[self-care:open-source-failures] Fix recurring live-mode target-authority precompute failures in githubnext/gh-aw-cao

Recurring live-mode target-authority failures are blocking multiple `githubnext/gh-aw-cao` workflows before activation. In the bounded seven-day snapshot, the same evidence-backed message recurred across AW Doctor and Dependabot workflows: `Target authority missing: add .github/workflows/cao.json to the target default branch for live mode`.

**Action:** CAO maintainers should inspect the live-mode target-authority lookup for self-targeting workflows and correct the default-branch authority path or enrollment state; accept when the listed workflows pass control precompute in a new run without this message.

Critical findings:
- Severity: P1
- Normalized signature: `Target authority missing: add .github/workflows/cao.json to the target default branch for live mode`
- Affected repository: `githubnext/gh-aw-cao`
- Affected workflows: `AW Doctor / Failures`, `AW Doctor / Compiler Security`, `AW Doctor / Upgrade`, `Dependabot / Release Trains`
- Representative run: https://github.com/githubnext/gh-aw-cao/actions/runs/34063871833
- Confidence: high

Agent prompt
Investigate why live-mode workflows in `githubnext/gh-aw-cao` are failing control precompute with `Target authority missing: add .github/workflows/cao.json to the target default branch for live mode`. Compare the authority lookup inputs used by self-targeting workflows against the committed `.github/workflows/cao.json` on the default branch, identify the mismatch in target-authority resolution or enrollment assumptions, implement the smallest safe fix, and validate by rerunning one affected workflow until control precompute succeeds.

Evidence

Representative runs in the bounded snapshot:
- https://github.com/githubnext/gh-aw-cao/actions/runs/34063871833
- https://github.com/githubnext/gh-aw-cao/actions/runs/34062115824
- https://github.com/githubnext/gh-aw-cao/actions/runs/34062096653
- https://github.com/githubnext/gh-aw-cao/actions/runs/34059124630

Probable cause supported by evidence: control precompute could not confirm required live-mode target authority for the target repository. More specific root cause still requires workflow/source inspection.

### Control Plane
- Correlation ID: `34064919273-314`
- Central repository: `githubnext/gh-aw-cao`
- Control-plane run: https://github.com/githubnext/gh-aw-cao/actions/runs/34064919273

> Generated by [SelfCare / Open Source Failures](https://github.com/githubnext/gh-aw-cao/actions/runs/34065100762) · pi · gpt54 · 19.8 AIC · ⌖ 8.55 AIC · ⊞ 8.3K · [◷](https://github.com/search?q=repo%3Agithubnext%2Fgh-aw-cao+is%3Aissue+%22gh-aw-workflow-call-id%3A+githubnext%2Fgh-aw-cao%2Fself-care-open-source-failures%22&type=issues)
> - [x] expires on Sep 20, 2026, 10:54 PM UTC

Start with the live-mode target-authority lookup and the committed .github/workflows/cao.json on the default branch, then compare the inputs used by the listed self-targeting workflows. Use representative run 34063871833 and the other affected runs to trace control precompute. Done means a new run of an affected workflow passes control precompute without the target-authority error.

Written by the indexing model from the issue text.