Collapse the relationships card by default, with an honest count

The triage band now answers the common case above the fold - what an asset
serves, or what drives it - so the full edge list below is reference rather than
the first thing to read. On a machine showing two relationships, both already
appeared in the band and the banner, and the card was a third copy.

Collapsed by default, using the same caret, roles and keyboard handling as the
collapsible panels on the same page rather than a second, slightly different
one.

The card is NOT redundant with the band and is not removed. It is the only
surface with Add Relationship and the per-row delete, and it keeps each edge's
exact type where the band deduplicates by asset. The printer page is the proof:
four rows there - usesprinter and defaultprinter, for each of two machines -
collapse to two band chips, and which machine DEFAULTS to that printer is a
distinction the band deliberately drops. Fine for triage, wrong for management.

THE COUNT IS OF WHAT IS RENDERED, NOT OF EDGES

A collapsed card that just says "Relationships" cannot be told from an empty
one, so the header carries a count. The first version counted raw edges and was
wrong on the first machine it was shown: a symmetric Dualpath link is stored in
BOTH directions in the legacy shape, so it is two rows in outgoing+incoming and
one "Connected" entry on screen. The badge read 3 above a card that shows 2 when
expanded - exactly the mismatch the badge exists to prevent. It now counts
rendered items.

Add Relationship expands the card before opening the modal, so a new row cannot
land behind a closed caret and read as the add having failed.

Verified on screen: machine reads 2 and matches, printer reads 4 and matches.
Frontend 280 passed.

frontend/src/components/AssetRelationships.vue

@@ -1,16 +1,32 @@
 <template>
   <div class="relationships-section">
     <div class="section-header">
-      <h3 class="section-title">Relationships</h3>
+      <h3
+        class="section-title rel-header"
+        role="button"
+        tabindex="0"
+        :aria-expanded="!collapsed"
+        @click="collapsed = !collapsed"
+        @keydown.enter.prevent="collapsed = !collapsed"
+        @keydown.space.prevent="collapsed = !collapsed"
+      >
+        <span class="rel-caret" :class="{ collapsed }" aria-hidden="true">&#9662;</span>
+        <span>Relationships</span>
+        <!-- The count is what makes a collapsed card honest: without it, "no
+             relationships" and "four relationships you cannot see" look the
+             same from the outside. -->
+        <span v-if="!loading && relationshipCount" class="rel-count">{{ relationshipCount }}</span>
+      </h3>
       <button
         v-if="authStore.isAuthenticated"
         class="btn btn-sm btn-primary"
-        @click="showAddModal = true"
+        @click.stop="openAdd"
       >
         Add Relationship
       </button>
     </div>
 
+    <div v-show="!collapsed">
     <div v-if="loading" class="loading">Loading relationships...</div>
 
     <div v-else-if="!hasRelationships" class="empty-state">
@@ -96,6 +112,7 @@
         </div>
       </div>
     </template>
+    </div>
 
     <!-- Add Relationship Modal -->
     <div v-if="showAddModal" class="modal-overlay">
@@ -225,6 +242,22 @@ function colorForType(name) {
 }
 const showAddModal = ref(false)
 
+// Collapsed by default. The triage band above answers the common case - what
+// this asset serves or is driven by - so the full edge list is reference rather
+// than the first thing to read. It is NOT redundant with the band and is not
+// removed: it is the only surface with Add and delete, and it keeps each edge's
+// exact type where the band deduplicates by asset. On a printer, four rows
+// (usesprinter and defaultprinter, per machine) collapse to two band chips, and
+// which machine DEFAULTS to it is a distinction the band deliberately drops.
+const collapsed = ref(true)
+
+function openAdd() {
+  // Adding from a collapsed card would otherwise drop the new row behind the
+  // caret, which reads as the add having failed.
+  collapsed.value = false
+  showAddModal.value = true
+}
+
 // New relationship form
 const newRel = ref({
   direction: 'outgoing',
@@ -305,6 +338,14 @@ const directionalItems = computed(() => {
   return items
 })
 
+// Counts what the card RENDERS, not raw edges. A symmetric link (Dualpath) is
+// stored in both directions in the legacy shape, so it is two rows in
+// outgoing+incoming and one "Connected" entry on screen. Counting edges
+// displayed 3 on a machine that shows 2 when you expand it, which is precisely
+// the mismatch this badge exists to avoid.
+const relationshipCount = computed(
+  () => connectedItems.value.length + directionalItems.value.length)
+
 const canSave = computed(() => {
   return newRel.value.relationshiptypeid && newRel.value.targetAssetId && resolvedAssetId.value
 })
@@ -502,6 +543,38 @@ const getAssetRoute = assetRoute
   font-weight: 600;
 }
 
+/* Matches the collapsible panels on the same page (PluginAssetPanels) so a
+   card does not gain a second, slightly different caret. */
+.rel-header {
+  display: flex;
+  align-items: center;
+  gap: 0.5rem;
+  cursor: pointer;
+  user-select: none;
+}
+
+.rel-caret {
+  font-size: 0.7rem;
+  color: var(--text-light);
+  transition: transform 0.15s ease;
+}
+
+.rel-caret.collapsed { transform: rotate(-90deg); }
+
+.rel-count {
+  font-size: 0.75rem;
+  font-weight: 600;
+  color: var(--text-light);
+  background: var(--bg-hover, rgba(127, 127, 127, 0.14));
+  border-radius: 999px;
+  padding: 0.05rem 0.45rem;
+  font-variant-numeric: tabular-nums;
+}
+
+@media (prefers-reduced-motion: reduce) {
+  .rel-caret { transition: none; }
+}
+
 .empty-state {
   color: var(--text-light);
   font-style: italic;