fix(stream): TEST-tab chat — resolve liveChatId from snippet, poll until the broadcast is live

The open creator report ('I still cannot post a chat message in the TEST tab',
feedback 'Chat polling couldn't start — Mock Chat Input is disabled') was a single
cause: the liveChatId never resolved. Two YouTube API facts:
1. The id lives in snippet.liveChatId — contentDetails has no such property
   (TASK 44 read part=contentDetails: could never resolve).
   https://developers.google.com/youtube/v3/live/docs/liveBroadcasts
2. It only exists once the broadcast is LIVE — the official sample lists
   broadcastStatus=active, and our fetch ran before the frame pump pushed RTMP
   (enableAutoStart flips ready→live). A ready-state list legitimately returns
   no id.
   https://github.com/youtube/api-samples/blob/master/java/src/main/java/com/google/api/services/samples/youtube/cmdline/live/GetLiveChatId.java

Fix: GetBroadcastLiveChatIdAsync reads part=snippet and polls with a bounded
retry (10x/2s); PrepareAndStartLiveAsync starts the frame pump FIRST, then
resolves the id. Chat stays non-fatal. Docs ai.md/TASKS.md/HANDOFF.md +
MyMistakes recipe updated same commit. TASK 44.

Good Dog: ONE integration test (GetBroadcastLiveChatIdAsync_Polls_Snippet_...
) drives a broadcast that gains its id mid-retry and asserts every request used
part=snippet. Gate: clean build 0 warnings, vstest 318/318.

TASKS/task-44-test-chat-fix.md

@@ -0,0 +1,58 @@
+# TASK 44 — TEST-tab chat fix: resolve the liveChatId (snippet + live-timing)
+
+> Catalog: [`TASKS.md`](../TASKS.md). Status: ✅ **SHIPPED 2026-09-25** — the open
+> creator report from HANDOFF ("I still cannot post a chat message in the TEST tab")
+> diagnosed and fixed as its own scoped unit.
+
+## Provenance
+
+- **2026-09-24, HANDOFF** filed the creator report as open: "I still cannot post a chat
+  message in the TEST tab." The TEST drawer's Mock Chat Input posts a real
+  `liveChat/messages.insert`; the drawer only enables it when a liveChatId is resolved.
+- **2026-09-25 diagnosis:** the reported feedback line "Chat polling couldn't start —
+  Mock Chat Input is disabled" is `TestSessionViewModel.BeginTest` running with a **null
+  liveChatId**. That id comes from `YouTubeStreamService.GetBroadcastLiveChatIdAsync`,
+  and two API facts (from the official liveBroadcasts reference + the
+  `GetLiveChatId.java` sample) pin why it never resolved:
+  1. **It lives under `snippet.liveChatId`** — `contentDetails` has no such property at
+     all. The code requested `part=contentDetails` and read `contentDetails.liveChatId`,
+     a path that can NEVER resolve.
+  2. **YouTube only populates it once the broadcast is `live`** (the official sample
+     lists `broadcastStatus=active`). The fetch ran right after broadcast insert
+     (lifecycleStatus `ready`) AND before `_framePump.StartAsync()` pushed the RTMP
+     stream that `enableAutoStart` needs to flip the broadcast to live — so even the
+     correct property would have been empty at fetch time.
+
+## What shipped
+
+- `Services/YouTubeStreamService.cs`:
+  - `GetBroadcastLiveChatIdAsync(broadcastId, maxAttempts = 10, delayMs = 2000)` now
+    requests **`part=snippet`** and reads **`snippet.liveChatId`**, and **polls with a
+    bounded retry** until the id appears or the attempts exhaust (never throws). The
+    default retry (~20s) gives the encoder time to connect and the broadcast to go live.
+  - New private `FetchLiveChatIdAsync` holds the single read; the public method owns
+    the retry loop.
+- `ViewModels/MainViewModel.Streaming.Operations.cs`:
+  - `PrepareAndStartLiveAsync` **starts the frame pump FIRST**, then resolves the
+    liveChatId — the reorder is required for the id to ever exist. Chat remains
+    non-fatal: failure logs and the stream continues.
+  - TEST drawer wiring preserved: `TestSession.BeginTest(broadcastId, liveChatId)`
+    still opens the dock once the id resolves → Mock Chat Input becomes sendable.
+- Docs: `ai.md` (Test Stream section), `TASKS.md` (row 44), `HANDOFF.md` rewritten.
+
+## Validation (Good Dog: ONE integration test)
+
+`ytLive.Tests/YouTubeStreamServiceTests.cs`:
+`GetBroadcastLiveChatIdAsync_Polls_Snippet_Until_Id_Appears` — a fake broadcast that
+starts WITHOUT a liveChatId (not yet live) and gains it on a later poll; asserts the
+retry eventually resolves `Cg0LIVECHAT42`, that it polled ≥3 times, and that every
+request asked for **snippet** (never contentDetails) — the exact seam that was broken.
+
+**Gate: clean build 0 warnings; full vstest 318/318 passed.**
+
+## Reference
+
+- https://developers.google.com/youtube/v3/live/docs/liveBroadcasts (snippet.liveChatId;
+  contentDetails has no liveChatId)
+- https://github.com/youtube/api-samples/blob/master/java/src/main/java/com/google/api/services/samples/youtube/cmdline/live/GetLiveChatId.java
+  (sample lists broadcastStatus=active — id only present on live broadcasts)