High-contrast homogenization: Entry.Response.Limit.DescendantLimitDomination #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Passing to the descendant limit of the cutoff-mean row by domination #
The cutoff-mean rows of the cutoff estimate pair the cutoff's within-cell oscillation with the
crossed pairing ⟨Y₂,∇v⟩ + ⟨Y₁,a∇v⟩ of the optimizer state with the annealed mean, a quantity with
no sign; the exact partition identity underlying the cell decomposition still applies to this
signed integrand, and refining the terminal cell down to generation t-(H+N) leaves a defect
bounded by 32 d^2 Θ 3^{-(H+N)} times the terminal cell average of its modulus. Passing N to
infinity does not need an annealed bound on this remainder: it is enough that the remainder tends
to zero along every sample, which it does, since the bound above is a fixed constant times
3^{-N}, and this pathwise vanishing is exactly the hypothesis a dominated-convergence argument
needs to replace the depth-(H+N) cell part, in the limit, by the full oscillation half of the
row. A Cauchy-Schwarz estimate over the generations splits the geometric weight 3^{-n} of the
descendant sum as 3^{-n/4}·3^{-3n/4} so as to pair it against the source-load weights.
This is the descendant-limit step of the cutoff-mean row of the response transfer, and it serves
p.response.transfer.
The cutoff cell decomposition against a signed integrand #
The cutoff-mean rows of p.response.transfer pair the cutoff fluctuation with the crossed
pairing ⟨Y₂, ∇v⟩ + ⟨Y₁, a∇v⟩ of the optimizer state with the annealed mean, which has no sign.
The exact partition identity still writes the (φ - 1)-weighted average of a density f as the
normalized sum of the products of the cell means of φ - 1 and of f, and on each depth-n
descendant cell the cutoff deviates from its own cell mean by at most
32 d² responseCutoffProfileConst 3^{-n}. Because the integrand is signed, the cellwise
oscillation bound is applied against the cell average of |f|, so the remainder is controlled by
the terminal cell average of the absolute integrand rather than by the average of f itself.
The cutoff cell decomposition against a signed integrand. Replacing the cutoff by its
cell averages on the depth-n triadic subdivision of the terminal cell costs at most the cutoff
oscillation 32 d² Θ 3^{-n} times the terminal cell average of the absolute integrand. This is
the cell decomposition of the cutoff-mean rows of p.response.transfer.
The vanishing remainder of the descendant refinement #
The oscillation half of the cutoff-mean row of p.response.transfer refines the terminal cell
down to the generation t - (H + N) and keeps the defect of the depth-(H+N) cell part as a
remainder. The cell decomposition against a signed integrand bounds that defect by
32 d² Θ 3^{-(H+N)} times the terminal cell average of the modulus of the integrand; splitting the
power as 3^{-H} · 3^{-N} exhibits the printed factor 3^{-H} of the row and leaves a remainder
that vanishes geometrically in the refinement depth N. The (φ-1)-weighted integrand is
integrable wherever the integrand is, a response cutoff taking values in [0, 2].
The (φ-1)-weighted integrand is integrable wherever the integrand is. A response cutoff
is smooth and takes values in [0, 2], so |φ - 1| ≤ 1 and multiplying by φ - 1 preserves
integrability on any measurable set of finite volume.
The remainder of the descendant refinement carries the printed 3^{-H}. Replacing the
cutoff by its cell means on the depth-(H+N) triadic subdivision of the terminal cell costs at
most (32 d² Θ 3^{-H}) · 3^{-N} times the terminal cell average of the modulus of the integrand.
This is the remainder of the oscillation half of p.response.transfer, which vanishes as the
refinement depth N grows.
The descendant limit by domination instead of a remainder bound #
The descendant telescoping of the cutoff-mean row of p.response.transfer writes the annealed
functional, at every refinement depth, as its cell part plus the first N generation increments
plus a remainder. The passage to the limit does NOT need an annealed bound on the remainder: it
needs only that the remainder tends to zero along every sample, together with an integrable
envelope for the whole series of increments. Since the increments are flat averages of pairings
of a deterministic dual variable with cell averages of the optimizer state, such an envelope is
supplied by the scale-average seminorm of those cell averages, whereas a bound on the remainder
would need the annealed mean of a pointwise modulus. This module records the limit in that form,
and concludes the integrability of the annealed functional as well as the bound.
The descendant limit from a pathwise vanishing remainder and an integrable envelope. If a
sample functional Phi splits, at every refinement depth N, into a cell part Tcell, the first
N generation increments and a remainder rem N; if the remainder tends to zero along every
sample; if the increments are dominated by a nonnegative family g whose sample sum is
P-integrable; and if every partial sum of the annealed increments is at most B; then Phi is
P-integrable and its annealed mean differs from that of the cell part by at most B.
This replaces the annealed remainder bound of
abs_integral_sub_integral_le_of_descendant_bound by hypotheses that the carriers supply: the
pathwise vanishing of the remainder needs only ellipticity along each sample, and the envelope
needs only the cell averages of the state, never a pointwise modulus.
A geometrically small pathwise remainder vanishes. If the modulus of rem N a is at most
C a · 3^{-N} for a finite sample constant C a, then rem N a → 0 along every sample. This is
the hrem0 hypothesis of
integrable_and_abs_integral_sub_integral_le_of_dominated_descendant, and it is all that the
descendant remainder of the cutoff-mean row has to supply: the sample constant is the cell average
of the modulus of the crossed pairing, which is finite along every sample by pathwise ellipticity
but has no annealed bound.
The descendant remainder of the cutoff-mean row vanishes along every sample #
The passage to the limit in the oscillation half of the cutoff-mean row of p.response.transfer
needs only that the depth-(H+N) remainder tends to zero along every sample. It does: the
remainder is at most (32 d² Θ 3^{-H}) 3^{-N} times the terminal-cell average of the modulus of
the crossed pairing, and that average is a finite constant of the sample — finite by the pathwise
ellipticity of the coefficient, with no bound uniform in the sample and no integrability in the
law. This module records the resulting vanishing for both signs, which is the hrem0 hypothesis
of the dominated descendant limit.
The descendant remainder vanishes along every sample, minus sign. For the terminal
optimizer family of the recentred coefficient a_- = a - g and a deterministic dual variable Y,
the difference between the (φ-1)-weighted terminal cell average of the crossed pairing and its
depth-(H+N) cell part tends to zero as N → ∞, for every sample.
The descendant remainder vanishes along every sample, plus sign. For the terminal
optimizer family of the recentred coefficient a_- = a - g and a deterministic dual variable Y,
the difference between the (φ-1)-weighted terminal cell average of the crossed pairing and its
depth-(H+N) cell part tends to zero as N → ∞, for every sample.
The descendant sum against the source-load weights #
For the response transfer of p.response.transfer, the within-cell oscillation of the cutoff is
paired with the gradient and the flux at every descendant generation, the generation-n term
carrying the factor 3^{-H-n}. Summing the descendants against the source-load weights and
applying Cauchy--Schwarz across the generations splits 3^{-n} = 3^{-n/4} · 3^{-3n/4} and bounds
the descendant sum by the square root of the load times the square root of a convergent geometric
series.
Cauchy--Schwarz across the descendant generations bounds the descendant sum
∑ n, 3^{-n} √(L n) by the square root of the source-load weighted sum ∑ n, 3^{-3n/2} L n,
multiplied by the square root of the convergent geometric series ∑ n, 3^{-n/2}. Splitting
3^{-n} = 3^{-n/4} · 3^{-3n/4} is what keeps the printed factor 3^{-H}(E[J_t]𝓛_s)^{1/2} of
p.response.transfer.