High-contrast homogenization: Entry.Response.Rows.CellHalfFinalBound #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The Cell Half of the Cutoff-Mean Row with No Sample-Side Premises #
The carrier-level cell half of the cutoff-mean row is proved subject to eighteen sample-side
integrability premises: the subcell responses and response-integrand averages, the quadratic
readouts of the pathwise coarse block, the flat averages entering the annealed Cauchy-Schwarz
step, and the coordinates of the two optimizer cell means. Every one follows from data the row
already carries, together with the measurability of the subcell average of the terminal
optimizer's energy density obtained from a subcell indicator weight; this file discharges all
eighteen premises and restates the cell half with them removed. It also begins the oscillation
half: the crossed pairing of its (φ - (φ)_{V_w})-weighted vector of subcell means against the
dual variable is a scalar annealed average, and the five P-integrabilities its dominated form
needs at every depth reduce to three per-cell integrabilities. All of this is the cutoff-mean row
of p.response.transfer.
The cell half of the cutoff-mean row with no sample-side residue #
CellHalfCarrierBound proves the cell half of the cutoff-mean row of p.response.transfer on the
response carriers, but carries eighteen sample-side integrability premises: the subcell responses
and response-integrand averages, the quadratic readouts of the pathwise coarse block, the three
flat averages entering the annealed Cauchy--Schwarz step, and the coordinates of the two optimizer
cell means. All of them follow from the annealed data the row already carries — the coarse block
is integrable on every aligned cell and the terminal responses at the two scales are integrable —
together with the measurable canonical selection of the optimizer state.
This module discharges them and restates the cell half with those premises removed, and with the subcell maximizer family constructed internally rather than supplied.
The cell half of the cutoff-mean row on the carriers with no sample-side residue, minus
sign. Under the annealed data of the row — stationarity, the invertible grid, the coarse block
on every aligned subcell of the coarse scale, and the integrability of the two terminal responses —
the modulus of the crossed pairing of the cell part Ncell against Y^- is at most
√(L_s^-) √(2 tau^-). No sample-side integrability is assumed.
The cell half of the cutoff-mean row on the carriers with no sample-side residue, plus
sign. The adjoint twin of abs_vecDot_cellPart_respYMinus_le_clean, for the adjoint recentred
coefficient a_+ = aᵀ + g, the dual load q^+, the dual variable Y^+ and the terminal maximizer
family uP.
The crossed pairing of the oscillation part is a scalar annealed average #
The oscillation part of the cutoff-mean defect of p.response.transfer is the vector whose
coordinates are the annealed flat averages, over the coarse subcells V_w of the terminal cell, of
the (φ - (φ)_{V_w})-weighted subcell means of the optimizer state. Because the dual variable Y
is deterministic, its crossed pairing with that vector commutes with all three averages — the
expectation, the flat average over the subcells and the subcell mean — so the whole row is a
statement about the single SCALAR field ⟨Y₂, ∇u⟩ + ⟨Y₁, a∇u⟩, which is the form the Fenchel probe
of AK.HC (A.4) consumes.
The crossed pairing of the oscillation part is a scalar annealed average. The pairing of
the deterministic dual variable Y with the vector of annealed flat averages of the
(φ - (φ)_{V_w})-weighted subcell means of a block field is the annealed flat average of the
(φ - (φ)_{V_w})-weighted subcell means of the scalar crossed pairing. This is the scalar form of
the oscillation half of the cutoff-mean row of p.response.transfer.
The per-depth annealed readouts of the descendant sum are integrable #
The oscillation half of the cutoff-mean row of p.response.transfer, in its dominated form, asks
for five P-integrabilities at every refinement depth: the flat average of the squared pathwise
head, the flat average of the doubled cell energies, the product of the square roots of those two
flat averages, the weighted flat average of their per-cell product, and the generation increment
itself. Each is a finite normalized sum of per-cell readouts, and the only non-algebraic step is
the crossed product of square roots, which is dominated by the arithmetic mean of the moduli.
This module records the five reductions once and for all, so that the carrier instantiation has to supply only the three per-cell integrabilities — the squared head, the cell energy and the crossed pairing.
The flat average of the squared pathwise head is integrable.
The flat average of the doubled cell energies is integrable.
The Cauchy--Schwarz middle term is integrable. The product of the square roots of the two flat averages is dominated by the arithmetic mean of their moduli.
The weighted flat average of the per-cell products is integrable.
The generation increment is integrable. It is the weighted flat average of the per-cell crossed pairings.