perf(chat): batch live thinking rendering and bound timer updates (#5931)

* perf(chat): batch live thinking DOM updates

* test(chat): cover live thinking scheduler lifecycle

* fix(chat): guard background stop-state, restore live thinking text, drop source-text tests

- _closeOpenThinkingMarkup no longer overwrites currentAccumulated for
  backgrounded streams. It now mirrors the guard the delta path already uses
  (`if (!_isBg) currentAccumulated = accumulated`). Without it a backgrounded
  stream's text is written into the foreground session's stop-state, which
  abortCurrentRequest and detachCurrentStream then put in the wrong bubble.

- Split _extractLiveThinkingText into _liveThinkingText (strip every think tag)
  and _closedThinkingText (via extractThinkingBlocks). Slicing from the first
  <think> to the first </think> pinned the live box to "The" for the rest of the
  stream on the `<think>The</think>` + untagged-thinking pattern that the
  hasUnclosedThink detection deliberately keeps streaming through.

- The background transition now flushes with rich:true, so a stream that
  backgrounds mid-thinking isn't left as pre-wrap plain text permanently.

- Move the throttle to static/js/liveThinkingThrottle.js and import it. The
  .mjs suite imports the module instead of slicing it out of chat.js with
  vm.runInNewContext and marker comments.

- Replace the source-text assertions in tests/test_live_thinking_scheduler_js.py
  with behavioral coverage, per tests/TESTING_STANDARD.md. The .mjs suite grows
  from 3 to 6 cases.

- Collapse the duplicated tool_start/agent_step finalizers into one
  _endLiveThinkingSection().

* fix(chat): hoist thinking teardown out of the try block so catch can reach it

In an ES module a function declared inside `try { }` is scoped to that block,
and `catch` is a sibling scope rather than a nested one. _closeOpenThinkingMarkup
was declared inside the try and called from catch, so the call threw
ReferenceError and killed the rest of the error path: the stream never
finalized and the thinking block was never torn down.

Declare _closeOpenThinkingMarkup and a new _endThinkingOnTerminalPath next to
the existing _flushLiveThinking / _cancelLiveThinkingWork outer lets and assign
them inside the try, which is the pattern those two already use for exactly
this reason.

Verified against a live stream in a browser: before, clicking stop mid-thinking
logged "_closeOpenThinkingMarkup is not defined" and left no finalized thinking
section; after, the block collapses to "View thinking process" correctly.

* perf(chat): extract live thinking at commit cadence

* fix(chat): bound live thinking work

* test(chat): update stream invariant assertions

---------

Co-authored-by: Léo <leograndcontact@gmail.com>

static/js/MODULE_SUMMARY.md

@@ -61,6 +61,7 @@ The largest and most central subsystem. Chat submission → backend SSE → prog
 | **`chatRenderer.js`** | Message DOM construction: `addMessage`, role labels, model route labels, color coding, footers, metrics, code blocks, sources boxes (`web`/`research`/`RAG`), findings box, images, report links, ask-user cards, welcome screen, and transcript utilities. |
 | **`streamingRenderer.js`** | Incremental streaming renderer used by `chat.js`. Freezes finalized DOM blocks and only re-renders the growing tail to avoid flicker and O(N²) re-parsing. |
 | **`streamingSegmenter.js`** | Splits a token stream into display units (text vs code fences) for `streamingRenderer.js`. |
+| **`liveThinkingThrottle.js`** | Trailing-edge coalescer for the live thinking block in `chat.js`: one DOM commit per 100 ms carrying the latest reasoning text, with `flush`/`cancel` for terminal and session-switch paths. |
 | **`slashCommands.js`** | Slash-command registry (`/help`, `/setup`, etc.), parsing, and dispatch handlers. Exported functions are consumed by `chat.js` and `slashAutocomplete.js`. |
 | **`slashAutocomplete.js`** | Composer autocomplete popup for `/` commands. |
 | **`composerArrowUpRecall.js`** | Recall last user message with `↑` on an empty composer. |