Add Next page assignment save parity

[hta218]

Summary

• Add an explicit WeaverseNextPageAssignment type and export it from @weaverse/next.
• Normalize /api/public/project page assignments in the Next server client before they reach loader/runtime data.
• Preserve save-compatible fields (projectId, type, locale, handle) plus inherited/fallback meta; normalize nullish locale to ''; drop malformed assignments instead of fabricating one.
• Add tests for server loader payload shape and Studio runtime reuse updates.
• Add the Claude handoff + work-log entry under the existing Next Studio bridge spec.

Verification

• pnpm --filter @weaverse/next test -- __tests__/next-server.test.tsx __tests__/next-adapter.test.tsx
• pnpm --filter @weaverse/next typecheck
• pnpm --filter @weaverse/next build
• pnpm exec biome check packages/next/src packages/next/__tests__ packages/next/README.md --diagnostic-level=error
• git diff --check
• autoreview-copilot --mode local — clean, no accepted/actionable findings

Notes

Claude started the SDK implementation from the handoff; Hermes completed the missing tests/work-log, reviewed the diff, and ran verification.

Tracker: Weaverse/builder#2659