feat(09-01): add theme column to user table + Better Auth additionalField

- migrations/084_add_user_theme.sql: ALTER TABLE user ADD COLUMN theme TEXT NOT NULL DEFAULT 'system'
- Defensive backfill UPDATE for in-flight NULL rows on managed Postgres
- COMMENT ON COLUMN documents allowed values (light|dark|system)
- lib/auth.ts: adds theme additionalField with defaultValue 'system' after timezone
- session.user.theme now exposed via Better Auth same as session.user.timezone

lib/auth.ts

@@ -96,6 +96,10 @@ export const auth = betterAuth({
         type: "string",
         defaultValue: process.env.DEFAULT_TIMEZONE || "UTC",
       },
+      theme: {
+        type: "string",
+        defaultValue: "system",
+      },
     },
   },
 });

migrations/084_add_user_theme.sql

@@ -0,0 +1,20 @@
+-- =============================================================================
+-- Per-user theme preference (Phase 9 — THEME-01, THEME-05)
+-- =============================================================================
+-- Adds a `theme` column to the Better Auth "user" table so light/dark/system
+-- preference persists server-side and applies on sign-in across devices.
+--
+-- Allowed values: 'light' | 'dark' | 'system' (validated in API layer).
+-- Default 'system' means next-themes detects OS preference at render time —
+-- zero behavior change for existing signed-in users.
+-- =============================================================================
+
+ALTER TABLE "user"
+  ADD COLUMN IF NOT EXISTS theme TEXT NOT NULL DEFAULT 'system';
+
+-- Defensive backfill (DEFAULT covers new inserts, but on managed Postgres a
+-- column added with DEFAULT may briefly show NULL in flight on some replicas).
+UPDATE "user" SET theme = 'system' WHERE theme IS NULL;
+
+COMMENT ON COLUMN "user".theme IS
+  'User theme preference: light | dark | system. Applied app-wide via next-themes; server is canonical.';