feat(brain): route real-time search by live broadcast state

STREAM_BROWSER becomes the broadcast *capability* master flag; the live
screen-share state (new ToolContext.broadcasting, passed per turn by the bot)
decides the backend:
  - master off            -> broadcast disabled, always Gemini
  - master on + live on    -> on-screen Chrome (visible on the stream)
  - master on + live off   -> Gemini
context.broadcasting is None outside the voice path (evals, text entry, older
bot) and falls back to the master flag, so current behaviour is unchanged.
This is the brain-side foundation; bot-side wiring (bridge passes broadcast
state, auto-broadcast on voice join, voice on/off toggle) follows.

Specs + docs/llm_contexts.md updated. Covered by
tests/test_web_search_broadcast_routing.py.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

src/jarvis/tools/base.py

@@ -22,6 +22,7 @@ class ToolContext:
         max_retries: int,
         user_print: Callable[[str], None],
         language: Optional[str] = None,
+        broadcasting: Optional[bool] = None,
     ):
         self.db = db
         self.cfg = cfg
@@ -36,6 +37,13 @@ class ToolContext:
         # treat absence as "no signal" and fall back to their own default
         # rather than assuming English.
         self.language = language
+        # Live broadcast (screen-share / Go-Live) state for THIS turn, passed in
+        # by the bot per request. Controls real-time search routing when the
+        # master flag ``cfg.stream_browser`` is on: broadcasting -> on-screen
+        # Chrome search (visible on the stream), not broadcasting -> Gemini.
+        # ``None`` means "no signal" (evals, text entry, older bot): callers
+        # fall back to the master flag so behaviour is unchanged.
+        self.broadcasting = broadcasting
 
 
 class Tool(ABC):