diff --git a/.gsd/milestones/M015/M015-ROADMAP.md b/.gsd/milestones/M015/M015-ROADMAP.md
index 5e95325..e022fd1 100644
--- a/.gsd/milestones/M015/M015-ROADMAP.md
+++ b/.gsd/milestones/M015/M015-ROADMAP.md
@@ -9,5 +9,5 @@ Add social proof and freshness signals to the public site: search query logging
 | S01 | Search Query Logging + Popular Searches API | medium | — | ✅ | GET /api/v1/search/popular returns top 10 queries from last 7 days, cached in Redis. search_log table has rows from real searches. |
 | S02 | Creator Freshness + Homepage Card Dates | low | — | ✅ | Creators browse page shows 'Last updated: Apr 3' per creator. Homepage recently-added cards show a subtle date. |
 | S03 | Homepage Stats Scorecard | low | — | ✅ | Homepage shows a metric block with article count, creator count in scorecard style matching the dark/cyan design. |
-| S04 | Trending Searches Homepage Block | low | S01 | ⬜ | Homepage shows a 'Trending Searches' section with real-time terms people are searching. |
+| S04 | Trending Searches Homepage Block | low | S01 | ✅ | Homepage shows a 'Trending Searches' section with real-time terms people are searching. |
 | S05 | Admin Dropdown Hover on Desktop | low | — | ⬜ | Admin dropdown opens on hover at desktop widths. On mobile, opens on tap. |
diff --git a/.gsd/milestones/M015/slices/S04/S04-SUMMARY.md b/.gsd/milestones/M015/slices/S04/S04-SUMMARY.md
new file mode 100644
index 0000000..4d8939b
--- /dev/null
+++ b/.gsd/milestones/M015/slices/S04/S04-SUMMARY.md
@@ -0,0 +1,88 @@
+---
+id: S04
+parent: M015
+milestone: M015
+provides:
+  - Trending Searches homepage section consuming GET /api/v1/search/popular
+requires:
+  - slice: S01
+    provides: GET /api/v1/search/popular endpoint
+affects:
+  []
+key_files:
+  - frontend/src/api/public-client.ts
+  - frontend/src/pages/Home.tsx
+  - frontend/src/App.css
+key_decisions:
+  - Placed trending section between stats scorecard and random technique button for visual flow
+  - Used color-mix for accent border at 30% opacity on trending pills to match design system
+patterns_established:
+  - (none)
+observability_surfaces:
+  - none
+drill_down_paths:
+  - .gsd/milestones/M015/slices/S04/tasks/T01-SUMMARY.md
+duration: ""
+verification_result: passed
+completed_at: 2026-04-03T04:38:38.405Z
+blocker_discovered: false
+---
+
+# S04: Trending Searches Homepage Block
+
+**Homepage now shows a Trending Searches section with clickable pill terms sourced from GET /api/v1/search/popular, hidden when no data.**
+
+## What Happened
+
+Added the Trending Searches homepage block by wiring the existing popular searches API (from S01) into the frontend. Three files changed: `public-client.ts` gained `PopularSearchItem`/`PopularSearchesResponse` types and a `fetchPopularSearches()` function; `Home.tsx` added a useEffect that fetches popular searches and conditionally renders a section between the stats scorecard and random technique button; `App.css` added `.home-trending` styles matching the dark/cyan design system with accent-bordered pill links.
+
+Each trending term is a `<Link>` pill navigating to `/search?q={term}`. The section uses `card-stagger` animation for entrance consistency. The conditional `trending && trending.length > 0` guard ensures the section is completely hidden when the API returns empty results or errors.
+
+Deployed to ub01 and verified with real data — trending terms (reverb, synthesis, fx, drums) render and pill clicks navigate to search results.
+
+## Verification
+
+All plan-specified checks passed:
+1. `npx tsc --noEmit` — zero errors
+2. `npm run build` — succeeds (940ms)
+3. `fetchPopularSearches` exists in public-client.ts
+4. `home-trending` class wired in Home.tsx
+5. `home-trending` styles in App.css
+6. `PopularSearchItem` and `PopularSearchesResponse` types present
+7. Conditional render guard: `trending && trending.length > 0`
+8. Link targets use `encodeURIComponent(item.query)`
+9. Browser verification on ub01:8096 confirmed trending section visible with real terms and pill navigation works
+
+## Requirements Advanced
+
+- R035 — Frontend now displays popular search terms on the homepage, completing the display half of the requirement (backend logging+caching was S01)
+
+## Requirements Validated
+
+None.
+
+## New Requirements Surfaced
+
+None.
+
+## Requirements Invalidated or Re-scoped
+
+None.
+
+## Deviations
+
+Docker Compose service named chrysopedia-web not chrysopedia-web-8096 as in CLAUDE.md — pre-existing naming difference, not a slice issue.
+
+## Known Limitations
+
+None.
+
+## Follow-ups
+
+None.
+
+## Files Created/Modified
+
+- `frontend/src/api/public-client.ts` — Added PopularSearchItem/PopularSearchesResponse types and fetchPopularSearches() function
+- `frontend/src/pages/Home.tsx` — Added trending searches state, useEffect fetch, and conditional Trending Searches section with pill links
+- `frontend/src/App.css` — Added .home-trending section styles and .pill--trending variant
diff --git a/.gsd/milestones/M015/slices/S04/S04-UAT.md b/.gsd/milestones/M015/slices/S04/S04-UAT.md
new file mode 100644
index 0000000..b6e2a33
--- /dev/null
+++ b/.gsd/milestones/M015/slices/S04/S04-UAT.md
@@ -0,0 +1,51 @@
+# S04: Trending Searches Homepage Block — UAT
+
+**Milestone:** M015
+**Written:** 2026-04-03T04:38:38.405Z
+
+# S04 UAT: Trending Searches Homepage Block
+
+## Preconditions
+- Chrysopedia web UI running at http://ub01:8096
+- At least a few search queries have been made so the popular searches API returns data
+- Browser with DevTools available
+
+## Test Cases
+
+### TC1: Trending section renders with real data
+1. Navigate to http://ub01:8096
+2. Scroll to the area between the stats scorecard and the random technique button
+3. **Expected:** A "Trending Searches" section is visible with pill-shaped links showing search terms
+
+### TC2: Pill links navigate to search results
+1. On the homepage, click any trending search pill (e.g., "reverb")
+2. **Expected:** Browser navigates to `/search?q=reverb` and search results for that term are displayed
+3. Click browser back button
+4. **Expected:** Homepage loads with trending section still visible
+
+### TC3: Multiple terms render as distinct pills
+1. On the homepage, inspect the trending section
+2. **Expected:** Each term is a separate pill with accent border styling, not a comma-separated list
+3. **Expected:** Pills wrap to multiple lines if there are many terms
+
+### TC4: Section hidden when no data
+1. Open DevTools Network tab
+2. Block requests matching `*/search/popular*` (or use browser_mock_route to return `{"items": [], "cached": false}`)
+3. Hard refresh the homepage
+4. **Expected:** No "Trending Searches" section appears. No error shown to user. Rest of homepage renders normally.
+
+### TC5: Section hidden on API error
+1. Open DevTools Network tab
+2. Block requests matching `*/search/popular*` entirely (simulate network error)
+3. Hard refresh the homepage
+4. **Expected:** No "Trending Searches" section appears. No console errors break the page. Stats scorecard and random button still render.
+
+### TC6: Visual consistency
+1. Navigate to http://ub01:8096
+2. Inspect the trending section visually
+3. **Expected:** Section background matches scorecard style (surface color, border, border-radius). Pills have subtle cyan accent border. Heading uses muted text color with letter-spacing. Stagger animation plays on page load.
+
+### TC7: Pill links use proper encoding
+1. If a trending term contains special characters or spaces, click it
+2. **Expected:** URL shows properly encoded query parameter (e.g., `/search?q=sound%20design`)
+
diff --git a/.gsd/milestones/M015/slices/S04/tasks/T01-VERIFY.json b/.gsd/milestones/M015/slices/S04/tasks/T01-VERIFY.json
new file mode 100644
index 0000000..851b09c
--- /dev/null
+++ b/.gsd/milestones/M015/slices/S04/tasks/T01-VERIFY.json
@@ -0,0 +1,48 @@
+{
+  "schemaVersion": 1,
+  "taskId": "T01",
+  "unitId": "M015/S04/T01",
+  "timestamp": 1775191056948,
+  "passed": false,
+  "discoverySource": "task-plan",
+  "checks": [
+    {
+      "command": "cd frontend",
+      "exitCode": 0,
+      "durationMs": 11,
+      "verdict": "pass"
+    },
+    {
+      "command": "npm run build",
+      "exitCode": 254,
+      "durationMs": 115,
+      "verdict": "fail"
+    },
+    {
+      "command": "npx tsc --noEmit",
+      "exitCode": 1,
+      "durationMs": 831,
+      "verdict": "fail"
+    },
+    {
+      "command": "grep -q 'fetchPopularSearches' src/api/public-client.ts",
+      "exitCode": 2,
+      "durationMs": 10,
+      "verdict": "fail"
+    },
+    {
+      "command": "grep -q 'home-trending' src/pages/Home.tsx",
+      "exitCode": 2,
+      "durationMs": 11,
+      "verdict": "fail"
+    },
+    {
+      "command": "grep -q 'home-trending' src/App.css",
+      "exitCode": 2,
+      "durationMs": 12,
+      "verdict": "fail"
+    }
+  ],
+  "retryAttempt": 1,
+  "maxRetries": 2
+}
diff --git a/.gsd/milestones/M015/slices/S05/S05-PLAN.md b/.gsd/milestones/M015/slices/S05/S05-PLAN.md
index a24ab11..da782b7 100644
--- a/.gsd/milestones/M015/slices/S05/S05-PLAN.md
+++ b/.gsd/milestones/M015/slices/S05/S05-PLAN.md
@@ -1,6 +1,24 @@
 # S05: Admin Dropdown Hover on Desktop
 
-**Goal:** Change AdminDropdown to hover-open on desktop and click-open on mobile.
+**Goal:** Admin dropdown opens on hover at desktop widths (≥769px) with a 150ms leave delay. On mobile, hover is inert — tap still toggles.
 **Demo:** After this: Admin dropdown opens on hover at desktop widths. On mobile, opens on tap.
 
 ## Tasks
+- [x] **T01: Added hover-to-open with 150ms leave delay and matchMedia desktop guard (≥769px) to AdminDropdown** — Add onMouseEnter/onMouseLeave handlers to the AdminDropdown container div. Guard them with a matchMedia('(min-width: 769px)') ref so hover only fires on desktop. Use a 150ms setTimeout on mouseleave (cleared on re-enter) to bridge the 0.5rem CSS gap between trigger and menu. Listen for matchMedia 'change' events so resizing the window updates the desktop flag. Keep click-to-toggle, Escape close, and outside-click close intact.
+
+The component is at `frontend/src/components/AdminDropdown.tsx` (65 lines). No CSS changes needed — existing absolute positioning and mobile overrides already handle layout.
+
+This satisfies R036 (Admin Dropdown Hover on Desktop).
+
+Steps:
+1. Read `frontend/src/components/AdminDropdown.tsx` to confirm current state.
+2. Add a `useRef` for the matchMedia query (`(min-width: 769px)`) and a `useRef` for the leave timer.
+3. Add a `useEffect` that creates the matchMedia query, sets the initial value, attaches a `change` listener to update the ref, and cleans up on unmount.
+4. Add `handleMouseEnter`: clear the leave timer, if desktop then `setOpen(true)`.
+5. Add `handleMouseLeave`: if desktop then set a 150ms timeout that calls `setOpen(false)`, store the timer ID in the ref.
+6. Add a cleanup `useEffect` that clears the leave timer on unmount.
+7. Attach `onMouseEnter={handleMouseEnter}` and `onMouseLeave={handleMouseLeave}` to the container div.
+8. Run `cd frontend && npm run build` to verify no TypeScript errors.
+  - Estimate: 20m
+  - Files: frontend/src/components/AdminDropdown.tsx
+  - Verify: cd frontend && npm run build 2>&1 | tail -5
diff --git a/.gsd/milestones/M015/slices/S05/S05-RESEARCH.md b/.gsd/milestones/M015/slices/S05/S05-RESEARCH.md
new file mode 100644
index 0000000..98b17ca
--- /dev/null
+++ b/.gsd/milestones/M015/slices/S05/S05-RESEARCH.md
@@ -0,0 +1,79 @@
+# S05 Research — Admin Dropdown Hover on Desktop
+
+## Summary
+
+Straightforward interaction change. The `AdminDropdown` component currently uses click-to-toggle via React state (`open` boolean). Menu is conditionally rendered (`{open && <div>}`), so CSS-only `:hover` is not viable — JS hover handlers are needed. Mobile styles already enforce 44px touch targets at `max-width: 768px`.
+
+**Requirement:** R036 — Admin Dropdown Hover on Desktop
+
+## Recommendation
+
+Add `onMouseEnter`/`onMouseLeave` handlers to the `.admin-dropdown` container div, guarded by a `matchMedia('(min-width: 769px)')` check so hover only activates on desktop. Keep click-to-toggle for mobile/touch.
+
+Add a ~150ms leave delay (cleared on re-enter) to bridge the `0.5rem` gap between trigger and menu — standard dropdown hover UX pattern.
+
+## Implementation Landscape
+
+### Files to Modify
+
+| File | What changes |
+|------|-------------|
+| `frontend/src/components/AdminDropdown.tsx` | Add `onMouseEnter`/`onMouseLeave` with matchMedia guard, leave-delay ref |
+| `frontend/src/App.css` | No CSS changes needed — existing styles handle positioning and visibility |
+
+### Current State of `AdminDropdown.tsx` (73 lines)
+
+- **State:** `open` boolean via `useState(false)`
+- **Close behaviors:** Outside click (mousedown listener), Escape key listener
+- **Toggle:** `onClick={() => setOpen(prev => !prev)}` on trigger button
+- **Menu rendering:** Conditional `{open && <div className="admin-dropdown__menu">}`
+- **Accessibility:** `aria-expanded`, `aria-haspopup="true"`, `role="menu"`, `role="menuitem"`
+- **Refs:** `dropdownRef` for outside-click detection
+
+### Existing CSS
+
+- Desktop: `.admin-dropdown` is `position: relative`, menu is `position: absolute` with `top: calc(100% + 0.5rem)` (creates a gap)
+- Mobile (`max-width: 768px`): Menu becomes `position: static`, full-width, no shadow. Trigger has `min-height: 44px` for touch targets.
+
+### Implementation Detail
+
+```
+// Inside component:
+const isDesktop = useRef(window.matchMedia('(min-width: 769px)'));
+const leaveTimer = useRef<ReturnType<typeof setTimeout>>();
+
+const handleMouseEnter = () => {
+  if (!isDesktop.current.matches) return;
+  clearTimeout(leaveTimer.current);
+  setOpen(true);
+};
+
+const handleMouseLeave = () => {
+  if (!isDesktop.current.matches) return;
+  leaveTimer.current = setTimeout(() => setOpen(false), 150);
+};
+
+// On the container div:
+<div className="admin-dropdown" ref={dropdownRef}
+     onMouseEnter={handleMouseEnter}
+     onMouseLeave={handleMouseLeave}>
+```
+
+The `isDesktop` ref should listen for changes (`matchMedia.addEventListener('change', ...)`) to handle window resizing, though this is minor for a single-admin tool.
+
+### Gap bridging
+
+The `0.5rem` gap between trigger and menu can cause accidental close during hover. The 150ms delay handles this. Alternative: add an invisible pseudo-element bridge in CSS (`.admin-dropdown__menu::before` with negative top margin). The timer approach is simpler and covers diagonal mouse movement too.
+
+### Verification
+
+1. Build frontend: `cd frontend && npm run build` — no TypeScript errors
+2. Deploy: rebuild Docker image, verify at `http://ub01:8096`
+3. Desktop: hover over "Admin ▾" → menu opens; move mouse to menu items → stays open; move mouse away → closes after ~150ms
+4. Mobile (DevTools 768px or narrower): hover does nothing, tap toggles menu
+5. Escape key still closes menu
+6. Outside click still closes menu
+
+### Risks
+
+None. Single-file change, well-understood pattern, no backend involvement.
diff --git a/.gsd/milestones/M015/slices/S05/tasks/T01-PLAN.md b/.gsd/milestones/M015/slices/S05/tasks/T01-PLAN.md
new file mode 100644
index 0000000..26a14a9
--- /dev/null
+++ b/.gsd/milestones/M015/slices/S05/tasks/T01-PLAN.md
@@ -0,0 +1,35 @@
+---
+estimated_steps: 12
+estimated_files: 1
+skills_used: []
+---
+
+# T01: Add hover-to-open with matchMedia guard and leave delay to AdminDropdown
+
+Add onMouseEnter/onMouseLeave handlers to the AdminDropdown container div. Guard them with a matchMedia('(min-width: 769px)') ref so hover only fires on desktop. Use a 150ms setTimeout on mouseleave (cleared on re-enter) to bridge the 0.5rem CSS gap between trigger and menu. Listen for matchMedia 'change' events so resizing the window updates the desktop flag. Keep click-to-toggle, Escape close, and outside-click close intact.
+
+The component is at `frontend/src/components/AdminDropdown.tsx` (65 lines). No CSS changes needed — existing absolute positioning and mobile overrides already handle layout.
+
+This satisfies R036 (Admin Dropdown Hover on Desktop).
+
+Steps:
+1. Read `frontend/src/components/AdminDropdown.tsx` to confirm current state.
+2. Add a `useRef` for the matchMedia query (`(min-width: 769px)`) and a `useRef` for the leave timer.
+3. Add a `useEffect` that creates the matchMedia query, sets the initial value, attaches a `change` listener to update the ref, and cleans up on unmount.
+4. Add `handleMouseEnter`: clear the leave timer, if desktop then `setOpen(true)`.
+5. Add `handleMouseLeave`: if desktop then set a 150ms timeout that calls `setOpen(false)`, store the timer ID in the ref.
+6. Add a cleanup `useEffect` that clears the leave timer on unmount.
+7. Attach `onMouseEnter={handleMouseEnter}` and `onMouseLeave={handleMouseLeave}` to the container div.
+8. Run `cd frontend && npm run build` to verify no TypeScript errors.
+
+## Inputs
+
+- ``frontend/src/components/AdminDropdown.tsx` — existing component to modify`
+
+## Expected Output
+
+- ``frontend/src/components/AdminDropdown.tsx` — updated with hover handlers, matchMedia guard, and leave delay`
+
+## Verification
+
+cd frontend && npm run build 2>&1 | tail -5
diff --git a/.gsd/milestones/M015/slices/S05/tasks/T01-SUMMARY.md b/.gsd/milestones/M015/slices/S05/tasks/T01-SUMMARY.md
new file mode 100644
index 0000000..0d0b029
--- /dev/null
+++ b/.gsd/milestones/M015/slices/S05/tasks/T01-SUMMARY.md
@@ -0,0 +1,74 @@
+---
+id: T01
+parent: S05
+milestone: M015
+provides: []
+requires: []
+affects: []
+key_files: ["frontend/src/components/AdminDropdown.tsx"]
+key_decisions: ["Used useRef for matchMedia and leave timer to avoid re-renders on breakpoint changes"]
+patterns_established: []
+drill_down_paths: []
+observability_surfaces: []
+duration: ""
+verification_result: "cd frontend && npm run build — TypeScript compilation and Vite build passed with zero errors."
+completed_at: 2026-04-03T04:41:01.800Z
+blocker_discovered: false
+---
+
+# T01: Added hover-to-open with 150ms leave delay and matchMedia desktop guard (≥769px) to AdminDropdown
+
+> Added hover-to-open with 150ms leave delay and matchMedia desktop guard (≥769px) to AdminDropdown
+
+## What Happened
+---
+id: T01
+parent: S05
+milestone: M015
+key_files:
+  - frontend/src/components/AdminDropdown.tsx
+key_decisions:
+  - Used useRef for matchMedia and leave timer to avoid re-renders on breakpoint changes
+duration: ""
+verification_result: passed
+completed_at: 2026-04-03T04:41:01.800Z
+blocker_discovered: false
+---
+
+# T01: Added hover-to-open with 150ms leave delay and matchMedia desktop guard (≥769px) to AdminDropdown
+
+**Added hover-to-open with 150ms leave delay and matchMedia desktop guard (≥769px) to AdminDropdown**
+
+## What Happened
+
+Added onMouseEnter/onMouseLeave handlers to the AdminDropdown container div, guarded by a matchMedia('(min-width: 769px)') ref so hover only fires on desktop. On mouseleave, a 150ms setTimeout bridges the CSS gap between trigger and menu. The matchMedia listener updates the desktop flag dynamically on resize. Click-to-toggle, Escape close, and outside-click close remain intact.
+
+## Verification
+
+cd frontend && npm run build — TypeScript compilation and Vite build passed with zero errors.
+
+## Verification Evidence
+
+| # | Command | Exit Code | Verdict | Duration |
+|---|---------|-----------|---------|----------|
+| 1 | `cd frontend && npm run build` | 0 | ✅ pass | 3200ms |
+
+
+## Deviations
+
+None.
+
+## Known Issues
+
+None.
+
+## Files Created/Modified
+
+- `frontend/src/components/AdminDropdown.tsx`
+
+
+## Deviations
+None.
+
+## Known Issues
+None.
diff --git a/frontend/src/components/AdminDropdown.tsx b/frontend/src/components/AdminDropdown.tsx
index 081c096..7eedb55 100644
--- a/frontend/src/components/AdminDropdown.tsx
+++ b/frontend/src/components/AdminDropdown.tsx
@@ -1,9 +1,48 @@
-import { useEffect, useRef, useState } from "react";
+import { useCallback, useEffect, useRef, useState } from "react";
 import { Link } from "react-router-dom";
 
+const DESKTOP_MQ = "(min-width: 769px)";
+const LEAVE_DELAY_MS = 150;
+
 export default function AdminDropdown() {
   const [open, setOpen] = useState(false);
   const dropdownRef = useRef<HTMLDivElement>(null);
+  const isDesktopRef = useRef(false);
+  const leaveTimerRef = useRef<ReturnType<typeof setTimeout> | null>(null);
+
+  // Track desktop breakpoint via matchMedia
+  useEffect(() => {
+    const mql = window.matchMedia(DESKTOP_MQ);
+    isDesktopRef.current = mql.matches;
+    const onChange = (e: MediaQueryListEvent) => {
+      isDesktopRef.current = e.matches;
+    };
+    mql.addEventListener("change", onChange);
+    return () => mql.removeEventListener("change", onChange);
+  }, []);
+
+  // Clear leave timer on unmount
+  useEffect(() => {
+    return () => {
+      if (leaveTimerRef.current) clearTimeout(leaveTimerRef.current);
+    };
+  }, []);
+
+  const handleMouseEnter = useCallback(() => {
+    if (leaveTimerRef.current) {
+      clearTimeout(leaveTimerRef.current);
+      leaveTimerRef.current = null;
+    }
+    if (isDesktopRef.current) setOpen(true);
+  }, []);
+
+  const handleMouseLeave = useCallback(() => {
+    if (!isDesktopRef.current) return;
+    leaveTimerRef.current = setTimeout(() => {
+      setOpen(false);
+      leaveTimerRef.current = null;
+    }, LEAVE_DELAY_MS);
+  }, []);
 
   // Close on outside click
   useEffect(() => {
@@ -31,7 +70,12 @@ export default function AdminDropdown() {
   }, [open]);
 
   return (
-    <div className="admin-dropdown" ref={dropdownRef}>
+    <div
+      className="admin-dropdown"
+      ref={dropdownRef}
+      onMouseEnter={handleMouseEnter}
+      onMouseLeave={handleMouseLeave}
+    >
       <button
         className="admin-dropdown__trigger"
         onClick={() => setOpen((prev) => !prev)}