[BUG] Cloud routine sessions ignore CLAUDE_CODE_EFFORT_LEVEL; effort only controllable via undocumented session_context.effort_level

Summary

Scheduled cloud sessions (routines at claude.ai/code/routines) always launch at the model's default effort. Every documented mechanism for raising effort fails in cloud sessions, even though the model-config docs state CLAUDE_CODE_EFFORT_LEVEL "takes precedence over all other methods." The only control that works is an undocumented effort_level field in the triggers API's session_context — not exposed anywhere in the claude.ai UI.

Environment

• Claude Code on the web / scheduled routines, observed 2026-06-11
• Model: claude-fable-5 (default effort: high); effort verified via the session footer badge

What I tried (each verified by the badge on real runs)

| Mechanism | Result |
|---|---|
| "env": {"CLAUDE_CODE_EFFORT_LEVEL": "max"} in the repo's committed .claude/settings.json | Var IS present in the session shell (echo prints max); badge stays High |
| CLAUDE_CODE_EFFORT_LEVEL=max in the cloud environment's Environment variables (the documented channel for harness vars like CLAUDE_AUTOCOMPACT_PCT_OVERRIDE) | Same — var reaches the shell, badge stays High |
| effortLevel settings key | Rejects "max" by design; known-flaky locally (#65651, #45453) |
| Triggers API: session_context.effort_level: "max" | Works — badge shows Max. The strict CreateSessionRequest proto accepts effort_level while rejecting effort/effortLevel as unknown fields, so the field is real but undocumented |

Repro

1. Create a routine via POST /v1/code/triggers with model: claude-fable-5.
2. Add CLAUDE_CODE_EFFORT_LEVEL=max to the cloud environment's env vars.
3. Run it with a prompt containing echo $CLAUDE_CODE_EFFORT_LEVEL → prints max, while the session badge shows High.
4. Update the trigger with session_context.effort_level: "max" → next run's badge shows Max.

Asks

1. Honor CLAUDE_CODE_EFFORT_LEVEL in cloud sessions per documented precedence — or document that cloud sessions resolve effort from session config only.
2. Document effort_level in the triggers API and/or expose effort in the routines UI.

Related: #55598, #65651