feat(apes-sidebar): Phase 2 — badge 计数 + auto-collapse（visible 门控暂停）

- apes:tasks badge = running 会话数；apes:agents badge = subagent 数
  （纯 (ctx)=>number|null 函数，0→null 隐藏，host 封顶 99+）
- useSessionList 接受 visible：非活动 tab 切 no-op subscribe，
  停止 live session-list churn 重渲染隐藏面板（better-sidebar
  'live views pause otherwise' 契约 = auto-collapse）
- tests/tabs.spec.ts：新增 5 个 badge 契约测试（6→11），mock
  sessions feed 直测 badge 纯函数
- AGENTS.md：Phase 2 标 partial（per-tab state / scratch 冒烟 /
  apps·messages·profile feed 延后，better-sidebar 0.18.0 无相应 API/数据源）

门禁：typecheck rc=0 · build rc=0（client.js 15.74→20K）· test 25/25

tests/tabs.spec.ts

@@ -28,6 +28,7 @@ interface Registered {
   order?: number
   single?: boolean
   component: unknown
+  badge?: (ctx: unknown, scope: unknown, state: unknown) => string | number | null | undefined
 }
 
 /**
@@ -154,3 +155,80 @@ describe('apes-sidebar client tab registration (Phase 1)', () => {
     }
   })
 })
+
+/**
+ * Build a mock `ctx.sessions.list.getSnapshot()` feed for badge assertions.
+ * `byId` maps session id → a summary with optional `running` / `origin`.
+ */
+function makeMockSessionsCtx(byId: Record<string, { running?: boolean; origin?: string }>) {
+  const snapshot = { current: undefined as string | undefined, byId }
+  return {
+    sessions: {
+      list: {
+        getSnapshot: () => snapshot,
+        subscribe: () => () => undefined,
+      },
+    },
+  }
+}
+
+/**
+ * Phase 2 badge contract: `apes:tasks` shows the running-session count,
+ * `apes:agents` shows the subagent count, and BOTH hide (return null) when
+ * their count is 0. The badge is a pure `(ctx) => …` function, so it is
+ * exercised directly against a mock sessions feed — no DOM, no rendering.
+ */
+describe('apes-sidebar tab badges (Phase 2)', () => {
+  /** Register into a fresh mock registry and return the captured descriptors. */
+  function registeredDescriptors() {
+    const { registry, registered } = makeMockRegistry()
+    const eff = makeEffectHarness()
+    apply({ betterSidebar: registry, locale: makeMockLocale(), ...eff } as never)
+    return registered
+  }
+
+  function badgeOf(registered: Registered[], id: string) {
+    return registered.find((d) => d.id === id)?.badge
+  }
+
+  it('apes:tasks badge is the running-session count (2 running → 2)', () => {
+    const badge = badgeOf(registeredDescriptors(), 'apes:tasks')
+    expect(badge).toBeTypeOf('function')
+    const ctx = makeMockSessionsCtx({
+      a: { running: true },
+      b: { running: true },
+      c: { running: false },
+    })
+    expect(badge!(ctx as never, {} as never, {} as never)).toBe(2)
+  })
+
+  it('apes:tasks badge is null when nothing is running (hidden pill)', () => {
+    const badge = badgeOf(registeredDescriptors(), 'apes:tasks')
+    const ctx = makeMockSessionsCtx({ a: { running: false }, b: {} })
+    expect(badge!(ctx as never, {} as never, {} as never)).toBeNull()
+  })
+
+  it('apes:agents badge is the subagent count (1 subagent → 1)', () => {
+    const badge = badgeOf(registeredDescriptors(), 'apes:agents')
+    expect(badge).toBeTypeOf('function')
+    const ctx = makeMockSessionsCtx({
+      parent: {},
+      child: { origin: 'subagent' },
+      other: { running: true }, // running but NOT a subagent → not counted
+    })
+    expect(badge!(ctx as never, {} as never, {} as never)).toBe(1)
+  })
+
+  it('apes:agents badge is null when there are no subagents (hidden pill)', () => {
+    const badge = badgeOf(registeredDescriptors(), 'apes:agents')
+    const ctx = makeMockSessionsCtx({ a: {}, b: { running: true } })
+    expect(badge!(ctx as never, {} as never, {} as never)).toBeNull()
+  })
+
+  it('apps/messages/profile have NO badge (information placeholders, no count)', () => {
+    const registered = registeredDescriptors()
+    for (const id of ['apes:apps', 'apes:messages', 'apes:profile']) {
+      expect(badgeOf(registered, id)).toBeUndefined()
+    }
+  })
+})