diff --git a/dot_local/bin/executable_brave-new-tab b/dot_local/bin/executable_brave-new-tab
new file mode 100755
index 0000000..8bed772
--- /dev/null
+++ b/dot_local/bin/executable_brave-new-tab
@@ -0,0 +1,24 @@
+#!/bin/sh
+# brave-new-tab URL
+# Open URL in a new tab of the running Brave Origin (its last active window),
+# or start Brave Origin with it. Detached, so the caller never blocks.
+# Without a scheme, localhost, *.localhost, *.local, and 127.x get http://;
+# anything else gets https://.
+set -eu
+
+[ $# -eq 1 ] || { echo "usage: brave-new-tab URL" >&2; exit 2; }
+
+url=$1
+case $url in
+*://* | about:*) ;;
+*)
+    host=${url%%[/?#]*}
+    host=$(printf '%s' "${host%:*}" | tr '[:upper:]' '[:lower:]')
+    case $host in
+    localhost | *.localhost | *.local | 127.*) url="http://$url" ;;
+    *) url="https://$url" ;;
+    esac
+    ;;
+esac
+
+exec setsid -f brave-origin --new-tab "$url" >/dev/null 2>&1
diff --git a/dot_local/bin/executable_umbriel-float-square b/dot_local/bin/executable_umbriel-float-square
new file mode 100755
index 0000000..59c1137
--- /dev/null
+++ b/dot_local/bin/executable_umbriel-float-square
@@ -0,0 +1,40 @@
+#!/bin/sh
+# umbriel-float-square
+# Floating focused window: square, as tall as the usable height (below
+# Noctalia's bar) minus PAD px above and below, centered on its output.
+# Tiled: center its column (the old Mod+C). Extents are fractions of the
+# usable area, so both sides are computed from the output's usable size.
+set -eu
+
+PAD=4
+
+window=$(umbriel windows --json | jq -c '[.[] | select(.active)][0] // empty')
+[ -n "$window" ] || exit 0
+
+if [ "$(printf '%s' "$window" | jq -r .floating)" != true ]; then
+    exec umbriel msg column-center
+fi
+
+id=$(printf '%s' "$window" | jq -r .id)
+output=$(printf '%s' "$window" | jq -r .output)
+extents=$(umbriel outputs --json | jq -r --arg o "$output" --argjson pad "$PAD" '
+    def frac: . * 10000 | floor / 10000;
+    .[] | select(.name == $o) | .usable_area
+    | (.height - 2 * $pad) as $side
+    | "\($side) \([$side / .width, 1] | min | frac) \($side / .height | frac)"')
+set -- $extents
+side=$1
+
+umbriel msg "window-set-secondary-extent:$3"
+umbriel msg "window-set-primary-extent:$2"
+
+# window-center uses the size the client has committed; centering before the
+# resize lands centers the old width. Wait up to 1 s for the new width.
+i=0
+while [ $i -lt 20 ]; do
+    w=$(umbriel windows --json | jq -r --arg id "$id" '.[] | select(.id == $id) | .w')
+    [ $((w - side)) -ge -1 ] && [ $((w - side)) -le 1 ] && break
+    sleep 0.05
+    i=$((i + 1))
+done
+exec umbriel msg window-center
diff --git a/dot_local/bin/executable_umbriel-launch b/dot_local/bin/executable_umbriel-launch
index b719afa..eead14d 100755
--- a/dot_local/bin/executable_umbriel-launch
+++ b/dot_local/bin/executable_umbriel-launch
@@ -11,11 +11,12 @@ pattern=$1
 shift
 
 # Matches in a stable order (window id); the one after the focused match, or
-# the first when focus is elsewhere.
+# the first when focus is elsewhere. `.active` is the one window with keyboard
+# focus; `.focused` is set on every output's last-focused window.
 target=$(umbriel windows --json | jq -r --arg p "$pattern" '
     [.[] | select(.scratchpad == "" and (.app_id | test($p; "i")))] | sort_by(.id)
     | if length == 0 then empty
-      else (map(.focused) | index(true)) as $i
+      else (map(.active) | index(true)) as $i
         | if $i == null then .[0].id else .[($i + 1) % length].id end
       end')