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_holdem.go

@@ -109,8 +109,8 @@ func viewHoldem(g holdem.State, viewer int) holdemView {
 		ToAct:    g.ToAct,
 		Phase:    string(g.Phase),
 		Stack:    g.Seats[viewer].Stack,
-		BoughtIn: g.BoughtIn,
-		Rake:     g.Paid, // the part players actually paid, not the part the table lifted
+		BoughtIn: g.BoughtIn, // a table total; the caller overrides it with this seat's own stake
+		Rake:     g.Seats[viewer].Paid, // this seat's rake alone, not the table's — see the ledger line on the felt
 		Payout:   g.Payout,
 	}
 	for _, p := range g.Side {
@@ -543,6 +543,11 @@ func (s *Server) handleHoldemMove(w http.ResponseWriter, r *http.Request) {
 		acting := seats[seat]
 		acting.Away = false
 		acting.LastSeen = time.Now().Unix()
+		// A top-up is more real money crossing the border, so the seat's staked row —
+		// the record of what they brought and can still take home — has to grow with
+		// it. Without this the storage invariant (stacks + pot == staked - rake) drifts
+		// by every top-up, and the felt under-reports what they bought in for.
+		acting.Staked += topped
 
 		t.State, t.Phase, t.HandNo, t.Deadline = st.State, st.Phase, st.HandNo, st.Deadline
 		if err := storage.CommitTable(storage.TableCommit{