# Jamming exponent identity SM-1-H0-to-H4

## 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 H4: 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
\[
F_\tau=d(\tau)F_{hh}+v(\tau,h)F_h+r(\tau)F(1-F),\qquad
0\le F(0,h)\le1\quad(h\in\mathbb R).
\]

Then \(0\le F(\tau,h)\le1\) for every \(\tau\ge0\) and \(h\in\mathbb R\).

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.