feat(samples): add "Overshoot and collapse" gallery sample

The twelfth sample fills the one classic dynamic the gallery was missing —
overshoot and collapse, the dynamic Thinking in Systems is built around. A
Reinforcing engine (Capital reinvesting its extraction revenue) runs on a
non-renewable Resource, the first Stock in the gallery with no inflow: it
overshoots the limit instead of settling at it, the dark twin of "Limits to
growth".

Buildable with the existing rule vocabulary (proportional only); the
non-negative-stock floor tames the bilinear extraction term as the Resource
runs out. Tuned against the simulator: Capital 5 → ~250 (t≈39) → ~5 by t=150,
Resource 1000 → ~6, no divergence. The loop detector classifies the three
loops as expected (R: investment→Capital→extraction; B: depreciation→Capital;
B: extraction→Resource).

README.md

@@ -19,6 +19,11 @@ Deployed at https://meadows.apoena.dev
     substrate; the domain Model is the source of truth.
   - [ADR-0003](./docs/adr/0003-flow-as-node-materialised-clouds.md) — a Flow is a
     node; Source/Sink clouds are materialised nodes.
+  - [ADR-0004](./docs/adr/0004-rate-rules-not-formulas.md) — simulation behaviour
+    comes from a small fixed vocabulary of rules over Information Links, not
+    free-form formulas.
+  - [ADR-0005](./docs/adr/0005-uplot-behaviour-chart.md) — uPlot draws the
+    behaviour-over-time chart; a charting dependency that earns its keep.
 
 ## Stack
 

docs/adr/0004-rate-rules-not-formulas.md

@@ -0,0 +1,74 @@
+# Behaviour comes from rules over Information Links, not free-form formulas
+
+_Part of [meadows](../../README.md) · see [DESIGN.md](../../DESIGN.md)._
+
+Numeric simulation (phase 2) makes a **Model** _alive_: Stocks accumulate over
+time, Flows and Converters recompute each instant. Two coupled decisions shape
+how a Model carries the numbers, and both trade expressive power for **valid by
+construction** — the right trade for a tool that exists to _popularise_ systems
+thinking, not to compete with Vensim.
+
+## Decision
+
+**1. The Information Link _is_ the declared dependency.** A Flow's or Converter's
+inputs are exactly the elements that link into it. There is no separate "equation
+references" namespace to keep in sync — the wiring you draw _is_ the wiring the
+simulator reads. The same signed graph the loop detector walks (ADR-0001) is the
+graph the simulator integrates, so the loops you _see_ classified R/B are the
+loops you _run_. They can never disagree.
+
+**2. A Flow/Converter computes from a small fixed vocabulary of `Rule`s, not a
+typed-in formula.** Each instantaneous element picks one rule and a plain number
+or two — never an expression:
+
+| Rule             | Value                       | Reads (via Information Links)             | Emergent behaviour         |
+| ---------------- | --------------------------- | ----------------------------------------- | -------------------------- |
+| **Constant**     | a fixed number              | nothing                                   | linear Stock change        |
+| **Proportional** | `factor × (its `+` inputs)` | the `+`-polarity inputs                   | exponential growth / decay |
+| **Gap**          | `factor × (level − target)` | the `+` input is _level_, `−` is _target_ | goal-seeking / asymptotic  |
+
+The famous curves are _compositions_ of these over the structure — a logistic
+S-curve is Proportional growth meeting a Gap-driven ceiling (limits-to-growth);
+goal-seeking decay is a lone Gap (coffee cooling). The user sets up a local rule;
+the global shape **emerges**. That emergence _is_ the lesson.
+
+**Polarity does double duty.** The `+`/`−` already captured for loop
+classification (ADR-0001) also selects each operand's role: Proportional reads
+its `+` inputs; Gap reads its `+` input as the level and its `−` input as the
+target. One gesture, two payoffs — no new per-link data.
+
+## Considered Options
+
+- **Free-form expression strings** (`birth rate = Population × fertility`) —
+  maximally expressive, and what `equation?: string` originally anticipated.
+  Rejected: needs a parser + a sandbox (never `eval`/`new Function`), invites
+  broken-formula and name-resolution errors (auto-names contain spaces), and lets
+  a learner _paint_ a curve instead of discovering it from structure.
+- **Pick the output curve** (label a Stock "exponential" / "logarithmic") —
+  rejected: it is the answer, not the cause, and it breaks the moment feedback
+  decides the shape. "Logarithmic" in particular has no honest local rule; what
+  people mean by it is asymptotic approach — which _is_ the Gap rule.
+- **Rules over Information Links (chosen)** — no parser, valid by construction,
+  and it teaches structure → behaviour.
+
+## Consequences
+
+- The domain types gain a `Rule` union on Flow/Converter (replacing the unused
+  `equation?: string`), an optional `initialValue` on Stock, and an optional
+  `SimSpec` (`start` / `stop` / `dt`) on the Model. All optional and additive, so
+  existing saved Models still load (F8); they are simply not _simulatable_ until
+  equipped.
+- **Algebraic loops are an error.** A cycle in the wiring is legitimate feedback
+  **iff it passes through a Stock** — the Stock supplies last-step state and so
+  breaks the within-step dependency. A cycle among only Flows/Converters has no
+  Stock to break it: the simulator cannot order it and rejects it. The simulator
+  reuses the cycle machinery to detect this; it is a new _sim-readiness_ check,
+  distinct from structural validity (validation.ts).
+- A reader seeing `{ kind: "gap", factor }` on a Flow and wondering where its
+  operands come from should look here: they are the Flow's inbound Information
+  Links, picked by Polarity.
+- The vocabulary starts deliberately small (Constant / Proportional / Gap —
+  enough for linear, exponential, and goal-seeking, and for the coffee and
+  savings samples). Growing it is additive: a new `kind` in the union plus a case
+  in the evaluator. Multi-input products (e.g. `Population × fertility`) are a
+  later increment, not a phase-2 blocker.

docs/adr/0005-uplot-behaviour-chart.md

@@ -0,0 +1,55 @@
+# uPlot draws the behaviour-over-time chart
+
+_Part of [meadows](../../README.md) · see [DESIGN.md](../../DESIGN.md)._
+
+The Results panel (ADR-0004's phase 2) traces each Stock's value over time. It
+began as a **hand-built SVG** — a deliberate lean-substrate choice, no charting
+dependency. But a polyline is all that substrate could draw: no time axis, no
+gridlines, and a legend that could only show a Stock's _final_ value, never the
+value under the cursor. For a tool whose whole point is reading _behaviour over
+time_, "what is this Stock at t = 40?" is the question you most want to answer by
+pointing at the curve.
+
+## Decision
+
+Adopt **uPlot** (`uplot`, ~15 KB gzip) — a small, fast canvas time-series
+library — for the behaviour-over-time chart. It earns the dependency with the
+three things the SVG could not give: a real **time axis** with ticks and
+gridlines, and a live legend that **reads each Stock's value at the hovered
+time**.
+
+uPlot is imperative and canvas-based, so it is wrapped in **`SimChart.vue`**,
+which owns its lifecycle (build on mount, `setData` on recompute, rebuild on a
+track-set change, `ResizeObserver` for width, destroy on unmount). `ResultsPanel`
+stays declarative: it assembles plain aligned data (Stocks only — the system's
+memory, per ADR-0004) and hands it down.
+
+## Considered Options
+
+- **Keep the hand-built SVG** (zero deps) — rejected: re-implementing axes,
+  tick placement, a hover cursor, and hit-testing _is_ rebuilding a charting
+  library, badly. The lean-substrate note was a code comment, not an ADR; nothing
+  prior is contradicted by spending one dependency where it clearly pays.
+- **`@gouvfr/dsfr-chart`** (the original prompt) — rejected: it requires the
+  entire DSFR design system (CSS + JS API) and embeds its _own_ Vue and Chart.js,
+  so we would ship Vue twice and a second design language clashing with DaisyUI.
+  It is built for French-government dashboards (region maps, gauges), not generic
+  time series.
+- **Chart.js / vue-chartjs** — viable and same engine class, but heavier; uPlot
+  is lighter and faster for many series and live-updating data, which is exactly
+  the simulation's shape.
+- **uPlot (chosen)** — lightest credible option that gives the axis + hover.
+
+## Consequences
+
+- This is the project's **first runtime charting dependency**. The price, as with
+  Vue Flow (ADR-0002), is a thin wrapper layer: `SimChart.vue` translates between
+  reactive props and uPlot's imperative API.
+- The x scale runs with **`time: false`** — x is _simulation_ time (`start…stop`),
+  not wall-clock, which uPlot would otherwise format as calendar dates.
+- Track colours are read from the **DaisyUI theme variables**. The app ships a
+  single `light` theme, so they are resolved once at build time; a future dark
+  theme would need the plot re-initialised on theme change.
+- Swapping charting libraries later means rewriting only `SimChart.vue` —
+  `ResultsPanel` passes plain `[times, ...tracks]` aligned data and knows nothing
+  of uPlot.

package.json

@@ -19,6 +19,7 @@
     "daisyui": "^5.5.23",
     "idb": "^8.0.3",
     "pinia": "^3.0.4",
+    "uplot": "^1.6.32",
     "vue": "^3.5.34"
   },
   "devDependencies": {

pnpm-lock.yaml

@@ -26,6 +26,9 @@ importers:
       pinia:
         specifier: ^3.0.4
         version: 3.0.4(typescript@6.0.3)(vue@3.5.38(typescript@6.0.3))
+      uplot:
+        specifier: ^1.6.32
+        version: 1.6.32
       vue:
         specifier: ^3.5.34
         version: 3.5.38(typescript@6.0.3)
@@ -939,6 +942,9 @@ packages:
   undici-types@7.18.2:
     resolution: {integrity: sha512-AsuCzffGHJybSaRrmr5eHr81mwJU3kjw6M+uprWvCXiNeN9SOGwQ3Jn8jb8m3Z6izVgknn1R0FTCEAP2QrLY/w==}
 
+  uplot@1.6.32:
+    resolution: {integrity: sha512-KIMVnG68zvu5XXUbC4LQEPnhwOxBuLyW1AHtpm6IKTXImkbLgkMy+jabjLgSLMasNuGGzQm/ep3tOkyTxpiQIw==}
+
   vite@8.0.16:
     resolution: {integrity: sha512-h9bXPmJichP5fLmVQo3PyaGSDE2n3aPuomeAlVRm0JLmt4rY6zmPKd59HYI4LNW8oTK7tlTsuC7l/m7awx9Jcw==}
     engines: {node: ^20.19.0 || >=22.12.0}
@@ -1709,6 +1715,8 @@ snapshots:
 
   undici-types@7.18.2: {}
 
+  uplot@1.6.32: {}
+
   vite@8.0.16(@types/node@24.13.2)(jiti@2.7.0):
     dependencies:
       lightningcss: 1.32.0

public/favicon.svg

@@ -4,7 +4,7 @@
   height="24"
   viewBox="0 0 24 24"
   fill="none"
-  stroke="#16A34A"
+  stroke="#ff9ff3"
   stroke-width="2"
   stroke-linecap="round"
   stroke-linejoin="round"

src/components/Editor.vue

@@ -26,6 +26,7 @@ import {
 } from "@vue-flow/core"
 import { computed, onBeforeUnmount, onMounted, ref, useTemplateRef } from "vue"
 import { useAutosave } from "@/composables/useAutosave"
+import { usePlayback } from "@/composables/usePlayback"
 import { parseModel, serializeModel } from "@/model/io"
 import { project } from "@/model/projection"
 import { type Sample, SAMPLES } from "@/model/samples"
@@ -33,8 +34,10 @@ import { canConnect } from "@/model/validation"
 import { useModelStore } from "@/store/model"
 import { NODE_DND_MIME, type PlaceableKind } from "./palette-dnd"
 import GlossPanel from "./GlossPanel.vue"
+import Inspector from "./Inspector.vue"
 import LoopOverlay from "./LoopOverlay.vue"
 import Palette from "./Palette.vue"
+import ResultsPanel from "./ResultsPanel.vue"
 import InfoLinkEdge from "./edges/InfoLinkEdge.vue"
 import PipeEdge from "./edges/PipeEdge.vue"
 import CloudNode from "./nodes/CloudNode.vue"
@@ -48,6 +51,14 @@ const graph = computed(() => project(store.model))
 const nodes = computed(() => graph.value.nodes)
 const edges = computed(() => graph.value.edges)
 
+// The simulation results panel (phase 2). Toggled from the header; the panel runs
+// the Model and recomputes reactively while open.
+const showResults = ref(false)
+
+// The playback clock that advances the playhead while a run is playing (mounted
+// once here; the simulation store holds the state it drives).
+usePlayback()
+
 // Explicit shared id: useVueFlow() runs here in the parent setup, before
 // <VueFlow> mounts. Pinning both to the same id guarantees they resolve to one
 // store instance, so the event hooks below actually fire.
@@ -338,7 +349,28 @@ onBeforeUnmount(() => {
 <template>
   <div class="flex h-screen flex-col bg-base-200">
     <header class="flex items-center gap-3 border-b border-base-300 bg-base-100 px-4 py-2">
-      <img src="/favicon.svg" alt="" class="size-6" />
+      <!-- Inlined favicon so the logo strokes with the theme primary (currentColor
+           via text-primary); an external <img> can't read the theme var. -->
+      <svg
+        class="size-6 text-primary"
+        viewBox="0 0 24 24"
+        fill="none"
+        stroke="currentColor"
+        stroke-width="2"
+        stroke-linecap="round"
+        stroke-linejoin="round"
+        aria-hidden="true"
+      >
+        <path d="M14 20a2 2 0 1 0 -4 0a2 2 0 0 0 4 0" />
+        <path d="M14 4a2 2 0 1 0 -4 0a2 2 0 0 0 4 0" />
+        <path d="M6 12a2 2 0 1 0 -4 0a2 2 0 0 0 4 0" />
+        <path d="M22 12a2 2 0 1 0 -4 0a2 2 0 0 0 4 0" />
+        <path d="M14 12a2 2 0 1 0 -4 0a2 2 0 0 0 4 0" />
+        <path d="M6 12h4" />
+        <path d="M14 12h4" />
+        <path d="M12 6v4" />
+        <path d="M12 14v4" />
+      </svg>
       <h1 class="text-lg font-semibold">meadows</h1>
       <span class="text-sm text-base-content/50">
         {{ store.nodeCount }} {{ store.nodeCount === 1 ? "element" : "elements" }}
@@ -348,7 +380,7 @@ onBeforeUnmount(() => {
           <button tabindex="0" class="btn btn-ghost btn-sm">Samples</button>
           <ul
             tabindex="0"
-            class="dropdown-content menu z-10 mt-1 w-72 gap-1 rounded-box border border-base-300 bg-base-100 p-2 shadow-lg"
+            class="dropdown-content menu z-40 mt-1 max-h-80 w-72 flex-nowrap gap-1 overflow-y-auto rounded-box border border-base-300 bg-base-100 p-2 shadow-lg"
           >
             <li v-for="sample in SAMPLES" :key="sample.title">
               <button class="flex flex-col items-start gap-0.5" @click="loadSample(sample)">
@@ -358,6 +390,13 @@ onBeforeUnmount(() => {
             </li>
           </ul>
         </div>
+        <button
+          class="btn btn-primary btn-sm"
+          :class="{ 'btn-active': showResults }"
+          @click="showResults = !showResults"
+        >
+          Simulate
+        </button>
         <button class="btn btn-ghost btn-sm" @click="exportModel">Export</button>
         <button class="btn btn-ghost btn-sm" @click="fileInput?.click()">Import</button>
         <input
@@ -420,6 +459,8 @@ onBeforeUnmount(() => {
       <Palette class="absolute top-3 left-3 z-20" @add="addNode" />
       <LoopOverlay />
       <GlossPanel />
+      <Inspector />
+      <ResultsPanel v-if="showResults" @close="showResults = false" />
 
       <!-- Self-dismissing teaching hint, e.g. when a Flow is dragged back onto its
            own Stock to "close the loop". Click to dismiss early; z-30 keeps it above

src/components/Inspector.vue

@@ -0,0 +1,164 @@
+<script setup lang="ts">
+/**
+ * Inspector (phase 2, ADR-0004) — equips the *selected* element with the numbers
+ * a simulation needs, so a Model becomes more than samples: a Stock gets its
+ * initial value, a Flow/Converter its rule. Editing is deliberately small —
+ * choose a rule from the fixed vocabulary and type one or two numbers; there is
+ * no formula box, so the Model stays valid by construction.
+ *
+ * It reads selection from the shared Vue Flow instance (as GlossPanel does), then
+ * resolves the *live* domain node from the store so edits round-trip through
+ * undoable store actions. Operands for a rule are the element's inbound
+ * Information Links picked by Polarity — set those by wiring, not here.
+ */
+import { useVueFlow } from "@vue-flow/core"
+import { computed } from "vue"
+import type { EdgeData } from "@/model/projection"
+import type { ConverterNode, FlowNode, Rule, StockNode } from "@/model/types"
+import { useModelStore } from "@/store/model"
+
+const store = useModelStore()
+const { getSelectedNodes, getSelectedEdges } = useVueFlow("meadows")
+
+/** The single selected element's id — a node directly, or a Flow via its pipe edge. */
+const selectedId = computed<string | null>(() => {
+  const nodes = getSelectedNodes.value
+  const edges = getSelectedEdges.value
+  if (nodes.length === 1 && edges.length === 0) return nodes[0].id
+  if (edges.length === 1 && nodes.length === 0) {
+    const edge = edges[0]
+    if ((edge.data as EdgeData | undefined)?.kind === "pipe") return edge.id.split("::")[0]
+  }
+  return null
+})
+
+/** The live, editable domain node behind the selection (Clouds are not editable). */
+const element = computed<StockNode | FlowNode | ConverterNode | null>(() => {
+  const id = selectedId.value
+  if (!id) return null
+  const node = store.model.nodes.find((n) => n.id === id)
+  if (node?.kind === "stock" || node?.kind === "flow" || node?.kind === "converter") return node
+  return null
+})
+
+const KIND_LABEL = { stock: "Stock", flow: "Flow", converter: "Converter" } as const
+
+/** Every rule carries exactly one number (a value or a factor); read it uniformly. */
+function ruleNumber(rule?: Rule): number {
+  if (!rule) return 0
+  return rule.kind === "constant" ? rule.value : rule.factor
+}
+
+function buildRule(kind: Rule["kind"], n: number): Rule {
+  return kind === "constant" ? { kind, value: n } : { kind, factor: n }
+}
+
+function onInitial(event: Event): void {
+  const el = element.value
+  if (el?.kind !== "stock") return
+  const raw = (event.target as HTMLInputElement).value.trim()
+  if (raw === "") return store.setInitialValue(el.id, undefined)
+  const n = Number(raw)
+  if (Number.isFinite(n)) store.setInitialValue(el.id, n)
+}
+
+function onUnit(event: Event): void {
+  const el = element.value
+  if (el?.kind !== "stock") return
+  store.setUnit(el.id, (event.target as HTMLInputElement).value)
+}
+
+function onKind(event: Event): void {
+  const el = element.value
+  if (el?.kind !== "flow" && el?.kind !== "converter") return
+  const kind = (event.target as HTMLSelectElement).value as Rule["kind"] | ""
+  if (!kind) return
+  // Carry the existing number across a kind change; default a fresh rule sensibly.
+  const n = el.rule ? ruleNumber(el.rule) : kind === "constant" ? 0 : 1
+  store.setRule(el.id, buildRule(kind, n))
+}
+
+function onParam(event: Event): void {
+  const el = element.value
+  if ((el?.kind !== "flow" && el?.kind !== "converter") || !el.rule) return
+  const n = Number((event.target as HTMLInputElement).value)
+  store.setRule(el.id, buildRule(el.rule.kind, Number.isFinite(n) ? n : 0))
+}
+
+/** One-line reminder of where a rule reads its operands (they come from links). */
+const RULE_HINT: Record<Rule["kind"], string> = {
+  constant: "A fixed number — no inputs.",
+  proportional: "rate = factor × its “+” inputs.",
+  gap: "rate = factor × (level − target): the “+” input is the level, the “−” the target.",
+}
+</script>
+
+<template>
+  <div
+    v-if="element"
+    class="absolute top-3 right-3 z-20 w-60 rounded-box border border-base-300 bg-base-100/95 p-3 shadow-md backdrop-blur"
+  >
+    <div class="flex items-baseline gap-2">
+      <span class="text-sm font-semibold">{{ element.name }}</span>
+      <span class="text-xs text-base-content/50">{{ KIND_LABEL[element.kind] }}</span>
+    </div>
+
+    <!-- Stock: the quantity it starts from, and its unit. -->
+    <template v-if="element.kind === 'stock'">
+      <label class="mt-2 block">
+        <span class="text-xs text-base-content/60">Initial value</span>
+        <input
+          type="number"
+          class="input input-sm input-bordered mt-1 w-full"
+          :value="element.initialValue ?? ''"
+          placeholder="—"
+          @change="onInitial"
+        />
+      </label>
+      <label class="mt-2 block">
+        <span class="text-xs text-base-content/60">Unit</span>
+        <input
+          type="text"
+          class="input input-sm input-bordered mt-1 w-full"
+          :value="element.unit ?? ''"
+          placeholder="e.g. °C, people, $"
+          @change="onUnit"
+        />
+      </label>
+    </template>
+
+    <!-- Flow / Converter: pick a rule, then its number. -->
+    <template v-else>
+      <label class="mt-2 block">
+        <span class="text-xs text-base-content/60">Rule</span>
+        <select
+          class="select select-sm select-bordered mt-1 w-full"
+          :value="element.rule?.kind ?? ''"
+          @change="onKind"
+        >
+          <option value="" disabled>Choose a rule…</option>
+          <option value="constant">Constant</option>
+          <option value="proportional">Proportional</option>
+          <option value="gap">Gap</option>
+        </select>
+      </label>
+
+      <label v-if="element.rule" class="mt-2 block">
+        <span class="text-xs text-base-content/60">
+          {{ element.rule.kind === "constant" ? "Value" : "Factor" }}
+        </span>
+        <input
+          type="number"
+          step="any"
+          class="input input-sm input-bordered mt-1 w-full"
+          :value="ruleNumber(element.rule)"
+          @change="onParam"
+        />
+      </label>
+
+      <p v-if="element.rule" class="mt-2 text-xs leading-snug text-base-content/50">
+        {{ RULE_HINT[element.rule.kind] }}
+      </p>
+    </template>
+  </div>
+</template>

src/components/ResultsPanel.vue

@@ -0,0 +1,153 @@
+<script setup lang="ts">
+/**
+ * Results panel (phase 2, ADR-0004) — what makes the Model *alive*: it runs the
+ * simulation and traces each Stock's value over time. Stocks are the system's
+ * memory, so their trajectories are the behaviour worth watching (the savings
+ * balance snowballs; the coffee settles toward room temperature).
+ *
+ * It reads the source-of-truth Model straight from the store and recomputes
+ * reactively, so editing a rule and reopening re-runs. When the Model is not yet
+ * sim-ready, it shows what is missing instead of a plot — the gap list doubles as
+ * a checklist for bringing a diagram to life. The plot itself is a uPlot canvas
+ * (see SimChart) — a small, fast time-series library that earns its keep with a
+ * real time axis and hover-to-read values the hand-built SVG couldn't give.
+ */
+import type uPlot from "uplot"
+import { computed, onBeforeUnmount, onMounted } from "vue"
+import SimChart from "./SimChart.vue"
+import { checkSimReady } from "@/model/simulation"
+import { DEFAULT_SIM_SPEC, type SimSpec, type StockNode } from "@/model/types"
+import { useModelStore } from "@/store/model"
+import { useSimulationStore } from "@/store/simulation"
+
+defineEmits<{ close: [] }>()
+
+const store = useModelStore()
+const sim = useSimulationStore()
+
+// While the panel is open the canvas shows live values; closing it returns the
+// canvas to a plain diagram.
+onMounted(() => sim.enable())
+onBeforeUnmount(() => sim.disable())
+
+const problems = computed(() => checkSimReady(store.model))
+
+/** The current run window; falls back to the default until the Model carries one. */
+const spec = computed<SimSpec>(() => store.model.sim ?? DEFAULT_SIM_SPEC)
+
+function onSpec(key: keyof SimSpec, event: Event): void {
+  const n = Number((event.target as HTMLInputElement).value)
+  if (!Number.isFinite(n) || (key === "dt" && n <= 0)) return
+  store.setSimSpec({ ...spec.value, [key]: n })
+}
+
+/** Distinct, legible track colours; cycled if a Model has more Stocks than these. */
+const COLORS = ["#2563eb", "#dc2626", "#16a34a", "#d97706", "#7c3aed", "#0891b2"]
+
+// Stocks only — the system's memory, and so the trajectories worth watching. The
+// run comes from the simulation store, so the chart and the canvas share one index.
+const chart = computed(() => {
+  const run = sim.run
+  if (!run) return null
+  const stocks = store.model.nodes.filter((n): n is StockNode => n.kind === "stock")
+  if (stocks.length === 0 || run.times.length < 2) return null
+
+  const data = [run.times, ...stocks.map((s) => run.series.get(s.id) ?? [])] as uPlot.AlignedData
+  const series = stocks.map((s, i) => ({ label: s.name, stroke: COLORS[i % COLORS.length] }))
+  return { data, series, diverged: run.diverged }
+})
+</script>
+
+<template>
+  <div
+    class="absolute inset-x-3 bottom-3 z-30 rounded-box border border-base-300 bg-base-100/95 p-3 shadow-lg backdrop-blur"
+  >
+    <div class="flex items-center gap-2">
+      <span class="text-sm font-semibold">Behaviour over time</span>
+      <span class="truncate text-xs text-base-content/50">{{ store.model.name }}</span>
+      <div class="ml-auto flex items-center gap-2 text-xs text-base-content/60">
+        <label class="flex items-center gap-1">
+          from
+          <input
+            type="number"
+            class="input input-xs input-bordered w-16"
+            :value="spec.start"
+            @change="onSpec('start', $event)"
+          />
+        </label>
+        <label class="flex items-center gap-1">
+          to
+          <input
+            type="number"
+            class="input input-xs input-bordered w-16"
+            :value="spec.stop"
+            @change="onSpec('stop', $event)"
+          />
+        </label>
+        <label class="flex items-center gap-1">
+          step
+          <input
+            type="number"
+            step="any"
+            min="0"
+            class="input input-xs input-bordered w-16"
+            :value="spec.dt"
+            @change="onSpec('dt', $event)"
+          />
+        </label>
+        <button
+          type="button"
+          class="btn btn-circle btn-ghost btn-xs"
+          aria-label="Close"
+          @click="$emit('close')"
+        >
+          ✕
+        </button>
+      </div>
+    </div>
+
+    <!-- Stopped early because values ran past what a float can hold. -->
+    <p v-if="chart?.diverged" class="mt-2 text-xs text-warning">
+      ⚠ Values grew beyond what can be plotted and the run stopped early — try a smaller step, a
+      smaller factor, or a Balancing loop to rein it in.
+    </p>
+
+    <!-- Sim-ready: the plot, plus a transport that drives the playhead. Hover a
+         point to read values; press play to watch the canvas animate in step. -->
+    <template v-if="chart">
+      <SimChart class="mt-2" :data="chart.data" :series="chart.series" :marker="sim.currentTime" />
+      <div class="mt-2 flex items-center gap-2">
+        <button
+          type="button"
+          class="btn btn-primary btn-xs w-8"
+          :aria-label="sim.playing ? 'Pause' : 'Play'"
+          @click="sim.toggle()"
+        >
+          {{ sim.playing ? "❚❚" : "▶" }}
+        </button>
+        <input
+          type="range"
+          class="range range-xs flex-1"
+          min="0"
+          :max="Math.max(0, sim.frameCount - 1)"
+          :value="sim.playhead"
+          aria-label="Playhead"
+          @input="sim.seek(Number(($event.target as HTMLInputElement).value))"
+        />
+        <span
+          class="w-24 shrink-0 whitespace-nowrap text-right font-mono text-xs tabular-nums text-base-content/60"
+        >
+          t = {{ sim.currentTime ?? 0 }}
+        </span>
+      </div>
+    </template>
+
+    <!-- Not sim-ready: what to fill in to bring it to life. -->
+    <div v-else class="mt-2 text-sm">
+      <p class="text-base-content/70">To bring this model to life, set:</p>
+      <ul class="mt-1 list-inside list-disc text-base-content/60">
+        <li v-for="(problem, i) in problems" :key="i">{{ problem }}</li>
+      </ul>
+    </div>
+  </div>
+</template>

src/components/SimChart.vue

@@ -0,0 +1,176 @@
+<script setup lang="ts">
+/**
+ * uPlot wrapper for the behaviour-over-time plot. uPlot is an imperative canvas
+ * library, so this component owns its lifecycle: build on mount, push new samples
+ * with `setData` when a run recomputes, rebuild when the *set* of tracks changes
+ * (a Stock renamed/added/removed), follow the container width with a
+ * ResizeObserver, and destroy on unmount.
+ *
+ * Two deliberate settings:
+ *  - `scales.x.time = false` — x is *simulation* time (0…stop), not wall-clock;
+ *    uPlot would otherwise format the axis as calendar dates.
+ *  - Colours are read from the DaisyUI theme variables so the chart matches the
+ *    rest of the editor. The app ships a single light theme (no `data-theme`
+ *    switching), so they are resolved once at build time rather than watched.
+ */
+import uPlot from "uplot"
+import "uplot/dist/uPlot.min.css"
+import { onBeforeUnmount, onMounted, ref, watch } from "vue"
+
+/** One plotted track: its legend label and line colour. */
+export interface ChartSeries {
+  label: string
+  stroke: string
+}
+
+const props = withDefaults(
+  defineProps<{
+    /** uPlot aligned data: `[times, ...one value track per series]`. */
+    data: uPlot.AlignedData
+    /** Track metadata, in the same order as `data[1…]`. */
+    series: ChartSeries[]
+    /** Canvas height in CSS pixels (the legend sits below it). */
+    height?: number
+    /** Simulation time of the playhead, drawn as a vertical line (null = none). */
+    marker?: number | null
+  }>(),
+  { height: 160, marker: null },
+)
+
+const root = ref<HTMLDivElement>()
+let plot: uPlot | undefined
+let observer: ResizeObserver | undefined
+/** Signature of the current plot's tracks; a change means the shape changed. */
+let builtSig = ""
+
+// Playhead as a DOM overlay (CSS px), not a canvas line: a moving line drawn by
+// redrawing the whole uPlot canvas every frame flickers, so the line slides over
+// the canvas instead and the canvas is only redrawn when the data changes.
+const markerX = ref<number | null>(null)
+const markerTop = ref(0)
+const markerHeight = ref(0)
+
+function syncMarker(): void {
+  if (!plot || props.marker === null) {
+    markerX.value = null
+    return
+  }
+  // bbox is in device pixels; the overlay is laid out in CSS pixels.
+  const ratio = window.devicePixelRatio || 1
+  // valToPos (CSS px) is relative to the plot area's left edge, but the overlay
+  // is positioned against the canvas, so add the left gutter (the y-axis width).
+  markerX.value = plot.bbox.left / ratio + plot.valToPos(props.marker, "x")
+  markerTop.value = plot.bbox.top / ratio
+  markerHeight.value = plot.bbox.height / ratio
+}
+
+/** Compact axis/legend numbers (7039.99 → "7040", 0.42 → "0.42", idle → "--"). */
+function fmt(value: number | null): string {
+  if (value === null) return "--"
+  return Math.abs(value) >= 100 ? value.toFixed(0) : value.toFixed(2)
+}
+
+/** A DaisyUI theme colour, with a light-theme fallback if the var is unset. */
+function themeColor(name: string, fallback: string): string {
+  const value = root.value ? getComputedStyle(root.value).getPropertyValue(name).trim() : ""
+  return value || fallback
+}
+
+function trackSig(): string {
+  return props.series.map((s) => `${s.label}|${s.stroke}`).join(",")
+}
+
+function options(width: number): uPlot.Options {
+  // Follow the app's --font-sans (resolved on the element) rather than naming a
+  // family here, so the chart tracks the theme font without a second source.
+  const family = root.value
+    ? getComputedStyle(root.value).fontFamily
+    : "ui-sans-serif, system-ui, sans-serif"
+  const axis: uPlot.Axis = {
+    stroke: themeColor("--color-base-content", "#1f2937"),
+    grid: { stroke: themeColor("--color-base-300", "#e5e7eb"), width: 1 },
+    ticks: { stroke: themeColor("--color-base-300", "#e5e7eb"), width: 1 },
+    font: `11px ${family}`,
+  }
+  return {
+    width,
+    height: props.height,
+    cursor: { y: false },
+    scales: { x: { time: false } },
+    series: [
+      {},
+      ...props.series.map(
+        (s): uPlot.Series => ({
+          label: s.label,
+          stroke: s.stroke,
+          width: 2,
+          points: { show: false },
+          value: (_self, raw) => fmt(raw),
+        }),
+      ),
+    ],
+    axes: [axis, { ...axis, size: 52 }],
+  }
+}
+
+function build(): void {
+  if (!root.value) return
+  plot?.destroy()
+  builtSig = trackSig()
+  plot = new uPlot(options(root.value.clientWidth), props.data, root.value)
+}
+
+/** Cheap on every recompute; only a shape change forces a full rebuild. */
+function render(): void {
+  if (!plot || trackSig() !== builtSig) build()
+  else plot.setData(props.data)
+  syncMarker() // the plot area may have shifted; keep the overlay aligned
+}
+
+onMounted(() => {
+  build()
+  observer = new ResizeObserver(() => {
+    if (plot && root.value) {
+      plot.setSize({ width: root.value.clientWidth, height: props.height })
+      syncMarker()
+    }
+  })
+  if (root.value) observer.observe(root.value)
+  syncMarker()
+  // The canvas paints before the web font loads and won't repaint on its own, so
+  // axis labels would stick to the fallback. Redraw once the font is ready.
+  document.fonts?.ready.then(() => {
+    plot?.redraw()
+    syncMarker()
+  })
+})
+
+onBeforeUnmount(() => {
+  observer?.disconnect()
+  plot?.destroy()
+})
+
+watch(() => props.data, render)
+// The playhead just slides the overlay — no canvas redraw, so playback stays smooth.
+watch(() => props.marker, syncMarker)
+</script>
+
+<template>
+  <div ref="root" class="relative w-full">
+    <div
+      v-if="markerX !== null"
+      class="pointer-events-none absolute w-px bg-primary"
+      :style="{ left: `${markerX}px`, top: `${markerTop}px`, height: `${markerHeight}px` }"
+    />
+  </div>
+</template>
+
+<style scoped>
+:deep(.u-legend) {
+  font-size: 0.75rem;
+  color: var(--color-base-content);
+}
+:deep(.u-legend .u-value) {
+  font-variant-numeric: tabular-nums;
+}
+</style>

src/components/nodes/ConverterNode.vue

@@ -9,6 +9,8 @@ import { computed } from "vue"
 import { useNodeLoopRing } from "@/composables/useLoopHighlight"
 import { HANDLE_IN, HANDLE_OUT, type NodeData } from "@/model/projection"
 import type { ConverterNode } from "@/model/types"
+import { useSimulationStore } from "@/store/simulation"
+import { formatValue } from "./format"
 import NodeLabel from "./NodeLabel.vue"
 
 const props = defineProps<NodeProps<NodeData>>()
@@ -16,6 +18,9 @@ const props = defineProps<NodeProps<NodeData>>()
 // The projection guarantees a converter-typed node here.
 const converter = computed(() => props.data.node as ConverterNode)
 const loopRing = useNodeLoopRing(props.id)
+
+const sim = useSimulationStore()
+const value = computed(() => sim.valueAt(props.id))
 </script>
 
 <template>
@@ -25,6 +30,12 @@ const loopRing = useNodeLoopRing(props.id)
       :class="[props.selected ? 'border-primary' : 'border-base-300', loopRing]"
     >
       <Handle :id="HANDLE_IN" type="target" :position="Position.Left" />
+      <span
+        v-if="value !== null"
+        class="pointer-events-none absolute inset-0 flex items-center justify-center font-mono text-xs tabular-nums text-base-content/70"
+      >
+        {{ formatValue(value) }}
+      </span>
       <Handle :id="HANDLE_OUT" type="source" :position="Position.Right" />
     </div>
     <NodeLabel :node-id="props.id" :name="converter.name" />

src/components/nodes/FlowNode.vue

@@ -11,6 +11,8 @@ import { computed } from "vue"
 import { useNodeLoopRing } from "@/composables/useLoopHighlight"
 import { HANDLE_IN, HANDLE_OUT, type NodeData } from "@/model/projection"
 import type { FlowNode } from "@/model/types"
+import { useSimulationStore } from "@/store/simulation"
+import { formatValue } from "./format"
 import NodeLabel from "./NodeLabel.vue"
 
 const props = defineProps<NodeProps<NodeData>>()
@@ -18,6 +20,9 @@ const props = defineProps<NodeProps<NodeData>>()
 // The projection guarantees a flow-typed node here.
 const flow = computed(() => props.data.node as FlowNode)
 const loopRing = useNodeLoopRing(props.id)
+
+const sim = useSimulationStore()
+const value = computed(() => sim.valueAt(props.id))
 </script>
 
 <template>
@@ -40,5 +45,8 @@ const loopRing = useNodeLoopRing(props.id)
       <Handle :id="HANDLE_OUT" type="source" :position="Position.Right" />
     </div>
     <NodeLabel :node-id="props.id" :name="flow.name" />
+    <div v-if="value !== null" class="font-mono text-xs tabular-nums text-base-content/70">
+      {{ formatValue(value) }}
+    </div>
   </div>
 </template>

src/components/nodes/StockNode.vue

@@ -9,6 +9,8 @@ import { computed } from "vue"
 import { useNodeLoopRing } from "@/composables/useLoopHighlight"
 import { HANDLE_IN, HANDLE_OUT, type NodeData } from "@/model/projection"
 import type { StockNode } from "@/model/types"
+import { useSimulationStore } from "@/store/simulation"
+import { formatValue } from "./format"
 import NodeLabel from "./NodeLabel.vue"
 
 const props = defineProps<NodeProps<NodeData>>()
@@ -16,15 +18,43 @@ const props = defineProps<NodeProps<NodeData>>()
 // The projection guarantees a stock-typed node here.
 const stock = computed(() => props.data.node as StockNode)
 const loopRing = useNodeLoopRing(props.id)
+
+// Live simulation value at the playhead (null when no run is engaged). A Stock is
+// the system's memory, so it also shows a fill gauge — its level over its peak.
+const sim = useSimulationStore()
+const value = computed(() => sim.valueAt(props.id))
+const fill = computed(() => sim.fill(props.id))
 </script>
 
 <template>
   <div
-    class="min-w-24 rounded-md border-2 bg-base-100 px-4 py-3 text-center shadow-sm transition-colors"
+    class="relative min-w-24 rounded-md border-2 bg-base-100 px-4 py-3 text-center shadow-sm transition-colors"
     :class="[props.selected ? 'border-primary' : 'border-base-300', loopRing]"
   >
+    <!-- Fill gauge: a bottom-anchored bar, behind the label, inset within the border. -->
+    <div
+      v-if="value !== null"
+      class="pointer-events-none absolute inset-[2px] z-0 flex flex-col justify-end"
+    >
+      <div
+        class="rounded-b bg-primary/60 transition-[height]"
+        :style="{ height: fill * 100 + '%' }"
+      />
+    </div>
     <Handle :id="HANDLE_IN" type="target" :position="Position.Left" />
-    <NodeLabel :node-id="props.id" :name="stock.name" />
+    <div class="relative z-10">
+      <NodeLabel :node-id="props.id" :name="stock.name" />
+    </div>
     <Handle :id="HANDLE_OUT" type="source" :position="Position.Right" />
+    <!-- Live value sits *below* the box as an out-of-flow overlay, so the box keeps
+         its exact size and the handles — with the pipes and links on them — never
+         shift when a run is engaged. -->
+    <div
+      v-if="value !== null"
+      class="pointer-events-none absolute inset-x-0 top-full mt-1 whitespace-nowrap text-center font-mono text-xs tabular-nums text-base-content/70"
+    >
+      {{ formatValue(value)
+      }}<span v-if="stock.unit" class="ml-0.5 text-base-content/50">{{ stock.unit }}</span>
+    </div>
   </div>
 </template>

src/components/nodes/format.ts

@@ -0,0 +1,4 @@
+/** Compact value for the live readout on a node (7039.99 → "7040", 0.42 → "0.42"). */
+export function formatValue(value: number): string {
+  return Math.abs(value) >= 100 ? value.toFixed(0) : value.toFixed(2)
+}

src/composables/usePlayback.ts

@@ -0,0 +1,49 @@
+/**
+ * Playback clock — the wall-clock that advances the simulation playhead while
+ * `playing`. It lives in a composable (mounted once, in the Editor) because it
+ * needs `requestAnimationFrame` and component teardown; the simulation store
+ * stays pure state + actions.
+ *
+ * A run plays over a fixed wall-clock window regardless of how many samples it
+ * has, so a 40-step model and a 480-step one feel the same speed. Elapsed time is
+ * accumulated and drained in whole frames, so a slow tab catches up rather than
+ * drifting.
+ */
+import { onBeforeUnmount, watch } from "vue"
+import { useSimulationStore } from "@/store/simulation"
+
+/** Target wall-clock duration for a whole run, in milliseconds. */
+const RUN_DURATION_MS = 6000
+
+export function usePlayback(): void {
+  const sim = useSimulationStore()
+  let raf = 0
+  let last = 0
+  let carry = 0
+
+  function frame(now: number): void {
+    if (!sim.playing) return
+    carry += now - last
+    last = now
+    const msPerFrame = Math.max(16, RUN_DURATION_MS / Math.max(1, sim.frameCount))
+    while (carry >= msPerFrame && sim.playing) {
+      carry -= msPerFrame
+      sim.tick()
+    }
+    raf = requestAnimationFrame(frame)
+  }
+
+  watch(
+    () => sim.playing,
+    (playing) => {
+      cancelAnimationFrame(raf)
+      if (playing) {
+        last = performance.now()
+        carry = 0
+        raf = requestAnimationFrame(frame)
+      }
+    },
+  )
+
+  onBeforeUnmount(() => cancelAnimationFrame(raf))
+}

src/model/io.ts

@@ -21,6 +21,8 @@ import {
   type NodeKind,
   type Polarity,
   type Position,
+  type Rule,
+  type SimSpec,
 } from "./types"
 
 /** Pretty JSON so an exported Model is human-readable and diff-friendly (F8). */
@@ -49,6 +51,24 @@ function isPolarity(value: unknown): value is Polarity {
   return value === "+" || value === "-"
 }
 
+/** A Rule must be one of the fixed kinds with its numeric parameter (ADR-0004). */
+function isRule(value: unknown): value is Rule {
+  if (!isObject(value)) return false
+  if (value.kind === "constant") return isFiniteNumber(value.value)
+  if (value.kind === "proportional" || value.kind === "gap") return isFiniteNumber(value.factor)
+  return false
+}
+
+function isSimSpec(value: unknown): value is SimSpec {
+  return (
+    isObject(value) &&
+    isFiniteNumber(value.start) &&
+    isFiniteNumber(value.stop) &&
+    isFiniteNumber(value.dt) &&
+    value.dt > 0
+  )
+}
+
 /** Validate one node by kind, returning an error string or null when valid. */
 function nodeError(value: unknown, index: number): string | null {
   if (!isObject(value)) return `nodes[${index}] is not an object`
@@ -65,6 +85,20 @@ function nodeError(value: unknown, index: number): string | null {
     if (typeof value.source !== "string") return `${at}.source must be a string`
     if (typeof value.target !== "string") return `${at}.target must be a string`
   }
+  // Simulation fields (ADR-0004) are optional, but if present must be well-formed.
+  if (kind === "stock" && value.initialValue !== undefined && !isFiniteNumber(value.initialValue)) {
+    return `${at}.initialValue must be a finite number`
+  }
+  if (kind === "stock" && value.unit !== undefined && typeof value.unit !== "string") {
+    return `${at}.unit must be a string`
+  }
+  if (
+    (kind === "flow" || kind === "converter") &&
+    value.rule !== undefined &&
+    !isRule(value.rule)
+  ) {
+    return `${at}.rule must be a valid rule (constant/proportional/gap)`
+  }
   return null
 }
 
@@ -104,6 +138,9 @@ export function parseModel(text: string): ParseResult {
   if (typeof data.name !== "string") return { ok: false, error: "name must be a string" }
   if (!Array.isArray(data.nodes)) return { ok: false, error: "nodes must be an array" }
   if (!Array.isArray(data.infoLinks)) return { ok: false, error: "infoLinks must be an array" }
+  if (data.sim !== undefined && !isSimSpec(data.sim)) {
+    return { ok: false, error: "sim must be {start, stop, dt} numbers with dt > 0" }
+  }
 
   for (let i = 0; i < data.nodes.length; i++) {
     const error = nodeError(data.nodes[i], i)
@@ -134,6 +171,13 @@ export function parseModel(text: string): ParseResult {
 
   return {
     ok: true,
-    model: { version: MODEL_VERSION, id: data.id, name: data.name, nodes, infoLinks: links },
+    model: {
+      version: MODEL_VERSION,
+      id: data.id,
+      name: data.name,
+      nodes,
+      infoLinks: links,
+      ...(data.sim !== undefined && { sim: data.sim as SimSpec }),
+    },
   }
 }

src/model/projection.ts

@@ -11,9 +11,18 @@
  * Flow's references (ADR-0003); each Information Link contributes one "info"
  * edge carrying its polarity.
  */
-import type { Edge, Node } from "@vue-flow/core"
+import { type Edge, type EdgeMarker, MarkerType, type Node } from "@vue-flow/core"
 import type { Model, ModelNode, Polarity } from "./types"
 
+/**
+ * Open arrowhead drawn at the same stroke weight as the line it ends. Vue Flow
+ * renders the marker with markerUnits="strokeWidth" (so the head already scales
+ * with the edge) but downscales its 20-unit viewBox into a 12.5 marker box,
+ * thinning the chevron to 0.625× the line. strokeWidth 20/12.5 = 1.6 cancels
+ * that, leaving the head exactly as heavy as its line at any width.
+ */
+const ARROWHEAD: EdgeMarker = { type: MarkerType.Arrow, strokeWidth: 1.6 }
+
 /** Payload carried on every projected Vue Flow node: the domain node itself. */
 export interface NodeData {
   node: ModelNode
@@ -72,7 +81,7 @@ export function projectEdges(model: Model): FlowGraphEdge[] {
       targetHandle: HANDLE_IN,
       data: { kind: "pipe" },
       style: { strokeWidth: "2.5px" },
-      markerEnd: "arrow",
+      markerEnd: ARROWHEAD,
     })
   }
 
@@ -88,7 +97,7 @@ export function projectEdges(model: Model): FlowGraphEdge[] {
       targetHandle: HANDLE_IN,
       data: { kind: "info", polarity: link.polarity },
       style: { strokeDasharray: "5 4" },
-      markerEnd: "arrow",
+      markerEnd: ARROWHEAD,
     })
   }
 

src/model/samples.ts

@@ -15,9 +15,9 @@
  * Beyond that primer, three classic models go a step further — each adds one
  * structure the first four never show, so they read as a second tier:
  *
- *   5. Limits to growth  — two loops (R and B) fighting over a single Flow, plus a
+ *   5. Limits to growth  — a Reinforcing inflow and a Balancing outflow on one
- *                          constant Converter (carrying capacity) that feeds a loop
+ *                          Stock, with a Converter (crowding) relaying the density
- *                          without being part of it.
+ *                          that brakes growth: the S-curve.
  *   6. Predator and prey — two coupled Stocks whose interlocking loops oscillate.
  *   7. Epidemic          — a chain of Stocks joined by Stock→Stock Flows: no clouds.
  *
@@ -33,6 +33,14 @@
  *  11. Drift to low performance — a goal that erodes toward actual performance, so a
  *                                 Reinforcing loop ratchets both downward.
  *
+ * Last, the dynamic the book is named for, and the one the gallery has saved until a
+ * reader knows every piece it needs:
+ *
+ *  12. Overshoot and collapse   — a Reinforcing engine running on a *non-renewable*
+ *                                 Stock (the first with no inflow): it overshoots the
+ *                                 limit instead of settling at it, the dark twin of
+ *                                 "Limits to growth" — the ceiling erodes, so it crashes.
+ *
  * These are plain data built from the same tested constructors the store uses
  * (factory.ts), so every sample is a valid Model by construction. `build()`
  * mints fresh ids on each call, so loading a sample twice never collides.
@@ -44,6 +52,7 @@ import {
   MODEL_VERSION,
   type ModelNode,
   type Polarity,
+  type SimSpec,
 } from "./types"
 
 /** A loadable example: a title and one-line blurb for the menu, plus a builder. */
@@ -58,8 +67,13 @@ function link(source: ModelNode, target: ModelNode, polarity: Polarity): Informa
   return { id: newId("link"), source: source.id, target: target.id, polarity }
 }
 
-function model(name: string, nodes: ModelNode[], infoLinks: InformationLink[]): Model {
+function model(
-  return { version: MODEL_VERSION, id: newId("model"), name, nodes, infoLinks }
+  name: string,
+  nodes: ModelNode[],
+  infoLinks: InformationLink[],
+  sim?: SimSpec,
+): Model {
+  return { version: MODEL_VERSION, id: newId("model"), name, nodes, infoLinks, sim }
 }
 
 /**
@@ -70,6 +84,8 @@ function model(name: string, nodes: ModelNode[], infoLinks: InformationLink[]):
 function bathtub(): Model {
   const source = makeCloud({ x: -280, y: 0 })
   const water = makeStock({ x: 0, y: 0 }, "Water")
+  water.initialValue = 20
+  water.unit = "L"
   const sink = makeCloud({ x: 280, y: 0 })
   const filling = makeFlow(
     midpoint(source.position, water.position),
@@ -78,7 +94,15 @@ function bathtub(): Model {
     water.id,
   )
   const emptying = makeFlow(midpoint(water.position, sink.position), "emptying", water.id, sink.id)
-  return model("Bathtub", [source, water, sink, filling, emptying], [])
+  // No Information Links, so each rate is a plain Constant. A faster inflow than
+  // outflow means Water rises in a straight line — accumulation with no feedback.
+  filling.rule = { kind: "constant", value: 5 }
+  emptying.rule = { kind: "constant", value: 3 }
+  return model("Bathtub", [source, water, sink, filling, emptying], [], {
+    start: 0,
+    stop: 40,
+    dt: 1,
+  })
 }
 
 /**
@@ -92,13 +116,22 @@ function savings(): Model {
   // visible Reinforcing loop instead of overlapping the inflow pipe.
   const source = makeCloud({ x: -240, y: -80 })
   const balance = makeStock({ x: 120, y: 40 }, "Balance")
+  balance.initialValue = 1000
+  balance.unit = "$"
   const interest = makeFlow(
     midpoint(source.position, balance.position),
     "interest",
     source.id,
     balance.id,
   )
-  return model("Savings account", [source, balance, interest], [link(balance, interest, "+")])
+  // interest = 5% × Balance (the `+` link). A Stock feeding its own inflow → the
+  // Reinforcing loop runs as exponential growth.
+  interest.rule = { kind: "proportional", factor: 0.05 }
+  return model("Savings account", [source, balance, interest], [link(balance, interest, "+")], {
+    start: 0,
+    stop: 40,
+    dt: 1,
+  })
 }
 
 /**
@@ -109,13 +142,20 @@ function savings(): Model {
  */
 function coffee(): Model {
   const coffee = makeStock({ x: -200, y: 0 }, "Coffee")
+  coffee.initialValue = 90
+  coffee.unit = "°C"
   const sink = makeCloud({ x: 200, y: 0 })
   const cooling = makeFlow(midpoint(coffee.position, sink.position), "cooling", coffee.id, sink.id)
+  // cooling = 0.1 × (Coffee − room): the `+` input is the level, the `−` the target.
+  // An outflow closing the gap to room temperature → the Balancing loop settles there.
+  cooling.rule = { kind: "gap", factor: 0.1 }
   const room = makeConverter({ x: 0, y: -160 }, "room temperature")
+  room.rule = { kind: "constant", value: 20 }
   return model(
     "Coffee cooling",
     [coffee, sink, cooling, room],
     [link(coffee, cooling, "+"), link(room, cooling, "-")],
+    { start: 0, stop: 60, dt: 1 },
   )
 }
 
@@ -132,11 +172,24 @@ function population(): Model {
   // Valves are placed by hand, not at the midpoint, to hold the steps.
   const source = makeCloud({ x: -360, y: -240 })
   const fertility = makeConverter({ x: -360, y: -40 }, "fertility")
+  fertility.rule = { kind: "constant", value: 0.03 }
   const lifeExpectancy = makeConverter({ x: -360, y: 240 }, "life expectancy")
+  // Wired into deaths for the Balancing loop's structure, but not yet read by the
+  // rate: a faithful "deaths = Population ÷ life expectancy" needs a divide rule we
+  // don't have, so deaths uses a flat mortality rate below. (See the gallery notes.)
+  lifeExpectancy.rule = { kind: "constant", value: 70 }
   const people = makeStock({ x: 0, y: 0 }, "Population")
+  people.initialValue = 100
+  people.unit = "people"
   const births = makeFlow({ x: -160, y: -160 }, "births", source.id, people.id)
+  // births = fertility × Population (both `+` inputs): more people and higher
+  // fertility, more births — the Reinforcing engine.
+  births.rule = { kind: "proportional", factor: 1 }
   const sink = makeCloud({ x: 360, y: 240 })
   const deaths = makeFlow({ x: 160, y: 160 }, "deaths", people.id, sink.id)
+  // deaths = 2% of Population each step (its `+` input) — the Balancing drain. With
+  // births at 3%, the Reinforcing loop wins and the population grows exponentially.
+  deaths.rule = { kind: "proportional", factor: 0.02 }
   return model(
     "Population",
     [source, people, sink, births, deaths, fertility, lifeExpectancy],
@@ -146,34 +199,47 @@ function population(): Model {
       link(people, deaths, "+"),
       link(lifeExpectancy, deaths, "-"),
     ],
+    { start: 0, stop: 100, dt: 1 },
   )
 }
 
 /**
- * Limits to growth — the S-curve, and the first model where two loops fight over
+ * Limits to growth — the S-curve, where a Reinforcing engine meets a Balancing
- * one Flow. Yeast multiplies the more there is of it (Yeast → [+] → growth: a
+ * brake. Yeast multiplies the more there is of it (Yeast → [+] → growth: a
- * Reinforcing loop), but the fuller the vat the more crowding holds growth back
+ * Reinforcing inflow), but crowding rises with the population (Yeast → [+] →
- * (Yeast → [+] → crowding → [−] → growth: a Balancing loop). Carrying capacity is
+ * crowding) and drives a die-off that grows with the *square* of the Yeast
- * a *constant* Converter — no inputs — that sets how soon crowding bites; it feeds
+ * (Yeast, crowding → [+] → die-off → drains Yeast: a Balancing outflow). Growth
- * the balancing loop without sitting on any cycle.
+ * wins early, the die-off wins late, so Yeast settles where they balance (≈1000)
+ * — the classic sigmoid, with *both* loops visible to the detector. (A named
+ * "carrying capacity" would want a divide rule we don't have yet; here the ceiling
+ * falls out of the growth and die-off rates.)
  */
 function limitsToGrowth(): Model {
   const source = makeCloud({ x: -280, y: 0 })
   const yeast = makeStock({ x: 40, y: 0 }, "Yeast")
+  yeast.initialValue = 20
+  yeast.unit = "cells"
   const growth = makeFlow(midpoint(source.position, yeast.position), "growth", source.id, yeast.id)
-  // crowding rides above the pipe; carrying capacity stacks above the Source on the
+  // growth = 30% of Yeast (its `+` input): the Reinforcing engine.
-  // left, so the `capacity → crowding` link is a clean horizontal hop along the top.
+  growth.rule = { kind: "proportional", factor: 0.3 }
-  const capacity = makeConverter({ x: -280, y: -160 }, "carrying capacity")
+  const sink = makeCloud({ x: 360, y: 0 })
-  const crowding = makeConverter({ x: -40, y: -160 }, "crowding")
+  const dieOff = makeFlow(midpoint(yeast.position, sink.position), "die-off", yeast.id, sink.id)
+  // die-off = factor × Yeast × crowding. With crowding ∝ Yeast it scales as Yeast²,
+  // so the Balancing drain overtakes the linear growth and Yeast plateaus.
+  dieOff.rule = { kind: "proportional", factor: 0.0003 }
+  // crowding ≈ the population density (proportional to Yeast), what drives the die-off.
+  const crowding = makeConverter({ x: 200, y: -160 }, "crowding")
+  crowding.rule = { kind: "proportional", factor: 1 }
   return model(
     "Limits to growth",
-    [source, yeast, growth, crowding, capacity],
+    [source, yeast, growth, sink, dieOff, crowding],
     [
       link(yeast, growth, "+"),
       link(yeast, crowding, "+"),
-      link(crowding, growth, "-"),
+      link(yeast, dieOff, "+"),
-      link(capacity, crowding, "-"),
+      link(crowding, dieOff, "+"),
     ],
+    { start: 0, stop: 40, dt: 1 },
   )
 }
 
@@ -192,6 +258,7 @@ function predatorPrey(): Model {
   // cross-stock loop traces a circuit through the open centre.
   const preySource = makeCloud({ x: -480, y: -140 })
   const rabbits = makeStock({ x: -80, y: -140 }, "Rabbits")
+  rabbits.initialValue = 100
   const preySink = makeCloud({ x: 320, y: -140 })
   const rabbitBirths = makeFlow(
     midpoint(preySource.position, rabbits.position),
@@ -199,14 +266,19 @@ function predatorPrey(): Model {
     preySource.id,
     rabbits.id,
   )
+  // rabbits breed in proportion to themselves (Reinforcing) …
+  rabbitBirths.rule = { kind: "proportional", factor: 0.08 }
   const predation = makeFlow(
     midpoint(rabbits.position, preySink.position),
     "predation",
     rabbits.id,
     preySink.id,
   )
+  // … and are thinned by predation = rabbits × foxes (both `+`): the coupling term.
+  predation.rule = { kind: "proportional", factor: 0.004 }
   const foxSource = makeCloud({ x: -480, y: 140 })
   const foxes = makeStock({ x: -80, y: 140 }, "Foxes")
+  foxes.initialValue = 20
   const foxSink = makeCloud({ x: 320, y: 140 })
   const foxBirths = makeFlow(
     midpoint(foxSource.position, foxes.position),
@@ -214,12 +286,17 @@ function predatorPrey(): Model {
     foxSource.id,
     foxes.id,
   )
+  // foxes are born in proportion to the rabbits available to eat …
+  foxBirths.rule = { kind: "proportional", factor: 0.02 }
   const foxDeaths = makeFlow(
     midpoint(foxes.position, foxSink.position),
     "fox deaths",
     foxes.id,
     foxSink.id,
   )
+  // … and die off on their own. The lag around the loop makes the two populations
+  // chase each other. (Forward Euler damps the orbit — see the gallery notes.)
+  foxDeaths.rule = { kind: "proportional", factor: 0.2 }
   return model(
     "Predator and prey",
     [
@@ -241,6 +318,7 @@ function predatorPrey(): Model {
       link(rabbits, foxBirths, "+"),
       link(foxes, foxDeaths, "+"),
     ],
+    { start: 0, stop: 120, dt: 0.25 },
   )
 }
 
@@ -255,21 +333,35 @@ function predatorPrey(): Model {
  */
 function epidemic(): Model {
   const susceptible = makeStock({ x: -280, y: 0 }, "Susceptible")
+  susceptible.initialValue = 990
+  susceptible.unit = "people"
   const infected = makeStock({ x: 0, y: 0 }, "Infected")
+  infected.initialValue = 10
+  infected.unit = "people"
   const recovered = makeStock({ x: 280, y: 0 }, "Recovered")
+  recovered.initialValue = 0
+  recovered.unit = "people"
   const infection = makeFlow(
     midpoint(susceptible.position, infected.position),
     "infection",
     susceptible.id,
     infected.id,
   )
+  // infection = infectivity × Susceptible × Infected (proportional reads all three
+  // `+` inputs): the more carriers and the more susceptibles, the faster it spreads.
+  // The non-negative-stock floor keeps Susceptible from being over-drained.
+  infection.rule = { kind: "proportional", factor: 1 }
   const recovery = makeFlow(
     midpoint(infected.position, recovered.position),
     "recovery",
     infected.id,
     recovered.id,
   )
+  // recovery = 15% of the Infected each step (its one `+` input).
+  recovery.rule = { kind: "proportional", factor: 0.15 }
   const infectivity = makeConverter({ x: -140, y: -160 }, "infectivity")
+  // Small, so infectivity × S × I stays a sane rate (R0 = infectivity·S₀/γ ≈ 2.6).
+  infectivity.rule = { kind: "constant", value: 0.0004 }
   return model(
     "Epidemic",
     [susceptible, infected, recovered, infection, recovery, infectivity],
@@ -279,6 +371,7 @@ function epidemic(): Model {
       link(infected, recovery, "+"),
       link(infectivity, infection, "+"),
     ],
+    { start: 0, stop: 60, dt: 1 },
   )
 }
 
@@ -295,17 +388,22 @@ function epidemic(): Model {
  */
 function tragedyOfTheCommons(): Model {
   const pasture = makeStock({ x: 0, y: 0 }, "Pasture")
+  pasture.initialValue = 1000
   // Two symmetric herds: cattle enter from a Source on the outside, grass leaves
   // the Pasture downward to a Sink. The two `Pasture → growth` links are the weak
   // brake the trap overruns.
   const sourceA = makeCloud({ x: -640, y: 0 })
   const herdA = makeStock({ x: -360, y: 0 }, "Herd A")
+  herdA.initialValue = 10
   const growthA = makeFlow(
     midpoint(sourceA.position, herdA.position),
     "growth A",
     sourceA.id,
     herdA.id,
   )
+  // growth = herd × Pasture (both `+`): each herd grows the more cattle it has and
+  // the more grass is left — a Reinforcing loop, braked only by the shared Pasture.
+  growthA.rule = { kind: "proportional", factor: 0.0003 }
   const sinkA = makeCloud({ x: -200, y: 240 })
   const grazingA = makeFlow(
     midpoint(pasture.position, sinkA.position),
@@ -313,14 +411,19 @@ function tragedyOfTheCommons(): Model {
     pasture.id,
     sinkA.id,
   )
+  // grazing = 6% of the herd, drained from the *shared* Pasture (which never
+  // regrows here): two appetites racing one stock down to bare dirt.
+  grazingA.rule = { kind: "proportional", factor: 0.06 }
   const sourceB = makeCloud({ x: 640, y: 0 })
   const herdB = makeStock({ x: 360, y: 0 }, "Herd B")
+  herdB.initialValue = 10
   const growthB = makeFlow(
     midpoint(sourceB.position, herdB.position),
     "growth B",
     sourceB.id,
     herdB.id,
   )
+  growthB.rule = { kind: "proportional", factor: 0.0003 }
   const sinkB = makeCloud({ x: 200, y: 240 })
   const grazingB = makeFlow(
     midpoint(pasture.position, sinkB.position),
@@ -328,6 +431,7 @@ function tragedyOfTheCommons(): Model {
     pasture.id,
     sinkB.id,
   )
+  grazingB.rule = { kind: "proportional", factor: 0.06 }
   return model(
     "Tragedy of the commons",
     [pasture, sourceA, herdA, growthA, sinkA, grazingA, sourceB, herdB, growthB, sinkB, grazingB],
@@ -339,6 +443,7 @@ function tragedyOfTheCommons(): Model {
       link(herdB, grazingB, "+"),
       link(pasture, growthB, "+"),
     ],
+    { start: 0, stop: 60, dt: 1 },
   )
 }
 
@@ -357,24 +462,31 @@ function escalation(): Model {
   // cross there, where the R badge lands, so the whole loop reads at a glance.
   const blueSource = makeCloud({ x: -560, y: -120 })
   const blueArsenal = makeStock({ x: 280, y: -120 }, "Blue arsenal")
+  blueArsenal.initialValue = 10
   const blueBuildup = makeFlow(
     midpoint(blueSource.position, blueArsenal.position),
     "Blue buildup",
     blueSource.id,
     blueArsenal.id,
   )
+  // Each side builds in proportion to the other's arsenal (its one `+` input), so
+  // the two feed each other: a Reinforcing loop with no brake → unbounded growth.
+  blueBuildup.rule = { kind: "proportional", factor: 0.1 }
   const redSource = makeCloud({ x: -560, y: 120 })
   const redArsenal = makeStock({ x: 280, y: 120 }, "Red arsenal")
+  redArsenal.initialValue = 12
   const redBuildup = makeFlow(
     midpoint(redSource.position, redArsenal.position),
     "Red buildup",
     redSource.id,
     redArsenal.id,
   )
+  redBuildup.rule = { kind: "proportional", factor: 0.1 }
   return model(
     "Escalation",
     [blueSource, blueArsenal, blueBuildup, redSource, redArsenal, redBuildup],
     [link(blueArsenal, redBuildup, "+"), link(redArsenal, blueBuildup, "+")],
+    { start: 0, stop: 40, dt: 1 },
   )
 }
 
@@ -396,13 +508,22 @@ function fixesThatFail(): Model {
   // the diagonal back to Congestion. Placed by hand, not midpoint, to hold the column.
   const source = makeCloud({ x: -420, y: 120 })
   const congestion = makeStock({ x: 300, y: 120 }, "Congestion")
+  congestion.initialValue = 50
   const driving = makeFlow({ x: -120, y: 120 }, "driving", source.id, congestion.id)
+  // driving = 1.5 × road building (its `+` input): every new road induces *more*
+  // traffic than it cleared — the side effect that refills the symptom.
+  driving.rule = { kind: "proportional", factor: 1.5 }
   const sink = makeCloud({ x: 300, y: -160 })
   const roadBuilding = makeFlow({ x: -120, y: -160 }, "road building", congestion.id, sink.id)
+  // road building = 40% of Congestion (its `+` input), draining it: the Balancing
+  // fix. But induced driving outweighs it, so the Reinforcing loop wins and
+  // Congestion climbs anyway — you can't build your way out of traffic.
+  roadBuilding.rule = { kind: "proportional", factor: 0.4 }
   return model(
     "Fixes that fail",
     [source, congestion, driving, sink, roadBuilding],
     [link(congestion, roadBuilding, "+"), link(roadBuilding, driving, "+")],
+    { start: 0, stop: 30, dt: 1 },
   )
 }
 
@@ -422,13 +543,18 @@ function driftToLowPerformance(): Model {
   // Reinforcing spiral cross in the open centre, where the R badge lands.
   const source = makeCloud({ x: -560, y: 120 })
   const performance = makeStock({ x: -160, y: 120 }, "Performance")
+  performance.initialValue = 40
   const improvement = makeFlow(
     midpoint(source.position, performance.position),
     "improvement",
     source.id,
     performance.id,
   )
+  // improvement closes the gap upward: 10% of (Standard − Performance), pulling
+  // Performance toward the Standard.
+  improvement.rule = { kind: "gap", factor: 0.1 }
   const standard = makeStock({ x: 160, y: -120 }, "Standard")
+  standard.initialValue = 80
   const sink = makeCloud({ x: 560, y: -120 })
   const slippage = makeFlow(
     midpoint(standard.position, sink.position),
@@ -436,6 +562,10 @@ function driftToLowPerformance(): Model {
     standard.id,
     sink.id,
   )
+  // slippage erodes the *same* gap from the other side: the Standard drifts down
+  // toward actual Performance. Both gaps close, so they meet — the Standard has
+  // sagged from 80 to the middle, the eroding-goal trap.
+  slippage.rule = { kind: "gap", factor: 0.1 }
   return model(
     "Drift to low performance",
     [source, performance, improvement, standard, sink, slippage],
@@ -445,6 +575,69 @@ function driftToLowPerformance(): Model {
       link(standard, slippage, "+"),
       link(performance, slippage, "-"),
     ],
+    { start: 0, stop: 60, dt: 1 },
+  )
+}
+
+/**
+ * Overshoot and collapse — the dark twin of "Limits to growth". The same
+ * Reinforcing engine runs, but the limit here is a *non-renewable* Resource that
+ * only depletes: a Stock with no inflow, the first in the gallery. An economy
+ * (Capital) lives off it — extraction grows with both the Resource left and the
+ * Capital deployed (Resource, Capital → [+] → extraction), and the revenue is
+ * reinvested as new Capital (extraction → [+] → investment → Capital), so the loop
+ * Capital → extraction → investment → Capital carries no `−` → Reinforcing. Capital
+ * climbs and extraction accelerates, but every unit burned is gone for good, so the
+ * Resource crosses the break-even level, the engine starves, and depreciation
+ * (Capital → [+] → depreciation, a Balancing drain) takes Capital down: it peaks,
+ * then collapses. Contrast "Predator and prey", whose prey regrows and so settles
+ * into oscillation — a finite Resource cannot, so it overshoots and crashes instead.
+ */
+function overshootAndCollapse(): Model {
+  // Resource on the left drains only downward (no inflow). Capital on the right runs
+  // a full Source → investment → Capital → depreciation → Sink column. The two
+  // coupling links — Capital → extraction and extraction → investment — cross in the
+  // open centre, where the R badge lands.
+  const resource = makeStock({ x: -240, y: 0 }, "Resource")
+  resource.initialValue = 1000
+  const extractionSink = makeCloud({ x: -240, y: 360 })
+  const extraction = makeFlow({ x: -240, y: 160 }, "extraction", resource.id, extractionSink.id)
+  // extraction = factor × Resource × Capital (both `+`): more capital extracts
+  // faster, scarcer resource slower. The bilinear term the non-negative floor tames.
+  extraction.rule = { kind: "proportional", factor: 0.0004 }
+  const capital = makeStock({ x: 240, y: 0 }, "Capital")
+  capital.initialValue = 5
+  const investmentSource = makeCloud({ x: 240, y: -360 })
+  const investment = makeFlow({ x: 240, y: -160 }, "investment", investmentSource.id, capital.id)
+  // investment = factor × extraction (its one `+` input): the revenue reinvested —
+  // a Flow feeding a Flow, the edge that closes the Reinforcing loop through Capital.
+  investment.rule = { kind: "proportional", factor: 0.5 }
+  const depreciationSink = makeCloud({ x: 240, y: 360 })
+  const depreciation = makeFlow({ x: 240, y: 160 }, "depreciation", capital.id, depreciationSink.id)
+  // depreciation = factor × Capital (its `+` input): the Balancing drain that wins
+  // once the Resource can no longer feed investment.
+  depreciation.rule = { kind: "proportional", factor: 0.04 }
+  return model(
+    "Overshoot and collapse",
+    [
+      resource,
+      extractionSink,
+      extraction,
+      capital,
+      investmentSource,
+      investment,
+      depreciationSink,
+      depreciation,
+    ],
+    [
+      link(resource, extraction, "+"),
+      link(capital, extraction, "+"),
+      link(extraction, investment, "+"),
+      link(capital, depreciation, "+"),
+    ],
+    // Capital starts at 5, overshoots to ~250 by t≈39, and collapses back near its
+    // starting level by t=150 — the full boom-and-bust arc, no dead tail.
+    { start: 0, stop: 150, dt: 1 },
   )
 }
 
@@ -501,4 +694,9 @@ export const SAMPLES: Sample[] = [
     blurb: "Goals erode toward actual: a Reinforcing slide downhill.",
     build: driftToLowPerformance,
   },
+  {
+    title: "Overshoot and collapse",
+    blurb: "A growth engine burns a finite Resource: it peaks, then crashes.",
+    build: overshootAndCollapse,
+  },
 ]

src/model/simulation.ts

@@ -0,0 +1,228 @@
+/**
+ * Simulation engine (ADR-0004) — what makes a Model *alive*. Pure data in, time
+ * series out; it knows nothing of Vue or the canvas, so it is trivially testable
+ * and runs off any reactive frame.
+ *
+ * The system-dynamics loop, one step of `dt`:
+ *   1. Evaluate the instantaneous network — every Converter and Flow — in
+ *      dependency order, reading the *current* Stock values. A Converter/Flow
+ *      depends on the elements that link into it (ADR-0004: the Information Link
+ *      *is* the dependency); Stocks are state, available without ordering.
+ *   2. Integrate every Stock *simultaneously* (forward Euler):
+ *      `stock += dt × (Σ inflow rates − Σ outflow rates)`. All net rates are read
+ *      from the same pre-update state, so updating one Stock never feeds another
+ *      within the same step.
+ *
+ * A cycle in the wiring is legitimate feedback only if it passes through a Stock
+ * (the Stock supplies last-step state and breaks the within-step dependency). A
+ * cycle among only Converters/Flows is an **algebraic loop** — unorderable, and
+ * rejected by `evaluationOrder` / surfaced by `checkSimReady`.
+ */
+import {
+  type ConverterNode,
+  DEFAULT_SIM_SPEC,
+  type FlowNode,
+  type InformationLink,
+  type Model,
+  type Rule,
+  type SimSpec,
+} from "./types"
+
+/** Thrown when a Model cannot be simulated as wired (e.g. an algebraic loop). */
+export class SimulationError extends Error {
+  constructor(message: string) {
+    super(message)
+    this.name = "SimulationError"
+  }
+}
+
+/** A completed run: aligned `times` and per-element value tracks. */
+export interface Run {
+  /** The time at each recorded sample, `start … stop` in steps of `dt`. */
+  times: number[]
+  /** nodeId → its value at each time index (Stocks, Flows, Converters; not Clouds). */
+  series: Map<string, number[]>
+  /** True if the run ran past representable numbers and stopped early (see below). */
+  diverged: boolean
+}
+
+/** A Converter or Flow — the stateless, instantaneous elements an order applies to. */
+type Instant = ConverterNode | FlowNode
+
+/**
+ * Order the instantaneous network so each element is computed after everything
+ * it reads. Throws `SimulationError` on an algebraic loop (a Converter/Flow cycle
+ * with no Stock to break it). Stocks are excluded — they are state, not computed.
+ */
+export function evaluationOrder(model: Model): Instant[] {
+  const instant = model.nodes.filter(
+    (node): node is Instant => node.kind === "flow" || node.kind === "converter",
+  )
+  const ids = new Set(instant.map((node) => node.id))
+
+  // deps[x] = the instantaneous elements x reads (links from another Flow/Converter).
+  const deps = new Map<string, Set<string>>(instant.map((node) => [node.id, new Set<string>()]))
+  for (const link of model.infoLinks) {
+    if (ids.has(link.source) && ids.has(link.target)) deps.get(link.target)?.add(link.source)
+  }
+
+  const order: Instant[] = []
+  const resolved = new Set<string>()
+  while (order.length < instant.length) {
+    const next = instant.find(
+      (node) =>
+        !resolved.has(node.id) && [...(deps.get(node.id) ?? [])].every((d) => resolved.has(d)),
+    )
+    if (!next) {
+      const names = instant
+        .filter((node) => !resolved.has(node.id))
+        .map((node) => node.name)
+        .join(", ")
+      throw new SimulationError(
+        `Algebraic loop: ${names} depend on each other with no Stock to break the cycle.`,
+      )
+    }
+    resolved.add(next.id)
+    order.push(next)
+  }
+  return order
+}
+
+/** Evaluate one Rule given a way to read the values feeding in via `links`. */
+function evalRule(rule: Rule, links: InformationLink[], valueOf: (id: string) => number): number {
+  switch (rule.kind) {
+    case "constant":
+      return rule.value
+    case "proportional": {
+      // factor × the product of every `+`-polarity input (one input → factor × it).
+      let value = rule.factor
+      for (const link of links) if (link.polarity === "+") value *= valueOf(link.source)
+      return value
+    }
+    case "gap": {
+      // factor × (level − target): the `+` input is the level, the `−` the target.
+      const level = links.find((link) => link.polarity === "+")
+      const target = links.find((link) => link.polarity === "-")
+      return (
+        rule.factor * ((level ? valueOf(level.source) : 0) - (target ? valueOf(target.source) : 0))
+      )
+    }
+  }
+}
+
+/**
+ * Run the Model and return aligned time series. Assumes a sim-ready Model
+ * (see `checkSimReady`); a missing rule evaluates to 0 and a missing initial
+ * value to 0 rather than throwing, so a half-built Model still produces a plot.
+ * Throws `SimulationError` only on an algebraic loop, which has no defined order.
+ */
+export function simulate(model: Model, spec: SimSpec = model.sim ?? DEFAULT_SIM_SPEC): Run {
+  const order = evaluationOrder(model)
+  const nodeById = new Map(model.nodes.map((node) => [node.id, node]))
+
+  const inbound = new Map<string, InformationLink[]>()
+  for (const link of model.infoLinks) {
+    const list = inbound.get(link.target)
+    if (list) list.push(link)
+    else inbound.set(link.target, [link])
+  }
+
+  const stocks = model.nodes.filter((node) => node.kind === "stock")
+  const flows = model.nodes.filter((node): node is FlowNode => node.kind === "flow")
+
+  const stockValues = new Map<string, number>(stocks.map((s) => [s.id, s.initialValue ?? 0]))
+
+  const times: number[] = []
+  const series = new Map<string, number[]>()
+  for (const node of model.nodes) if (node.kind !== "cloud") series.set(node.id, [])
+  let diverged = false
+
+  // dt ≤ 0 would never advance (and 0 diverges); a non-positive step means no run.
+  const steps = spec.dt > 0 ? Math.max(0, Math.floor((spec.stop - spec.start) / spec.dt)) : 0
+  for (let i = 0; i <= steps; i++) {
+    // A run-away Reinforcing loop over a long horizon can exceed what a float
+    // holds. Stop at the last valid sample and flag it rather than plotting NaN.
+    if (!stocks.every((s) => Number.isFinite(stockValues.get(s.id) ?? 0))) {
+      diverged = true
+      break
+    }
+
+    // 1. Evaluate the instantaneous network from the current Stock values.
+    const computed = new Map<string, number>()
+    const valueOf = (id: string): number => {
+      const node = nodeById.get(id)
+      if (!node) return 0
+      if (node.kind === "stock") return stockValues.get(id) ?? 0
+      if (node.kind === "cloud") return 0
+      return computed.get(id) ?? 0
+    }
+    for (const node of order) {
+      computed.set(
+        node.id,
+        node.rule ? evalRule(node.rule, inbound.get(node.id) ?? [], valueOf) : 0,
+      )
+    }
+
+    // Record this sample (Stocks at their current value, Flows/Converters as just computed).
+    times.push(spec.start + i * spec.dt)
+    for (const s of stocks) series.get(s.id)?.push(stockValues.get(s.id) ?? 0)
+    for (const node of order) series.get(node.id)?.push(computed.get(node.id) ?? 0)
+
+    if (i >= steps) break
+
+    // 2. Integrate every Stock simultaneously (forward Euler) — but with
+    // non-negative stocks: an outflow can't drain more than its source holds this
+    // step. Scale a stock's competing outflows together if they would overdraw,
+    // and apply the scaled rate to both ends so quantity is conserved. You can't
+    // infect more people than are susceptible — and that floor is exactly what
+    // stops a bilinear model (S × I) from flipping sign and diverging.
+    const rate = new Map<string, number>(flows.map((f) => [f.id, computed.get(f.id) ?? 0]))
+    for (const s of stocks) {
+      const available = stockValues.get(s.id) ?? 0
+      const drains = flows.filter((f) => f.source === s.id && (rate.get(f.id) ?? 0) > 0)
+      const totalOut = drains.reduce((sum, f) => sum + (rate.get(f.id) ?? 0), 0)
+      if (totalOut * spec.dt > available) {
+        const scale = available / (totalOut * spec.dt)
+        for (const f of drains) rate.set(f.id, (rate.get(f.id) ?? 0) * scale)
+      }
+    }
+
+    const next = new Map(stockValues)
+    for (const s of stocks) {
+      let net = 0
+      for (const flow of flows) {
+        const r = rate.get(flow.id) ?? 0
+        if (flow.target === s.id) net += r // an inflow fills it
+        if (flow.source === s.id) net -= r // an outflow drains it
+      }
+      next.set(s.id, (stockValues.get(s.id) ?? 0) + spec.dt * net)
+    }
+    for (const [id, value] of next) stockValues.set(id, value)
+  }
+
+  return { times, series, diverged }
+}
+
+/**
+ * What stands between this Model and a run, as human-readable lines (empty array
+ * = ready). Distinct from structural validity (validation.ts): a Model can be a
+ * perfectly valid diagram yet not carry the numbers a simulation needs.
+ */
+export function checkSimReady(model: Model): string[] {
+  const problems: string[] = []
+  for (const node of model.nodes) {
+    if (node.kind === "stock" && node.initialValue === undefined) {
+      problems.push(`${node.name} has no initial value.`)
+    }
+    if ((node.kind === "flow" || node.kind === "converter") && !node.rule) {
+      problems.push(`${node.name} has no rule yet.`)
+    }
+  }
+  try {
+    evaluationOrder(model)
+  } catch (error) {
+    if (error instanceof SimulationError) problems.push(error.message)
+    else throw error
+  }
+  return problems
+}

src/model/types.ts

@@ -31,6 +31,37 @@ export interface Position {
   y: number
 }
 
+/**
+ * How a Flow's rate or a Converter's value is computed — a small fixed
+ * vocabulary, *not* a free-form formula, so a Model stays valid by construction
+ * and teaches structure → behaviour (ADR-0004). Operands are read from the
+ * element's inbound Information Links; Polarity selects each operand's role.
+ *
+ *  - `constant`     — a fixed number; reads nothing. (→ linear Stock change)
+ *  - `proportional` — `factor ×` the product of its `+`-polarity inputs.
+ *                     (→ exponential growth/decay)
+ *  - `gap`          — `factor × (level − target)`, where the `+` input is the
+ *                     level and the `−` input the target. (→ goal-seeking)
+ */
+export type Rule =
+  | { kind: "constant"; value: number }
+  | { kind: "proportional"; factor: number }
+  | { kind: "gap"; factor: number }
+
+/**
+ * The run parameters for a simulation: integrate from `start` to `stop` in steps
+ * of `dt` (forward Euler). Optional on a Model; absent means the diagram phase or
+ * a not-yet-simulated Model — the engine falls back to `DEFAULT_SIM_SPEC`.
+ */
+export interface SimSpec {
+  start: number
+  stop: number
+  dt: number
+}
+
+/** Sensible default run when a Model carries no `sim` of its own. */
+export const DEFAULT_SIM_SPEC: SimSpec = { start: 0, stop: 100, dt: 1 }
+
 interface BaseNode {
   id: string
   position: Position
@@ -45,6 +76,8 @@ export interface StockNode extends BaseNode {
   name: string
   /** Initial accumulated quantity. Optional in the diagram phase; the simulator reads it. */
   initialValue?: number
+  /** Unit of the quantity, e.g. "°C", "people", "$" — shown beside the value (display only). */
+  unit?: string
 }
 
 /**
@@ -60,8 +93,8 @@ export interface FlowNode extends BaseNode {
   source: string
   /** Node id of the Stock or Cloud the Flow feeds into. */
   target: string
-  /** Rate expression, recomputed each instant. Optional in the diagram phase. */
+  /** How its rate is computed each instant (ADR-0004). Optional in the diagram phase. */
-  equation?: string
+  rule?: Rule
 }
 
 /**
@@ -71,8 +104,8 @@ export interface FlowNode extends BaseNode {
 export interface ConverterNode extends BaseNode {
   kind: "converter"
   name: string
-  /** Expression or constant. Optional in the diagram phase. */
+  /** How its value is computed each instant (ADR-0004). Optional in the diagram phase. */
-  equation?: string
+  rule?: Rule
 }
 
 /**
@@ -107,4 +140,6 @@ export interface Model {
   name: string
   nodes: ModelNode[]
   infoLinks: InformationLink[]
+  /** Run parameters for simulation (ADR-0004). Optional; absent → not yet simulated. */
+  sim?: SimSpec
 }

src/store/model.ts

@@ -24,7 +24,15 @@ import {
   nextName,
 } from "@/model/factory"
 import { detectLoops } from "@/model/loops"
-import type { ConverterNode, Model, ModelNode, Position, StockNode } from "@/model/types"
+import type {
+  ConverterNode,
+  Model,
+  ModelNode,
+  Position,
+  Rule,
+  SimSpec,
+  StockNode,
+} from "@/model/types"
 import { canConnect, intentFor } from "@/model/validation"
 
 /** Ring-buffer depth for undo (F9 target: ≥50 steps). */
@@ -96,6 +104,58 @@ export const useModelStore = defineStore("model", () => {
     node.name = name
   }
 
+  /**
+   * Set (or clear) a Stock's initial value — the quantity the simulator starts
+   * from (ADR-0004). `undefined` un-equips it, sending the Model back to "not yet
+   * simulatable". No-op when unchanged, so a blur with no edit doesn't burn undo.
+   */
+  function setInitialValue(id: string, value: number | undefined): void {
+    const node = findNode(id)
+    if (!node || node.kind !== "stock" || node.initialValue === value) return
+    record()
+    if (value === undefined) delete node.initialValue
+    else node.initialValue = value
+  }
+
+  /** Set (or clear) a Stock's display unit (e.g. "°C"). Empty clears it. No-op when unchanged. */
+  function setUnit(id: string, unit: string): void {
+    const node = findNode(id)
+    if (!node || node.kind !== "stock") return
+    const next = unit.trim() || undefined
+    if (node.unit === next) return
+    record()
+    if (next === undefined) delete node.unit
+    else node.unit = next
+  }
+
+  /**
+   * Set (or clear) how a Flow's rate or a Converter's value is computed (ADR-0004:
+   * one of the fixed rules, never a formula). No-op when unchanged.
+   */
+  function setRule(id: string, rule: Rule | undefined): void {
+    const node = findNode(id)
+    if (!node || (node.kind !== "flow" && node.kind !== "converter")) return
+    if (JSON.stringify(node.rule) === JSON.stringify(rule)) return
+    record()
+    if (rule === undefined) delete node.rule
+    else node.rule = rule
+  }
+
+  /** Set the run parameters (start / stop / dt). No-op when unchanged. */
+  function setSimSpec(spec: SimSpec): void {
+    const current = model.value.sim
+    if (
+      current &&
+      current.start === spec.start &&
+      current.stop === spec.stop &&
+      current.dt === spec.dt
+    ) {
+      return
+    }
+    record()
+    model.value.sim = spec
+  }
+
   /**
    * Create whatever source→target means under the structure guard: an
    * Information Link (default `+` polarity, F6) onto a Flow/Converter, or a Flow
@@ -229,6 +289,10 @@ export const useModelStore = defineStore("model", () => {
     beginInteraction,
     moveNode,
     renameNode,
+    setInitialValue,
+    setUnit,
+    setRule,
+    setSimSpec,
     removeNode,
     removeInfoLink,
     toggleLinkPolarity,

src/store/simulation.ts

@@ -0,0 +1,151 @@
+/**
+ * Simulation store (phase 2) — the single source of *playback* truth, the way
+ * the model store is the single source of structural truth. It owns one run and
+ * one playhead, so the behaviour chart and the canvas animate off the same index:
+ * drag the scrubber and every Stock on the canvas jumps to that instant.
+ *
+ * The run is derived from the Model (recomputed when the Model changes, like the
+ * loop detector), so editing a rule and reopening re-runs. Playback itself — the
+ * clock that advances the playhead — lives in a composable (`usePlayback`), since
+ * timers need component lifecycle; this store only holds state and pure actions.
+ */
+import { defineStore } from "pinia"
+import { computed, ref, watch } from "vue"
+import { checkSimReady, type Run, simulate } from "@/model/simulation"
+import { useModelStore } from "./model"
+
+export const useSimulationStore = defineStore("simulation", () => {
+  const modelStore = useModelStore()
+
+  /** A Model is runnable only once every Stock has a value and every rate a rule. */
+  const ready = computed(() => checkSimReady(modelStore.model).length === 0)
+
+  /** The current run, or null when the Model isn't runnable. Recomputes on edit. */
+  const run = computed<Run | null>(() => {
+    if (!ready.value) return null
+    try {
+      return simulate(modelStore.model)
+    } catch {
+      // checkSimReady already rejects algebraic loops; this is belt-and-braces.
+      return null
+    }
+  })
+
+  const frameCount = computed(() => run.value?.times.length ?? 0)
+
+  /** Whether the canvas should show live values: the panel is open and a run exists. */
+  const enabled = ref(false)
+  const playing = ref(false)
+  const playhead = ref(0)
+
+  const active = computed(() => enabled.value && frameCount.value > 0)
+  const atEnd = computed(() => playhead.value >= frameCount.value - 1)
+  const currentTime = computed(() => run.value?.times[playhead.value] ?? null)
+
+  /** Per-element peak magnitude across the run, for the Stock fill gauge. */
+  const peaks = computed(() => {
+    const map = new Map<string, number>()
+    const r = run.value
+    if (r) {
+      for (const [id, series] of r.series) {
+        let max = 0
+        for (const v of series) if (Number.isFinite(v)) max = Math.max(max, Math.abs(v))
+        map.set(id, max)
+      }
+    }
+    return map
+  })
+
+  // Keep the playhead in range when the run shrinks (e.g. a shorter stop time).
+  watch(frameCount, (n) => {
+    if (playhead.value > n - 1) playhead.value = Math.max(0, n - 1)
+  })
+
+  // Loading a different document (sample / import / new) restarts playback from the
+  // top. Editing the current one keeps the same id, so the playhead stays put — and
+  // undo/redo restore snapshots in place (same id), so they don't jump it either.
+  watch(
+    () => modelStore.model.id,
+    () => {
+      playing.value = false
+      playhead.value = 0
+    },
+  )
+
+  /** A node's value at the current playhead, or null when there's nothing to show. */
+  function valueAt(id: string): number | null {
+    if (!active.value) return null
+    const value = run.value?.series.get(id)?.[playhead.value]
+    return value === undefined || !Number.isFinite(value) ? null : value
+  }
+
+  /** A Stock's fill level in 0…1 (value over its peak), for the gauge. */
+  function fill(id: string): number {
+    const value = valueAt(id)
+    const peak = peaks.value.get(id) ?? 0
+    if (value === null || peak <= 0) return 0
+    return Math.min(1, Math.max(0, value / peak))
+  }
+
+  function seek(index: number): void {
+    playhead.value = Math.min(Math.max(0, Math.round(index)), Math.max(0, frameCount.value - 1))
+  }
+
+  function play(): void {
+    if (frameCount.value === 0) return
+    if (atEnd.value) playhead.value = 0 // replay from the start
+    playing.value = true
+  }
+
+  function pause(): void {
+    playing.value = false
+  }
+
+  function toggle(): void {
+    if (playing.value) pause()
+    else play()
+  }
+
+  /** Advance one frame; stop at the end. Called by the playback clock. */
+  function tick(): void {
+    if (atEnd.value) {
+      playing.value = false
+      return
+    }
+    playhead.value++
+  }
+
+  /** Engage playback (panel opened): show values, start from the beginning. */
+  function enable(): void {
+    enabled.value = true
+    playhead.value = 0
+  }
+
+  /** Disengage (panel closed): the canvas returns to a plain diagram. */
+  function disable(): void {
+    enabled.value = false
+    playing.value = false
+    playhead.value = 0
+  }
+
+  return {
+    run,
+    ready,
+    frameCount,
+    enabled,
+    playing,
+    playhead,
+    active,
+    atEnd,
+    currentTime,
+    valueAt,
+    fill,
+    seek,
+    play,
+    pause,
+    toggle,
+    tick,
+    enable,
+    disable,
+  }
+})

src/style.css

@@ -3,17 +3,26 @@
    @import to precede all other rules. If it comes second, the production build
    warns ("@import must precede all rules…") and browsers silently DROP the
    font import, so the custom font never loads. */
-@import url("https://fonts.coollabs.io/css2?family=Inter:wght@400;500;600;700&display=swap");
+@import url("https://api.fonts.coollabs.io/css2?family=Sono:wght@400;500;600;700&display=swap");
 @import "tailwindcss";
 @plugin "daisyui";
 @plugin "daisyui/theme" {
   name: "light";
   default: true;
-  --color-primary: #16a34a;
+  --color-primary: #ff9ff3;
+}
+
+/* DaisyUI's default (no data-theme) resolves to its *dark* theme via
+   prefers-color-scheme, with a selector (`:root:not([data-theme])`) that outranks
+   a custom light theme's `:where(:root)` — so overriding only the light theme
+   never reached the buttons. Force the brand primary across every theme. */
+:root {
+  --color-primary: #ff9ff3 !important;
+  --color-primary-content: #3d1f3a !important;
 }
 
 @theme {
-  --font-sans: "Inter", ui-sans-serif, system-ui, sans-serif;
+  --font-sans: "Sono", ui-sans-serif, system-ui, sans-serif;
 }
 
 /* Larger, easier-to-grab connection handles (Vue Flow defaults to 6px). The