fix: apply the review items for #691 at merge

test: pin the double-quoted PATH literal of withRemoteTmuxPath() (dash 0.5.11+, bash and busybox pass the shell cases unquoted)
docs: note in architecture-invariants that a new remote pane inherits the widened PATH from the creating tmux client

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

test/remote-tmux-path.test.ts

@@ -8,6 +8,7 @@ import {
   buildRemoteTmuxCheckCommand,
   buildRemoteSessionAliveCommand,
   buildRemoteListSessionsCommand,
+  withRemoteTmuxPath,
 } from '../src/remote-hosts.js';
 import {
   buildRemoteLaunchCommand,
@@ -106,6 +107,15 @@ esac
   });
   afterAll(() => rmSync(root, { recursive: true, force: true }));
 
+  // Literal pin for the quotes. dash 0.5.11+, bash and busybox do not field-split
+  // `export` arguments, so the real-shell cases below also pass with an unquoted
+  // PATH, and the other builder tests compute their expectation from the wrapper.
+  it('double-quotes the whole exported PATH value', () => {
+    expect(withRemoteTmuxPath('tmux -V')).toBe(
+      'export PATH="$HOME/.opencode/bin:$HOME/.local/bin:$HOME/bin:/opt/homebrew/bin:/usr/local/bin:/usr/bin:/bin:/usr/sbin:/sbin:$PATH"; tmux -V'
+    );
+  });
+
   for (const shell of ['/bin/sh', ...(existsSync('/bin/dash') ? ['/bin/dash'] : [])]) {
     for (const item of cases) {
       it(`${item.name} finds the remote-user tmux with ${shell} and preserves targeting/options`, () => {