The configured request timeout (API_TIMEOUT_MS) only feeds the SDK client
timeout, which on a streaming request is cleared the moment response headers
arrive. Slow local/3P models return 200 headers fast but spend minutes
prefilling before the first SSE token, so that window was governed by the idle
watchdog (CLAUDE_STREAM_IDLE_TIMEOUT_MS, 240s) -- a separate env users cannot
configure. The configured timeout therefore never applied to slow prefill, so
raising it had no effect (#826, follow-up to #449).
Split the stream watchdog into phases: use a first-token budget (tied to the
user's request timeout via CLAUDE_STREAM_FIRST_TOKEN_TIMEOUT_MS) until the first
chunk arrives, then switch to the shorter mid-stream idle budget. The overall
max-duration cap (#766) still backstops trickling bodies; CLI defaults fall back
to prior behavior.
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