The every-cell bound of the origin carrier, at the margin scale #
The origin-carrier cell output asks for a bound on every Morrey cell of the
selected pressure gradient, at every centre and every positive radius. The
one-sided transfer supplies it from two inputs: a growth bound on cells below a
fixed scale ρ₀, and the total integral on the carrier cylinder. The scale
ρ₀ is a free parameter of that transfer, and choosing it is exactly the choice
of where the two regimes meet.
Choosing ρ₀ = R₁, the radius of the carrier itself, makes the small-cell input
impossible to supply, for two independent reasons.
- In time, a cell centred at
zwithz.2 ∈ Ioc (-(3/4) ^ 2) 0and radius up toR₁ < 3/4has windowIoc (z.2 - R₁ ^ 2) z.2, which reaches below-9/16 - R₁ ^ 2and hence below-1. Nothing in the hypotheses of the estimate controls the solution before time-1. - In space, the slice estimate at a cell of radius
ris applied on a cylinder of radius2 * r, and forrcomparable withR₁that doubled ball leaves the ball of radiusR₀on which the pressure slice data lives.
Choosing instead the margin scale ρ₀ = (1 - R₁) / 4 removes both. The
region the domain hypothesis controls is the closed unit cylinder, so the
governing margin is the one between the carrier radius R₁ and 1, not the one
between R₁ and R₀. At that scale 3 * r ≤ 1 - R₁, so a cell meeting the
carrier has its doubled ball inside the unit ball; and r ≤ 1/4, so
(2 * r) ^ 2 ≤ 1/4 and every doubled window stays inside Ioc (-1) 0. Cells at
or above the margin scale are not reached by the slice estimate at all and are
bounded by the total integral on the carrier, with the explicit scale factor that
the one-sided transfer constant carries.
The margin scale must not be allowed to degenerate. Because R₁ < 3/4, the
scale (1 - R₁) / 4 lies in [1/16, 1/4), bounded away from zero uniformly in
the admissible data. A scale proportional to R₀ - R₁ would not be: by
oneSidedMorreyBound_scale_eq, measuring the transfer constant at a scale ρ₀
instead of at R₁ inflates the whole-carrier constant by exactly
(R₁ / ρ₀) ^ (5 (1 - (6/5)/κ)), and with ρ₀ = (R₀ - R₁) / 3 that factor is
unbounded as R₁ ↑ R₀, which would make the comparison with the explicit
majorant of the estimate unsatisfiable at the top of the admissible range. At
ρ₀ = (1 - R₁) / 4 the factor is at most 12 ^ (5 (1 - (6/5)/κ)).
Nothing else in the estimate changes: the transfer, the carrier, the field and the conclusion are the established ones.
The margin scale of the origin carrier is bounded below uniformly in the admissible data. This is what keeps the comparison with the explicit majorant of the estimate from degenerating.
Every window of a margin-scale cell lies inside the unit time interval.
This is the satisfiability certificate of the small-cell regime: the data
hypotheses of the estimate control the solution only on Ioc (-1) 0, and a cell
centred anywhere in the cylinder of admissible centres, at a radius at most the
margin scale, never reaches outside that interval.
The origin-carrier cell output from the two regimes, with the small-cell
input taken only below a free scale ρ₀. The centres are the ones the one-sided
transfer consumes, so the statement composes with the established transfer without
change; only the radius threshold moves from R₁ to ρ₀.
The origin-carrier cell producer from margin-scale past-cylinder slice data. This is the established past-slice route with the growth clause taken only at the margin scale, where it is supplyable, and with the resulting transfer constant compared against the explicit majorant of the estimate. The field is bound before every clause that mentions it, and the two integral constants are bound with it.