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/games/holdem/eval.go

@@ -187,7 +187,12 @@ func (s *State) payPot(pot Pot, live []ranked, evs *[]Event) {
 			// climbed while every human folded would be telling them it had.
 			for _, w := range winners {
 				if !s.Seats[w.seat].Bot {
+					// The table total (for the audit's delta) and the winner's own
+					// running tally (for the ledger line the felt shows them). At a
+					// table with two humans these are different numbers: each player is
+					// only ever quoted the rake that came out of a pot they won.
 					s.Paid += rake / int64(len(winners))
+					s.Seats[w.seat].Paid += rake / int64(len(winners))
 				}
 			}
 			*evs = append(*evs, Event{Kind: "rake", Seat: -1, Amount: rake})

internal/games/holdem/holdem.go

@@ -101,6 +101,7 @@ type Seat struct {
 	Bet   int64         `json:"bet"`   // put in on this street
 	Total int64         `json:"total"` // put in across this hand
 	Won   int64         `json:"won"`   // taken out of the pot this hand
+	Paid  int64         `json:"paid"`  // rake lifted from pots this seat has won, all session
 	State SeatState     `json:"state"`
 	Acted bool          `json:"acted"` // has chosen to do something this street
 }

internal/games/holdem/holdem_test.go

@@ -458,6 +458,40 @@ func TestYouOnlyPayRakeOnPotsYouWin(t *testing.T) {
 	}
 }
 
+// At a table with two humans the session-rake line each is shown is their own,
+// not the table's: the house took it out of a pot one of them won, and quoting it
+// to the other says the house has taken money off them that it has not. So the
+// per-seat tally only ever moves on the seat that won the pot.
+func TestRakeIsChargedToTheSeatThatWonIt(t *testing.T) {
+	s := State{Tier: Tiers[1], Flopped: true,
+		Seats: []Seat{{Name: "Reala"}, {Name: "Bob"}}}
+	var evs []Event
+
+	// Reala wins a 400 pot. The rake on it (5%, 20) is Reala's to have paid.
+	s.payPot(Pot{Amount: 400, Eligible: []int{0}}, []ranked{{seat: 0}}, &evs)
+	if s.Seats[0].Paid != 20 {
+		t.Errorf("the seat that won the pot paid %d in rake, want 20", s.Seats[0].Paid)
+	}
+	if s.Seats[1].Paid != 0 {
+		t.Errorf("the other player was charged %d in rake for a pot they were not in", s.Seats[1].Paid)
+	}
+	if s.Paid != 20 {
+		t.Errorf("the table total is %d, want 20 — it is still the sum for the audit", s.Paid)
+	}
+
+	// Bob wins the next one. His tally moves; Reala's stays where it was.
+	s.payPot(Pot{Amount: 200, Eligible: []int{1}}, []ranked{{seat: 1}}, &evs)
+	if s.Seats[0].Paid != 20 {
+		t.Errorf("Reala's tally moved on a pot Bob won: %d, want 20", s.Seats[0].Paid)
+	}
+	if s.Seats[1].Paid != 10 {
+		t.Errorf("Bob paid %d in rake on a 200 pot he won, want 10", s.Seats[1].Paid)
+	}
+	if s.Paid != 30 {
+		t.Errorf("the table total is %d, want 30", s.Paid)
+	}
+}
+
 func TestASplitPotSplits(t *testing.T) {
 	s := State{Tier: Tiers[1], Seats: []Seat{{Name: "You"}, {Name: "A", Bot: true}}}
 	var evs []Event