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/terminal-ui.js

@@ -1427,6 +1427,7 @@ Object.assign(CodemanApp.prototype, {
     if (this.shouldUseMobileOverview?.()) {
       const overlay = document.getElementById('welcomeOverlay');
       if (overlay) overlay.classList.remove('visible');
+      this.hideHomeSessions?.();
       this.showMobileOverview();
       this._updateCjkInputState?.();
       return;
@@ -1439,6 +1440,9 @@ Object.assign(CodemanApp.prototype, {
       this.applyWelcomeCliVisibility();
       this.loadHistorySessions();
       this.initSearchPanel();
+      // Open tabs down the left gutter. Self-gating: a window too narrow to hold
+      // the column without overlapping the content leaves it hidden.
+      this.showHomeSessions?.();
     }
     // Home screen has no input target — hide the CJK textarea (activeSessionId
     // is null by the time we get here). Guarded: defined on the app object.
@@ -1447,6 +1451,7 @@ Object.assign(CodemanApp.prototype, {
 
   hideWelcome() {
     this.hideMobileOverview?.();
+    this.hideHomeSessions?.();
     const overlay = document.getElementById('welcomeOverlay');
     if (overlay) {
       overlay.classList.remove('visible');