The home page's forge-name field says "free" or "taken" as you type #11239

Part of #11153.

Done:

• [ ] Typing a name shows "free" or "taken" (or why it is not allowed) beside the field, from the same rules the door applies (src/core/reserved.ts, existing children)
• [ ] Without JavaScript the form still works; the check is a small same-origin GET, rate-limited like the door
• [ ] Tests: the check answers free / taken / reserved; no layout shift on the home page

Parked by the crew: the check stayed red through 2 automatic tries (pull #11273 closed; its branch scout/11239-the-home-page-s-forge-name-field-says-fr at 20470c755d2f keeps the work so far). Planner: split this smaller, or rewrite what done looks like; a person can also say what to try.

try 1: docs/design/areas/front.md, docs/features/start-my-forge.md, src/http/server.ts, src/http/strangers.ts, src/spawn/door.ts, src/spawn/name_check.ts, src/… try 2: no files changed try 3: docs/features/selling-funnel.md, docs/features/start-my-forge.md, src/contract/openapi/routes/getApiV1AdminBillingFunnel.ts, src/contract/openapi/routes… try 4: src/web/studio/home.ts

Parked by the crew: the check stayed red through 2 automatic tries (pull #11283 closed; its branch mason/11239-the-home-page-s-forge-name-field-says-fr at 9836a5ea93d4 keeps the work so far).

Parked twice since a person last spoke here, so it is the planner's now. Every reason:

1. the check stayed red through 2 automatic tries (pull #11273 closed; its branch scout/11239-the-home-page-s-forge-name-field-says-fr at 20470c755d2f keeps the work so far)
2. the check stayed red through 2 automatic tries (pull #11283 closed; its branch mason/11239-the-home-page-s-forge-name-field-says-fr at 9836a5ea93d4 keeps the work so far)

The planner splits it, replaces it with a fresh issue, or defers it; a person can also say what to try. Nothing starts here until then, and the board lists it.

try 1: docs/design/areas/front.md, docs/features/start-my-forge.md, scripts/unreached-allow.json, src/actions/spawn.ts, src/billing/connection.ts, src/core/mig… try 2: test/design-audit.test.ts try 3: docs/MCP.md, docs/design/areas/front.md, docs/design/areas/render.md, docs/design/audit.md, docs/features/file-graphql.md, docs/features/file-package-js…

Split by the planner: both pulls sprawled far past the field (billing, OpenAPI, design docs). On main the field already exists but reads the /ready probe's 404 as "free", so admin shows free on production. Now #11311 (one name verdict and GET /forges/name/:name, startable) then #11312 (the home script uses it, start-forge.ts only).