure Poisson equation together with Riesz-kernel estimates, it proves the Far-Pressure Finite-Dimensionalization Lemma (Theorem 19.1): pressure sources beyond $\kappa R$ from the core are harmonic inside the core, and their Hessian can be compressed into a position-independent, 5-dimensional constant symmetric traceless matrix $H_0$ plus a spatially varying remainder suppressed by a power of $\kappa^{-1}$ — but the document explicitly stresses that finite-dimensionalization is not the same as smallness (§21); the genuine decoupling condition is that the rescaled enstrophy number $\mathfrak E_R=R\|\nabla u\|_2^2/\nu^2$ satisfies $\kappa^{-3}\mathfrak E_R\to0$, not simply $\kappa\to\infty$ — if the rescaled enstrophy diverges fast enough, the far-field pressure can still retain a critical-size effect. The document explicitly lays out three mutually non-substitutable observational interfaces: operator escape (the Miller ratio), local strain balance (C3-O's $\rho$), and far-field pressure (the harmonic matrix $H_0$). The research frontier turns to C3-Q: whether these three channels can be simultaneously sustained, under one and the same set of exact N–S constraints, out to infinite scale.