feat(v0.6): add Home (⌂) button to reach Sessions/Projects from a session

Once a project/session tab was open, the home chooser (Sessions/Projects
segmented control) was hidden and only reachable by closing every tab — no
way back to the Projects view to open another project.

Add a ⌂ Home button to the tab bar that overlays the chooser on top of the
open tabs (homeForced flag; updateHomeView shows home when no tab OR forced,
hiding the terminal panes). Clicking ⌂ again, clicking a tab (activate resets
homeForced), or opening another session/project dismisses the overlay.
Closing the last tab resets the flag. The button highlights while active.

Tests: +3 (tabs.test.ts). Full suite 415 green; both typechecks clean;
verified in-browser (open tab → ⌂ → Projects panel reachable → ⌂ → terminal).

public/style.css

@@ -207,6 +207,26 @@ body {
   background: var(--surface-2);
 }
 
+/* v0.6: Home (⌂) button — opens the Sessions/Projects chooser over open tabs. */
+.tab-home {
+  background: none;
+  border: none;
+  color: var(--text-dim);
+  cursor: pointer;
+  font-size: 17px;
+  padding: 0 10px;
+  flex: none;
+  border-radius: 7px;
+}
+.tab-home:hover {
+  color: #fff;
+  background: var(--surface-2);
+}
+.tab-home.active {
+  color: var(--on-accent, #1a1408);
+  background: var(--accent);
+}
+
 /* toolbar */
 #toolbar {
   flex: 0 0 auto;

public/tabs.ts

@@ -59,6 +59,11 @@ export class TabApp {
   private readonly projects: ProjectsPanel
   private homeView: HomeView = 'sessions'
   private readonly segControl: HTMLElement
+  // When true, the home chooser is overlaid on top of the open tabs (so you can
+  // jump back to Sessions/Projects without closing anything). Reset whenever a
+  // tab is activated. Irrelevant while no tab is open (home shows regardless).
+  private homeForced = false
+  private homeBtn: HTMLButtonElement | null = null
 
   constructor(paneHost: HTMLElement, tabBar: HTMLElement) {
     this.paneHost = paneHost
@@ -104,7 +109,9 @@ export class TabApp {
    * When no tab is open: show the seg control and the selected sub-panel.
    */
   private updateHomeView(): void {
-    const empty = this.tabs.length === 0
+    // Home shows when there is no tab OR the user explicitly forced it (the ⌂
+    // button), letting you reach Sessions/Projects without closing a session.
+    const showHome = this.tabs.length === 0 || this.homeForced
 
     // Sync active-class on segmented control buttons.
     const btns = this.segControl.querySelectorAll<HTMLButtonElement>('.home-seg-btn')
@@ -112,8 +119,12 @@ export class TabApp {
       const view: HomeView = i === 0 ? 'sessions' : 'projects'
       btn.classList.toggle('active', view === this.homeView)
     })
+    // The ⌂ button is "pressed" only while overlaying home on top of open tabs.
+    this.homeBtn?.classList.toggle('active', this.homeForced && this.tabs.length > 0)
 
-    if (empty) {
+    if (showHome) {
+      // Overlay home: hide every terminal pane so the chooser is on top.
+      for (const t of this.tabs) t.session.hide()
       this.segControl.style.display = 'flex'
       this.launcher.setVisible(this.homeView === 'sessions')
       this.projects.setVisible(this.homeView === 'projects')
@@ -124,6 +135,19 @@ export class TabApp {
     }
   }
 
+  /** Toggle the home chooser overlay over the open tabs (the ⌂ button). No-op
+   *  when no tab is open (home is already showing). */
+  private toggleHome(): void {
+    if (this.tabs.length === 0) return
+    this.homeForced = !this.homeForced
+    if (this.homeForced) {
+      this.updateHomeView()
+    } else {
+      // Return to the active terminal (activate re-shows its pane + hides home).
+      this.activate(this.activeIndex)
+    }
+  }
+
   private saveHomeView(): void {
     try {
       localStorage.setItem(HOME_VIEW_KEY, this.homeView)
@@ -310,11 +334,13 @@ export class TabApp {
   activate(i: number): void {
     if (i < 0 || i >= this.tabs.length) return
     this.maybeAskNotify() // first switch is a user gesture — request notif permission
+    this.homeForced = false // showing a terminal dismisses any home overlay
     this.activeIndex = i
     const entry = this.tabs[i]
     if (entry) entry.hasActivity = false // viewing clears the unread dot
     this.tabs.forEach((t, idx) => (idx === i ? t.session.show() : t.session.hide()))
     this.tabs.forEach((t) => this.refreshTab(t)) // in-place class/text update, no rebuild
+    this.updateHomeView() // hide the seg control + home panels now a tab is active
     this.updateApprovalBar()
     this.persist()
   }
@@ -327,6 +353,7 @@ export class TabApp {
     if (this.tabs.length === 0) {
       // v0.5: closing the last tab returns to the home screen — no auto-blank tab.
       this.activeIndex = -1
+      this.homeForced = false // back to the natural "no tabs → home" state
       this.persist()
       this.rebuild() // updateHomeView() (in rebuild) shows the chooser
       this.updateApprovalBar()
@@ -459,6 +486,17 @@ export class TabApp {
   private rebuild(): void {
     this.tabBar.replaceChildren()
 
+    // Home (⌂): overlay the Sessions/Projects chooser over the open tabs so you
+    // can start another session/project without closing the current one.
+    const home = document.createElement('button')
+    home.className = 'tab-home'
+    home.textContent = '⌂'
+    home.title = 'Home — Sessions & Projects'
+    home.setAttribute('aria-label', 'Home — Sessions & Projects')
+    home.addEventListener('click', () => this.toggleHome())
+    this.tabBar.appendChild(home)
+    this.homeBtn = home
+
     this.tabs.forEach((entry, idx) => {
       const tabEl = document.createElement('div')
       entry.el = tabEl