Add debug logging to file and fix blank startup screen

- Added write_log Tauri command that writes to frontend.log in app data dir
- App.svelte now logs each startup step (Tauri import, sidecar check, launch)
- Startup overlays use inline styles as fallback so they're visible even if
  CSS variables fail to load
- Debug status shown on the checking/connecting screens
- Rust side logs startup info to app.log (resource dir, data dir)

Log files location: %APPDATA%/net.anhonesthost.local-transcription/ (Windows)
or ~/Library/Application Support/net.anhonesthost.local-transcription/ (macOS)

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

src-tauri/src/lib.rs

@@ -3,6 +3,26 @@ mod sidecar;
 use std::sync::Mutex;
 use tauri::Manager;
 
+/// App log directory, set during setup.
+static LOG_DIR: std::sync::OnceLock<std::path::PathBuf> = std::sync::OnceLock::new();
+
+/// Write a log message to the app's log file (for debugging).
+#[tauri::command]
+fn write_log(message: String) {
+    if let Some(log_dir) = LOG_DIR.get() {
+        let log_path = log_dir.join("frontend.log");
+        use std::io::Write;
+        if let Ok(mut f) = std::fs::OpenOptions::new()
+            .create(true)
+            .append(true)
+            .open(&log_path)
+        {
+            let _ = writeln!(f, "[{}] {}", chrono::Local::now().format("%H:%M:%S%.3f"), message);
+        }
+    }
+    eprintln!("[frontend] {}", message);
+}
+
 #[cfg_attr(mobile, tauri::mobile_entry_point)]
 pub fn run() {
     tauri::Builder::default()
@@ -22,9 +42,22 @@ pub fn run() {
                 .app_data_dir()
                 .expect("failed to resolve app data dir");
 
-            // Ensure the data directory exists
             std::fs::create_dir_all(&data_dir).expect("failed to create app data dir");
 
+            // Set up logging
+            LOG_DIR.set(data_dir.clone()).ok();
+            let log_path = data_dir.join("app.log");
+            if let Ok(mut f) = std::fs::OpenOptions::new()
+                .create(true)
+                .append(true)
+                .open(&log_path)
+            {
+                use std::io::Write;
+                let _ = writeln!(f, "\n=== App started at {} ===", chrono::Local::now());
+                let _ = writeln!(f, "Resource dir: {}", resource_dir.display());
+                let _ = writeln!(f, "Data dir: {}", data_dir.display());
+            }
+
             sidecar::init_dirs(resource_dir, data_dir);
             Ok(())
         })
@@ -35,6 +68,7 @@ pub fn run() {
             sidecar::get_sidecar_port,
             sidecar::start_sidecar,
             sidecar::stop_sidecar,
+            write_log,
         ])
         .run(tauri::generate_context!())
         .expect("error while running tauri application");