feat: complete minute birth-time rectification flow

frontend/src/app/api/birth-rectification/route.ts

@@ -1,5 +1,6 @@
 import { NextResponse } from "next/server";
 import { chinaLocations } from "@/data/china-locations";
+import { finiteBirthNumber, resolveMissingBirthTimezoneOffset } from "@/lib/birth-profile-timezone";
 
 type Profile = {
   date?: string;
@@ -7,24 +8,32 @@ type Profile = {
   provinceCode?: string;
   cityCode?: string;
   districtCode?: string;
+  latitude?: number | null;
+  longitude?: number | null;
+  timezoneId?: string;
+  timezoneOffset?: number | null;
 };
 
 const jyotishApiBase = process.env.JYOTISH_API_BASE ?? "http://127.0.0.1:5200";
 
-function payloadFromProfile(profile: Profile) {
+export async function payloadFromProfile(profile: Profile) {
   if (!profile.date || !profile.time) return null;
+  const resolved = await resolveMissingBirthTimezoneOffset(profile, { preferredTime: profile.time }) as Profile;
   const province = chinaLocations.country.provinces.find((item) => item.code === profile.provinceCode);
   const city = province?.cities.find((item) => item.code === profile.cityCode);
   const district = city?.districts.find((item) => item.code === profile.districtCode);
   const location = district ?? city;
-  if (!location) return null;
+  const lat = finiteBirthNumber(resolved.latitude) ?? location?.center[1] ?? null;
+  const lon = finiteBirthNumber(resolved.longitude) ?? location?.center[0] ?? null;
+  const tz = finiteBirthNumber(resolved.timezoneOffset) ?? (location ? chinaLocations.country.timezone : null);
+  if (lat === null || lon === null || tz === null) return null;
   return {
     birth_time: `${profile.date} ${profile.time}`,
     uncertainty_minutes: 30,
     step_minutes: 2,
-    lat: location.center[1],
-    lon: location.center[0],
-    tz: chinaLocations.country.timezone,
+    lat,
+    lon,
+    tz,
   };
 }
 
@@ -48,7 +57,7 @@ async function fetchQuestionnaire(payload: Record<string, unknown>) {
 
 export async function POST(request: Request) {
   const body = await request.json().catch(() => null) as { profile?: Profile } | null;
-  const payload = payloadFromProfile(body?.profile ?? {});
+  const payload = await payloadFromProfile(body?.profile ?? {}).catch(() => null);
   if (!payload) {
     return NextResponse.json({
       status: "blocked",