# Jamming exponent identity SM-1-H0-to-H1

## Setting

You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.

## Problem

Prove H1: Let \(0<c<1\), \(b=(1+c)/2\), and \(H_-(t)=\mathbf1_{t<0}\) for \(t\ne0\).
Let \(J\in C^1(\mathbb R)\) be \(C^4\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy
\[
\frac c2J''=(-b+cH_-)tJ'+J-H_-+\frac{c^2}{4}(J')^2\qquad(t\ne0).
\]
For some \(C_0,q,\eta>0\), assume
\[
|J(t)-H_-(t)|+\sum_{j=1}^4|J^{(j)}(t)|
\le C_0(1+|t|)^qe^{-\eta t^2}
\]
for all sufficiently large \(|t|\). For \(t\ne0\), put \(f(t)=H_-(t)+cJ''(t)/2\).

The function \(f\) extends to \(C^2(\mathbb R)\) and satisfies
\[
\frac c2f''+\left((b-cH_-)t-\frac{c^2}{2}J'\right)f'+cf(1-f)=0
\]
on the entire real line, where the coefficient at \(t=0\) is defined by its continuous limit.

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.