feat(ads): carry the banner scroll position across navigation

Per the owner's idea: instead of moving the banner out of the per-screen header
(which would change its position), remember the banner's "life stage" and resume
it on the next screen. The engine already keeps the message + rotation timing;
this adds the scroll offset:

- bannerEngine tracks the in-flight scroll (target, duration, start). On attach,
  if a scroll is still running, it computes the current offset and calls the new
  host's resumeScroll(fromTx, toPx, remaining) — the view jumps to the carried
  offset and continues to the end over the remaining time, instead of restarting
  at the left.
- A finished scroll is left at its end; the rotator's own loop then takes over.

Verified: spot-checked in the browser (a long message at offset -785 resumes at
-788 on the next screen, not 0) and unit-tested (attach mid-scroll calls
resumeScroll with a partial offset and the remaining duration).

ui/src/lib/banner.test.ts

@@ -78,6 +78,7 @@ function recordingHost(overflow = 0): { host: BannerHost; calls: string[]; shown
       hide: () => calls.push('hide'),
       overflowPx: () => overflow,
       scrollTo: () => calls.push('scroll'),
+      resumeScroll: () => calls.push('resume'),
     },
   };
 }
@@ -175,7 +176,7 @@ describe('bannerEngine', () => {
     const shown: string[] = [];
     return {
       shown,
-      host: { show: (md) => shown.push(md), resetScroll() {}, hide() {}, overflowPx: () => 0, scrollTo() {} },
+      host: { show: (md) => shown.push(md), resetScroll() {}, hide() {}, overflowPx: () => 0, scrollTo() {}, resumeScroll() {} },
     };
   }
 
@@ -236,6 +237,7 @@ describe('bannerEngine', () => {
       hide() {},
       overflowPx: () => overflow,
       scrollTo: () => scrolled++,
+      resumeScroll() {},
     };
     configureBanner([{ weight: 1, messages: ['z0'] }], big);
     attachBannerHost(host);
@@ -251,4 +253,28 @@ describe('bannerEngine', () => {
     expect(scrolled).toBeGreaterThan(0);
     detachBannerHost(host);
   });
+
+  it('resumes a long message scroll mid-flight on a new host (navigation)', () => {
+    vi.useFakeTimers();
+    const big: BannerTimings = { holdMs: 10000, edgePauseMs: 100, scrollPxPerSec: 100, fadeOutMs: 100, gapMs: 50, fadeInMs: 100 };
+    const noop: BannerHost = { show() {}, resetScroll() {}, hide() {}, overflowPx: () => 500, scrollTo() {}, resumeScroll() {} };
+    let resumed: { fromTx: number; toPx: number; dur: number } | null = null;
+    const b: BannerHost = { ...noop, resumeScroll: (fromTx, toPx, dur) => (resumed = { fromTx, toPx, dur }) };
+
+    configureBanner([{ weight: 1, messages: ['scrollme'] }], big);
+    attachBannerHost(noop);
+    // fadeIn(100) + edgePause(100) -> scrollTo (dur = 500/100*1000 = 5000ms); then 2000ms into it.
+    vi.advanceTimersByTime(big.fadeInMs + big.edgePauseMs + 2000);
+
+    // Remount mid-scroll: the new host resumes from ~40% of the way, not from the start.
+    detachBannerHost(noop);
+    attachBannerHost(b);
+    expect(resumed).not.toBeNull();
+    expect(resumed!.toPx).toBe(500);
+    expect(resumed!.fromTx).toBeLessThan(0); // already scrolled left of 0
+    expect(resumed!.fromTx).toBeGreaterThan(-500); // but not yet at the end
+    expect(resumed!.dur).toBeGreaterThan(0);
+    expect(resumed!.dur).toBeLessThan(5000); // only the remaining time
+    detachBannerHost(b);
+  });
 });

ui/src/lib/banner.ts

@@ -68,6 +68,10 @@ export interface BannerHost {
   overflowPx(): number;
   /** Animate the horizontal scroll to toPx over durationMs. */
   scrollTo(toPx: number, durationMs: number): void;
+  /** Resume an in-flight scroll on a freshly-mounted view: jump to fromTx (px, signed) instantly,
+   *  then continue to toPx over durationMs. Used by the engine on attach to carry the scroll
+   *  position across a navigation; the rotator itself never calls it. */
+  resumeScroll(fromTx: number, toPx: number, durationMs: number): void;
 }
 
 export interface Rotator {

ui/src/lib/bannerEngine.ts

@@ -10,15 +10,20 @@ let rotator: Rotator | null = null;
 let mounted: BannerHost | null = null;
 let current = '';
 let key = '';
+// The in-flight horizontal scroll of the current message, so a view mounted by navigation can
+// resume it from the same offset instead of restarting at the left. Null when not scrolling.
+let activeScroll: { toPx: number; dur: number; start: number } | null = null;
 
-// proxy is the rotator's host: it records the current message (so a freshly-mounted view can
-// resync to it) and forwards every effect to the currently-attached DOM host, if any.
+// proxy is the rotator's host: it records the current message + scroll (so a freshly-mounted view
+// can resume them) and forwards every effect to the currently-attached DOM host, if any.
 const proxy: BannerHost = {
   show(md) {
     current = md;
+    activeScroll = null;
     mounted?.show(md);
   },
   resetScroll() {
+    activeScroll = null;
     mounted?.resetScroll();
   },
   hide(durationMs) {
@@ -28,8 +33,12 @@ const proxy: BannerHost = {
     return mounted?.overflowPx() ?? 0;
   },
   scrollTo(toPx, durationMs) {
+    activeScroll = { toPx, dur: durationMs, start: Date.now() };
     mounted?.scrollTo(toPx, durationMs);
   },
+  resumeScroll(fromTx, toPx, durationMs) {
+    mounted?.resumeScroll(fromTx, toPx, durationMs);
+  },
 };
 
 // bannerKey identifies a campaigns+timings set so configureBanner restarts the cycle only on a
@@ -70,6 +79,16 @@ export function bannerCurrent(): string {
  */
 export function attachBannerHost(host: BannerHost): void {
   mounted = host;
+  // Resume the current message's scroll from where it had reached, so navigation does not restart
+  // a long message at the left. Only while a scroll is still in flight; a finished scroll is left
+  // at its end and the rotator's own loop takes over.
+  if (activeScroll) {
+    const elapsed = Date.now() - activeScroll.start;
+    if (elapsed < activeScroll.dur) {
+      const progress = elapsed / activeScroll.dur;
+      host.resumeScroll(-activeScroll.toPx * progress, activeScroll.toPx, activeScroll.dur - elapsed);
+    }
+  }
 }
 
 /**