WebFetch/WebSearch fail with "400 adaptive thinking is not supported on this model" under Sonnet medium

Nobody has claimed this yet.

## Description
WebFetch and WebSearch consistently fail with the same API error, reproducible across multiple attempts and different URLs/queries:

API Error: 400 adaptive thinking is not supported on this model

## Environment
- Model: Sonnet, "medium" reasoning effort
- Reproduced 5+ times across WebFetch and WebSearch, different URLs and queries, same error every time

## What I ruled out
- Not a network/connectivity issue — `curl` to the same URLs succeeds (200 OK) from the same machine/session
- Not a specific-URL issue — reproduces across different domains and with plain search queries

## Suspected cause
WebFetch's own tool description states it "processes fetched content using a small, fast model" — a separate internal API call for summarization. The error strongly suggests that internal call is passing an "adaptive thinking" parameter the underlying model/tier doesn't support, specific to the Sonnet-medium configuration.

## Expected behavior
WebFetch/WebSearch should work normally under Sonnet medium, or fail with a clearer error if this reasoning-effort tier genuinely isn't supported for these tools.

No source file or test is named. Start by tracing the WebFetch and WebSearch entry points for the Sonnet medium configuration and the internal content-processing request, then reproduce the error with both tools. Done means both tools work under Sonnet medium or return a clear, intentional unsupported-tier error.

Written by the indexing model from the issue text.