# Periodic BBM NE-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 \(\mathbb T=\mathbb R/(2\pi\mathbb Z)\). Suppose \(u\in C(\mathbb R;H^1(\mathbb T;\mathbb R))\) satisfies
\[
(1-\partial_x^2)u_t+\partial_x(u+u^2/2)=0
\]
in distributions. If there exist \(C\in\mathbb R\), \(T_0>0\), and a nonempty open set \(\omega\subset\mathbb T\) such that \(u=C\) on \(\omega\times(0,T_0)\), then
\[
u(x,t)=C\qquad\text{for every }(x,t)\in\mathbb T\times\mathbb R.
\]

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.