CubeShaking #
Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.
The per-mesoscopic-cube shaking step #
Source: ABK26, the display
J(z+cu_{-h}, (mu s0)^{-1/2}e, (mu s0)^{1/2}e; a+h)
<= 6 mu^{-1} J(z+cu_{-h}, s0^{-1/2}e, s0^{1/2}e; a)
+ 4 mu^{-1}(mu-1)^2 s0 lambda_{1/2,2}^{-1}(z+cu_{-h}; a)
+ C mu^{-1} s0^{-1} lambda_{1/2,2}^{-1}(z+cu_{-h}; a) 3^{-2h} |h|^2_{W^{1,inf}}
+ C lambda_{1/2,2}^{-2}(z+cu_{-h}; a) 3^{-4h} |grad h|^2_{W^{1,inf}} ,
obtained from the conditional J-sensitivity at delta = 1 followed by
e.shaking.lambda at lam = mu and e.ellipticities.monotone.ordered.
This module contains the shaking half of that display, which is
unconditional, together with the fold that turns a conditional per-cube
(p,q)-sensitivity into exactly the hcube hypothesis of
Provider.ResponseUnconditional.Aggregation.
The remaining consumption point #
- the mesoscopic gate (which the mesoscopic depth supplies unconditionally,
exactly as
LambdaUnconditional.deep_cube_smallnessdoes for thelambdahalf), and - a Term A budget carrying the cube scale factor and an
L^2value budget rather than the rootW^{1,infinity}value norm.
Every declaration in this module is an internal helper for the Section 2.4 sensitivity providers.
The coarse gauge bounds the shaken quadratic form #
The sigma_*^{-1} quadratic form at the sigma0-normalized co-loading is
bounded by sigma0 lambda_{3/8,2}^{-1}(R; a). This is
e.ellipticities.monotone.ordered in the form used in ABK26.
The shaking step at a mesoscopic cube #
The shaking step of the per-cube display. At any triadic cube, the
response of the base field at the (mu sigma0)-normalized loading is bounded by
twice mu^{-1} times the response at the sigma0-normalized loading plus the
(mu-1)^2 gauge term.
This is e.shaking.lambda at lam = mu followed by
e.ellipticities.monotone.ordered; it is unconditional.
From a conditional per-cube sensitivity to the aggregation hypothesis #
The per-mesoscopic-cube estimate of the source display.
hsens is the display of ABK26, before shaking: the factor 3 = 1 + delta + (gate) at delta = 1, the Term A error |p|^2 λ_{3/8,2}^{-1}(R) (V R + 3^{-2H} ‖∇h‖^2) with |p|^2 = (μ σ0)^{-1}, and the Term BC error |p·q| 3^{-2H} λ_{3/8,2}^{-2}(R) ‖∇h‖^2 with |p·q| = 1.