games: the rake you pay, and the rake the table lifts

They are different numbers and the felt was quoting the wrong one. Every raked
pot has chips lifted off it, whoever wins — that has to stay true or the table
stops balancing. But the bots' chips are not real, so a pot a bot wins costs
you nothing, and the counter under your stack was climbing anyway while you sat
there folding.

Rake is now every chip off the table (so the chips conserve) and Paid is the
part that came out of a pot you won, which is the only part that is money and
the only part worth telling you about. A chop costs you half of it. The audit
log takes Paid too: the house's income is what it made off the player, not what
it lifted off a bot.

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

internal/games/holdem/holdem.go

@@ -204,7 +204,19 @@ type State struct {
 	// table; Payout is the stack that goes home, and is only set once you're up.
 	BoughtIn int64 `json:"bought_in"`
 	Payout   int64 `json:"payout"`
-	Rake     int64 `json:"rake"` // taken by the house across the whole session
+
+	// Two rakes, and they are different numbers.
+	//
+	// Rake is every chip the house has lifted off this table. It exists so the
+	// chips balance: a pot that is raked is a pot that pays out less than it holds,
+	// and the difference has to be somewhere.
+	//
+	// Paid is the part of that which came out of a pot *you* won — the only part
+	// that is real money, and the only part worth quoting you. Rake a pot a bot
+	// wins and you have paid nothing; a counter that climbed anyway while you sat
+	// folding would be lying to you.
+	Rake int64 `json:"rake"`
+	Paid int64 `json:"paid"`
 
 	// The seed rides in the state for the same reason it does in UNO: the bots
 	// choose and the deck is reshuffled every hand, so the engine needs randomness