[Candidate / unverified] Solution for 61434432-3fb5-4143-8d8e-671bcbdac253: Ensure budget_tokens >= 1024 and max_tokens > budget_tokens with headroom for an

## Support semantics This is a **candidate / unverified** Solution from the KFA Solution Factory Tier-2 gate. It is source-grounded research material, not an Outcome, and does **not** claim independent reproduction or confirmed field success. ## Cause Extended thinking docs require budget_tokens minimum 1024 and less than max_tokens except interleaved thinking; thinking tokens count toward max_tokens. ## Steps 1. Compute max_tokens strictly greater than budget_tokens before calling messages.create. 2. Enforce minimum budget_tokens 1024 per API rules. 3. Account for interleaved thinking exception where budget may exceed max_tokens. ## Expected result Requests pass budget validation without the equal max_tokens/budget_tokens 400. ## Misleading approaches - Setting max_tokens equal to budget_tokens from shared defaults. - Combining extended thinking with max_tokens:0 pre-warming. ## Limitations - Interleaved thinking changes the less-than rule. - Adaptive thinking models use effort instead of budget_tokens. ## Unknowns - Target deployment specifics were not executed in this worker pass. ## Evidence claims - (documented_platform_behavior) budget_tokens must be at least 1024 and less than max_tokens for extended thinking except interleaved thinking. — https://platform.claude.com/docs/en/build-with-claude/extended-thinking.md