perf(pump): slice 6 — break the 15.6ms sleep quantum (the REAL ceiling behind takes 7-9)

Take 9's numbers were decisive: paste cache moved work to ~25ms/frame but the
period stayed ~37ms. The missing ~12ms per tick is Task.Delay rounding every
sub-tick request up to the Windows system-clock tick (~15.6ms default —
documented: learn.microsoft.com/en-us/dotnet/api/system.threading.tasks.task.delay).
A frame finishing 3ms early requested 3ms and slept 15.6. Producer capped at
~27fps no matter how fast the compositor got — which is why two real render
fixes read as 'zero change' in playback. Game-loop/OBS canon for this
(stackoverflow.com/questions/5441464; learn.microsoft.com/en-us/windows/win32/
api/timeapi/nf-timeapi-timebeginperiod): raise the timer resolution for the
session, sleep only the bulk of the remainder, and SPIN the last ~2ms across
the deadline.

- FramePump: timeBeginPeriod(1) on entering the pump loop, timeEndPeriod(1) in
  the finally; pacing = bulk _pacingDelay(ahead - 2ms) + bounded Thread.SpinWait
  tail; blown deadlines rebase unchanged (never burst).
- Stats now report avg wait: render+submit+wait must equal the period — the
  accounting is closed, no stage can hide in an unmeasured gap again.
- Webcam dropped its IsOpaque paste-cache bypass: it re-sampled ~156k px every
  tick even between identical device frames; cached paste beats the sampler on
  hits, costs the same on misses.

52/52 per-class green (pacing + pixel suites unchanged — output byte-stable),
clean build 0 warnings. Docs same commit (ai.md slice 6, TASKS.md take-10
gate, MyMistakes #3 + renumber, HANDOFF). User's top-bar spec remains next in
queue (Unit B) — re-sent many times, captured, no open questions.

HANDOFF.md

@@ -52,11 +52,20 @@ RealApp boot-smoke. Scope-check passed.
    not cache keys. Tests: `PasteCache_...` + 85/85 across compositor/pump/chat/capture/session
    classes; clean build 0 warnings. (Cam bypasses the cache via IsOpaque; revisit if take 9 is
    borderline. Follow-ups unchanged: vertical-tier alloc, debounced chat re-render on bursts.)
-3. **Take 9 (user, ~30s record-only):** confirm the wordmark superscript matches the startup.log
-   Build line, then stats must show `≈300/300, avg render ≤ ~8ms` and playback 1x. If yes →
-   **recording saga CLOSED**, Unit B starts. If render ~10-15 → cam-resample cache (drop the
-   IsOpaque bypass); if n/300 sags only during chat bursts → debounced off-tick raster slice.
-4. **Unit B — the top bar + session logic (user spec 2026-09-04 re-sent twice + decisions settled in Q&A):**
+3. **Take 9 ran + slice 6 (2026-09-04, THE FINDING):** paste cache moved render 26.5→22.4ms yet
+   period stayed ~37ms — the gap is Task.Delay's ~15.6ms Windows sleep quantum padding every sub-tick
+   wait. **This is why takes 7→9 read as "zero change" despite real wins: the sleep floor dominated.**
+   Fixed (media-app canon, cited in code + MyMistakes #3): timeBeginPeriod(1) for the pump's life
+   (paired End in finally), sleep the bulk, SPIN the last 2ms across the deadline; stats gained
+   `avg wait` so render+submit+wait ≈ period (accounting closed — nothing can hide). Webcam now
+   routes through the paste cache too (bypass re-sampled 156k px even between identical device
+   frames). 52/52 per-class green, clean build 0 warnings, committed this slice.
+4. **Take 10 (user, ~30s record-only):** read the superscript; stats must show `≈300/300, avg render
+   ~22, wait ~0-3` — honest 60fps IF render+submit ≤ ~16.7. If wait is near zero and n/300 sits at
+   ~200, the remaining gap is pure render 22ms → next slice = per-phase compositor timing (the stats
+   can split blit phases the same way they split resolve; that is the honest path, not a guess).
+   If take 10 lands at ~300/300 → recording saga CLOSED.
+5. **Unit B — the top bar + session logic (user spec 2026-09-04 re-sent twice + decisions settled in Q&A):**
    - Two-line top bar. Line 1: center = REC + **LIVE** pills (text renamed from ON-AIR; pills become
      mutually-exclusive RADIOS — record-OR-stream ruling), right = avatar + **Login/Logout** button
      (no account status light). Line 2: centered primary **Start** (grayed while NO pill armed —