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.

ui/src/gen/fbs/scrabblefb.ts

@@ -42,6 +42,7 @@ export { FriendCode } from './scrabblefb/friend-code.js';
 export { FriendList } from './scrabblefb/friend-list.js';
 export { FriendRespondRequest } from './scrabblefb/friend-respond-request.js';
 export { GameActionRequest } from './scrabblefb/game-action-request.js';
+export { GameLimits } from './scrabblefb/game-limits.js';
 export { GameList } from './scrabblefb/game-list.js';
 export { GameOverEvent } from './scrabblefb/game-over-event.js';
 export { GameView } from './scrabblefb/game-view.js';

ui/src/gen/fbs/scrabblefb/game-limits.ts

@@ -0,0 +1,66 @@
+// automatically generated by the FlatBuffers compiler, do not modify
+
+import * as flatbuffers from 'flatbuffers';
+
+export class GameLimits {
+  bb: flatbuffers.ByteBuffer|null = null;
+  bb_pos = 0;
+  __init(i:number, bb:flatbuffers.ByteBuffer):GameLimits {
+  this.bb_pos = i;
+  this.bb = bb;
+  return this;
+}
+
+static getRootAsGameLimits(bb:flatbuffers.ByteBuffer, obj?:GameLimits):GameLimits {
+  return (obj || new GameLimits()).__init(bb.readInt32(bb.position()) + bb.position(), bb);
+}
+
+static getSizePrefixedRootAsGameLimits(bb:flatbuffers.ByteBuffer, obj?:GameLimits):GameLimits {
+  bb.setPosition(bb.position() + flatbuffers.SIZE_PREFIX_LENGTH);
+  return (obj || new GameLimits()).__init(bb.readInt32(bb.position()) + bb.position(), bb);
+}
+
+vsAi():number {
+  const offset = this.bb!.__offset(this.bb_pos, 4);
+  return offset ? this.bb!.readInt32(this.bb_pos + offset) : 0;
+}
+
+random():number {
+  const offset = this.bb!.__offset(this.bb_pos, 6);
+  return offset ? this.bb!.readInt32(this.bb_pos + offset) : 0;
+}
+
+friends():number {
+  const offset = this.bb!.__offset(this.bb_pos, 8);
+  return offset ? this.bb!.readInt32(this.bb_pos + offset) : 0;
+}
+
+static startGameLimits(builder:flatbuffers.Builder) {
+  builder.startObject(3);
+}
+
+static addVsAi(builder:flatbuffers.Builder, vsAi:number) {
+  builder.addFieldInt32(0, vsAi, 0);
+}
+
+static addRandom(builder:flatbuffers.Builder, random:number) {
+  builder.addFieldInt32(1, random, 0);
+}
+
+static addFriends(builder:flatbuffers.Builder, friends:number) {
+  builder.addFieldInt32(2, friends, 0);
+}
+
+static endGameLimits(builder:flatbuffers.Builder):flatbuffers.Offset {
+  const offset = builder.endObject();
+  return offset;
+}
+
+static createGameLimits(builder:flatbuffers.Builder, vsAi:number, random:number, friends:number):flatbuffers.Offset {
+  GameLimits.startGameLimits(builder);
+  GameLimits.addVsAi(builder, vsAi);
+  GameLimits.addRandom(builder, random);
+  GameLimits.addFriends(builder, friends);
+  return GameLimits.endGameLimits(builder);
+}
+}

ui/src/gen/fbs/scrabblefb/game-view.ts

@@ -113,8 +113,13 @@ unreadMessages():boolean {
   return offset ? !!this.bb!.readInt8(this.bb_pos + offset) : false;
 }
 
+kind():number {
+  const offset = this.bb!.__offset(this.bb_pos, 34);
+  return offset ? this.bb!.readInt32(this.bb_pos + offset) : 0;
+}
+
 static startGameView(builder:flatbuffers.Builder) {
-  builder.startObject(15);
+  builder.startObject(16);
 }
 
 static addId(builder:flatbuffers.Builder, idOffset:flatbuffers.Offset) {
@@ -189,12 +194,16 @@ static addUnreadMessages(builder:flatbuffers.Builder, unreadMessages:boolean) {
   builder.addFieldInt8(14, +unreadMessages, +false);
 }
 
+static addKind(builder:flatbuffers.Builder, kind:number) {
+  builder.addFieldInt32(15, kind, 0);
+}
+
 static endGameView(builder:flatbuffers.Builder):flatbuffers.Offset {
   const offset = builder.endObject();
   return offset;
 }
 
-static createGameView(builder:flatbuffers.Builder, idOffset:flatbuffers.Offset, variantOffset:flatbuffers.Offset, dictVersionOffset:flatbuffers.Offset, statusOffset:flatbuffers.Offset, players:number, toMove:number, turnTimeoutSecs:number, multipleWordsPerTurn:boolean, moveCount:number, endReasonOffset:flatbuffers.Offset, seatsOffset:flatbuffers.Offset, lastActivityUnix:bigint, vsAi:boolean, unreadChat:boolean, unreadMessages:boolean):flatbuffers.Offset {
+static createGameView(builder:flatbuffers.Builder, idOffset:flatbuffers.Offset, variantOffset:flatbuffers.Offset, dictVersionOffset:flatbuffers.Offset, statusOffset:flatbuffers.Offset, players:number, toMove:number, turnTimeoutSecs:number, multipleWordsPerTurn:boolean, moveCount:number, endReasonOffset:flatbuffers.Offset, seatsOffset:flatbuffers.Offset, lastActivityUnix:bigint, vsAi:boolean, unreadChat:boolean, unreadMessages:boolean, kind:number):flatbuffers.Offset {
   GameView.startGameView(builder);
   GameView.addId(builder, idOffset);
   GameView.addVariant(builder, variantOffset);
@@ -211,6 +220,7 @@ static createGameView(builder:flatbuffers.Builder, idOffset:flatbuffers.Offset,
   GameView.addVsAi(builder, vsAi);
   GameView.addUnreadChat(builder, unreadChat);
   GameView.addUnreadMessages(builder, unreadMessages);
+  GameView.addKind(builder, kind);
   return GameView.endGameView(builder);
 }
 }

ui/src/gen/fbs/scrabblefb/profile.ts

@@ -5,6 +5,7 @@ import * as flatbuffers from 'flatbuffers';
 import { AdsInfo } from '../scrabblefb/ads-info.js';
 import { BannerInfo } from '../scrabblefb/banner-info.js';
 import { DictVersion } from '../scrabblefb/dict-version.js';
+import { GameLimits } from '../scrabblefb/game-limits.js';
 
 
 export class Profile {
@@ -141,8 +142,13 @@ ads(obj?:AdsInfo):AdsInfo|null {
   return offset ? (obj || new AdsInfo()).__init(this.bb!.__indirect(this.bb_pos + offset), this.bb!) : null;
 }
 
+gameLimits(obj?:GameLimits):GameLimits|null {
+  const offset = this.bb!.__offset(this.bb_pos, 40);
+  return offset ? (obj || new GameLimits()).__init(this.bb!.__indirect(this.bb_pos + offset), this.bb!) : null;
+}
+
 static startProfile(builder:flatbuffers.Builder) {
-  builder.startObject(18);
+  builder.startObject(19);
 }
 
 static addUserId(builder:flatbuffers.Builder, userIdOffset:flatbuffers.Offset) {
@@ -241,6 +247,10 @@ static addAds(builder:flatbuffers.Builder, adsOffset:flatbuffers.Offset) {
   builder.addFieldOffset(17, adsOffset, 0);
 }
 
+static addGameLimits(builder:flatbuffers.Builder, gameLimitsOffset:flatbuffers.Offset) {
+  builder.addFieldOffset(18, gameLimitsOffset, 0);
+}
+
 static endProfile(builder:flatbuffers.Builder):flatbuffers.Offset {
   const offset = builder.endObject();
   return offset;