feat(flow): add Picture in Picture keep-alive mode

- Add FlowKeepAliveMode (Dynamic Island vs PiP) with iCloud sync
- Settings: keep-alive picker; inactivity timeout only for Live Activity
- PiP: waveform sample-buffer controller, mic released between utterances
- PiP sessions have no idle expiry; user closing PiP ends the session
- FlowSessionManager branches hostReady, session start, and utterance paths
- UIBackgroundModes picture-in-picture; bilingual strings and tests

Co-authored-by: Rocky <hkgood@users.noreply.github.com>

OSGKeyboardShared/Models/SyncedAppSettingsV2.swift

@@ -29,6 +29,7 @@ public struct SyncedAppSettingsV2: Codable, Equatable, Sendable {
     public var polishIntensity: SyncedField<PolishIntensity>
     public var llmThinkingEnabled: SyncedField<Bool>
     public var flowSkipAppSwitch: SyncedField<Bool>
+    public var flowKeepAliveMode: SyncedField<FlowKeepAliveMode>
     public var flowInactivityDuration: SyncedField<FlowInactivityDuration>
 
     public init(
@@ -50,6 +51,7 @@ public struct SyncedAppSettingsV2: Codable, Equatable, Sendable {
         polishIntensity: SyncedField<PolishIntensity>,
         llmThinkingEnabled: SyncedField<Bool>,
         flowSkipAppSwitch: SyncedField<Bool>,
+        flowKeepAliveMode: SyncedField<FlowKeepAliveMode>,
         flowInactivityDuration: SyncedField<FlowInactivityDuration>
     ) {
         self.schemaVersion = schemaVersion
@@ -70,6 +72,7 @@ public struct SyncedAppSettingsV2: Codable, Equatable, Sendable {
         self.polishIntensity = polishIntensity
         self.llmThinkingEnabled = llmThinkingEnabled
         self.flowSkipAppSwitch = flowSkipAppSwitch
+        self.flowKeepAliveMode = flowKeepAliveMode
         self.flowInactivityDuration = flowInactivityDuration
     }
 
@@ -92,6 +95,7 @@ public struct SyncedAppSettingsV2: Codable, Equatable, Sendable {
         case polishIntensity
         case llmThinkingEnabled
         case flowSkipAppSwitch
+        case flowKeepAliveMode
         case flowInactivityDuration
     }
 
@@ -124,6 +128,14 @@ public struct SyncedAppSettingsV2: Codable, Equatable, Sendable {
             forKey: .llmThinkingEnabled
         ) ?? SyncedField(value: false, updatedAt: polishIntensity.updatedAt, deviceID: polishIntensity.deviceID)
         flowSkipAppSwitch = try container.decode(SyncedField<Bool>.self, forKey: .flowSkipAppSwitch)
+        flowKeepAliveMode = try container.decodeIfPresent(
+            SyncedField<FlowKeepAliveMode>.self,
+            forKey: .flowKeepAliveMode
+        ) ?? SyncedField(
+            value: .liveActivity,
+            updatedAt: flowSkipAppSwitch.updatedAt,
+            deviceID: flowSkipAppSwitch.deviceID
+        )
         flowInactivityDuration = try container.decode(
             SyncedField<FlowInactivityDuration>.self,
             forKey: .flowInactivityDuration
@@ -171,6 +183,7 @@ public struct SyncedAppSettingsV2: Codable, Equatable, Sendable {
             polishIntensity.updatedAt,
             llmThinkingEnabled.updatedAt,
             flowSkipAppSwitch.updatedAt,
+            flowKeepAliveMode.updatedAt,
             flowInactivityDuration.updatedAt,
         ].max() ?? .distantPast
     }
@@ -208,6 +221,7 @@ public extension SyncedAppSettingsV2 {
             polishIntensity: field(configuration.polishIntensity),
             llmThinkingEnabled: field(configuration.llmThinkingEnabled),
             flowSkipAppSwitch: field(configuration.flowSkipAppSwitch),
+            flowKeepAliveMode: field(configuration.flowKeepAliveMode),
             flowInactivityDuration: field(configuration.flowInactivityDuration)
         )
     }
@@ -237,6 +251,7 @@ public extension SyncedAppSettingsV2 {
             polishIntensity: field(legacy.polishIntensity),
             llmThinkingEnabled: field(false),
             flowSkipAppSwitch: field(legacy.flowSkipAppSwitch),
+            flowKeepAliveMode: field(.liveActivity),
             flowInactivityDuration: field(legacy.flowInactivityDuration)
         )
     }
@@ -269,6 +284,7 @@ public extension SyncedAppSettingsV2 {
             polishIntensity: .merge(local: local.polishIntensity, remote: remote.polishIntensity),
             llmThinkingEnabled: .merge(local: local.llmThinkingEnabled, remote: remote.llmThinkingEnabled),
             flowSkipAppSwitch: .merge(local: local.flowSkipAppSwitch, remote: remote.flowSkipAppSwitch),
+            flowKeepAliveMode: .merge(local: local.flowKeepAliveMode, remote: remote.flowKeepAliveMode),
             flowInactivityDuration: .merge(
                 local: local.flowInactivityDuration,
                 remote: remote.flowInactivityDuration
@@ -294,6 +310,7 @@ public extension SyncedAppSettingsV2 {
         configuration.polishIntensity = polishIntensity.value
         configuration.llmThinkingEnabled = llmThinkingEnabled.value
         configuration.flowSkipAppSwitch = flowSkipAppSwitch.value
+        configuration.flowKeepAliveMode = flowKeepAliveMode.value
         configuration.flowInactivityDuration = flowInactivityDuration.value
     }
 
@@ -321,6 +338,7 @@ public extension SyncedAppSettingsV2 {
         patch(&copy.polishIntensity, value: configuration.polishIntensity)
         patch(&copy.llmThinkingEnabled, value: configuration.llmThinkingEnabled)
         patch(&copy.flowSkipAppSwitch, value: configuration.flowSkipAppSwitch)
+        patch(&copy.flowKeepAliveMode, value: configuration.flowKeepAliveMode)
         patch(&copy.flowInactivityDuration, value: configuration.flowInactivityDuration)
         return copy
     }
@@ -351,6 +369,7 @@ public extension SyncedAppSettingsV2 {
         touch(&copy.polishIntensity, value: configuration.polishIntensity)
         touch(&copy.llmThinkingEnabled, value: configuration.llmThinkingEnabled)
         touch(&copy.flowSkipAppSwitch, value: configuration.flowSkipAppSwitch)
+        touch(&copy.flowKeepAliveMode, value: configuration.flowKeepAliveMode)
         touch(&copy.flowInactivityDuration, value: configuration.flowInactivityDuration)
         return copy
     }