High-contrast homogenization: Provider.Quenched.SmallContrastEchoWitness #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The echo witness: the construction recursion admits a staircase solution #
gate branch (c). The two-component scheduler's below-range
echo re-excites, and the certificate is stronger than an envelope-class
kill: the construction's hypothesis set — the recursion at the preserved-lag
anchor shiftedBaseIndex ns available from ns, the a-priori floor, the
origin initial datum (the stated condition), antitonicity, and a source that
still exceeds the floor on the pre-base block — admits an explicit
solution, the echo witness, whose value on the j-th octave of the inverse
anchor orbit is the tower echoTower (j+1) = A^(2^(j+1)-1)·δ₀^(2^(j+1)).
The octaves stretch geometrically (echoBoundary (j+1) − ns is at least
(4/3)^(j+1) times the burn-in m₀+1−ns), so at the fixed octave j₀
with (D+1)·3^(j₀) ≤ 4^(j₀) the witness value is a law-free positive
constant while any target M·3^(−mu·(n−T₀)) with T₀ ≤ ns + D·(m₀+1−ns)
is smaller once the burn-in is large, with the smallness threshold made
explicit by the logarithmic echo gap. Hence
NO method consuming only the construction hypotheses — the late-start/original-
anchor two-component scheduler included — can conclude a law-free rate at
a linear threshold. The first echo damps in magnitude (δ₀ → A·δ₀²),
exactly as the scheduler's design intuited; the tower records that the
demand at the octaves outruns the damping against any geometric target.
The tower of levels: each octave squares the previous level.