# Jamming exponent identity SM-1-H1-to-H5

## Setting

You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.

## Problem

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.

Prove H5: 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 \(f(t)=H_-(t)+cJ''(t)/2\).

Let \(a\in\mathbb R\). Suppose \(p\in C^1(\mathbb R)\) is strictly positive, is \(C^2\) on each open half-line, and satisfies
\[
\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\frac{c^2}{2}(p'J'+pJ'')\qquad(t\ne0).
\]
Assume \(p,p'\) grow at most polynomially.

Suppose there are \(\gamma_\infty>0\), positive scalars \(\gamma(y),A_y\), and real scalars \(I_y\ne0,Q_y\), for \(y\ge1\), such that
\[
\gamma(y)y^c\longrightarrow\gamma_\infty,\qquad
yI_y=\frac{\gamma(y)}2Q_y,
\]
\[
\frac{I_y}{A_y}\longrightarrow\int_{\mathbb R}pf^2(1-f)\,dt,\qquad
\frac{\gamma_\infty Q_y}{A_y y^{2b}}\longrightarrow\int_{\mathbb R}p(f')^2\,dt.
\]
The integrals are improper integrals over \(\mathbb R\setminus\{0\}\), and both limits are finite real numbers.

Let \(d,r:[0,\infty)\to\mathbb R\) be continuous, with \(d>0\) and \(r\ge0\). Let \(v:[0,\infty)\times\mathbb R\to\mathbb R\) be continuous and locally Lipschitz in space, and satisfy \(|v(\tau,h)|\le C_T(1+|h|)\) for \(0\le\tau\le T\), for each finite \(T\).
Suppose \(F\in C([0,\infty)\times\mathbb R)\cap C^{1,2}((0,\infty)\times\mathbb R)\) is bounded on each strip \([0,T]\times\mathbb R\) and satisfies
\[
F_\tau=d(\tau)F_{hh}+v(\tau,h)F_h+r(\tau)F(1-F),\qquad
0\le F(0,h)\le1\quad(h\in\mathbb R).
\]

Suppose there are \(\tau_n\ge0\) and maps \(h_n:\mathbb R\setminus\{0\}\to\mathbb R\) with
\[
F(\tau_n,h_n(t))\longrightarrow f(t)\qquad(t\ne0).
\]
Then
\[
a+b=1.
\]

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.