diff --git a/web/src/App.vue b/web/src/App.vue
index 675c0b6..c183b7e 100644
--- a/web/src/App.vue
+++ b/web/src/App.vue
@@ -163,15 +163,15 @@ onMounted(async () => {
     grid-column: 1;
     grid-row: 1;
     min-height: 0;
-    display: flex;
-    flex-direction: column;
+    position: relative;
   }
 
-  /* The settings card fills the sidebar so its height matches the two-row
-     graph grid; it scrolls internally only if its own content is taller. */
+  /* The card absolutely fills the sidebar, so its own content never grows the
+     row: the graph grid defines the height, the sidebar matches it, and the
+     card scrolls internally if its content (e.g. Advanced expanded) is taller. */
   .sidebar :deep(.card) {
-    flex: 1;
-    min-height: 0;
+    position: absolute;
+    inset: 0;
     overflow-y: auto;
   }
 
diff --git a/web/src/components/AdvancedFields.vue b/web/src/components/AdvancedFields.vue
index b0dc716..25e84ba 100644
--- a/web/src/components/AdvancedFields.vue
+++ b/web/src/components/AdvancedFields.vue
@@ -79,21 +79,6 @@ function setField(field: keyof Config, event: Event) {
   if (Number.isFinite(nextValue)) store.setBaseField(field, nextValue)
 }
 
-// Seeds are just names for worlds; a small range keeps the fields and
-// shared URLs readable while still giving plenty of distinct worlds.
-function freshSeed(): number {
-  const buffer = new Uint32Array(1)
-  crypto.getRandomValues(buffer)
-  return (buffer[0] ?? 0) % 10000
-}
-
-function reroll(field: keyof Config) {
-  store.setBaseField(field, freshSeed())
-}
-
-function rerollWorld() {
-  for (const { field } of seedFields) store.setBaseField(field, freshSeed())
-}
 </script>
 
 <template>
@@ -120,6 +105,13 @@ function rerollWorld() {
           @change="setField(def.field, $event)"
         />
       </label>
+    </div>
+
+    <div class="seeds-head">
+      <span class="seeds-title">Seeds</span>
+      <span class="seeds-note">labels for a random world, not counts</span>
+    </div>
+    <div class="grid">
       <span
         v-for="def in seedFields"
         :key="def.field"
@@ -128,20 +120,23 @@ function rerollWorld() {
       >
         <label class="seed">
           <span class="fl">{{ def.label }}</span>
-          <input
-            type="number"
-            min="0"
-            step="1"
-            :value="store.state.base[def.field]"
-            :aria-invalid="offendingField === def.field || undefined"
-            @change="setField(def.field, $event)"
-          />
+          <span class="seedval">
+            <span class="hash" aria-hidden="true">#</span>
+            <input
+              type="number"
+              min="0"
+              step="1"
+              :value="store.state.base[def.field]"
+              :aria-invalid="offendingField === def.field || undefined"
+              @change="setField(def.field, $event)"
+            />
+          </span>
         </label>
         <button
           class="dice"
           type="button"
           :aria-label="`Reroll ${def.label}`"
-          @click="reroll(def.field)"
+          @click="store.rerollSeed(def.field)"
         >
           <svg class="ic" viewBox="0 0 24 24">
             <rect x="4" y="4" width="16" height="16" rx="3" />
@@ -150,7 +145,6 @@ function rerollWorld() {
         </button>
       </span>
     </div>
-    <button class="reroll-world" type="button" @click="rerollWorld()">Reroll world</button>
   </details>
 </template>
 
@@ -226,6 +220,44 @@ details[open] .chevron {
   min-width: 0;
 }
 
+.seeds-head {
+  display: flex;
+  align-items: baseline;
+  flex-wrap: wrap;
+  gap: 8px;
+  margin-top: 18px;
+}
+
+.seeds-title {
+  font-size: 12px;
+  font-weight: 600;
+  letter-spacing: 0.04em;
+  text-transform: uppercase;
+  color: var(--ink-4);
+}
+
+.seeds-note {
+  font-size: 11.5px;
+  font-weight: 400;
+  color: var(--ink-4);
+}
+
+.seedval {
+  display: flex;
+  align-items: center;
+  gap: 3px;
+}
+
+.hash {
+  color: var(--ink-4);
+  font-weight: 600;
+  font-size: 13px;
+}
+
+.seedval input[type='number'] {
+  width: 52px;
+}
+
 input[type='number'] {
   width: 72px;
   font: 600 13px/1.4 inherit;
@@ -258,23 +290,6 @@ input[type='number'] {
   height: 14px;
 }
 
-.reroll-world {
-  margin-top: 12px;
-  font-size: 12.5px;
-  font-weight: 600;
-  color: var(--ink-3);
-  background: none;
-  border: 1px solid var(--border);
-  border-radius: 8px;
-  padding: 5px 12px;
-  cursor: pointer;
-}
-
-.reroll-world:hover {
-  border-color: var(--ink-4);
-  color: var(--ink);
-}
-
 /* Single column when the controls container is narrow (sidebar / mobile). */
 @container controls (max-width: 380px) {
   .grid {
diff --git a/web/src/components/ControlsCard.vue b/web/src/components/ControlsCard.vue
index 12885bf..9caeb38 100644
--- a/web/src/components/ControlsCard.vue
+++ b/web/src/components/ControlsCard.vue
@@ -33,7 +33,16 @@ const offendsProgramEffect = offends('programEffect')
 
 <template>
   <section class="card ctl" aria-label="World settings">
-    <h3>World</h3>
+    <div class="ctl-head">
+      <h3>World</h3>
+      <button class="reroll" type="button" title="Draw a new random world" @click="store.rerollSeeds()">
+        <svg class="ic" viewBox="0 0 24 24" aria-hidden="true">
+          <rect x="4" y="4" width="16" height="16" rx="3" />
+          <path d="M9 9h.01M15 15h.01M15 9h.01M9 15h.01" />
+        </svg>
+        Reroll world
+      </button>
+    </div>
     <p v-if="store.state.validationError" class="invalid-note" role="alert">
       {{ store.state.validationError }}
     </p>
@@ -122,6 +131,13 @@ const offendsProgramEffect = offends('programEffect')
   container-name: controls;
 }
 
+.ctl-head {
+  display: flex;
+  align-items: center;
+  justify-content: space-between;
+  gap: 8px;
+}
+
 h3 {
   font-size: 13px;
   font-weight: 600;
@@ -130,6 +146,35 @@ h3 {
   color: var(--ink-4);
 }
 
+.reroll {
+  display: inline-flex;
+  align-items: center;
+  gap: 6px;
+  font-size: 12px;
+  font-weight: 600;
+  color: var(--ink-3);
+  background: var(--bg);
+  border: 1px solid var(--border);
+  border-radius: 999px;
+  padding: 4px 11px;
+  cursor: pointer;
+}
+
+.reroll:hover {
+  border-color: var(--ink-4);
+  color: var(--ink);
+}
+
+.reroll svg.ic {
+  width: 13px;
+  height: 13px;
+  fill: none;
+  stroke: currentColor;
+  stroke-width: 2;
+  stroke-linecap: round;
+  stroke-linejoin: round;
+}
+
 .group + .group {
   margin-top: 22px;
   border-top: 1px solid var(--border-soft);
diff --git a/web/src/components/PanelGrid.vue b/web/src/components/PanelGrid.vue
index 5f20267..3e2ad17 100644
--- a/web/src/components/PanelGrid.vue
+++ b/web/src/components/PanelGrid.vue
@@ -40,7 +40,12 @@ function syncRowHeight() {
   const element = gridElement.value
   if (!element) return
   if (oneScreen.matches) {
-    const rowHeight = Math.max(150, Math.floor((element.clientHeight - ROW_GAP_PX) / 2))
+    // clientHeight includes the container's padding (which gives the card
+    // shadows room); subtract it so two rows fit the content area exactly.
+    const style = getComputedStyle(element)
+    const padV = parseFloat(style.paddingTop) + parseFloat(style.paddingBottom)
+    const available = element.clientHeight - padV
+    const rowHeight = Math.max(150, Math.floor((available - ROW_GAP_PX) / 2))
     element.style.setProperty('--row-h', `${rowHeight}px`)
   } else {
     element.style.removeProperty('--row-h')
@@ -166,9 +171,10 @@ onBeforeUnmount(() => {
 @media (min-width: 761px) {
   .panelgrid {
     overflow-y: auto;
-    /* Reserve room so the scrollbar (overlay or classic) never sits over the
-       right column of graphs when a 5th scenario makes the grid scroll. */
-    padding-right: 14px;
+    /* Room so the card shadows are not clipped at the scroll edges; the right
+       also leaves the scrollbar gutter. syncRowHeight subtracts this padding
+       so the 2x2 still fits exactly. */
+    padding: 4px 14px 14px 6px;
     scrollbar-gutter: stable;
   }
 }
diff --git a/web/src/composables/useLayout.ts b/web/src/composables/useLayout.ts
index 9954f40..b1424cb 100644
--- a/web/src/composables/useLayout.ts
+++ b/web/src/composables/useLayout.ts
@@ -16,8 +16,8 @@ export interface LayoutPoint {
   y: number
 }
 
-export const NETWORK_VIEW_WIDTH = 380
-export const NETWORK_VIEW_HEIGHT = 230
+export const NETWORK_VIEW_WIDTH = 340
+export const NETWORK_VIEW_HEIGHT = 270
 const LAYOUT_PADDING = 14
 const LAYOUT_TICKS = 300
 
@@ -38,29 +38,31 @@ export function layoutForGraph(config: Config, edges: number[][]): LayoutPoint[]
 
   const random = mulberry32(Number(config.graphSeed) >>> 0)
   const nodes: LayoutNode[] = Array.from({ length: config.numStudents }, (_, index) => {
-    // Seeded start positions in a wide disc; the forces do the rest.
+    // Seeded start positions in a balanced disc; the forces do the rest.
     const angle = random() * Math.PI * 2
     const radius = Math.sqrt(random())
-    return { index, x: Math.cos(angle) * radius * 150, y: Math.sin(angle) * radius * 60 }
+    return { index, x: Math.cos(angle) * radius * 120, y: Math.sin(angle) * radius * 90 }
   })
   const links: SimulationLinkDatum<LayoutNode>[] = edges.map((edge) => ({
     source: edge[0] ?? 0,
     target: edge[1] ?? 0,
   }))
 
-  // The vertical pull is stronger than the horizontal one, so the cloud
-  // settles into the panel's wide ellipse by itself; fitting then scales
-  // both axes uniformly, which keeps the spacing organic (anisotropic
-  // stretching reads as line patterns).
+  // A gentle horizontal pull with a stronger vertical one settles the cloud
+  // into a slightly wide blob (a softer version of the original ellipse) while
+  // staying organic: its hubs and clusters show, not a uniform disc. The
+  // charge, link and collision forces do the structural work; fitting then
+  // scales both axes uniformly, so the spacing stays organic (anisotropic
+  // stretching would read as line patterns).
   forceSimulation(nodes)
     .randomSource(random)
     .force(
       'link',
       forceLink<LayoutNode, SimulationLinkDatum<LayoutNode>>(links).distance(16).strength(0.3),
     )
-    .force('charge', forceManyBody().strength(-15))
-    .force('x', forceX(0).strength(0.028))
-    .force('y', forceY(0).strength(0.22))
+    .force('charge', forceManyBody().strength(-16))
+    .force('x', forceX(0).strength(0.04))
+    .force('y', forceY(0).strength(0.085))
     .force('collide', forceCollide(5.5))
     .stop()
     .tick(LAYOUT_TICKS)
@@ -82,6 +84,10 @@ export function layoutForGraph(config: Config, edges: number[][]): LayoutPoint[]
     x: offsetX + ((node.x ?? 0) - minX) * scale,
     y: offsetY + ((node.y ?? 0) - minY) * scale,
   }))
-  layoutCache.set(cacheKey, points)
+  // Only cache once the real edges are in. A reroll changes the config (and
+  // this cache key) immediately, before the new edges arrive from the API;
+  // computing then would cache a link-less, uniform-circle layout under this
+  // key and never recompute it. Without edges, recompute next time instead.
+  if (edges.length > 0) layoutCache.set(cacheKey, points)
   return points
 }
diff --git a/web/src/composables/usePlayback.ts b/web/src/composables/usePlayback.ts
index 83d53ab..3c73f8f 100644
--- a/web/src/composables/usePlayback.ts
+++ b/web/src/composables/usePlayback.ts
@@ -1,3 +1,4 @@
+import { watch } from 'vue'
 import { useSimStore, type SimStore } from './useSimStore'
 
 // Playback (spec 5.4): one global round drives every panel, the chart
@@ -142,6 +143,21 @@ export function createPlayback(store: SimStore = useSimStore()) {
     play()
   }
 
+  // A reroll (rerollSeed / rerollSeeds) asks for a fresh playthrough: once its
+  // debounced run lands (running -> idle), restart the spread from round 0.
+  watch(
+    () => state.runState,
+    (now, before) => {
+      if (!state.pendingReplay) return
+      if (now === 'idle' && before === 'running') {
+        state.pendingReplay = false
+        replay()
+      } else if (now === 'error') {
+        state.pendingReplay = false
+      }
+    },
+  )
+
   return {
     play,
     pause,
diff --git a/web/src/composables/useSimStore.ts b/web/src/composables/useSimStore.ts
index ab4e543..7e49d6c 100644
--- a/web/src/composables/useSimStore.ts
+++ b/web/src/composables/useSimStore.ts
@@ -3,6 +3,7 @@ import { ApiError, fetchDefaultConfig, runScenario } from '@/lib/api'
 import { clampConfig, clampConfigField } from '@/lib/bounds'
 import { roundPct } from '@/lib/format'
 import { graphKey } from '@/lib/graph'
+import { freshSeed } from '@/lib/seed'
 import { parseUrlState, serializeUrlState } from '@/lib/urlState'
 import { deepfakeSchoolPreset } from '@/presets/deepfake-school'
 import { MAX_PANELS, type PanelSpec, type StudyPreset } from '@/presets/types'
@@ -44,6 +45,7 @@ export interface SimState {
   validationError: string | null // 400 from the engine, shown inline under the controls
   urlStateInvalid: boolean // the opened link held malformed state; preset shown instead
   announcement: string // text for the single polite live region (spec section 8)
+  pendingReplay: boolean // a reroll asked playback to restart from round 0 once the run lands
 }
 
 function createPanelId(): string {
@@ -74,6 +76,7 @@ export function createSimStore(preset: StudyPreset = deepfakeSchoolPreset) {
     validationError: null,
     urlStateInvalid: false,
     announcement: '',
+    pendingReplay: false,
   })
 
   let debounceTimer: ReturnType<typeof setTimeout> | null = null
@@ -195,6 +198,20 @@ export function createSimStore(preset: StudyPreset = deepfakeSchoolPreset) {
     scheduleRun()
   }
 
+  // Seeds name the random world. Rerolling one (or all) draws fresh values
+  // and asks playback to replay the spread from round 0 once the rerun lands.
+  const SEED_FIELDS = ['graphSeed', 'thresholdSeed', 'educationSeed'] as const
+
+  function rerollSeed(field: keyof Config) {
+    state.pendingReplay = true
+    setBaseField(field, freshSeed())
+  }
+
+  function rerollSeeds() {
+    state.pendingReplay = true
+    for (const field of SEED_FIELDS) setBaseField(field, freshSeed())
+  }
+
   // Re-clamp the base config and every panel override; needed when
   // numStudents changes (the relational ceilings move) and when the
   // bounds first arrive while the URL may have carried wild values.
@@ -351,6 +368,8 @@ export function createSimStore(preset: StudyPreset = deepfakeSchoolPreset) {
     initialize,
     retry,
     setBaseField,
+    rerollSeed,
+    rerollSeeds,
     addPanel,
     duplicatePanel,
     removePanel,
diff --git a/web/src/lib/seed.ts b/web/src/lib/seed.ts
new file mode 100644
index 0000000..f707256
--- /dev/null
+++ b/web/src/lib/seed.ts
@@ -0,0 +1,8 @@
+// Seeds are just names for a random world. A small range (0-9999) keeps the
+// fields and shared URLs readable while still giving plenty of distinct
+// worlds (amended 2026-06-10: a full uint32 made fields and URLs unreadable).
+export function freshSeed(): number {
+  const buffer = new Uint32Array(1)
+  crypto.getRandomValues(buffer)
+  return (buffer[0] ?? 0) % 10000
+}