Sync the bottom bar to screen transitions and scroll

## Summary

- Hook the bottom bar up to react-native-screens' transition progress. Every `Layout.Screen` now exposes a `presence` shared value (via `useScreenPresence`) that follows the native push/pop/swipe frame by frame, so screens with `minimalShell` hide and reveal the bar in sync with the transition instead of after it. Fixes the delayed tab bar when leaving a conversation.
- Replace the minimal-shell and hide-border refcounts with a small registry that sums 0..1 contributions on the UI thread. The bottom bar border now fades with the screen rather than toggling.
- Bring back the scroll-linked bottom bar on Home (removed in #5203). The header and footer share the same drag-linked value, and the bar is revealed on feed change, blur, and app foreground so badges aren't missed. Web included.

## Test plan

- [ ] Swipe back from a conversation: bar tracks the swipe, including a cancelled swipe
- [ ] Back button from a conversation: bar animates with the pop
- [ ] Push a profile over a conversation and pop back
- [ ] Post thread: bottom bar border fades in/out with the screen
- [ ] Home: scroll down hides header and bar together, scroll up shows them
- [ ] Home: scrolled down, swipe into a convo and back - no jump
- [ ] Home: bar reappears on feed switch, on leaving Home, and on foregrounding the app
- [ ] Deep link / notification into a conversation from another tab
- [ ] Mobile web: bar hides/shows on scroll and on conversation screen
- [ ] Rapid swipe-back and swipe-back while scrolling (software-mansion/react-native-reanimated#9402, software-mansion/react-native-screens#2659)

Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01JWQciPDiipBhqiW922Nivs

src/view/com/util/MainScrollProvider.tsx

@@ -13,6 +13,7 @@ import {useSafeAreaInsets} from 'react-native-safe-area-context'
 import {EventEmitter} from 'eventemitter3'
 
 import {ScrollProvider} from '#/lib/ScrollContext'
+import {useMinimalShellScrollMode} from '#/state/shell/minimal-mode'
 import {useShellLayout} from '#/state/shell/shell-layout'
 import {IS_LIQUID_GLASS, IS_NATIVE, IS_WEB} from '#/env'
 
@@ -95,6 +96,7 @@ export function useHomeHeaderTransform() {
 export function MainScrollProvider({children}: {children: React.ReactNode}) {
   const {headerHeight} = useShellLayout()
   const headerMode = useHomeHeaderMode()
+  const footerScrollMode = useMinimalShellScrollMode()
   const {top: topInset} = useSafeAreaInsets()
   const headerPinnedHeight = IS_LIQUID_GLASS ? topInset : 0
   const startDragOffset = useSharedValue<number | null>(null)
@@ -104,14 +106,14 @@ export function MainScrollProvider({children}: {children: React.ReactNode}) {
   const setMode = useCallback(
     (v: boolean) => {
       'worklet'
-      headerMode.set(
-        withSpring(v ? 1 : 0, {
-          ...Reanimated3DefaultSpringConfig,
-          overshootClamping: true,
-        }),
-      )
+      const target = withSpring(v ? 1 : 0, {
+        ...Reanimated3DefaultSpringConfig,
+        overshootClamping: true,
+      })
+      headerMode.set(target)
+      footerScrollMode.set(target)
     },
-    [headerMode],
+    [headerMode, footerScrollMode],
   )
 
   useEffect(() => {
@@ -216,6 +218,7 @@ export function MainScrollProvider({children}: {children: React.ReactNode}) {
         if (newValue !== headerMode.get()) {
           // Manually adjust the value. This won't be (and shouldn't be) animated.
           headerMode.set(newValue)
+          footerScrollMode.set(newValue)
         }
       } else {
         if (didJustRestoreScroll.get()) {
@@ -239,6 +242,7 @@ export function MainScrollProvider({children}: {children: React.ReactNode}) {
       headerHeight,
       headerPinnedHeight,
       headerMode,
+      footerScrollMode,
       setMode,
       startDragOffset,
       startMode,