# Jamming exponent identity — H3

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 \(p:\mathbb R\to(0,\infty)\) be continuous and 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.

Then
\[
\int_{\mathbb R}p(f')^2\,dt=2\int_{\mathbb R}pf^2(1-f)\,dt>0.
\]