feat(home): open-tab column on the desktop home screen; bigger phone home button

The welcome overlay centers ~560px of content in a ~1400px window, so both
gutters are dead space. The left one now carries the open tabs as a vertical
list (home-sessions.js): one row per live session plus saved web tabs, in TAB
order rather than by urgency, because the row badges are the Alt+1..9 indices.
Clicking a row enters that session.

Working state is deliberately the phone's, exactly: a pulsing green dot ringed
by the same tab-load-spin the tab strip uses while a tab loads, now with a green
halo added on both surfaces so "working" reads identically wherever you see it.

The column is position:absolute so the centered content never moves, which is
why it needs a width gate in two places (HOME_SESSIONS_MIN_WIDTH = 1180 in JS,
a max-width: 1179px media query as the backstop for a resize that outruns the
matchMedia listener). A test pins the two equal. State classification is reused
from mobile-overview.js rather than re-derived, so the two home screens cannot
disagree about what counts as needing you.

Phones keep the mobile overview, and their brand "C" was a 0.85rem inline span,
roughly a 12x13px target on the one control that gets you back to that screen.
It is now a 44px-wide button filling the full header height, with the glyph
scaled to match. 44 is horizontal only: the phone header is pinned to 36px and
clips overflow, so a true 44x44 would mean taking height off the terminal.

Verified end to end against a real isolated instance (own tmux socket + data
dir): 18 browser checks covering render, live update through the tab renderer,
the working dot's animation/glow/ring, row click, the narrow-window gate, the
phone fallback, and a real touch tap on the far corner of the new hit box.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>

src/web/public/index.html

@@ -322,6 +322,11 @@
 
       <!-- Welcome Overlay (shown when no session active) -->
       <div class="welcome-overlay" id="welcomeOverlay">
+        <!-- Open tabs as a vertical column in the left gutter (home-sessions.js).
+             Absolutely positioned so the centered content below never moves, and
+             therefore only rendered where the gutter is wider than the column;
+             ships `hidden` and only that module reveals it. -->
+        <aside class="home-sessions" id="homeSessions" hidden></aside>
         <div class="welcome-content">
           <h1 class="welcome-title">Codeman</h1>
           <p class="welcome-desc">Manage AI Coding tools in persistent tmux sessions.</p>
@@ -2769,6 +2774,7 @@
   <script defer src="session-ui.js"></script>
   <script defer src="webview-tabs.js"></script>
   <script defer src="mobile-overview.js"></script>
+  <script defer src="home-sessions.js"></script>
   <script defer src="entrance-animations.js"></script>
   <script defer src="ralph-wizard.js"></script>
   <script defer src="api-client.js"></script>