claimroutineinferred_dep

Current and density bounds with wall decay

Bounds current by

s^{-2d}\sigma^mP_*^d

, with

e^{-c_0\sigma}

wall decay

pointwise current bounds for energy and wall flux

Claim·§5 Estimating the energy on a fixed fiber·inside propEnergy barrier on a fixed fiber

Statement (paper)

There are constants

C,m,c_0>0

, with

c_0

independent of

\gamma

, such that the wave current satisfies

\|A_{\mathrm w}\|\le Cs^{-2d}\sigma^m P_*^d

for

r\le0

and

\|A_{\mathrm w}\|\le Cs^{-2d}\sigma^m e^{-c_0\sigma}P_*^d

for

r=0

, where

\|A_{\mathrm w}\|

is the matrix operator norm. Consequently the regularized density satisfies

\mu\le C s^{-2d}\sigma^m P_*^d

for

r\le0

In plain English

The wave current's matrix norm satisfies

\|A_{\mathrm w}\|\le Cs^{-2d}\sigma^m P_*^d

for

r\le0

, with an extra factor

e^{-c_0\sigma}

on the wall

r=0

, where

c_0

is independent of

\gamma

. Consequently the regularized density obeys

\mu\le Cs^{-2d}\sigma^m P_*^d

for

r\le0

Why it's here

This turns the positive, divergence-free wave current into pointwise bounds in terms of the smoothed density

P_*

, using the fiber Hessian and Jacobian control for the geometry. The polynomial bound on

r\le0

and the Gaussian decay at the wall are the inputs for the later quotient bound

u/r

, the energy and wall-flux bounds, and the interior density lower bound.

Classified routine:

A technical step inside the fixed-fiber energy barrier proof. It bounds the wave current pointwise by combining the positivity and divergence-free structure from prop-current, the Jacobian control from lem-fiber-geometry, and the regularity of the smoothed density P_* from the previous claim, so it is bookkeeping built on earlier nodes.

Uses

propWave current is positive and divergence-freelemFiber Hessian geometry and JacobianclaimSmoothed density P_* is regular and controlled

Used by

claimUpper bound on boundary quotient u/rclaimEnergy and wall flux boundsclaimInterior density lower bound by Gaussian inversion

Figures

Panel (a) draws the fiber in modulus coordinates: the horizontal axis is $|W|$ with tick $w_0$, and the vertical axis is $|Q|=|\Phi|/a$ with ticks $1/20$ and $1/10$. A quarter-ellipse arc labeled $r=0$ runs from $(0,1/10)$ to $(w_0,0)$ and bounds the shaded region $r<0$. A dotted line at height $1/20$ meets the arc, and the part of the arc below it is drawn heavy and labeled "flux wall". A small dashed box near the origin, containing a smaller solid box, is marked (b). Panel (b) enlarges that corner: a dashed rectangle $|W|\le2\gamma$, $|Q|\le1/100$, labeled "Gaussian inversion, $g(Q)=1$", contains a darker solid rectangle $|W|\le\gamma$, $|Q|\le1/400$, labeled "centers". The figure shows that the region where the wall flux lives is separated from the central region where the Gaussian multiplier is inverted. — Figure 1schematicp. 19
The fixed fiber in modulus coordinates. The wave flux through $r=0$ is supported on the heavy arc, where $|Q|\le1/20$ and hence $|W|\ge\sqrt{3}\,w_0/2$ . This separation gives Gaussian decay $e^{-cs/h}$ even for $|v|\le w_0/3$ . In the enlargement (b), once $K\gamma$ is large, the radius- $h$ balls about the marked centers project into the dashed patch. There $|W-v|\le4\gamma$ on the enlarged cap $|v|\le2\gamma$ , so Gaussian inversion costs at most $e^{C\gamma^2s/h}$ . The aperture $\gamma$ is fixed independently of $h$ ; its size relative to $w_0$ is schematic.
What it shows
Panel (a) draws the fiber in modulus coordinates: the horizontal axis is $|W|$ with tick $w_0$ , and the vertical axis is $|Q|=|\Phi|/a$ with ticks $1/20$ and $1/10$ . A quarter-ellipse arc labeled $r=0$ runs from $(0,1/10)$ to $(w_0,0)$ and bounds the shaded region $r<0$ . A dotted line at height $1/20$ meets the arc, and the part of the arc below it is drawn heavy and labeled "flux wall". A small dashed box near the origin, containing a smaller solid box, is marked (b). Panel (b) enlarges that corner: a dashed rectangle $|W|\le2\gamma$ , $|Q|\le1/100$ , labeled "Gaussian inversion, $g(Q)=1$ ", contains a darker solid rectangle $|W|\le\gamma$ , $|Q|\le1/400$ , labeled "centers". The figure shows that the region where the wall flux lives is separated from the central region where the Gaussian multiplier is inverted.
Read from the image by claude-opus-5.5 · confidence 85%
SupportspropEnergy barrier on a fixed fiberclaimCurrent and density bounds with wall decayclaimInterior density lower bound by Gaussian inversionclaimInterior energy lower bound via ball quadraticclaimChoosing gamma and K yields the barrier inequalities

Sources

PaperOpen

Section: §5 Estimating the energy on a fixed fiber
Page: —
Label: —
Line: sections/barrier.tex:66

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Current and density bounds with wall Gaussian decay" rests on claim-prop-fixed-barrier-1