fix(health): stop startup wait at its timeout

The startup probe slept a full backoff step even when the deadline was closer. A 31 second timeout ended after about 41 seconds. The probe now sleeps no longer than the time left.

backend_health.go

@@ -118,7 +118,7 @@ func (bhm *BackendHealthManager) WaitForBackendReady(timeout time.Duration) erro
 		}
 
 		// Exponential backoff (1.5x growth, capped at maxDelay), no jitter
-		time.Sleep(currentDelay)
+		time.Sleep(min(currentDelay, time.Until(deadline)))
 		currentDelay = time.Duration(float64(currentDelay) * 1.5)
 		if currentDelay > maxDelay {
 			currentDelay = maxDelay

backend_health_deadline_test.go

@@ -0,0 +1,32 @@
+package main
+
+import (
+	"net"
+	"testing"
+	"time"
+
+	libpack_logger "github.com/lukaszraczylo/graphql-monitoring-proxy/logging"
+	"github.com/valyala/fasthttp"
+)
+
+// Pins theorems/Theorems/ReadyBackoff.lean: the startup wait never sleeps past its timeout.
+func TestWaitForBackendReady_HonoursTimeout(t *testing.T) {
+	ln, err := net.Listen("tcp", "127.0.0.1:0")
+	if err != nil {
+		t.Fatal(err)
+	}
+	addr := ln.Addr().String()
+	_ = ln.Close() // closed port: every probe is refused immediately
+
+	bhm := NewBackendHealthManager(&fasthttp.Client{}, "http://"+addr, "", libpack_logger.New())
+	defer bhm.cancel()
+
+	const timeout = 3 * time.Second
+	start := time.Now()
+	if err := bhm.WaitForBackendReady(timeout); err == nil {
+		t.Fatal("want error for unreachable backend")
+	}
+	if elapsed := time.Since(start); elapsed > timeout+500*time.Millisecond {
+		t.Fatalf("waited %v, want <= %v", elapsed, timeout+500*time.Millisecond)
+	}
+}