Add codex-core public API listing (#20243)

Summary:
- Add a checked-in codex-core public API listing generated by
cargo-public-api.
- Add scripts/regen-public-api.sh with an embedded crate list,
auto-install for cargo-public-api 0.51.0, pinned nightly, and --check
mode.
- Add Rust CI jobs on the codex Linux x64 runner pool to verify the
listing stays up to date.

Testing:
- bash -n scripts/regen-public-api.sh
- just regen-public-api --check
- yq '.' .github/workflows/rust-ci.yml
.github/workflows/rust-ci-full.yml
- git diff --check

codex-rs/Cargo.lock

@@ -2479,6 +2479,23 @@ dependencies = [
  "zstd 0.13.3",
 ]
 
+[[package]]
+name = "codex-core-api"
+version = "0.0.0"
+dependencies = [
+ "codex-analytics",
+ "codex-arg0",
+ "codex-config",
+ "codex-core",
+ "codex-exec-server",
+ "codex-features",
+ "codex-login",
+ "codex-model-provider-info",
+ "codex-models-manager",
+ "codex-protocol",
+ "codex-utils-absolute-path",
+]
+
 [[package]]
 name = "codex-core-plugins"
 version = "0.0.0"
@@ -3498,18 +3515,7 @@ version = "0.0.0"
 dependencies = [
  "anyhow",
  "clap",
- "codex-arg0",
- "codex-config",
- "codex-core",
- "codex-exec-server",
- "codex-features",
- "codex-login",
- "codex-model-provider-info",
- "codex-models-manager",
- "codex-protocol",
- "codex-rollout",
- "codex-thread-store",
- "codex-utils-absolute-path",
+ "codex-core-api",
  "tracing",
 ]