Add GitHub Actions Workflow for TypeScript Type-Check Across SDK, API, and Web

[vjuliaife]

Description:
Without a tsc --noEmit step in CI, TypeScript type errors in the SDK, Express API, or Next.js web app only surface at runtime or during manual developer builds. A dedicated type-check workflow ensures that any PR introducing type unsafety is blocked before merge, enforcing the type contract shared across all three TypeScript packages through tsconfig.base.json.

Acceptance Criteria:

• Create .github/workflows/typecheck.yml that triggers on pull_request targeting main and on direct pushes to main
• Workflow installs Node.js 20 via actions/setup-node and runs npm ci from the workspace root to install all package dependencies in one step
• A typecheck job runs npx tsc --noEmit -p packages/sdk/tsconfig.json, npx tsc --noEmit -p apps/api/tsconfig.json, and npx tsc --noEmit -p apps/web/tsconfig.json as separate named steps so failure is attributable to a specific package
• The workflow fails if any tsc invocation exits with a non-zero code and surfaces the compiler errors in the job log
• npm package cache is restored using actions/cache keyed on package-lock.json hash to keep cold-start times under 90 seconds
• tsconfig.base.json is validated to have "strict": true so the type-check gate is meaningful; the workflow step asserts this with a jq check
• Add a badge in README.md linking to the type-check workflow status

Relevant Files:

• .github/workflows/typecheck.yml — new workflow file to create
• tsconfig.base.json — shared TypeScript base config extended by all packages
• packages/sdk/tsconfig.json — SDK TypeScript config
• apps/api/tsconfig.json — API TypeScript config
• apps/web/tsconfig.json — web TypeScript config
• package.json — npm workspace root

[dahmeezy]

@dahmeezy has applied to work on this issue as part of the Stellar Wave Program's 6th wave.

I would love to work on this issue, please assign it to me.

ℹ️ Repo Maintainers: To accept this application, review their application or assign @dahmeezy to this issue.