feat(ui): per-kind active-game limit lock on the New Game screen

Carry the caller's per-tier active-game caps and each game's kind on the
wire (Profile.game_limits + GameView.kind, additive FBS + gateway transcode
+ client codec, committed regen). The New Game screen counts the player's
active games per kind from the lobby and locks a capped start: an outline
button with a lock that opens a funnel modal instead of a game -- a sign-in
prompt for a guest, a "finish a current game first" notice for a signed-in
account (native Telegram popup, in-app modal elsewhere). The lock lifts via
the existing profile refetch after a guest->durable upgrade.

Remove the lobby's old at_game_limit New-Game tab disable + notice: the flag
(now the random-kind cap) conflicted with the per-kind lock -- it hid the
screen where the lock lives and wrongly blocked an unfulfilled kind. The New
Game tab is always enabled; the per-kind start lock is the only gate. The
at_game_limit wire field stays (unused by the client) for a later cleanup.

Tests: client lock logic + codec kind/game_limits roundtrip + gateway
transcode encode + native popup builders (unit); a mock e2e for the lock
badge and the modal.

pkg/fbs/scrabble.fbs

@@ -80,6 +80,11 @@ table GameView {
   // message (not just a nudge). With unread_chat it colours the badge — both set is the regular
   // badge, unread_chat alone is the softer nudge-only badge.
   unread_messages:bool;
+  // kind is the game's origin for the active-game limits: 0 unknown (a pre-existing game, never
+  // gated), 1 vs_ai, 2 random, 3 friends. The lobby counts active games per kind against the
+  // caller's per-kind limits (Profile.game_limits) to lock a capped New-Game start (added
+  // trailing — backward-compatible).
+  kind:int;
 }
 
 // MoveRecord is one decoded move (a committed play, or a hint preview).
@@ -267,6 +272,10 @@ table Profile {
   // ads carries the post-move interstitial config (cooldowns + suppressed) for the client-mirrored
   // gate (added trailing — backward-compatible).
   ads:AdsInfo;
+  // game_limits carries the caller's tier active-game caps per kind (-1 = unlimited); the client
+  // counts its active games per kind from the lobby and locks a capped New-Game start (added
+  // trailing — backward-compatible).
+  game_limits:GameLimits;
 }
 
 // AdsInfo is the post-move interstitial config: the client-mirrored cooldowns (seconds) and whether
@@ -278,6 +287,15 @@ table AdsInfo {
   suppressed:bool;
 }
 
+// GameLimits is the caller's tier active-game caps per kind (-1 = unlimited), resolved to the
+// guest or durable tier server-side. It rides Profile.game_limits for the client's per-kind
+// New-Game lock.
+table GameLimits {
+  vs_ai:int;
+  random:int;
+  friends:int;
+}
+
 // BlockStatus reports the caller's current manual block. The UI fetches it after any operation
 // returns the "account_blocked" result code, then replaces every screen with the terminal blocked
 // screen and stops all push/poll. permanent is false for a temporary block; until is an RFC3339

pkg/fbs/scrabblefb/GameLimits.go

@@ -0,0 +1,94 @@
+// Code generated by the FlatBuffers compiler. DO NOT EDIT.
+
+package scrabblefb
+
+import (
+	flatbuffers "github.com/google/flatbuffers/go"
+)
+
+type GameLimits struct {
+	_tab flatbuffers.Table
+}
+
+func GetRootAsGameLimits(buf []byte, offset flatbuffers.UOffsetT) *GameLimits {
+	n := flatbuffers.GetUOffsetT(buf[offset:])
+	x := &GameLimits{}
+	x.Init(buf, n+offset)
+	return x
+}
+
+func FinishGameLimitsBuffer(builder *flatbuffers.Builder, offset flatbuffers.UOffsetT) {
+	builder.Finish(offset)
+}
+
+func GetSizePrefixedRootAsGameLimits(buf []byte, offset flatbuffers.UOffsetT) *GameLimits {
+	n := flatbuffers.GetUOffsetT(buf[offset+flatbuffers.SizeUint32:])
+	x := &GameLimits{}
+	x.Init(buf, n+offset+flatbuffers.SizeUint32)
+	return x
+}
+
+func FinishSizePrefixedGameLimitsBuffer(builder *flatbuffers.Builder, offset flatbuffers.UOffsetT) {
+	builder.FinishSizePrefixed(offset)
+}
+
+func (rcv *GameLimits) Init(buf []byte, i flatbuffers.UOffsetT) {
+	rcv._tab.Bytes = buf
+	rcv._tab.Pos = i
+}
+
+func (rcv *GameLimits) Table() flatbuffers.Table {
+	return rcv._tab
+}
+
+func (rcv *GameLimits) VsAi() int32 {
+	o := flatbuffers.UOffsetT(rcv._tab.Offset(4))
+	if o != 0 {
+		return rcv._tab.GetInt32(o + rcv._tab.Pos)
+	}
+	return 0
+}
+
+func (rcv *GameLimits) MutateVsAi(n int32) bool {
+	return rcv._tab.MutateInt32Slot(4, n)
+}
+
+func (rcv *GameLimits) Random() int32 {
+	o := flatbuffers.UOffsetT(rcv._tab.Offset(6))
+	if o != 0 {
+		return rcv._tab.GetInt32(o + rcv._tab.Pos)
+	}
+	return 0
+}
+
+func (rcv *GameLimits) MutateRandom(n int32) bool {
+	return rcv._tab.MutateInt32Slot(6, n)
+}
+
+func (rcv *GameLimits) Friends() int32 {
+	o := flatbuffers.UOffsetT(rcv._tab.Offset(8))
+	if o != 0 {
+		return rcv._tab.GetInt32(o + rcv._tab.Pos)
+	}
+	return 0
+}
+
+func (rcv *GameLimits) MutateFriends(n int32) bool {
+	return rcv._tab.MutateInt32Slot(8, n)
+}
+
+func GameLimitsStart(builder *flatbuffers.Builder) {
+	builder.StartObject(3)
+}
+func GameLimitsAddVsAi(builder *flatbuffers.Builder, vsAi int32) {
+	builder.PrependInt32Slot(0, vsAi, 0)
+}
+func GameLimitsAddRandom(builder *flatbuffers.Builder, random int32) {
+	builder.PrependInt32Slot(1, random, 0)
+}
+func GameLimitsAddFriends(builder *flatbuffers.Builder, friends int32) {
+	builder.PrependInt32Slot(2, friends, 0)
+}
+func GameLimitsEnd(builder *flatbuffers.Builder) flatbuffers.UOffsetT {
+	return builder.EndObject()
+}

pkg/fbs/scrabblefb/GameView.go

@@ -209,8 +209,20 @@ func (rcv *GameView) MutateUnreadMessages(n bool) bool {
 	return rcv._tab.MutateBoolSlot(32, n)
 }
 
+func (rcv *GameView) Kind() int32 {
+	o := flatbuffers.UOffsetT(rcv._tab.Offset(34))
+	if o != 0 {
+		return rcv._tab.GetInt32(o + rcv._tab.Pos)
+	}
+	return 0
+}
+
+func (rcv *GameView) MutateKind(n int32) bool {
+	return rcv._tab.MutateInt32Slot(34, n)
+}
+
 func GameViewStart(builder *flatbuffers.Builder) {
-	builder.StartObject(15)
+	builder.StartObject(16)
 }
 func GameViewAddId(builder *flatbuffers.Builder, id flatbuffers.UOffsetT) {
 	builder.PrependUOffsetTSlot(0, flatbuffers.UOffsetT(id), 0)
@@ -260,6 +272,9 @@ func GameViewAddUnreadChat(builder *flatbuffers.Builder, unreadChat bool) {
 func GameViewAddUnreadMessages(builder *flatbuffers.Builder, unreadMessages bool) {
 	builder.PrependBoolSlot(14, unreadMessages, false)
 }
+func GameViewAddKind(builder *flatbuffers.Builder, kind int32) {
+	builder.PrependInt32Slot(15, kind, 0)
+}
 func GameViewEnd(builder *flatbuffers.Builder) flatbuffers.UOffsetT {
 	return builder.EndObject()
 }

pkg/fbs/scrabblefb/Profile.go

@@ -244,8 +244,21 @@ func (rcv *Profile) Ads(obj *AdsInfo) *AdsInfo {
 	return nil
 }
 
+func (rcv *Profile) GameLimits(obj *GameLimits) *GameLimits {
+	o := flatbuffers.UOffsetT(rcv._tab.Offset(40))
+	if o != 0 {
+		x := rcv._tab.Indirect(o + rcv._tab.Pos)
+		if obj == nil {
+			obj = new(GameLimits)
+		}
+		obj.Init(rcv._tab.Bytes, x)
+		return obj
+	}
+	return nil
+}
+
 func ProfileStart(builder *flatbuffers.Builder) {
-	builder.StartObject(18)
+	builder.StartObject(19)
 }
 func ProfileAddUserId(builder *flatbuffers.Builder, userId flatbuffers.UOffsetT) {
 	builder.PrependUOffsetTSlot(0, flatbuffers.UOffsetT(userId), 0)
@@ -307,6 +320,9 @@ func ProfileStartDictVersionsVector(builder *flatbuffers.Builder, numElems int)
 func ProfileAddAds(builder *flatbuffers.Builder, ads flatbuffers.UOffsetT) {
 	builder.PrependUOffsetTSlot(17, flatbuffers.UOffsetT(ads), 0)
 }
+func ProfileAddGameLimits(builder *flatbuffers.Builder, gameLimits flatbuffers.UOffsetT) {
+	builder.PrependUOffsetTSlot(18, flatbuffers.UOffsetT(gameLimits), 0)
+}
 func ProfileEnd(builder *flatbuffers.Builder) flatbuffers.UOffsetT {
 	return builder.EndObject()
 }