apollo_proof_manager: measure cache lock times in log

[einat-starkware]

Note

Low Risk
Mostly observability and a small ordering change in set_proof; low functional risk but could increase log volume and affect performance in hot cache paths.

Overview
Adds tracing-based timing logs around ProofCache mutex acquisition in get, insert, and contains, emitting lock_duration_ms for observability.

Changes ProofManager::set_proof to write the proof to filesystem storage before inserting into the in-memory cache, avoiding cached-only state on storage failure, and adds the tracing dependency.

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[reviewable-StarkWare]

This change is 

[einat-starkware]

• apollo_proof_manager: measure cache lock times in log #12702 👈 (View in Graphite)
• main-v0.14.2

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Artifacts upload workflows:

• Started at 2026-02-18T12:25:44Z
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[noaov1]

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-- commits line 2 at r1:

Suggestion:

- 497380c: apollo_proof_manager: measure cache lock times in log and store in proof manager before caching 

[cursor]

Cursor Bugbot has reviewed your changes and found 1 potential issue.

[einat-starkware]

@einat-starkware reviewed 2 files and all commit messages, and resolved 2 discussions.
Reviewable status: complete! all files reviewed, all discussions resolved (waiting on einat-starkware).

[noaov1]

@noaov1 reviewed 1 file and all commit messages.
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[noaov1]

@noaov1 reviewed 1 file.
Reviewable status: complete! all files reviewed, all discussions resolved (waiting on einat-starkware).