games: the buy-in and the rake each player sees are their own, not the table's

The two-browser pass found it: at a table two humans share, the felt quoted
each of them the pair's total. "Bought in for 200" to a player who put in 100,
and a session-rake line that climbed on a pot the other one won.

Both were table totals the view read straight off the engine — correct while a
table had one human, wrong the moment it had two. Fixed along the border it
already draws: bought_in is border accounting, so it comes from the viewer's own
game_seats.staked; session rake is a within-table event, so it rides a new
per-seat Seat.Paid beside the audit's table-total s.Paid.

And top-up never grew game_seats.staked, so the storage invariant drifted by
every top-up and the felt under-reported the buy-in — it does now.

Claude-Session: https://claude.ai/code/session_013M5nD7PgUboJXoDcYHzpuJ

internal/web/games_play.go

@@ -303,7 +303,7 @@ func (s *Server) table(user string) (tableView, error) {
 		if live.TableID == "" {
 			return s.dropUnreadable(user, v, fmt.Errorf("holdem row with no table"))
 		}
-		t, _, err := storage.LoadTable(live.TableID)
+		t, tableSeats, err := storage.LoadTable(live.TableID)
 		if errors.Is(err, storage.ErrNoSuchTable) {
 			// The table closed under them (reaped, or the last hand cashed them out).
 			// Their claim is stale; clear it so they can sit down again.
@@ -321,6 +321,15 @@ func (s *Server) table(user string) (tableView, error) {
 			return s.dropUnreadable(user, v, err)
 		}
 		hv := viewHoldem(g, seat)
+		// bought_in is a per-player figure — "you bought in for X" — but the engine's
+		// BoughtIn is the table's total across every human. The player's own stake is
+		// border accounting, which lives in storage; take it from their seat row.
+		for _, ts := range tableSeats {
+			if ts.Seat == seat {
+				hv.BoughtIn = ts.Staked
+				break
+			}
+		}
 		v.Holdem = &hv
 	default:
 		return s.dropUnreadable(user, v, fmt.Errorf("unknown game %q", live.Game))