bug(ai-gateway): document that provider API key env vars must be ABSENT (not empty) for auto-routing

What I observed

Migrating a classifier from @anthropic-ai/sdk to the AI SDK + Gateway pattern (per the ai-gateway skill) produced 401 invalid x-api-key from api.anthropic.com on every call, even after:

• Enabling AI Gateway in the project's Vercel dashboard
• Updating the model string to "anthropic/claude-haiku-4.5"
• Confirming VERCEL_OIDC_TOKEN is auto-injected on deploys

Vercel runtime logs showed the request hit api.anthropic.com directly (cookie Domain confirmed upstream), not ai-gateway.vercel.sh.

Cause: ANTHROPIC_API_KEY was still set in Vercel env (with an empty-string value, leftover from the previous direct-provider integration). The AI SDK detects the env-var presence (regardless of value) and chooses direct-provider mode over gateway routing. Empty key then 401s upstream.

What I expected

The current skill says:

Pass a "provider/model" string to the model parameter — the AI SDK automatically routes it through the AI Gateway
No gateway() wrapper or additional package needed.

I expected the model-string approach to route through the gateway once the dashboard had it enabled. Env-var absence wasn't a precondition I'd noticed.

Proposed fix

Add a gotcha note in the Setup or Authentication (OIDC) section:

Gotcha: provider-specific API key env vars must be ABSENT for auto-routing

If your Vercel project has a provider's static API key in env (e.g. ANTHROPIC_API_KEY, OPENAI_API_KEY), the AI SDK detects it and uses direct-provider mode instead of gateway routing — even when the value is an empty string. To force gateway routing, REMOVE the env var entirely:

vercel env rm ANTHROPIC_API_KEY production --yes
vercel env rm ANTHROPIC_API_KEY preview --yes
vercel env rm ANTHROPIC_API_KEY development --yes

Then trigger a redeploy so warm function instances cold-start with the new env. Verification: successful gateway-routed requests do NOT hit api.anthropic.com. Symptom of the misroute is 401 invalid x-api-key from the upstream provider despite AI Gateway showing as enabled.

Why this matters

Bites projects migrating from direct-provider SDK to AI Gateway — the most common migration the skill is helping with. The upstream-provider 401 is misleading because it looks like a key problem, not a routing problem, and the cure (REMOVE the env var, don't reset it) is non-obvious.

Context

• Plugin: vercel @ 0.40.0
• Skill: skills/ai-gateway/SKILL.md
• AI SDK: ai@^6.0.191
• Time debugging: ~20 minutes

Open skills/ai-gateway/SKILL.md and read the Setup and Authentication (OIDC) sections first. Add a note explaining that provider API key variables must be absent, not empty, for automatic gateway routing, including the removal commands and redeploy guidance. Done means the migration precondition, misleading 401 symptom, and verification step are documented clearly.