diff --git a/ui/frontend/src/map/render.ts b/ui/frontend/src/map/render.ts
index e65e601..3c6461b 100644
--- a/ui/frontend/src/map/render.ts
+++ b/ui/frontend/src/map/render.ts
@@ -213,7 +213,81 @@ const EMPTY_HIDDEN_IDS: ReadonlySet<PrimitiveID> = new Set();
 // lighter than the dark theme background (`0x0a0e1a`) so it reads
 // as a faint fog without contrasting against the rest of the map.
 // The colour is tunable in Phase 35 polish.
-const FOG_COLOR = 0x12162a;
+export const FOG_COLOR = 0x12162a;
+
+/**
+ * FogPaintOp is one item in the ordered draw sequence produced by
+ * `fogPaintOps`. The renderer dispatches each op directly onto a
+ * Pixi `Graphics`; the indirection exists so the Phase 29 layered
+ * overpaint (fog rect then background-coloured circles on top) can
+ * be unit-tested without a Pixi context.
+ */
+export type FogPaintOp =
+	| {
+			readonly kind: "fillRect";
+			readonly x: number;
+			readonly y: number;
+			readonly width: number;
+			readonly height: number;
+			readonly color: number;
+			readonly alpha: number;
+	  }
+	| {
+			readonly kind: "fillCircle";
+			readonly x: number;
+			readonly y: number;
+			readonly radius: number;
+			readonly color: number;
+			readonly alpha: number;
+	  };
+
+/**
+ * fogPaintOps returns the ordered sequence of paint operations that
+ * draw the Phase 29 visible-hyperspace overlay on a single torus
+ * copy. The first op is the fog-coloured rectangle covering the
+ * full world; subsequent ops are background-coloured circles, one
+ * per visibility circle, painted on top of the fog rectangle. The
+ * natural rendering order unions overlapping circles for free —
+ * earlier iterations relied on Pixi v8's `Graphics.cut()` to
+ * subtract holes, but `cut()` produced incorrect unions for
+ * overlapping circles (the symptom was a handful of disconnected
+ * arc segments instead of a clean union).
+ *
+ * Empty `circles` returns an empty list — the caller skips fog
+ * rendering entirely. Width/height ≤ 0 also returns empty so a
+ * degenerate world cannot produce a non-emit op set.
+ */
+export function fogPaintOps(
+	world: { width: number; height: number },
+	circles: ReadonlyArray<{ x: number; y: number; radius: number }>,
+	fogColor: number,
+	bgColor: number,
+): FogPaintOp[] {
+	if (circles.length === 0) return [];
+	if (world.width <= 0 || world.height <= 0) return [];
+	const ops: FogPaintOp[] = [
+		{
+			kind: "fillRect",
+			x: 0,
+			y: 0,
+			width: world.width,
+			height: world.height,
+			color: fogColor,
+			alpha: 1,
+		},
+	];
+	for (const c of circles) {
+		ops.push({
+			kind: "fillCircle",
+			x: c.x,
+			y: c.y,
+			radius: c.radius,
+			color: bgColor,
+			alpha: 1,
+		});
+	}
+	return ops;
+}
 
 export async function createRenderer(opts: RendererOptions): Promise<RendererHandle> {
 	const theme = opts.theme ?? DARK_THEME;
@@ -712,23 +786,22 @@ export async function createRenderer(opts: RendererOptions): Promise<RendererHan
 				g.destroy();
 			}
 			fogGraphics = [];
-			if (circles.length === 0) return;
-			// Layered overpaint: a fog-tinted rectangle covers the
-			// world, then opaque background-coloured circles drawn on
-			// top reveal the visible-hyperspace area. The natural
-			// rendering order handles overlapping circles correctly —
-			// Pixi v8's `Graphics.cut()` produces inconsistent
-			// results for unions of holes (the previous Phase 29
-			// implementation hit this), and the overpaint approach
-			// avoids the geometry calculation entirely.
-			const bg = theme.background;
+			const ops = fogPaintOps(
+				opts.world,
+				circles,
+				FOG_COLOR,
+				theme.background,
+			);
+			if (ops.length === 0) return;
 			for (const copy of copies) {
 				const g = new Graphics();
-				g.rect(0, 0, opts.world.width, opts.world.height);
-				g.fill({ color: FOG_COLOR, alpha: 1 });
-				for (const c of circles) {
-					g.circle(c.x, c.y, c.radius);
-					g.fill({ color: bg, alpha: 1 });
+				for (const op of ops) {
+					if (op.kind === "fillRect") {
+						g.rect(op.x, op.y, op.width, op.height);
+					} else {
+						g.circle(op.x, op.y, op.radius);
+					}
+					g.fill({ color: op.color, alpha: op.alpha });
 				}
 				// Fog sits below every primitive on the same copy so
 				// planet glyphs paint on top. `addChildAt(g, 0)` keeps
diff --git a/ui/frontend/tests/fog-paint-ops.test.ts b/ui/frontend/tests/fog-paint-ops.test.ts
new file mode 100644
index 0000000..6da4a7e
--- /dev/null
+++ b/ui/frontend/tests/fog-paint-ops.test.ts
@@ -0,0 +1,130 @@
+// Phase 29 unit coverage for the Phase 29 fog overlay's layered
+// overpaint logic. `fogPaintOps` lives in `src/map/render.ts` next
+// to its sole consumer (`RendererHandle.setVisibilityFog`) — the
+// renderer dispatches each op straight onto a Pixi `Graphics`, so
+// the unit test exercises the public ordering contract: a single
+// fog-coloured rectangle followed by one background-coloured
+// circle per visibility entry. The natural rendering order unions
+// overlapping circles for free, replacing the earlier `cut()`
+// implementation that produced disconnected arc segments.
+
+import { describe, expect, test } from "vitest";
+
+import { FOG_COLOR, fogPaintOps } from "../src/map/render";
+
+const BG_COLOR = 0x0a0e1a;
+const WORLD = { width: 1000, height: 800 };
+
+describe("fogPaintOps", () => {
+	test("empty input returns no ops", () => {
+		expect(fogPaintOps(WORLD, [], FOG_COLOR, BG_COLOR)).toEqual([]);
+	});
+
+	test("single circle emits fog rect + one bg circle in that order", () => {
+		const ops = fogPaintOps(
+			WORLD,
+			[{ x: 100, y: 200, radius: 50 }],
+			FOG_COLOR,
+			BG_COLOR,
+		);
+		expect(ops).toEqual([
+			{
+				kind: "fillRect",
+				x: 0,
+				y: 0,
+				width: 1000,
+				height: 800,
+				color: FOG_COLOR,
+				alpha: 1,
+			},
+			{
+				kind: "fillCircle",
+				x: 100,
+				y: 200,
+				radius: 50,
+				color: BG_COLOR,
+				alpha: 1,
+			},
+		]);
+	});
+
+	test("multiple circles produce one fog rect followed by N bg circles", () => {
+		const ops = fogPaintOps(
+			WORLD,
+			[
+				{ x: 100, y: 100, radius: 50 },
+				{ x: 300, y: 200, radius: 80 },
+				{ x: 500, y: 600, radius: 30 },
+			],
+			FOG_COLOR,
+			BG_COLOR,
+		);
+		expect(ops.length).toBe(4);
+		expect(ops[0].kind).toBe("fillRect");
+		for (let i = 1; i < ops.length; i++) {
+			expect(ops[i].kind).toBe("fillCircle");
+			// Background-coloured circles paint on top of the fog rect.
+			const op = ops[i];
+			if (op.kind === "fillCircle") {
+				expect(op.color).toBe(BG_COLOR);
+				expect(op.alpha).toBe(1);
+			}
+		}
+	});
+
+	test("overlapping circles are emitted independently — the rendering order unions them", () => {
+		// Two overlapping circles around adjacent LOCAL planets — the
+		// op list keeps both circles. The renderer relies on the
+		// overpaint to merge them visually; `cut()` (the previous
+		// implementation) miscomputed the union.
+		const ops = fogPaintOps(
+			WORLD,
+			[
+				{ x: 200, y: 200, radius: 100 },
+				{ x: 250, y: 200, radius: 100 },
+			],
+			FOG_COLOR,
+			BG_COLOR,
+		);
+		expect(ops.length).toBe(3);
+		expect(ops[1]).toMatchObject({ x: 200, y: 200, radius: 100 });
+		expect(ops[2]).toMatchObject({ x: 250, y: 200, radius: 100 });
+	});
+
+	test("the fog rect always covers the full world rectangle", () => {
+		const ops = fogPaintOps(
+			{ width: 3200, height: 1600 },
+			[{ x: 0, y: 0, radius: 10 }],
+			FOG_COLOR,
+			BG_COLOR,
+		);
+		expect(ops[0]).toEqual({
+			kind: "fillRect",
+			x: 0,
+			y: 0,
+			width: 3200,
+			height: 1600,
+			color: FOG_COLOR,
+			alpha: 1,
+		});
+	});
+
+	test("zero or negative world dimensions return no ops", () => {
+		expect(
+			fogPaintOps(
+				{ width: 0, height: 800 },
+				[{ x: 0, y: 0, radius: 10 }],
+				FOG_COLOR,
+				BG_COLOR,
+			),
+		).toEqual([]);
+		expect(
+			fogPaintOps(
+				{ width: 1000, height: -1 },
+				[{ x: 0, y: 0, radius: 10 }],
+				FOG_COLOR,
+				BG_COLOR,
+			),
+		).toEqual([]);
+	});
+});