Close the strip: type-I Navier-Stokes blowup and the direction-regularity exponent in [-1/2, 0]

Setting. For a Navier-Stokes velocity field with vorticity $\omega$, direction $\xi = \omega/|\omega|$, and peak set $P(t)$ (the high-vorticity region), define the scale-invariant direction-regularity observable

$$\Lambda(t) = \sup_{P(t)} |\nabla \xi|\, |\omega|^{-1/2}, \qquad \sigma_\Lambda = \frac{d \ln \Lambda}{d \ln \|\omega\|_\infty}.$$

Under the Navier-Stokes rescaling $u_s(x,t) = s\,u(sx, s^2t)$ one has $|\omega| \to s^2|\omega|$ and $|\nabla\xi| \to s|\nabla\xi|$, so $\Lambda$ is invariant and $\sigma_\Lambda$ is a pure number.

What is known. (i) Unconditionally, from Constantin's decomposition of the vorticity equation into magnitude and direction: at any growing spatial maximum of $|\omega|$, the Laplacian term is nonpositive, hence $\nu|\nabla\xi|^2 \le \alpha$ there, where $\alpha = \xi \cdot S\xi$ is the stretching rate. (ii) Qualitatively, under a type-I condition, direction continuity where vorticity is large excludes blowup (Giga-Miura 2011; Barker-Prange 2020 in the half-space with no-slip boundary). (iii) Quantified, conditional on a coherence hypothesis (H) (the peak set is a single coherent structure on which the near-field gradient bound holds out to the optimizing radius; measured constant $\lambda_0 = 5.0$): splitting the Biot-Savart kernel at radius $\rho$ and optimizing gives

$$\alpha \lesssim \left(\Lambda\, \|\omega\|_\infty^{3/2}\right)^{3/5} \|\omega\|_{L^2}^{2/5},$$

and feeding the energy dissipation identity with a type-I rate $\|\omega\|_\infty \sim (T-t)^{-\gamma}$, $\gamma \ge 1$: at $\sigma_\Lambda \le -1/6$ geometric depletion begins to act, and $\sigma_\Lambda < -1/2$ excludes type-I blowup by geometry and energy alone.

The problem: close the strip $-1/2 \le \sigma_\Lambda \le 0$. The corollary in (i) forbids the top under viscosity at a growing maximum; the conditional chain in (iii) supplies the bottom. Concretely:

(a) Establish the bound in (iii), or any bound sufficient for the $\sigma_\Lambda < -1/2$ exclusion, without Hypothesis (H), or exhibit the coherence-violating peak structure that defeats it.

(b) Show that a type-I singular solution cannot sustain $\sigma_\Lambda \in (-1/2, 0)$ as $t \to T$. With (a) and the endpoint corollary, type-I Navier-Stokes blowup is finished for this route.

Measured motivation. On the inviscid Luo-Hou corner flow, where blowup is proven in the Boussinesq analogue by Chen-Hou, the exponent calibrates at $\sigma_\Lambda = +1.00 \pm 0.03$ (full window; window-matched deep anchor $+0.589/+0.585$ on two grids). Under free-slip viscosity the deep-collapse exponent inverts to $\approx -1.2$ at $\nu \in \{10^{-4}, 10^{-3}\}$, with no resolved $\nu$-dependence at five tested viscosities down to $10^{-5}$. The measured mechanism sits below the strip at every tested viscosity. Measurement details, calibration, instrument checks, and the retraction record are in the referenced pages; the measurements are AI-assisted with a Leiden-format disclosure.