diff --git a/sw-block/prototype/enginev2/phase45_test.go b/sw-block/prototype/enginev2/phase45_test.go
index 3e0e96430..8b2eb2d5d 100644
--- a/sw-block/prototype/enginev2/phase45_test.go
+++ b/sw-block/prototype/enginev2/phase45_test.go
@@ -269,18 +269,25 @@ func TestSender_TargetFrozen_RejectsProgressBeyond(t *testing.T) {
 	}
 }
 
-func TestSender_NoBudget_TargetNotFrozen(t *testing.T) {
+func TestSender_NoBudget_TargetStillFrozen(t *testing.T) {
+	// Target freeze is intrinsic to catch-up, not budget-dependent.
 	s := NewSender("r1:9333", Endpoint{DataAddr: "r1:9333", Version: 1}, 1)
 	sess, _ := s.AttachSession(1, SessionCatchUp)
-	// No budget = no target freeze.
 
 	s.BeginConnect(sess.ID)
 	s.RecordHandshake(sess.ID, 50, 100)
 	s.BeginCatchUp(sess.ID)
 
-	// Without budget, no frozen target. Progress beyond 100 is allowed.
-	s.RecordCatchUpProgress(sess.ID, 100)
-	// Session target can be manually updated if needed (no freeze enforced).
+	// Frozen target enforced even without budget.
+	if sess.FrozenTargetLSN != 100 {
+		t.Fatalf("frozen target=%d, want 100", sess.FrozenTargetLSN)
+	}
+	if err := s.RecordCatchUpProgress(sess.ID, 100); err != nil {
+		t.Fatalf("at target: %v", err)
+	}
+	if err := s.RecordCatchUpProgress(sess.ID, 101); err == nil {
+		t.Fatal("beyond frozen target should be rejected even without budget")
+	}
 }
 
 // --- Sender-level rebuild test ---
diff --git a/sw-block/prototype/enginev2/sender.go b/sw-block/prototype/enginev2/sender.go
index a5b636569..55714744e 100644
--- a/sw-block/prototype/enginev2/sender.go
+++ b/sw-block/prototype/enginev2/sender.go
@@ -158,6 +158,10 @@ func (s *Sender) CompleteSessionByID(sessionID uint64) bool {
 		return false
 	}
 	sess := s.session
+	// Rebuild sessions must use CompleteRebuild, not this path.
+	if sess.Kind == SessionRebuild {
+		return false
+	}
 	// Truncation gate: if truncation was required, it must be recorded.
 	if sess.TruncateRequired && !sess.TruncateRecorded {
 		return false
@@ -310,7 +314,9 @@ func (s *Sender) BeginCatchUp(sessionID uint64, startTick ...uint64) error {
 		return fmt.Errorf("cannot begin catch-up: session phase=%s", s.session.Phase)
 	}
 	s.State = StateCatchingUp
-	// Freeze the target: catch-up will not chase beyond this.
+	// Freeze the target unconditionally: catch-up is a bounded (R, H0] contract.
+	// The session will not chase a moving head beyond this boundary.
+	s.session.FrozenTargetLSN = s.session.TargetLSN
 	if s.session.Budget != nil {
 		s.session.Budget.TargetLSNAtStart = s.session.TargetLSN
 	}
@@ -342,11 +348,10 @@ func (s *Sender) RecordCatchUpProgress(sessionID uint64, recoveredTo uint64, tic
 	if recoveredTo <= s.session.RecoveredTo {
 		return fmt.Errorf("progress regression: current=%d proposed=%d", s.session.RecoveredTo, recoveredTo)
 	}
-	// Enforce frozen target: reject progress beyond the contract boundary.
-	if s.session.Budget != nil && s.session.Budget.TargetLSNAtStart > 0 &&
-		recoveredTo > s.session.Budget.TargetLSNAtStart {
+	// Enforce frozen target unconditionally: catch-up is bounded to (R, H0].
+	if s.session.FrozenTargetLSN > 0 && recoveredTo > s.session.FrozenTargetLSN {
 		return fmt.Errorf("progress %d exceeds frozen target %d",
-			recoveredTo, s.session.Budget.TargetLSNAtStart)
+			recoveredTo, s.session.FrozenTargetLSN)
 	}
 	// Tick is mandatory when ProgressDeadlineTicks is configured.
 	if s.session.Budget != nil && s.session.Budget.ProgressDeadlineTicks > 0 && len(tick) == 0 {
diff --git a/sw-block/prototype/enginev2/session.go b/sw-block/prototype/enginev2/session.go
index f4eb3d5c4..745b80428 100644
--- a/sw-block/prototype/enginev2/session.go
+++ b/sw-block/prototype/enginev2/session.go
@@ -55,9 +55,10 @@ type RecoverySession struct {
 	InvalidateReason string // non-empty when invalidated
 
 	// Progress tracking.
-	StartLSN    uint64 // gap start (exclusive)
-	TargetLSN   uint64 // gap end (inclusive)
-	RecoveredTo uint64 // highest LSN recovered so far
+	StartLSN        uint64 // gap start (exclusive)
+	TargetLSN       uint64 // gap end (inclusive)
+	FrozenTargetLSN uint64 // frozen at BeginCatchUp — catch-up will not chase beyond this
+	RecoveredTo     uint64 // highest LSN recovered so far
 
 	// Truncation tracking: set when replica has divergent tail beyond committed.
 	TruncateRequired bool   // true if replica FlushedLSN > CommittedLSN at handshake