Give injected agent submit Enter a longer settle delay

The trailing CR that submits orchestrator-injected input was written
only 15ms after the body, inside TUI agents' paste-coalescing window,
so codex (and other paste-detecting agents) intermittently swallowed it
as a newline and left the message composed but unsent. Centralize the
per-piece timing in a pure pieceWriteDelay helper: keep 15ms between
body lines but give the final lone Enter a 100ms settle gap so the
agent closes the preceding burst and registers the CR as submit. Covers
send_input, send_message, timers, and the spawn initial prompt (all go
through writeInput).

Resolves the codex composer-submit TODO item.

internal/app/child_input_test.go

@@ -3,6 +3,7 @@ package app
 import (
 	"bytes"
 	"testing"
+	"time"
 )
 
 func TestInputWritePiecesOnlySplitAgentEnters(t *testing.T) {
@@ -27,3 +28,63 @@ func TestInputWritePiecesOnlySplitAgentEnters(t *testing.T) {
 		}
 	}
 }
+
+func TestPieceWriteDelay(t *testing.T) {
+	cases := []struct {
+		name  string
+		index int
+		total int
+		piece []byte
+		want  time.Duration
+	}{
+		{
+			name:  "first piece",
+			index: 0,
+			total: 3,
+			piece: []byte("body"),
+			want:  0,
+		},
+		{
+			name:  "middle body piece",
+			index: 1,
+			total: 3,
+			piece: []byte("body"),
+			want:  agentInterPieceDelay,
+		},
+		{
+			name:  "final carriage return submit",
+			index: 1,
+			total: 2,
+			piece: []byte("\r"),
+			want:  agentSubmitSettleDelay,
+		},
+		{
+			name:  "final newline submit",
+			index: 1,
+			total: 2,
+			piece: []byte("\n"),
+			want:  agentSubmitSettleDelay,
+		},
+		{
+			name:  "final non-enter piece",
+			index: 2,
+			total: 3,
+			piece: []byte("tail"),
+			want:  agentInterPieceDelay,
+		},
+		{
+			name:  "standalone enter fast path",
+			index: 0,
+			total: 1,
+			piece: []byte("\r"),
+			want:  0,
+		},
+	}
+	for _, tc := range cases {
+		t.Run(tc.name, func(t *testing.T) {
+			if got := pieceWriteDelay(tc.index, tc.total, tc.piece); got != tc.want {
+				t.Fatalf("pieceWriteDelay(%d, %d, %q) = %s, want %s", tc.index, tc.total, tc.piece, got, tc.want)
+			}
+		})
+	}
+}