Coarse-graining support: #
Support.Deterministic.CoarseCaccioppoli.SingleCubeToRaw.QuantitativeCutoffInputs.Setup.ScaleBounds
Imported from the Apache-2.0 CoarseGraining development at commit
c7ddd76c08ade64fed1b8d2ca51be14dfee8deb4.
Quantitative cutoff inputs: scale bounds #
Canonical L^∞ bound supplied by the quantitative cube cutoff gradient
estimate.
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Canonical derivative bound supplied by the quantitative cube cutoff Hessian estimate.
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The midpoint cutoff has twice the gradient scale of the full-gap cutoff.
The midpoint cutoff has four times the Hessian scale of the full-gap cutoff.
The small-cube Hessian contribution after multiplying the parent-cube
cutoff Hessian bound by a depth-j descendant scale. This is the formal
3^{-j} gain used in the LaTeX Caccioppoli proof.
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The constant-branch cutoff bracket, after multiplying by the exact negative Besov scale weight, splits into the local Hessian-scale term plus the gradient term.
On a depth-j descendant, cubeScaleFactor R converts the parent cutoff
Hessian bound into the descendant small-cube scale bound.
Inequality form of
cubeScaleFactor_mul_coarseCaccioppoliQuantitativeCutoffHessianBound_eq_descendant,
for feeding cutoff-product coefficient estimates.
On a depth-j descendant, a local length divided by the parent radius is
exactly the expected 3^{-j} factor, up to the radius normalization 1/2.
Parent cutoff constants rewritten in the depth-j descendant scale used by
the constant branch.
The parent cutoff gradient bound is controlled by the triadic gap scale, with the fixed parent radius carried as a front constant.
On a depth-j descendant, the parent cutoff-gradient bound gains the
small-cube length scale 3^{-j}. This is the centered-branch analogue of the
constant-branch cutoff-scale normalization.
After the triadic gap scale is chosen, the descendant-normalized parent
cutoff-gradient term is bounded by 3^{-j} 3^k times a fixed cutoff constant.
LaTeX-shaped version of
cubeBesovScaleWeight_neg_one_mul_parent_cutoffGradient_le_depth_pow.
Buffered LaTeX-shaped gradient cutoff bound. The midpoint cutoff doubles the full-gap gradient bound, while the triadic scale is still chosen from the full outer gap.
The Hessian gap factor gains one descendant scale. This is the
3^{-j} gap^{-2} ≤ 3^k line used after choosing the triadic gap scale and
taking descendants at depth j ≥ k.
Algebraic normal form for the descendant Hessian cutoff contribution:
parent Hessian bound times the small-cube volume scale is a fixed parent
front constant times 3^{-j} gap^{-2}.
The descendant Hessian cutoff contribution is controlled by the note's triadic gap scale, with only a fixed parent-radius front constant remaining.
The centered Hessian cutoff contribution has one extra small-cube scale,
so after the triadic gap choice it also has the 3^(k-j) normalization.
Buffered LaTeX-shaped Hessian cutoff bound. The midpoint cutoff quadruples the full-gap Hessian scale, while the triadic scale is still chosen from the full outer gap.
Combined small-cube cutoff contribution bounded by the note's triadic
scale. This packages the Hessian 3^{-j} gap^{-2} gain together with the
gradient gap^{-1} contribution.
Descendant form of the combined cutoff bound, in the scaled bracket used by the constant branch of the exact single-cube estimate.
Buffered version of coarseCaccioppoliDescendantCutoffTerms_le_radiusConst_mul_pow.
The midpoint cutoff inflates the Hessian contribution by 4 and the gradient
contribution by 2, while the triadic scale is still the full outer gap.
Descendant form of the buffered combined cutoff bound, in the scaled bracket used by the constant branch.