claimroutineinferred_dep

Interior energy lower bound via ball quadratic

Bounds

\mathcal F

below, losing only

\sigma^{-m}e^{-C_3\gamma^2\sigma}

denominator for the potential and the output bound

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

Statement (paper)

For every

T\in\mathbb R

and

2a\le s=|\tau|\le 3

, with

\sigma=s/h

, there are a fixed power

m

and constants

c_\gamma,C_3>0

such that

\mathcal F(T,\tau)\ge c_\gamma a^2\sigma^{-m}e^{-C_3\gamma^2\sigma}\|g_T\|_{\pc}^{\pc}

. Here at each point

X

the depth

-u

at the center of the ball of radius

h

satisfies

(-u)\mu\ge\epsilon h^2 m_X^{1+1/d}

, where

m_X

is the density lower bound .

In plain English

For every

T

and

2a\le s=|\tau|\le 3

, with

\sigma=s/h

, the fiber energy satisfies

\mathcal F(T,\tau)\ge c_\gamma a^2\sigma^{-m}e^{-C_3\gamma^2\sigma}\|g_T\|_{\pc}^{\pc}

for some fixed power

m

and constants

c_\gamma,C_3>0

Why it's here

This is the interior lower bound on the energy, which gives at

s=2a

and, divided into the wall-decay bound, gives the potential

V(s)

in . The idea is to compare

\mathcal F

with

X_0

by the energy comparison lemma, and to bound

X_0

below by integrating the density lower bound

m_X

on each

h

-ball. The depth

-u

at the ball center is controlled through the Dirichlet barrier bounds and the fiber geometry (Jacobian and determinant), giving

(-u)\mu\ge\epsilon h^2 m_X^{1+1/d}

, and summing over centers recovers

\|g_T\|_{\pc}^{\pc}

Classified routine:

A bookkeeping step inside the fixed-fiber barrier proof. It combines the energy comparison lemma, the Dirichlet barrier bounds and the fiber Jacobian/determinant geometry with the Gaussian density lower bound on h-balls, then sums over ball centers to recover the norm of g_T.

Uses

thmDirichlet problem for quaternionic Monge-AmpèrelemFiber energy comparable to X_0claimInterior density lower bound by Gaussian inversionlemFiber Hessian geometry and Jacobian

Used by

claimChoosing gamma and K yields the barrier inequalities

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:202

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Interior energy lower bound via ball quadratic" rests on claim-prop-fixed-barrier-6
info
inferred_dep
outline: "Interior energy lower bound via ball quadratic" rests on lem:fiber-geometry