[Merged by Bors] - chore(ci): wire MATHLIB_CACHE_BASE_URL from a repository variable

[marcelolynch]

This PR passes the repository variable MATHLIB_CACHE_BASE_URL to every workflow that reads the mathlib cache: build_template.yml (which serves the build, fork, bors, ci_dev, and release_cache workflows), olean_report.yaml, remove_deprecated_decls.yml, and the warm-start cache get in lake_cache_shadow.yml. The variable is normally unset, so reads keep their current path to Azure. An operator sets the variable to move cache read traffic to another host, and clears it to move the traffic back. Both changes take effect on the next job, with no deploy and no code change.

The tool side is #42657: the cache tool reads MATHLIB_CACHE_BASE_URL and treats an empty value as unset, which is the state ${{ vars.MATHLIB_CACHE_BASE_URL }} produces while the variable is undefined. The two PRs are safe in either merge order: before #42657 the tool ignores the variable, and after it the empty value keeps the default.

Out of scope: the nightly-testing repository has its own variable namespace, so a flip there is a separate switch. publish_tools.yml and cache_test.yml run no cache reads, so they stay unwired.

[github-actions]

PR summary c166c7c40d

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

No declarations were harmed in the making of this PR! 🐙

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean -- pending)

Computed after the build finishes.

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit c166c7c40d
Reference commit eb097a9288

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

---

⚠️ Workflow documentation reminder

This PR modifies files under .github/workflows/.
Please update docs/workflows.md if the workflow inventory, triggers, or behavior changed.

Modified workflow files:

• .github/workflows/build_template.yml
• .github/workflows/lake_cache_shadow.yml
• .github/workflows/olean_report.yaml
• .github/workflows/remove_deprecated_decls.yml

[bryangingechen]

Thanks!
bors r+

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• ci (staging) / Build
• ci (staging) / Post-CI job
• ci (staging) / Lint style
• ci (staging) / Test and lint
• ci (staging) / Post-Build Step