perf(gateway): pool backend conns; loadtest evaluate hot path

The loadtest harness never modelled game.evaluate — the debounced per-tile
play preview a real client fires several times per turn, the hottest gameplay
call. Model it (one evaluate per placed tile + reconsideration re-previews +
draft.save, human-paced; --eval / --eval-recon toggle it).

That realistic load surfaced the real bottleneck: the gateway's backend HTTP
client used the default transport (MaxIdleConnsPerHost=2), so every sync call
to the single backend host churned a fresh TCP connection — ~26500 TIME_WAIT
sockets at 500 players (near the ephemeral-port ceiling), burning ~1.75 gateway
cores while the backend sat near-idle. It was the unfixed root of the residual
transport_error the earlier passes chased on the client side.

Widen the keep-alive pool (backendMaxIdleConns=512, ~2x the observed 225-conn
peak). At 500 players the churn collapses to ~0 and peak gateway CPU drops ~7x
(~1.75 -> ~0.26 cores); postgres (~1.65 cores) becomes the busiest service.
This overturns the earlier "gateway is the binding constraint, scale it
horizontally" sizing — that was sizing around this bug, not a real floor.

Consolidate the loadtest trip reports into one loadtest/REPORT.md (drop the
R2/R7 split) and bake the finding into README / PRERELEASE / ARCHITECTURE /
TESTING.

gateway/internal/backendclient/client.go

@@ -22,6 +22,19 @@ import (
 	pushv1 "scrabble/pkg/proto/push/v1"
 )
 
+// backendMaxIdleConns sizes the REST keep-alive pool to the single backend host. The
+// default transport caps idle connections per host at 2 (http.DefaultMaxIdleConnsPerHost),
+// which — since every synchronous client call proxies to that one host — forces a fresh
+// TCP connection (and a lingering TIME_WAIT socket) for almost every request under load.
+// That connection churn burns gateway CPU and exhausts ephemeral ports at scale, all
+// while the backend itself sits near-idle. Pooling the connections lets them be reused.
+//
+// The stress harness measured the effect at 500 concurrent players: the churn collapsed
+// from ~26 500 TIME_WAIT sockets to ~0 and peak gateway CPU from ~1.75 to ~0.26 cores,
+// with the pool settling at ~225 live connections. 512 keeps ~2x headroom over that
+// observed peak so a burst never re-caps the pool. See loadtest/REPORT.md.
+const backendMaxIdleConns = 512
+
 // Client calls the backend's REST API and opens its push gRPC stream.
 type Client struct {
 	baseURL string
@@ -41,9 +54,14 @@ func New(httpURL, grpcAddr string, timeout time.Duration) (*Client, error) {
 	if err != nil {
 		return nil, fmt.Errorf("backendclient: dial push %s: %w", grpcAddr, err)
 	}
+	// Clone the default transport (keeping its proxy, dialer and timeouts) and widen the
+	// idle pool so REST calls to the backend reuse connections instead of churning them.
+	transport := http.DefaultTransport.(*http.Transport).Clone()
+	transport.MaxIdleConns = backendMaxIdleConns
+	transport.MaxIdleConnsPerHost = backendMaxIdleConns
 	return &Client{
 		baseURL: strings.TrimRight(httpURL, "/"),
-		http:    &http.Client{Timeout: timeout},
+		http:    &http.Client{Timeout: timeout, Transport: transport},
 		conn:    conn,
 		push:    pushv1.NewPushClient(conn),
 	}, nil