CubeSensitivity #
Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.
The conditional J-sensitivity at a mesoscopic cube, unconditionally #
Source: ABK26.
At the mesoscopic depth H = mesoscaleDepth s (1 + C₀ ‖∇h‖ λ_{s,2}^{-1}(□₀;a))
the dimensionless quantity ‖∇h‖ |R| λ_{3/8,2}^{-1}(R;a) is at most 1/C₀ at
every triadic descendant R of the unit cube of depth H, with no hypothesis
whatsoever on h. With C₀ chosen large enough this gate
- makes the mesoscopic coarse gauge at most double along the perturbation path
(
cubeGauge_le_two_mul_zero, theProvider.Lambda.Bootstrapengine run atRexactly as inProvider.LambdaUnconditional.CubeRatio), and - makes the Term BC coefficient of the general-loading
D_hbound at most1/4,
so the affine differential inequality of Provider.BigLambda.PathODE integrates
at δ = 1 to the per-cube estimate
J(R, a+h, p, q) ≤ 3 J(R, a, p, q)
+ K₁ |p|² λ_{3/8,2}^{-1}(R) (V(R) + 3^{-2H} ‖∇h‖²)
+ K₁ 3^{-2H} λ_{3/8,2}^{-2}(R) ‖∇h‖² |p·q| ,
with V(R) the cube-local L² value budget of
Provider.ResponseUnconditional.ValueLipschitz.
Every declaration in this module is an internal helper for the Section 2.4 sensitivity providers.
Two pieces of abstract path arithmetic #
The pointwise conversion of the general-loading D_h bound into the affine
differential inequality, with the path gauge replaced by twice its base value.
Exact integration of the affine differential inequality at δ = 1.
It is large enough both for the mesoscopic pure-flux spine (factor 4 C_flux)
and for the Term BC coefficient of the general-loading D_h bound (factor 4 C_BC).
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The mesoscopic depth attached to the base coefficient and the perturbation.
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The unconditional deep-cube gate #
The unconditional deep-cube gate. At the mesoscopic depth, the
dimensionless quantity governing both the mesoscopic gauge bootstrap and the
Term BC coefficient is at most 1/C₀, with no hypothesis on h.
The mesoscopic gauge along the path #
The mesoscopic coarse gauge at most doubles along the path.
The two ODE coefficients #
The Term A coefficient of the path ODE at a mesoscopic cube.
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The Term BC coefficient of the path ODE at a mesoscopic cube.
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The error constant #
The error constant of the per-mesoscopic-cube J-sensitivity.
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The per-mesoscopic-cube J-sensitivity at a general loading.