fix: load persisted capture region at launch; REGION-PERSIST selftest phase

Ben-reported: picker opened on every activation. AppModel.init set region = nil and never
read UserDefaults back; saving worked, every launch forgot it. init now loads via
loadPersistedRegion() (decode + isStillValid). Selftest phase 2 writes a known region,
reloads through the same path, asserts the rect, restores the user's stored value.
Coordinator ran it: PICKER-SELFTEST PASS + REGION-PERSIST PASS, 90/90 tests green.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

Sources/Shotdeck/PickerSelfTest.swift

@@ -111,9 +111,52 @@ enum PickerSelfTest {
 
         print("PICKER-SELFTEST PASS rect=\(format(got.rect))")
         fflush(stdout)
+
+        runRegionPersistPhase()
         exit(0)
     }
 
+    /// Phase 2: writes a known region under `CaptureRegion.defaultsKey`, reloads it through
+    /// `AppModel.loadPersistedRegion()` (the same path init uses), then restores whatever
+    /// value was stored before so a real picked region is untouched.
+    private static func runRegionPersistPhase() {
+        let defaults = UserDefaults.standard
+        let previous = defaults.data(forKey: CaptureRegion.defaultsKey)
+
+        let known = CaptureRegion(
+            displayID: CGMainDisplayID(),
+            rect: CGRect(x: 10, y: 10, width: 100, height: 100),
+            capturedScale: 2.0
+        )
+        var failure: String?
+        if let encoded = try? JSONEncoder().encode(known) {
+            defaults.set(encoded, forKey: CaptureRegion.defaultsKey)
+            if let loaded = AppModel.loadPersistedRegion() {
+                if loaded.rect != known.rect {
+                    failure = "expected=\(format(known.rect)) got=\(format(loaded.rect))"
+                }
+            } else {
+                failure = "loadPersistedRegion returned nil"
+            }
+        } else {
+            failure = "could not encode CaptureRegion"
+        }
+
+        if let previous {
+            defaults.set(previous, forKey: CaptureRegion.defaultsKey)
+        } else {
+            defaults.removeObject(forKey: CaptureRegion.defaultsKey)
+        }
+
+        if let failure {
+            print("REGION-PERSIST FAIL \(failure)")
+            fflush(stdout)
+            exit(1)
+        }
+        print("REGION-PERSIST PASS")
+        fflush(stdout)
+    }
+
     private static func interpolate(_ step: Int) -> NSPoint {
         let t = CGFloat(step) / CGFloat(dragSteps)
         return NSPoint(