Calibrating the Critical L¹-Morrey Program on a Forced Navier–Stokes Singularity: Quadratic Shear Cancellation, the Signed-Shell Profile, and the Absolute Volterra Window

We calibrate critical Morrey and vortex-stretching functionals on the forced Navier–Stokes singular-flow construction of OpenAI, conditional on the construction and quantitative hypotheses collected in (OA). Writing τ = 1 − t and 0 < h < 1/100, a finite-stage envelope gives ‖u(t)‖∞ comparable to τ^(−1/2−h) and evaluates the classical continuation quantities and the series’ absolute vorticity profiles. Velocity-based estimates are two-sided with constants; absolute vorticity estimates retain a fixed logarithmic factor. The signed heat–curl functional has the same constant-factor growth as the velocity, and the absolute Volterra window formula decays faster than the remaining time by a factor of order τ^(2h), up to logarithms. We introduce a signed-shell Constantin–Fefferman profile, taking the absolute value after integration on each sphere. It controls vortex stretching, is bounded by the absolute direction profile, and cancels exactly on axisymmetric columns and non-axisymmetric harmonics observed from the symmetry axis. On the constructed flow it is saturated at the axis at rate τ^(−1), whereas the absolute direction profile grows as τ^(−1−h) up to logarithms. Exact restricted-Hessian and azimuthal geometry yield a per-packet non-stationary-phase estimate inside a specified tube. A power gain requires an additional quantitative angular-frequency bound; absence of critical points alone is insufficient. A separate conditional averaging result gives the rate τ^(−1−h/2) up to logarithms under explicit uniform oscillatory-majorant and radius-sublevel hypotheses. These hypotheses are not established for the construction, and the full signed-profile condition (SP) remains open. We also give a Kato–Duhamel forcing-threshold formulation and the forced swirl maximum principle. The contribution is the profile comparison and its explicit cancellation geometry; no unconditional regularity or singularity theorem for the unforced equation is claimed.