[docs]: Go is not rendering the OpenAPI documentation, resulting in a discrepancy with the live documentation endpoint.

** Please make sure you read the contribution guide and file the issues in the right place. **
[Contribution guide.](https://google.github.io/adk-docs/contributing-guide/)

## 🔴 Required Information

The ADK documentation (`https://adk.dev/api-reference/rest/openapi.json`) is inconsistent with the actual API endpoints exposed by the Go implementation.

Example:

- go uses /api as the prefix by default
- go /health /apps/{app_name}/app-info is missing
- go has `/run_live`, but it's not available in the production environment.
- Go does not expose a `/docs` endpoint to render OpenAPI. -- https://adk.dev/runtime/api-server/#interactive-api-docs

**Steps to Reproduce:**
Please provide a numbered list of steps to reproduce the behavior:
1. Install '...'
2. Run '....'
3. Open '....'
4. Provide error or stacktrace

**Expected Behavior:**
A clear and concise description of what you expected to happen.

**Observed Behavior:**
What actually happened? Include error messages or crash stack traces here.

**Environment Details:**

- ADK Library Version: main
- OS: [e.g., macOS, Linux, Windows]
- Go Version: v1.26.5

**Model Information:**
- Which model is being used: (e.g., gemini-2.5-pro)

---

## 🟡 Optional Information
*Providing this information greatly speeds up the resolution process.*

**Regression:**
Did this work in a previous version of ADK? (Yes/No) If so, which one?

**Logs:**
Please attach relevant logs. Wrap them in code blocks (```) or attach a
text file.
```text
// Paste logs here
```

**Screenshots / Video:**
If applicable, add screenshots or screen recordings to help explain
your problem.

**Additional Context:**
Add any other context about the problem here.

**Minimal Reproduction Code:**
Please provide a code snippet or a link to a Gist/repo that isolates the issue.
```go
// Code snippet here
```

**How often has this issue occurred?:**

- Always (100%)
- Often (50%+)
- Intermittently (<50%)
- Once / Rare

Start by comparing the OpenAPI documentation at https://adk.dev/api-reference/rest/openapi.json with the routes exposed by the Go implementation, then review the interactive API documentation described at https://adk.dev/runtime/api-server/#interactive-api-docs. Done means the documented prefix, health and app-info routes, run_live availability, and /docs behavior accurately match the Go API and production environment.

Written by the indexing model from the issue text.