fix(pricing): add Azure GPT-5.6 cache write rates

[qdivan]

TLDR

Problem this solves:

• Azure GPT-5.6 cache writes are billed at zero

How it solves it:

• Adds cache-write rates for every Azure GPT-5.6 price entry

User Flow

Before: a platform team sees cached Azure prompt spend missing from proxy logs

1. They send POST https://litellm-domain/v1/chat/completions to an Azure GPT-5.6 deployment with a cached system prompt
2. The response returns HTTP 200 with cache_write_tokens in prompt_tokens_details
3. They open https://litellm-domain/ui/?page=logs and see the cache-write portion recorded at $0

After: the same request records cache writes at Azure's documented 1.25x input rate

1. They send the same POST https://litellm-domain/v1/chat/completions request to the same Azure GPT-5.6 deployment
2. The response returns HTTP 200 with the same cache_write_tokens usage details
3. They open https://litellm-domain/ui/?page=logs and see non-zero cache-write spend

Relevant issues

Fixes #37631

Linear ticket

Pre-Submission checklist

• I have added meaningful tests
• The handful of test files covering my change pass locally
• My PR passes required local formatting checks
• My PR's scope is as isolated as possible; it only solves 1 specific problem
• I have received a Greptile Confidence Score of at least 4/5 before requesting a maintainer review

Screenshots / Proof of Fix

A live Azure deployment is required for end-to-end proxy proof and was not available in this environment. The focused regression test validates all global, US, and EU Azure GPT-5.6 entries with cache-write token usage against their documented cache-write rates.

Type

🐛 Bug Fix

Caveats (if any)

• Live Azure proxy verification remains for CI or a credentialed environment

Final Attestation

• The regression test covers each affected Azure GPT-5.6 pricing entry

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.

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[codspeed-hq]

Merging this PR will not alter performance

✅ 31 untouched benchmarks

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_{Comparing qdivan:fix/azure-gpt56-cache-write-pricing (cadef9f) with litellm_internal_staging (6d47468)}

[qdivan]

I’m closing this draft because #35125 already covers the same 12 Azure GPT-5.6 global/US/EU cache-write pricing entries and regression coverage; #35841 also overlaps the global Terra/Luna portion. Withdrawing this duplicate to avoid splitting maintainer review.