High-contrast homogenization: Entry.Annealed.TransportSetup #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
This module assembles the two-grid transport estimates behind p.two.grid.transport.
For the rounded grids it derives grid-ratio and eccentricity control from the dimension, the
operator-norm identity ‖bridgeMap A B‖ ^ 2 = blockOpNorm (normalizedBlock A B) for positive
definite normalizers, the covariance of normalizedBlock under scaling, and the pointwise and
Schatten comparisons that exchange one normalizer for another at the cost of the grid proximity.
It then regroups a Whitney partition of an adapted cell into a capped row, a finite boundary
range and the fine cells, and shows that the resulting series orders the normalized coarse block
by the target mean, has finite moments, a positive full-matrix sum, finite rows and an exact row
reindexing; a fixed enlargement of the old coordinates contains the entire new-grid target, with
aligned parents at every higher generation.
Dimension alone fixes the grid ratio; projective proximity controls eccentricity.
The C-star identity for arbitrary positive definite normalizers on either grid.
p.two.grid.transport. The printed product is spectrally equivalent to the inverse of
normalizedBlock H F; its bounds are reciprocal, 1/(1+δ) and 1/(1-δ).
The pointwise normalization change needed inside the joint maximum, in both directions.
p.two.grid.transport. Old cells contribute the old eccentricity; normalization
on the new grid contributes the new one. The source envelope retains its full
fixed-Q bound, needed by transport beyond the bridge's first-moment estimate.
The Whitney row constant is Cw = Cwhitney * K₀ at the consumer.
The diagonal profile uses O8 to cancel the actual terminal mean, then O5.
Actual target blocks have finite moments, positive order, and the stationary normalized mean.
A Whitney-type countable sum has genuine finite moments. The auxiliary bounded-window envelope supplies finiteness only, with no uniform norm claim.
Response defects on open cells give the normalized countable partition order.
The convergent series is supplied by transport_whitney_sum_mem.
Instantiate Whitney conclusion (i) on the actual target. The finite row, partition and boundary mass clauses discharge all countable-sum guards. The geometric constant is selected from K₀ before the law.
Regroup a convergent Whitney series into the cap row, the finite boundary range, and all fine cells. This also supplies the exact empty-boundary case.
Positivity survives the actual infinite full-matrix sum.
Reindex one actual pair-indexed Whitney row by its finite center set. No averaging or cardinality estimate is used in this identity.
Normalize a positive source tail by the actual positive terminal mean.
A fixed enlargement in old coordinates contains the whole new-grid target. Its size is chosen before either grid; no enlarged source window is assumed.
Iterated aligned parents exist at every higher integer generation.
Every old cell below k inside the target lies in one of the fixed, finite scale-k parents of the enlarged old target. The index family has its exact count.