docs(ai-anthropic): add package README

[changbaebang]

🎯 Changes

@tanstack/ai-anthropic ships without a README.md, so its npm page is empty. This adds one in the shape of the other adapter READMEs (ai-groq, ai-mistral): setup, chat, explicit API key, tools, adaptive thinking (with a pointer to the per-model rules rather than restating them), prompt caching, summarization, the eight provider tools with the webSearchTool example, Claude on Vertex via @tanstack/ai-anthropic/vertex, a feature list, and links.

Every snippet is lifted from docs/adapters/anthropic.md; the factories and the eight tools are checked against src/index.ts and src/tools/index.ts. Same series as #1417 / #1434 / #1436 / #1437 / #1442.

Patch changeset included so the README reaches npm.

✅ Checklist

• I have followed the steps in the Contributing guide.
• I have tested code changes locally with pnpm run test:pr, or these tests do not apply to this pull request.
• I fully understand the code in this pull request, including any code generated with AI assistance.
• Docs: I updated docs/ for this change, or this change is not user-facing.
• Changeset: I added a changeset (pnpm changeset), or this PR does not change a published package.

🚀 Release Impact

• This change affects published code, and I have generated a changeset.
• This change is docs/CI/dev-only (no release).

Summary by CodeRabbit

• Documentation
  • Added comprehensive documentation for the Anthropic AI package.
  • Documented installation, environment setup, Claude chat usage, tool calling, thinking modes, prompt caching, summarization, and Vertex AI support.
  • Added examples and references for supported provider tools and configuration options.
  • Added links to related documentation.

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ℹ️ Recent review info

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📥 Commits

Reviewing files that changed from the base of the PR and between 645757a and 15d25b0.

📒 Files selected for processing (2)

• .changeset/ai-anthropic-readme.md
• packages/ai-anthropic/README.md

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📝 Walkthrough

Walkthrough

The pull request adds a README for @tanstack/ai-anthropic and a patch changeset. The README documents setup, chat, tools, thinking, prompt caching, summarization, provider tools, Vertex AI, and related links.

Changes

Anthropic package documentation

Layer / File(s)	Summary
README content and release metadata packages/ai-anthropic/README.md, .changeset/ai-anthropic-readme.md	The README documents installation, API-key setup, adapter usage, tool calling, thinking options, prompt caching, summarization, provider tools, and Claude on Vertex AI. The changeset records a patch release for the package.

Priority: ⬇️ Low

Estimated code review effort: 1 (Trivial) | ~5 minutes

Change: Other

🚥 Pre-merge checks | ✅ 5

✅ Passed checks (5 passed)

Check name	Status	Explanation
Title check	✅ Passed	The title clearly and concisely identifies the main change: adding the package README for @tanstack/ai-anthropic.
Description check	✅ Passed	The description follows the required template, explains the README content and rationale, records testing and checklist status, and documents the changeset and release impact.
Docstring Coverage	✅ Passed	No functions found in the changed files to evaluate docstring coverage. Skipping docstring coverage check. Docstring coverage is scoped to functions touched by this diff. Analyzed 0 functions across 0…
Linked Issues check	✅ Passed	Check skipped because no linked issues were found for this pull request.
Out of Scope Changes check	✅ Passed	Check skipped because no linked issues were found for this pull request.

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[github-actions]

Thanks for the PR, @changbaebang! 🙌 @tombeckenham will take a look.

Automated pre-review checks

• ✅ CI passing
• ✅ No merge conflicts
• ✅ Changeset present
• ✅ E2E test changes included

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