High-contrast homogenization: Entry.Response.Rows.CutoffOscillationHalfClosure #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Closing the oscillation half of the cutoff-mean row #
This module serves the cutoff estimate p.response.transfer. The oscillation half of the
cutoff-mean row of that estimate is bounded by the source load
once a pathwise vanishing descendant remainder and an increment envelope with four stated
properties are supplied; this file first records that abstract closure, then builds the required
envelope out of data the row already carries — the cutoff weights, the head energies, and the cell
energies. It then supplies that data for the terminal optimizer family of the response problem
itself: the refinement identity at every depth, the cutoff-weight increment, and the pathwise
crossed-pairing bound on each descendant cell. Together the three stages close the oscillation
half of the row for the negative sign directly on the response's own optimizer field, with no
analytic hypothesis left open.
The oscillation half of the cutoff-mean row, closed by domination #
The oscillation half of the second cutoff-mean row of
p.response.transfer is bounded by the source load, but the estimate asks for an annealed bound
on the descendant
remainder, and that bound would need the annealed mean of the cell average of a POINTWISE modulus
of the optimizer state, which the standing assumptions do not supply: the ellipticity constants of
a sample are not bounded in the law.
The passage to the limit does not need it. The remainder tends to zero along every sample from
the pathwise ellipticity alone, and the generation increments see only CELL AVERAGES of the state,
so the scale-average seminorm of those averages is an integrable envelope for the whole series.
This module restates the oscillation half through
integrable_and_abs_integral_sub_integral_le_of_dominated_descendant, which replaces the annealed
remainder bound by those two hypotheses, and additionally delivers the P-integrability of the
annealed functional.
The oscillation half of the cutoff-mean row is bounded by the source load, by domination.
The hypotheses of abs_integral_sub_cellPart_le_descendant_load with its annealed remainder bound
|∫ rem N| ≤ C₀ 3^{-N} M replaced by the pathwise vanishing rem N a → 0 and an integrable
envelope env for the generation increments, and with the integrability of Phi replaced by its
almost-everywhere strong measurability. The conclusion carries that integrability as well as the
bound.
The oscillation half of the cutoff-mean row with the envelope supplied #
CutoffOscillationHalfClosure bounds the oscillation half of the cutoff-mean row of
p.response.transfer
by the source load, from a pathwise vanishing remainder and an increment envelope with four
properties. DescendantEnvelopeBound builds that envelope out of the data the row already
carries: the
cutoff weights, the head energies and the cell energies of the generation cells. This module puts
the two together, so that the oscillation half consumes only the carriers' own inputs and no
envelope has to be exhibited at the call site.
The oscillation half of the cutoff-mean row, with the envelope built in. The hypotheses
of integrable_and_abs_integral_sub_cellPart_le_descendant_load with its four envelope conditions
removed: the Young splitting of the Cauchy--Schwarz product of the head energy and the cell energy
supplies them from the weight bound hθ, the pairing bound hbd, the annealed bounds hGsq and
hDval and the source-load summability hsumm that the row already carries.
The oscillation half of the cutoff-mean row at the carriers #
CutoffOscillationHalfClosure bounds the oscillation half of the cutoff-mean row of
p.response.transfer
by the source load from the carriers' own data. This module supplies that data for the terminal
optimizer family of the response cell: the refinement identity at every depth, the cutoff weight
increment, the pathwise crossed pairing bound on each descendant cell, the nonnegativity and the
annealed value of the descendant cell energies, the annealed head of each descendant generation,
and the pathwise vanishing of the remainder.
What is left as a hypothesis is one per-descendant-cell annealed readout, the crossed pairing, together with the measurability of the annealed functional and the integrability of its cell part.
The oscillation half of the cutoff-mean row at the carriers, minus sign. The
(φ-1)-weighted cell average of the crossed pairing of the deterministic dual variable with the
terminal optimizer state is P-integrable, and its annealed mean differs from that of its
depth-H cell part by at most 32 d² Θ 3^{-H} (3^{3/2} 𝓛_s^-)^{1/2} (8 E[J_t^-])^{1/2} times the
geometric factor.