diff --git a/docs/content/inference.md b/docs/content/inference.md
index 78540d43..0158b878 100644
--- a/docs/content/inference.md
+++ b/docs/content/inference.md
@@ -685,7 +685,8 @@ before anything else looks at the mask.
 **A polygon is the piece you pointed at, not the biggest one on the frame.** Which of the
 survivors you meant is a question only the points can answer, so the choice is made from the
 prompt: a point inside a piece picks that piece; several points inside several pieces pick the
-largest of *those*, because two positives describe one object rather than propose two; and a
+largest of *those*, because two positives describe one object rather than propose two, and two
+pieces of exactly the same size answer with the one whose earliest run comes first; and a
 point inside none of them picks the piece nearest to it, since a mask need not cover the exact
 pixel you clicked. Negative points never select - they say what the shape is not, and a piece is
 chosen before its shape is known.
diff --git a/frontend/annotator/src/core/geometry/simplify.ts b/frontend/annotator/src/core/geometry/simplify.ts
index e31bf000..4b1eb8a2 100644
--- a/frontend/annotator/src/core/geometry/simplify.ts
+++ b/frontend/annotator/src/core/geometry/simplify.ts
@@ -16,11 +16,20 @@
  * keeps it current, and neither half shares a toolchain with the other.
  *
  * **Exact equality is achievable and is what the gate asserts.** Both languages
- * hold IEEE-754 doubles, `Math.sqrt` and Python's `** 0.5` are the same
- * correctly-rounded operation, and every expression below is in the same order
- * as the Python it mirrors. Keeping that order is not style: the algorithm's
- * output is decided by comparisons, so a re-association that moves a value by
- * one unit in the last place can move a vertex.
+ * hold IEEE-754 doubles, and every expression below is in the same order as the
+ * Python it mirrors — that ordering is what the parity actually depends on.
+ * Keeping it is not style: the algorithm's output is decided by comparisons, so
+ * a re-association that moves a value by one unit in the last place can move a
+ * vertex.
+ *
+ * `Math.sqrt` and Python's `** 0.5` are *not* quite the same operation, though:
+ * CPython routes `** 0.5` through libm `pow`, which is not correctly rounded,
+ * and it disagrees with a correctly-rounded square root by one ulp on roughly a
+ * tenth of a percent of inputs. The mask pipeline avoids carrying that
+ * difference into nearest-piece selection by comparing squared distances there;
+ * `closingRadius` absorbs it through truncation. The existing simplification
+ * fixture continues to assert exact output for its covered contours. Making
+ * Python's side call `math.sqrt` would be a separate semantic change.
  *
  * ## Why a contour is the input rather than a mask
  *
diff --git a/frontend/annotator/src/core/mask/binaryMask.test.ts b/frontend/annotator/src/core/mask/binaryMask.test.ts
new file mode 100644
index 00000000..be7194ef
--- /dev/null
+++ b/frontend/annotator/src/core/mask/binaryMask.test.ts
@@ -0,0 +1,33 @@
+import { describe, expect, it } from "vitest";
+
+import { pythonRound } from "./binaryMask";
+
+describe("pythonRound", () => {
+  it("rounds a half to the even neighbour, as Python does", () => {
+    expect(pythonRound(0.5)).toBe(0);
+    expect(pythonRound(1.5)).toBe(2);
+    expect(pythonRound(2.5)).toBe(2);
+    expect(pythonRound(3.5)).toBe(4);
+  });
+
+  it("disagrees with Math.round on exactly the halves that decide a component", () => {
+    for (const half of [0.5, 2.5, 4.5, 6.5]) {
+      expect(pythonRound(half)).not.toBe(Math.round(half));
+    }
+  });
+
+  it("rounds a negative half to even too, and never to -0", () => {
+    expect(pythonRound(-0.5)).toBe(0);
+    expect(Object.is(pythonRound(-0.5), -0)).toBe(false);
+    expect(pythonRound(-1.5)).toBe(-2);
+    expect(pythonRound(-2.5)).toBe(-2);
+  });
+
+  it("rounds anything that is not a half to the nearest whole number", () => {
+    expect(pythonRound(0.4)).toBe(0);
+    expect(pythonRound(0.6)).toBe(1);
+    expect(pythonRound(-0.4)).toBe(0);
+    expect(pythonRound(-0.6)).toBe(-1);
+    expect(pythonRound(7)).toBe(7);
+  });
+});
diff --git a/frontend/annotator/src/core/mask/binaryMask.ts b/frontend/annotator/src/core/mask/binaryMask.ts
new file mode 100644
index 00000000..cb72d65b
--- /dev/null
+++ b/frontend/annotator/src/core/mask/binaryMask.ts
@@ -0,0 +1,59 @@
+/**
+ * What a segmenter answers with, in the asset's own pixels.
+ *
+ * ## The seam, and why it is structural rather than a dependency
+ *
+ * `@visionset/browser-inference` stops at a mask: model execution, an image
+ * embedding, a grid of bytes and a confidence. Turning that grid into a box or
+ * a polygon is a product decision — which pieces count as the object, how big a
+ * gap is an artefact, where the boundary of a pixel is — and it belongs here,
+ * beside the rest of the asset-pixel arithmetic, for the reason
+ * `visionset.inference.masks` lives above the segmentation adapter rather than
+ * inside it: it runs once for every model there will ever be.
+ *
+ * So `RawSegmentation` is *structurally* assignable to this type and neither
+ * package imports the other. A host composes them.
+ */
+export interface BinaryMask {
+  readonly width: number;
+  readonly height: number;
+  /**
+   * Row-major, `width * height` bytes, each **exactly 0 or 1**.
+   *
+   * Not "any non-zero is lit": the authoritative Python reads a row through
+   * `index(True)` and `index(False)`, which match the bytes 1 and 0 and nothing
+   * else, so a 255 already breaks the scan there. The contract is the contract
+   * rather than a convention this side is free to widen.
+   */
+  readonly mask: Uint8Array;
+}
+
+/**
+ * One connected piece of a mask, cropped to its own extent.
+ *
+ * `x` and `y` are where the crop sits in the asset, and every coordinate finally
+ * emitted has them added back. Cropped rather than carried at full size, which
+ * is what keeps a plural answer affordable: a 4K mask is eight million bytes,
+ * and one of those per piece would cost more than the forward pass.
+ */
+export interface Piece {
+  readonly x: number;
+  readonly y: number;
+  readonly mask: BinaryMask;
+}
+
+/**
+ * Python's `round()`: halves go to the even neighbour.
+ *
+ * `Math.round` takes them upward, and answers `-0` for `-0.5`. The difference
+ * decides which component a click at a half-pixel coordinate selects, and every
+ * test written at integer coordinates passes under either — which is exactly
+ * what makes it worth a function with a name.
+ */
+export function pythonRound(value: number): number {
+  const low = Math.floor(value);
+  const rest = value - low;
+  if (rest > 0.5) return low + 1;
+  if (rest < 0.5) return low;
+  return low % 2 === 0 ? low : low + 1;
+}
diff --git a/frontend/annotator/src/core/mask/closing.test.ts b/frontend/annotator/src/core/mask/closing.test.ts
new file mode 100644
index 00000000..53af99d8
--- /dev/null
+++ b/frontend/annotator/src/core/mask/closing.test.ts
@@ -0,0 +1,149 @@
+import { describe, expect, it } from "vitest";
+
+import { closingRadius, filled } from "./closing";
+import { bboxFrom, maskOf, runs } from "./runs";
+import type { Run } from "./runs";
+
+const lit = (mask: { width: number; mask: Uint8Array }): number =>
+  mask.mask.reduce((total: number, byte) => total + (byte === 0 ? 0 : 1), 0);
+
+/** A solid square with a one-row bite `depth` pixels deep, cut in from the right. */
+const notched = (depth: number, size = 64) => {
+  const runsOf: Run[] = [];
+  for (let y = 0; y < size; y += 1) {
+    if (y === Math.floor(size / 2) && depth > 0) runsOf.push([y, 0, size - depth - 1]);
+    else runsOf.push([y, 0, size - 1]);
+  }
+  return maskOf(size, size, runsOf);
+};
+
+/**
+ * A 64x64 square with an 8x8 bite out of its right edge.
+ *
+ * Wide in *both* directions, which is what makes it a bay rather than a notch:
+ * a one-row bite closes at radius 1 however deep it is, because the dimension
+ * the close has to bridge is its height.
+ */
+const bayed = () => {
+  const runsOf: Run[] = [];
+  for (let y = 0; y < 64; y += 1) {
+    if (y >= 28 && y < 36) runsOf.push([y, 0, 55]);
+    else runsOf.push([y, 0, 63]);
+  }
+  return maskOf(64, 64, runsOf);
+};
+
+/**
+ * A 64x64 square with a 2-px notch, inside an 80x80 canvas.
+ *
+ * Framed so background survives the close: `notched(2)` is the whole canvas but
+ * for two pixels, and closing it leaves nothing unlit at all.
+ */
+const framedNotch = () => {
+  const runsOf: Run[] = [];
+  for (let y = 8; y < 72; y += 1) {
+    if (y === 40) runsOf.push([y, 8, 69]);
+    else runsOf.push([y, 8, 71]);
+  }
+  return maskOf(80, 80, runsOf);
+};
+
+/** A 64x64 square with a centred square hole `hole` pixels on a side. */
+const holed = (hole: number, size = 64) => {
+  const low = Math.floor(size / 2) - Math.floor(hole / 2);
+  const runsOf: Run[] = [];
+  for (let y = 0; y < size; y += 1) {
+    if (y >= low && y < low + hole) {
+      runsOf.push([y, 0, low - 1], [y, low + hole, size - 1]);
+    } else {
+      runsOf.push([y, 0, size - 1]);
+    }
+  }
+  return maskOf(size, size, runsOf);
+};
+
+/** A `size`x`size` solid square, placed at the origin of a `canvas`x`canvas` frame. */
+const solidSquareIn = (canvas: number, size: number) =>
+  maskOf(canvas, canvas, Array.from({ length: size }, (_, y): Run => [y, 0, size - 1]));
+
+describe("closingRadius", () => {
+  it("scales with the piece's own area, not the frame", () => {
+    expect(closingRadius(solidSquareIn(64, 64))).toBe(closingRadius(solidSquareIn(200, 64)));
+    expect(closingRadius(notched(0, 200))).toBeGreaterThan(closingRadius(notched(0, 64)));
+  });
+
+  it("reaches nothing at all on a piece too small to have artefacts", () => {
+    expect(closingRadius(maskOf(8, 8, [[3, 3, 4]]))).toBe(0);
+  });
+
+  it("stops at the cap however large the piece", () => {
+    expect(
+      closingRadius(maskOf(800, 800, Array.from({ length: 800 }, (_, y): Run => [y, 0, 799]))),
+    ).toBe(6);
+  });
+
+  it("truncates rather than rounding, as Python's int() does", () => {
+    // 14400 lit pixels -> sqrt(28.8) / 2 = 2.6832..., which truncates to 2 and
+    // would round to 3. This is the assertion that catches a Math.round port.
+    expect(
+      closingRadius(maskOf(120, 120, Array.from({ length: 120 }, (_, y): Run => [y, 0, 119]))),
+    ).toBe(2);
+  });
+});
+
+describe("filled", () => {
+  it("hands back the very same mask when the reach works out at nothing", () => {
+    const small = maskOf(8, 8, [[3, 3, 4]]);
+    expect(filled(small)).toBe(small);
+  });
+
+  it("hands back the very same mask when the close changes nothing", () => {
+    const clean = notched(0);
+    expect(filled(clean)).toBe(clean);
+  });
+
+  it("closes a notch narrower than the reach", () => {
+    const narrow = notched(2);
+    const after = filled(narrow);
+    expect(after).not.toBe(narrow);
+    expect(lit(after)).toBeGreaterThan(lit(narrow));
+  });
+
+  it("leaves a bay wider than the reach alone", () => {
+    const wide = bayed();
+    expect(filled(wide)).toBe(wide);
+  });
+
+  it("closes a one-row bite however deep it is, because its height is the gap", () => {
+    const deep = notched(40);
+    expect(filled(deep)).not.toBe(deep);
+  });
+
+  it("fills a small enclosed hole", () => {
+    const withHole = holed(2);
+    expect(lit(withHole)).toBe(64 * 64 - 4);
+    expect(lit(filled(withHole))).toBe(64 * 64);
+  });
+
+  it("never moves the extent", () => {
+    for (const mask of [notched(2), notched(5), holed(2)]) {
+      expect(bboxFrom(filled(mask))).toEqual(bboxFrom(mask));
+    }
+  });
+
+  it("keeps the mask's own dimensions", () => {
+    const after = filled(notched(2));
+    expect(after.width).toBe(64);
+    expect(after.height).toBe(64);
+    expect(after.mask.length).toBe(64 * 64);
+  });
+
+  it("answers only 0 and 1", () => {
+    expect(new Set(filled(framedNotch()).mask)).toEqual(new Set([0, 1]));
+  });
+
+  it("leaves runs it did not need to touch exactly where they were", () => {
+    const after = filled(notched(2));
+    expect(runs(after)[0]).toEqual([0, 0, 63]);
+  });
+});
diff --git a/frontend/annotator/src/core/mask/closing.ts b/frontend/annotator/src/core/mask/closing.ts
new file mode 100644
index 00000000..7436fd4d
--- /dev/null
+++ b/frontend/annotator/src/core/mask/closing.ts
@@ -0,0 +1,149 @@
+/**
+ * Step 2 — close the small gaps in a piece, wherever they are.
+ *
+ * A morphological close: grow the shape, then shrink it back by the same amount.
+ * Anything narrower than the reach is bridged on the way out and not re-opened
+ * on the way back, and everything wider is left exactly as it was.
+ *
+ * ## Why bitsets, and why `BigInt`
+ *
+ * The two passes are whole-row shifts and ORs. Written as nested loops over
+ * pixels they would be a step per pixel per radius step, on the path somebody is
+ * waiting on after a click. `BigInt` rather than a number mask because a mask
+ * wider than 32 pixels would silently lose its top bits — and because Python's
+ * `int` and JavaScript's `BigInt` are the same arbitrary-precision
+ * two's-complement under `&`, `|`, `~`, `<<` and `>>`, which is what lets this
+ * be a port rather than a rewrite.
+ */
+
+import type { BinaryMask } from "./binaryMask";
+import { runs } from "./runs";
+
+/** The largest gap closed, as a share of the piece's own lit area. */
+export const CLOSING_REACH = 0.002;
+
+/** However large the piece, the reach stops here. */
+export const MAXIMUM_CLOSING_RADIUS = 6;
+
+/** The mask as one integer per row, offset by `pad` on every side. */
+export function bits(mask: BinaryMask, pad: number): { rows: bigint[]; width: number } {
+  const width = mask.width + 2 * pad;
+  const rows = new Array<bigint>(mask.height + 2 * pad).fill(0n);
+  for (const [y, first, last] of runs(mask)) {
+    const span = (1n << BigInt(last - first + 1)) - 1n;
+    rows[y + pad] = rows[y + pad]! | (span << BigInt(first + pad));
+  }
+  return { rows, width };
+}
+
+function joined(rows: readonly bigint[]): bigint {
+  let all = 0n;
+  for (const row of rows) all |= row;
+  return all;
+}
+
+function met(rows: readonly bigint[]): bigint {
+  let all = rows[0]!;
+  for (let index = 1; index < rows.length; index += 1) all &= rows[index]!;
+  return all;
+}
+
+/** Every lit pixel spread by `radius` in each direction — a square dilation. */
+function grown(rows: readonly bigint[], radius: number, width: number): bigint[] {
+  const frame = (1n << BigInt(width)) - 1n;
+  const across: bigint[] = [];
+  for (const row of rows) {
+    let spread = row;
+    for (let step = 1; step <= radius; step += 1) {
+      const shift = BigInt(step);
+      spread |= (row << shift) | (row >> shift);
+    }
+    across.push(spread & frame);
+  }
+  const height = rows.length;
+  const out: bigint[] = [];
+  for (let y = 0; y < height; y += 1) {
+    out.push(joined(across.slice(Math.max(0, y - radius), Math.min(height, y + radius + 1))));
+  }
+  return out;
+}
+
+/**
+ * The dual: a pixel survives only with its whole square neighbourhood lit.
+ *
+ * Outside the frame counts as unlit, which the shifts give for free — bits move
+ * off the end and zeros arrive — and which is right here because the frame was
+ * padded wide enough that nothing real sits against its edge.
+ */
+function shrunk(rows: readonly bigint[], radius: number, width: number): bigint[] {
+  const frame = (1n << BigInt(width)) - 1n;
+  const across: bigint[] = [];
+  for (const row of rows) {
+    let kept = row;
+    for (let step = 1; step <= radius; step += 1) {
+      const shift = BigInt(step);
+      kept &= (row << shift) & (row >> shift) & frame;
+    }
+    across.push(kept & frame);
+  }
+  const height = rows.length;
+  const out: bigint[] = [];
+  for (let y = 0; y < height; y += 1) {
+    out.push(
+      radius <= y && y < height - radius
+        ? met(across.slice(Math.max(0, y - radius), Math.min(height, y + radius + 1)))
+        : 0n,
+    );
+  }
+  return out;
+}
+
+/**
+ * How far to reach, for a piece of this size.
+ *
+ * A hole of area `a` needs a reach of about `sqrt(a) / 2` to be bridged, so the
+ * radius comes from the piece's own lit area rather than from a pixel count.
+ * Truncated — `Math.trunc` mirrors Python's `int()` — so the smallest shapes get
+ * no closing at all rather than one that would swallow a feature. That
+ * truncation is also what absorbs the one-ulp difference between `Math.sqrt`
+ * and Python's `** 0.5`: it never survives to change the truncated result.
+ */
+export function closingRadius(mask: BinaryMask): number {
+  let alight = 0;
+  for (const [, first, last] of runs(mask)) alight += last - first + 1;
+  return Math.min(Math.trunc(Math.sqrt(alight * CLOSING_REACH) / 2), MAXIMUM_CLOSING_RADIUS);
+}
+
+/**
+ * The piece with its narrow gaps closed.
+ *
+ * Returns the mask unchanged — the same object — when the reach works out at
+ * nothing or the shape has no gap that narrow, which is the common case and
+ * saves rebuilding a grid to say so.
+ *
+ * A close only ever adds pixels whose whole neighbourhood was already reachable,
+ * so it cannot push an edge outward and the extent does not move. That is why
+ * the box branch skips this step entirely.
+ */
+export function filled(mask: BinaryMask): BinaryMask {
+  const radius = closingRadius(mask);
+  if (radius < 1) return mask;
+  const { rows: before, width } = bits(mask, radius);
+  const after = shrunk(grown(before, radius, width), radius, width);
+  if (after.every((row, index) => row === before[index])) return mask;
+
+  const data = new Uint8Array(mask.width * mask.height);
+  for (let y = 0; y < mask.height; y += 1) {
+    // One binary rendering per row rather than a shift per pixel: reading bit
+    // `x` out of a wide BigInt costs the whole word, and a 4K row is 3840 of
+    // them. This is Python's `format(word, "0Nb")[::-1]` without the reversal.
+    const text = after[y + radius]!.toString(2);
+    const top = text.length - 1;
+    const base = y * mask.width;
+    for (let x = 0; x < mask.width; x += 1) {
+      const at = top - (x + radius);
+      data[base + x] = at >= 0 && text.charCodeAt(at) === 49 ? 1 : 0;
+    }
+  }
+  return { width: mask.width, height: mask.height, mask: data };
+}
diff --git a/frontend/annotator/src/core/mask/components.test.ts b/frontend/annotator/src/core/mask/components.test.ts
new file mode 100644
index 00000000..3b8c5e1f
--- /dev/null
+++ b/frontend/annotator/src/core/mask/components.test.ts
@@ -0,0 +1,114 @@
+import { describe, expect, it } from "vitest";
+
+import { components } from "./components";
+import { maskOf } from "./runs";
+import type { Run } from "./runs";
+
+/** A 1-px speck in the topmost row and a 36-px object below-left. */
+const speckled = () => {
+  const lit: Run[] = [[0, 9, 9]];
+  for (let y = 3; y < 9; y += 1) lit.push([y, 1, 6]);
+  return maskOf(10, 10, lit);
+};
+
+/** 25 px on the left, 16 px on the right, far apart. */
+const islands = () => {
+  const lit: Run[] = [];
+  for (let y = 2; y < 7; y += 1) lit.push([y, 1, 5]);
+  for (let y = 3; y < 7; y += 1) lit.push([y, 14, 17]);
+  return maskOf(24, 12, lit);
+};
+
+/** Two five-pixel components rooted at run indices 3 and 8 — the tie case. */
+const tied = () => {
+  const lit: Run[] = [];
+  for (const x of [0, 4, 8, 12, 16, 20, 24]) lit.push([0, x, x]);
+  lit.push([1, 0, 0], [1, 2, 2]);
+  for (const y of [1, 2, 3, 4]) lit.push([y, 12, 12]);
+  for (const y of [2, 3, 4, 5]) lit.push([y, 2, 2]);
+  return maskOf(30, 7, lit);
+};
+
+describe("components", () => {
+  it("has nothing to say about an empty mask", () => {
+    expect(components(maskOf(8, 8, []))).toEqual([]);
+  });
+
+  it("drops a speck before anything else looks at the mask", () => {
+    const pieces = components(speckled());
+    expect(pieces).toHaveLength(1);
+    expect(pieces[0]!.x).toBe(1);
+    expect(pieces[0]!.y).toBe(3);
+  });
+
+  it("leads with the piece under the click, not the topmost-leftmost one", () => {
+    expect(components(speckled(), [[3, 5]])[0]!.x).toBe(1);
+  });
+
+  it("leads with the pointed piece even when it is not the biggest", () => {
+    expect(components(islands(), [[15, 4]])[0]!.x).toBe(14);
+  });
+
+  it("falls back to the nearest piece for a click that is inside none", () => {
+    expect(components(islands(), [[20, 4]])[0]!.x).toBe(14);
+  });
+
+  it("compares squared distances when square roots would tie", () => {
+    // The second pixel is geometrically closer, although the two Math.sqrt
+    // results round equal on this input. Selection must not depend on that
+    // rounding tie or on the first run's reading order.
+    const nearTie = maskOf(64, 64, [
+      [12, 11, 11],
+      [26, 44, 44],
+    ]);
+    const pieces = components(nearTie, [[27.660028289416932, 18.62279046066009]]);
+    expect([pieces[0]!.x, pieces[0]!.y]).toEqual([44, 26]);
+  });
+
+  it("prefers the largest of the pieces several points land in", () => {
+    expect(
+      components(islands(), [
+        [15, 4],
+        [3, 4],
+      ])[0]!.x,
+    ).toBe(1);
+  });
+
+  it("settles an equal-area tie by reading order, whichever way the points arrive", () => {
+    expect(components(tied(), [[12, 2], [2, 3]])[0]!.x).toBe(12);
+    expect(components(tied(), [[2, 3], [12, 2]])[0]!.x).toBe(12);
+  });
+
+  it("returns every survivor biggest-first behind the pointed one", () => {
+    const pieces = components(islands(), [[15, 4]]);
+    expect(pieces.map((piece) => piece.x)).toEqual([14, 1]);
+  });
+
+  it("joins pixels that touch only at a corner", () => {
+    const diagonal = maskOf(4, 4, [
+      [1, 1, 1],
+      [2, 2, 2],
+    ]);
+    expect(components(diagonal)).toHaveLength(1);
+  });
+
+  it("selects the component Python's rounding points at, not Math.round's", () => {
+    // A single half-pixel click cannot tell the two roundings apart: the two
+    // candidate pixels are either one run or 8-connected, so they share a
+    // component, and a rounding that lands on nothing falls back to the piece
+    // the click is nearly inside anyway. It takes a *second* point, so that the
+    // largest-wins rule has something to choose between.
+    //
+    // A is 12 px across rows 1-2; B is 4 px, well away. The first point sits at
+    // y = 2.5: Python's half-to-even reads row 2, which is A, so `under` holds
+    // both and the larger wins. Math.round reads row 3, which is empty, so
+    // `under` holds only B and the answer is B.
+    const two = maskOf(12, 8, [
+      [1, 0, 5],
+      [2, 0, 5],
+      [1, 8, 9],
+      [2, 8, 9],
+    ]);
+    expect(components(two, [[2, 2.5], [8, 1]])[0]!.x).toBe(0);
+  });
+});
diff --git a/frontend/annotator/src/core/mask/components.ts b/frontend/annotator/src/core/mask/components.ts
new file mode 100644
index 00000000..0386e698
--- /dev/null
+++ b/frontend/annotator/src/core/mask/components.ts
@@ -0,0 +1,201 @@
+/**
+ * Step 1 — which pieces of a mask are worth turning into shapes.
+ *
+ * Union-find over *runs* rather than a flood fill over pixels, and the ordering
+ * at the end is the whole difference between the two geometries: a polygon
+ * takes the head of this list because a click asks about one object, and a
+ * box takes the union of all of it because a mask arriving in several pieces
+ * is nearly always one object seen around an occlusion.
+ */
+
+import type { Point } from "../types";
+import { pythonRound, type BinaryMask, type Piece } from "./binaryMask";
+import { runs, type Run } from "./runs";
+
+/**
+ * How big a piece has to be, against the biggest one, to survive the noise
+ * filter. Relative to the largest piece rather than to the frame, so it means
+ * the same thing on a mask covering everything and a mask covering a corner.
+ */
+export const MINIMUM_FRAGMENT_SHARE = 0.05;
+
+/** Do two runs on neighbouring rows touch, counting diagonals? */
+function adjacent(one: Run, other: Run): boolean {
+  return one[1] <= other[2] + 1 && other[1] <= one[2] + 1;
+}
+
+/** One label per run: which piece it belongs to, 8-connected. */
+function labelled(found: readonly Run[]): number[] {
+  const parent = found.map((_, index) => index);
+  const root = (index: number): number => {
+    let at = index;
+    while (parent[at] !== at) {
+      parent[at] = parent[parent[at]!]!;
+      at = parent[at]!;
+    }
+    return at;
+  };
+  const join = (left: number, right: number): void => {
+    const one = root(left);
+    const other = root(right);
+    // The earlier run wins, so a label is always its piece's first run.
+    if (one !== other) parent[Math.max(one, other)] = Math.min(one, other);
+  };
+
+  const rows = new Map<number, number[]>();
+  found.forEach((run, index) => {
+    const here = rows.get(run[0]);
+    if (here) here.push(index);
+    else rows.set(run[0], [index]);
+  });
+  for (const [y, here] of rows) {
+    const above = rows.get(y - 1);
+    if (!above) continue;
+    let mine = 0;
+    let prior = 0;
+    while (mine < here.length && prior < above.length) {
+      const ours = found[here[mine]!]!;
+      const theirs = found[above[prior]!]!;
+      if (adjacent(ours, theirs)) join(here[mine]!, above[prior]!);
+      if (ours[2] < theirs[2]) mine += 1;
+      else prior += 1;
+    }
+  }
+  return found.map((_, index) => root(index));
+}
+
+/** Squared distance from a point to a run, which is a horizontal segment of pixels. */
+function squaredGap(point: Point, run: Run): number {
+  const [x, y] = point;
+  const [row, first, last] = run;
+  const across = Math.max(0, first - x, x - last);
+  return across * across + (row - y) * (row - y);
+}
+
+/** Lit pixels per label, summed off the runs rather than counted. */
+function areas(found: readonly Run[], labels: readonly number[]): Map<number, number> {
+  const size = new Map<number, number>();
+  found.forEach((run, index) => {
+    const label = labels[index]!;
+    size.set(label, (size.get(label) ?? 0) + run[2] - run[1] + 1);
+  });
+  return size;
+}
+
+/**
+ * The label of the piece the prompt asks about.
+ *
+ * A point inside a piece picks that piece; several points inside several pieces
+ * pick the largest of them, and two of the same size pick the lowest label —
+ * the piece whose earliest run comes first, which is the `(-size, label)` order
+ * applied to the rest of the list below. A point inside none of them picks the
+ * piece nearest the point. Without any point the topmost-leftmost piece wins.
+ * Negatives never select: they say what the shape is not, and a piece is chosen
+ * before its shape is known.
+ */
+function pointedAt(
+  found: readonly Run[],
+  labels: readonly number[],
+  size: ReadonlyMap<number, number>,
+  at: readonly Point[],
+): number {
+  if (at.length === 0) return labels[0]!;
+
+  const under = new Set<number>();
+  for (const point of at) {
+    const x = pythonRound(point[0]);
+    const y = pythonRound(point[1]);
+    found.forEach((run, index) => {
+      if (run[0] === y && run[1] <= x && x <= run[2]) under.add(labels[index]!);
+    });
+  }
+  if (under.size > 0) {
+    let best = -1;
+    for (const label of under) {
+      if (best < 0) {
+        best = label;
+        continue;
+      }
+      const ours = size.get(label)!;
+      const theirs = size.get(best)!;
+      if (ours > theirs || (ours === theirs && label < best)) best = label;
+    }
+    return best;
+  }
+
+  let nearest = 0;
+  let closest = Infinity;
+  found.forEach((run, index) => {
+    let ours = Infinity;
+    for (const point of at) {
+      const found_ = squaredGap(point, run);
+      if (found_ < ours) ours = found_;
+    }
+    if (ours < closest) {
+      closest = ours;
+      nearest = index;
+    }
+  });
+  return labels[nearest]!;
+}
+
+/** That label's runs, painted into a mask the size of their own extent. */
+function cropped(label: number, found: readonly Run[], labels: readonly number[]): Piece {
+  const mine = found.filter((_, index) => labels[index] === label);
+  const top = mine[0]![0];
+  const bottom = mine[mine.length - 1]![0];
+  let left = Infinity;
+  let right = -Infinity;
+  for (const [, first, last] of mine) {
+    if (first < left) left = first;
+    if (last > right) right = last;
+  }
+  const width = right - left + 1;
+  const height = bottom - top + 1;
+  const data = new Uint8Array(width * height);
+  for (const [row, first, last] of mine) {
+    const base = (row - top) * width;
+    for (let x = first - left; x <= last - left; x += 1) data[base + x] = 1;
+  }
+  return { x: left, y: top, mask: { width, height, mask: data } };
+}
+
+/**
+ * The pieces of the mask worth turning into shapes.
+ *
+ * Everything below {@link MINIMUM_FRAGMENT_SHARE} of the largest piece is
+ * dropped as noise, first and unconditionally. What survives is ordered with the
+ * piece the prompt points at at the head and the rest biggest-first behind it,
+ * ties by the lowest label, so the answer is stable for a given mask.
+ *
+ * An empty mask answers with no pieces, which is an ordinary answer and not an
+ * error — the click landed on sky.
+ */
+export function components(mask: BinaryMask, at: readonly Point[] = []): Piece[] {
+  const found = runs(mask);
+  if (found.length === 0) return [];
+  const labels = labelled(found);
+  const size = areas(found, labels);
+
+  let largest = 0;
+  for (const area of size.values()) if (area > largest) largest = area;
+  const floor = largest * MINIMUM_FRAGMENT_SHARE;
+
+  const survived = new Set<number>();
+  for (const [label, area] of size) if (area >= floor) survived.add(label);
+
+  const keptRuns: Run[] = [];
+  const keptLabels: number[] = [];
+  found.forEach((run, index) => {
+    if (survived.has(labels[index]!)) {
+      keptRuns.push(run);
+      keptLabels.push(labels[index]!);
+    }
+  });
+
+  const first = pointedAt(keptRuns, keptLabels, size, at);
+  const rest = [...survived]
+    .filter((label) => label !== first)
+    .sort((one, other) => size.get(other)! - size.get(one)! || one - other);
+  return [first, ...rest].map((label) => cropped(label, found, labels));
+}
diff --git a/frontend/annotator/src/core/mask/maskGeometry.test.ts b/frontend/annotator/src/core/mask/maskGeometry.test.ts
new file mode 100644
index 00000000..b9b72a8a
--- /dev/null
+++ b/frontend/annotator/src/core/mask/maskGeometry.test.ts
@@ -0,0 +1,195 @@
+/**
+ * The TypeScript half of the mask-to-geometry parity gate.
+ *
+ * `tests/fixtures/mask_geometry.json` is written by `visionset.inference.masks`
+ * and kept current by `tests/inference/test_mask_geometry_fixture.py`. This
+ * proves the port reproduces it — every stage, exactly — which is what lets a
+ * host run a model in the browser and get the shapes the server would have
+ * given.
+ *
+ * **Exact equality, deliberately.** A tolerance on the comparison would let a
+ * genuine divergence through: the two implementations either run the same
+ * arithmetic in the same order or they will disagree about a vertex somewhere,
+ * and "somewhere" is what a golden fixture exists to find.
+ *
+ * The stages are asserted separately from the shapes so a failure says *which*
+ * step diverged rather than only that the answers differ.
+ */
+
+import { readFileSync } from "node:fs";
+import { describe, expect, it } from "vitest";
+
+import { DEFAULT_TOLERANCE, MINIMUM_TOLERANCE } from "../geometry/simplify";
+import type { Point } from "../types";
+import { CLOSING_REACH, MAXIMUM_CLOSING_RADIUS, closingRadius, filled } from "./closing";
+import { MINIMUM_FRAGMENT_SHARE, components } from "./components";
+import { contour, outline } from "./outline";
+import { maskOf, runs, type Run } from "./runs";
+import { shapesFromMask } from "./shapes";
+import type { BinaryMask } from "./binaryMask";
+import type { GeometryType } from "../types";
+
+interface Grid {
+  readonly width: number;
+  readonly height: number;
+  readonly runs: readonly (readonly number[])[];
+}
+
+interface Cropped extends Grid {
+  readonly x: number;
+  readonly y: number;
+}
+
+interface Shaped {
+  readonly geometry: Record<string, unknown>;
+  readonly contour: readonly (readonly number[])[];
+}
+
+interface Case {
+  readonly name: string;
+  readonly mask: Grid;
+  readonly at: readonly (readonly number[])[];
+  readonly components: readonly Cropped[];
+  readonly closing_radius: number | null;
+  readonly filled: Grid | null;
+  readonly outline: readonly (readonly number[])[] | null;
+  readonly contour: readonly (readonly number[])[] | null;
+  readonly shapes: Readonly<Record<string, Readonly<Record<string, readonly Shaped[]>>>>;
+}
+
+interface Fixture {
+  readonly minimum_fragment_share: number;
+  readonly closing_reach: number;
+  readonly maximum_closing_radius: number;
+  readonly minimum_tolerance: number;
+  readonly default_tolerance: number;
+  readonly tolerances: readonly number[];
+  readonly cases: readonly Case[];
+}
+
+const FIXTURE_URL = new URL("../../../../../tests/fixtures/mask_geometry.json", import.meta.url);
+const fixture = JSON.parse(readFileSync(FIXTURE_URL, "utf8")) as Fixture;
+
+/** The fixture keys a tolerance by Python's spelling of the float: `1.0`, not `1`. */
+const keyed = (tolerance: number): string =>
+  Number.isInteger(tolerance) ? `${tolerance}.0` : String(tolerance);
+
+const rebuilt = (grid: Grid): BinaryMask =>
+  maskOf(grid.width, grid.height, grid.runs.map((run) => [run[0]!, run[1]!, run[2]!] as Run));
+
+const points = (rows: readonly (readonly number[])[]): Point[] =>
+  rows.map((row) => [row[0]!, row[1]!] as Point);
+
+const plain = (runsOf: readonly Run[]): number[][] => runsOf.map((run) => [run[0], run[1], run[2]]);
+
+describe("the constants travel rather than being restated", () => {
+  it("holds the noise floor, the reach, the cap and the tolerances where the kernel does", () => {
+    expect(MINIMUM_FRAGMENT_SHARE).toBe(fixture.minimum_fragment_share);
+    expect(CLOSING_REACH).toBe(fixture.closing_reach);
+    expect(MAXIMUM_CLOSING_RADIUS).toBe(fixture.maximum_closing_radius);
+    expect(MINIMUM_TOLERANCE).toBe(fixture.minimum_tolerance);
+    expect(DEFAULT_TOLERANCE).toBe(fixture.default_tolerance);
+  });
+});
+
+describe.each(fixture.cases)("$name", (found) => {
+  const mask = rebuilt(found.mask);
+  const at = points(found.at);
+
+  it("reconstructs the same pixels the fixture describes", () => {
+    expect(plain(runs(mask))).toEqual(found.mask.runs);
+  });
+
+  it("survives the noise filter with the same pieces, in the same order", () => {
+    const pieces = components(mask, at);
+    expect(
+      pieces.map((piece) => ({
+        x: piece.x,
+        y: piece.y,
+        width: piece.mask.width,
+        height: piece.mask.height,
+        runs: plain(runs(piece.mask)),
+      })),
+    ).toEqual(
+      found.components.map((piece) => ({
+        x: piece.x,
+        y: piece.y,
+        width: piece.width,
+        height: piece.height,
+        runs: piece.runs.map((run) => [...run]),
+      })),
+    );
+  });
+
+  it("reaches the same distance into the head piece", () => {
+    const pieces = components(mask, at);
+    expect(pieces.length === 0 ? null : closingRadius(pieces[0]!.mask)).toBe(found.closing_radius);
+  });
+
+  it("closes the head piece into the same pixels", () => {
+    const pieces = components(mask, at);
+    if (found.filled === null) {
+      expect(pieces).toHaveLength(0);
+      return;
+    }
+    const whole = filled(pieces[0]!.mask);
+    expect(whole.width).toBe(found.filled.width);
+    expect(whole.height).toBe(found.filled.height);
+    expect(plain(runs(whole))).toEqual(found.filled.runs.map((run) => [...run]));
+  });
+
+  it("traces the same boundary, corner for corner", () => {
+    const pieces = components(mask, at);
+    if (found.outline === null) {
+      expect(pieces).toHaveLength(0);
+      return;
+    }
+    expect(outline(filled(pieces[0]!.mask))).toEqual(points(found.outline));
+  });
+
+  it("reduces to the same canonical contour, point for point", () => {
+    const pieces = components(mask, at);
+    if (found.contour === null) {
+      expect(pieces).toHaveLength(0);
+      return;
+    }
+    expect(contour(filled(pieces[0]!.mask))).toEqual(points(found.contour));
+  });
+
+  describe.each(Object.keys(found.shapes))("allowed=%s", (key) => {
+    const allowed = key.split(",") as GeometryType[];
+    it.each([...fixture.tolerances])("proposes the same shapes at %s px", (tolerance) => {
+      const expected = found.shapes[key]![keyed(tolerance)];
+      expect(expected).toBeDefined();
+      const actual = shapesFromMask(mask, { allowed, tolerance, at });
+      expect(actual).toEqual(
+        expected!.map((shape) => ({
+          geometry:
+            shape.geometry["type"] === "polygon"
+              ? { type: "polygon", points: points(shape.geometry["points"] as number[][]) }
+              : shape.geometry,
+          contour: points(shape.contour),
+        })),
+      );
+    });
+  });
+});
+
+describe("the gate would notice a port that ignored its input", () => {
+  it("has a case with more than one surviving component", () => {
+    expect(fixture.cases.some((found) => found.components.length > 1)).toBe(true);
+  });
+
+  it("has a case whose click is at a half pixel", () => {
+    expect(
+      fixture.cases.some((found) =>
+        found.at.some((point) => point.some((value) => !Number.isInteger(value))),
+      ),
+    ).toBe(true);
+  });
+
+  it("has a case the closing changes and a case it leaves alone", () => {
+    const withReach = fixture.cases.filter((found) => (found.closing_radius ?? 0) > 0);
+    expect(withReach.length).toBeGreaterThan(1);
+  });
+});
diff --git a/frontend/annotator/src/core/mask/outline.test.ts b/frontend/annotator/src/core/mask/outline.test.ts
new file mode 100644
index 00000000..0846e78a
--- /dev/null
+++ b/frontend/annotator/src/core/mask/outline.test.ts
@@ -0,0 +1,133 @@
+import { describe, expect, it } from "vitest";
+
+import { MINIMUM_TOLERANCE, simplified } from "../geometry/simplify";
+import { contour, outline, smoothed } from "./outline";
+import { maskOf } from "./runs";
+import type { Run } from "./runs";
+
+const square = (size: number, at = 0) =>
+  maskOf(size + at * 2, size + at * 2, Array.from({ length: size }, (_, y): Run => [y + at, at, at + size - 1]));
+
+describe("outline", () => {
+  it("is the pixels' edges, not their centres", () => {
+    expect(outline(maskOf(3, 3, [[1, 1, 1]]))).toEqual([
+      [1, 1],
+      [2, 1],
+      [2, 2],
+      [1, 2],
+    ]);
+  });
+
+  it("walks a rectangle's four corners clockwise from its top-left", () => {
+    expect(outline(square(3))).toEqual([
+      [0, 0],
+      [1, 0],
+      [2, 0],
+      [3, 0],
+      [3, 1],
+      [3, 2],
+      [3, 3],
+      [2, 3],
+      [1, 3],
+      [0, 3],
+      [0, 2],
+      [0, 1],
+    ]);
+  });
+
+  it("makes one ring of two pixels touching only at a corner", () => {
+    // Eight corners with only seven distinct: the shared corner (2,2) is
+    // visited twice, which is precisely what the left-turn-first rule produces
+    // — it crosses onto the second pixel rather than closing round the first.
+    // Asserting the sequence pins that rule; asserting distinctness would be
+    // false, and asserting a length would not tell the two turns apart.
+    expect(outline(maskOf(4, 4, [
+      [1, 1, 1],
+      [2, 2, 2],
+    ]))).toEqual([
+      [1, 1],
+      [2, 1],
+      [2, 2],
+      [3, 2],
+      [3, 3],
+      [2, 3],
+      [2, 2],
+      [1, 2],
+    ]);
+  });
+
+  it("never reaches an enclosed hole", () => {
+    const withHole = maskOf(5, 5, [
+      [1, 1, 3],
+      [2, 1, 1],
+      [2, 3, 3],
+      [3, 1, 3],
+    ]);
+    expect(outline(withHole)).toEqual(outline(maskOf(5, 5, [
+      [1, 1, 3],
+      [2, 1, 3],
+      [3, 1, 3],
+    ])));
+  });
+
+  it("has nothing to say about an empty mask", () => {
+    expect(outline(maskOf(4, 4, []))).toEqual([]);
+  });
+});
+
+describe("smoothed", () => {
+  it("cuts every corner at a quarter and three quarters of its edges", () => {
+    expect(smoothed([
+      [0, 0],
+      [2, 0],
+      [0, 2],
+    ])).toEqual([
+      [0.5, 0],
+      [1.5, 0],
+      [1.5, 0.5],
+      [0.5, 1.5],
+      [0, 1.5],
+      [0, 0.5],
+    ]);
+  });
+
+  it("leaves anything shorter than a ring alone, as a copy", () => {
+    const pair: [number, number][] = [
+      [0, 0],
+      [1, 1],
+    ];
+    expect(smoothed(pair)).toEqual(pair);
+    expect(smoothed(pair)).not.toBe(pair);
+  });
+
+  it("never leaves the ring it was given", () => {
+    const ring = outline(square(6));
+    for (const [x, y] of smoothed(ring)) {
+      expect(x).toBeGreaterThanOrEqual(0);
+      expect(x).toBeLessThanOrEqual(6);
+      expect(y).toBeGreaterThanOrEqual(0);
+      expect(y).toBeLessThanOrEqual(6);
+    }
+  });
+});
+
+describe("contour", () => {
+  it("is the smoothed trace reduced once at the floor", () => {
+    const mask = square(12);
+    expect(contour(mask)).toEqual(simplified(smoothed(outline(mask)), MINIMUM_TOLERANCE));
+  });
+
+  it("turns a staircase into one straight edge", () => {
+    const stair = maskOf(8, 8, [
+      [1, 1, 1],
+      [2, 1, 2],
+      [3, 1, 3],
+      [4, 1, 4],
+    ]);
+    expect(contour(stair).length).toBeLessThan(outline(stair).length);
+  });
+
+  it("has nothing to say about an empty mask", () => {
+    expect(contour(maskOf(4, 4, []))).toEqual([]);
+  });
+});
diff --git a/frontend/annotator/src/core/mask/outline.ts b/frontend/annotator/src/core/mask/outline.ts
new file mode 100644
index 00000000..4a775764
--- /dev/null
+++ b/frontend/annotator/src/core/mask/outline.ts
@@ -0,0 +1,131 @@
+/**
+ * Step 3 — the canonical boundary: traced along the pixels' edges, smoothed
+ * once, reduced once at the floor.
+ *
+ * The reduction is part of the definition rather than an optimisation:
+ * Douglas-Peucker is not nested — reducing at a quarter pixel and then at five
+ * does not give what reducing once at five gives — so the editor and the kernel
+ * can only be proved to agree if both start from the same points.
+ */
+
+import { MINIMUM_TOLERANCE, simplified } from "../geometry/simplify";
+import type { Point } from "../types";
+import type { BinaryMask } from "./binaryMask";
+import { bits } from "./closing";
+import { runs } from "./runs";
+
+type Corner = readonly [number, number];
+
+/** The indices of a non-negative integer's set bits, lowest first. */
+function* setBits(value: bigint): Generator<number> {
+  if (value <= 0n) return;
+  const text = value.toString(2);
+  const top = text.length - 1;
+  for (let index = top; index >= 0; index -= 1) {
+    if (text.charCodeAt(index) === 49) yield top - index;
+  }
+}
+
+/**
+ * Which way out of a corner that has more than one, left turn first.
+ *
+ * A corner with two ways out is where two lit pixels touch only diagonally. The
+ * left turn crosses onto the other pixel and keeps the 8-connected piece one
+ * ring; the right turn would close round the first pixel alone and cut the piece
+ * in two, which is not what `components` said the piece was.
+ */
+function turned(options: Corner[], at: Corner, heading: Corner): Corner {
+  const left: Corner = [heading[1], -heading[0]];
+  const right: Corner = [-heading[1], heading[0]];
+  for (const wanted of [left, heading, right]) {
+    for (let index = 0; index < options.length; index += 1) {
+      const candidate = options[index]!;
+      if (candidate[0] - at[0] === wanted[0] && candidate[1] - at[1] === wanted[1]) {
+        return options.splice(index, 1)[0]!;
+      }
+    }
+  }
+  throw new Error("a corner's ways out are its own edges");
+}
+
+/**
+ * The boundary of the piece this mask holds, along the pixels' edges.
+ *
+ * Vertices sit at pixel corners, so a lone lit pixel comes back as its unit
+ * square. Clockwise, starting at the top-left corner of the topmost-leftmost lit
+ * pixel — a corner with exactly one way in and one way out, which is what lets
+ * the walk stop on reaching it again.
+ *
+ * Only boundary pixels are ever visited, so the walk is linear in the perimeter
+ * rather than in the area. Holes have rings of their own and it never reaches
+ * them.
+ */
+export function outline(mask: BinaryMask): Point[] {
+  const found = runs(mask);
+  if (found.length === 0) return [];
+  const { rows } = bits(mask, 0);
+  const height = rows.length;
+  const stride = mask.width + 2;
+  const edges = new Map<number, Corner[]>();
+  const add = (sx: number, sy: number, ex: number, ey: number): void => {
+    const at = sy * stride + sx;
+    const here = edges.get(at);
+    if (here) here.push([ex, ey]);
+    else edges.set(at, [[ex, ey]]);
+  };
+
+  for (let y = 0; y < height; y += 1) {
+    const row = rows[y]!;
+    if (row === 0n) continue;
+    const above = y > 0 ? rows[y - 1]! : 0n;
+    const below = y + 1 < height ? rows[y + 1]! : 0n;
+    for (const x of setBits(row & ~above)) add(x, y, x + 1, y);
+    for (const x of setBits(row & ~(row >> 1n))) add(x + 1, y, x + 1, y + 1);
+    for (const x of setBits(row & ~below)) add(x + 1, y + 1, x, y + 1);
+    for (const x of setBits(row & ~(row << 1n))) add(x, y + 1, x, y);
+  }
+
+  const start: Corner = [found[0]![1], found[0]![0]];
+  const ring: Corner[] = [start];
+  let current = start;
+  let heading: Corner = [1, 0];
+  for (;;) {
+    const options = edges.get(current[1] * stride + current[0]);
+    if (options === undefined || options.length === 0) {
+      throw new Error("the outline walk left the piece it started in");
+    }
+    const following = options.length === 1 ? options.pop()! : turned(options, current, heading);
+    heading = [following[0] - current[0], following[1] - current[1]];
+    if (following[0] === start[0] && following[1] === start[1]) break;
+    ring.push(following);
+    current = following;
+  }
+  return ring.map(([x, y]): Point => [x, y]);
+}
+
+/**
+ * One pass of corner cutting over a closed ring.
+ *
+ * Every edge is replaced by the two points a quarter and three quarters of the
+ * way along it. On the unit-edge ring {@link outline} produces, that turns a
+ * staircase into a straight or gently curved line while moving no corner by more
+ * than half a pixel. Run on a ring whose straight runs had already been merged
+ * it would round the real corners of a rectangle, which is why it comes before
+ * any reduction.
+ */
+export function smoothed(ring: readonly Point[]): Point[] {
+  if (ring.length < 3) return [...ring];
+  const out: Point[] = [];
+  for (let index = 0; index < ring.length; index += 1) {
+    const [px, py] = ring[index]!;
+    const [qx, qy] = ring[(index + 1) % ring.length]!;
+    out.push([0.75 * px + 0.25 * qx, 0.75 * py + 0.25 * qy]);
+    out.push([0.25 * px + 0.75 * qx, 0.25 * py + 0.75 * qy]);
+  }
+  return out;
+}
+
+/** The canonical boundary: traced, smoothed, reduced once at the floor. */
+export function contour(mask: BinaryMask): Point[] {
+  return simplified(smoothed(outline(mask)), MINIMUM_TOLERANCE);
+}
diff --git a/frontend/annotator/src/core/mask/runs.test.ts b/frontend/annotator/src/core/mask/runs.test.ts
new file mode 100644
index 00000000..217262f2
--- /dev/null
+++ b/frontend/annotator/src/core/mask/runs.test.ts
@@ -0,0 +1,98 @@
+import { describe, expect, it } from "vitest";
+
+import { bboxFrom, maskOf, runs, spans } from "./runs";
+
+describe("runs", () => {
+  it("finds every maximal run in reading order", () => {
+    const mask = maskOf(6, 3, [
+      [0, 1, 2],
+      [0, 4, 5],
+      [2, 0, 0],
+    ]);
+    expect(runs(mask)).toEqual([
+      [0, 1, 2],
+      [0, 4, 5],
+      [2, 0, 0],
+    ]);
+  });
+
+  it("ends a run at the row's edge when the row ends lit", () => {
+    expect(runs(maskOf(4, 1, [[0, 2, 3]]))).toEqual([[0, 2, 3]]);
+  });
+
+  it("has nothing to say about an empty mask", () => {
+    expect(runs(maskOf(4, 4, []))).toEqual([]);
+  });
+});
+
+describe("spans", () => {
+  it("gives one entry per row, from its first lit pixel to its last", () => {
+    const mask = maskOf(6, 2, [
+      [0, 1, 1],
+      [0, 4, 5],
+      [1, 3, 3],
+    ]);
+    expect(spans(mask)).toEqual([
+      [0, 1, 5],
+      [1, 3, 3],
+    ]);
+  });
+});
+
+describe("bboxFrom", () => {
+  it("is the pixels' outer edge, so one lit pixel is one unit across", () => {
+    expect(bboxFrom(maskOf(5, 5, [[2, 3, 3]]))).toEqual({
+      type: "bbox",
+      x: 3,
+      y: 2,
+      width: 1,
+      height: 1,
+    });
+  });
+
+  it("spans every row it was given", () => {
+    const mask = maskOf(10, 10, [
+      [2, 1, 4],
+      [3, 6, 8],
+    ]);
+    expect(bboxFrom(mask)).toEqual({ type: "bbox", x: 1, y: 2, width: 8, height: 2 });
+  });
+
+  it("answers nothing for an empty mask rather than raising", () => {
+    expect(bboxFrom(maskOf(4, 4, []))).toBeNull();
+  });
+});
+
+describe("contract: Python index(True)/index(False) asymmetry", () => {
+  // runs() uses index(True) to find start (advances while byte !== 1) and
+  // index(False) to find end (advances while byte !== 0). This asymmetry
+  // means out-of-contract bytes (e.g. 255) are treated differently: they
+  // never start a run, but they extend a run. spans() uses index(True) for
+  // both directions, so it only counts 1 as lit. These tests pin that
+  // distinction — without them, the next reader will "simplify" back.
+
+  it("runs(): 255 inside a run extends the run", () => {
+    const mask = { width: 5, height: 1, mask: new Uint8Array([0, 1, 255, 1, 0]) };
+    expect(runs(mask)).toEqual([[0, 1, 3]]);
+  });
+
+  it("runs(): 255 alone never starts a run", () => {
+    const mask = { width: 2, height: 1, mask: new Uint8Array([255, 0]) };
+    expect(runs(mask)).toEqual([]);
+  });
+
+  it("runs(): run extends through 255 to row end", () => {
+    const mask = { width: 3, height: 1, mask: new Uint8Array([1, 1, 255]) };
+    expect(runs(mask)).toEqual([[0, 0, 2]]);
+  });
+
+  it("spans(): stops at the last actual 1, ignoring trailing 255", () => {
+    const mask = { width: 3, height: 1, mask: new Uint8Array([1, 1, 255]) };
+    expect(spans(mask)).toEqual([[0, 0, 1]]);
+  });
+
+  it("spans(): 255 alone is not a row", () => {
+    const mask = { width: 2, height: 1, mask: new Uint8Array([255, 0]) };
+    expect(spans(mask)).toEqual([]);
+  });
+});
diff --git a/frontend/annotator/src/core/mask/runs.ts b/frontend/annotator/src/core/mask/runs.ts
new file mode 100644
index 00000000..e8201df5
--- /dev/null
+++ b/frontend/annotator/src/core/mask/runs.ts
@@ -0,0 +1,97 @@
+/**
+ * A mask read as runs rather than pixels.
+ *
+ * Runs are what make every step after this one affordable on the interactive
+ * path: a megapixel mask of one clean object is a few hundred runs, and the
+ * loops below go round once per colour change in a row rather than once per
+ * pixel. This is the authoritative Python's own choice, kept — see
+ * `src/visionset/inference/masks.py`.
+ */
+
+import type { BinaryMask } from "./binaryMask";
+import type { BboxGeometry } from "../types";
+
+/** One maximal run of lit pixels: `[row, first, last]`, inclusive at both ends. */
+export type Run = readonly [y: number, first: number, last: number];
+
+/** Paint lit runs into a mask — the fixture's representation, and the tests'. */
+export function maskOf(width: number, height: number, lit: readonly Run[]): BinaryMask {
+  const data = new Uint8Array(width * height);
+  for (const [y, first, last] of lit) {
+    const base = y * width;
+    for (let x = first; x <= last; x += 1) data[base + x] = 1;
+  }
+  return { width, height, mask: data };
+}
+
+/** Every maximal run of lit pixels, in reading order. */
+export function runs(mask: BinaryMask): Run[] {
+  const found: Run[] = [];
+  const { width, height } = mask;
+  const data = mask.mask;
+  for (let y = 0; y < height; y += 1) {
+    const base = y * width;
+    let at = 0;
+    while (at < width) {
+      let first = at;
+      while (first < width && data[base + first] !== 1) first += 1;
+      if (first === width) break;
+      let cursor = first;
+      while (cursor < width && data[base + cursor] !== 0) cursor += 1;
+      const last = cursor - 1;
+      found.push([y, first, last]);
+      // +2 rather than +1: the pixel that ended the run is known unlit.
+      at = last + 2;
+    }
+  }
+  return found;
+}
+
+/** `[row, first, last]` for every row holding anything — the row's whole extent. */
+export function spans(mask: BinaryMask): Run[] {
+  const found: Run[] = [];
+  const { width, height } = mask;
+  const data = mask.mask;
+  for (let y = 0; y < height; y += 1) {
+    const base = y * width;
+    let first = -1;
+    for (let x = 0; x < width; x += 1) {
+      if (data[base + x] === 1) {
+        first = x;
+        break;
+      }
+    }
+    if (first < 0) continue;
+    let last = first;
+    for (let x = width - 1; x > first; x -= 1) {
+      if (data[base + x] === 1) {
+        last = x;
+        break;
+      }
+    }
+    found.push([y, first, last]);
+  }
+  return found;
+}
+
+/**
+ * The mask's extent, or `null` if it holds nothing.
+ *
+ * `null` rather than a throw: an empty mask is an ordinary answer from a model
+ * asked about an empty patch of sky, and the caller turns it into "no
+ * suggestion". The box is the pixels' outer edge, so a single lit pixel is one
+ * wide and one tall rather than zero.
+ */
+export function bboxFrom(mask: BinaryMask): BboxGeometry | null {
+  const rows = spans(mask);
+  if (rows.length === 0) return null;
+  const top = rows[0]![0];
+  const bottom = rows[rows.length - 1]![0];
+  let left = Infinity;
+  let right = -Infinity;
+  for (const [, first, last] of rows) {
+    if (first < left) left = first;
+    if (last > right) right = last;
+  }
+  return { type: "bbox", x: left, y: top, width: right - left + 1, height: bottom - top + 1 };
+}
diff --git a/frontend/annotator/src/core/mask/shapes.test.ts b/frontend/annotator/src/core/mask/shapes.test.ts
new file mode 100644
index 00000000..acd60d0e
--- /dev/null
+++ b/frontend/annotator/src/core/mask/shapes.test.ts
@@ -0,0 +1,83 @@
+import { describe, expect, it } from "vitest";
+
+import { maskOf } from "./runs";
+import type { Run } from "./runs";
+import { shapesFromMask } from "./shapes";
+
+/** 25 px on the left, 16 px on the right, far apart. */
+const islands = () => {
+  const lit: Run[] = [];
+  for (let y = 2; y < 7; y += 1) lit.push([y, 1, 5]);
+  for (let y = 3; y < 7; y += 1) lit.push([y, 14, 17]);
+  return maskOf(24, 12, lit);
+};
+
+const square = (size: number, at: number) =>
+  maskOf(size + at * 2, size + at * 2, Array.from({ length: size }, (_, y): Run => [y + at, at, at + size - 1]));
+
+describe("shapesFromMask", () => {
+  it("prefers a polygon wherever the class admits one", () => {
+    const shaped = shapesFromMask(square(12, 4), { allowed: ["bbox", "polygon"] });
+    expect(shaped[0]!.geometry.type).toBe("polygon");
+  });
+
+  it("does not let the caller's ordering decide the kind", () => {
+    const one = shapesFromMask(square(12, 4), { allowed: ["bbox", "polygon"] });
+    const other = shapesFromMask(square(12, 4), { allowed: ["polygon", "bbox"] });
+    expect(one).toEqual(other);
+  });
+
+  it("gives a box where that is all the class holds", () => {
+    const shaped = shapesFromMask(square(12, 4), { allowed: ["bbox"] });
+    expect(shaped[0]!.geometry).toEqual({ type: "bbox", x: 4, y: 4, width: 12, height: 12 });
+  });
+
+  it("offers nothing to a class admitting neither kind", () => {
+    expect(shapesFromMask(square(12, 4), { allowed: ["polyline", "keypoints"] })).toEqual([]);
+  });
+
+  it("carries the contour a polygon was reduced from, and nothing for a box", () => {
+    const polygon = shapesFromMask(square(12, 4), { allowed: ["polygon"] });
+    expect(polygon[0]!.contour.length).toBeGreaterThan(0);
+    const box = shapesFromMask(square(12, 4), { allowed: ["bbox"] });
+    expect(box[0]!.contour).toEqual([]);
+  });
+
+  it("unions every surviving piece for a box and traces only one for a polygon", () => {
+    const box = shapesFromMask(islands(), { allowed: ["bbox"], at: [[3, 4]] });
+    expect(box[0]!.geometry).toEqual({ type: "bbox", x: 1, y: 2, width: 17, height: 5 });
+    const polygon = shapesFromMask(islands(), { allowed: ["polygon"], at: [[3, 4]] });
+    const xs = (polygon[0]!.geometry as { points: readonly (readonly number[])[] }).points.map((p) => p[0]!);
+    expect(Math.max(...xs)).toBeLessThan(14);
+  });
+
+  it("does not move a box when the tolerance moves", () => {
+    const shapes = [0.25, 0.5, 1, 2, 4, 8, 16].map((tolerance) =>
+      shapesFromMask(islands(), { allowed: ["bbox"], tolerance, at: [[3, 4]] }),
+    );
+    for (const shaped of shapes) expect(shaped).toEqual(shapes[0]);
+  });
+
+  it("proposes nothing for a mask with nothing in it", () => {
+    expect(shapesFromMask(maskOf(8, 8, []), { allowed: ["polygon"] })).toEqual([]);
+    expect(shapesFromMask(maskOf(8, 8, []), { allowed: ["bbox"] })).toEqual([]);
+  });
+
+  it("refuses a piece too thin to have three distinct corners at a coarse tolerance", () => {
+    // 16 wide and one tall. The length matters: a 36-wide strip still keeps a
+    // vertex at this tolerance, because its cut-open ring deviates further from
+    // the closing chord than 16 px. Verified against the real Python.
+    const thin = maskOf(20, 8, [[3, 2, 17]]);
+    expect(shapesFromMask(thin, { allowed: ["polygon"], tolerance: 16 })).toEqual([]);
+  });
+
+  it("rejects a mask whose byte count does not match width times height", () => {
+    const malformed = { width: 4, height: 4, mask: new Uint8Array(8) };
+    expect(() => shapesFromMask(malformed, { allowed: ["bbox", "polygon"] })).toThrow(Error);
+  });
+
+  it("still answers normally for a well-formed mask", () => {
+    const shaped = shapesFromMask(square(12, 4), { allowed: ["bbox", "polygon"] });
+    expect(shaped[0]!.geometry.type).toBe("polygon");
+  });
+});
diff --git a/frontend/annotator/src/core/mask/shapes.ts b/frontend/annotator/src/core/mask/shapes.ts
new file mode 100644
index 00000000..46a7f494
--- /dev/null
+++ b/frontend/annotator/src/core/mask/shapes.ts
@@ -0,0 +1,144 @@
+/**
+ * A binary mask in, VisionSet geometry out.
+ *
+ * ## Where this sits
+ *
+ * Model execution stops at a binary mask. Everything below — which pieces of it
+ * count as the object, how big a gap is an artefact, where the boundary of a
+ * pixel is, whether the answer is a box or an outline — is a product decision in
+ * the asset's own pixels, and it runs once for every model there will ever be.
+ * That is why it lives here rather than in a runtime package, and why nothing
+ * here knows what produced the mask.
+ *
+ * `src/visionset/inference/masks.py` is authoritative. This reproduces it, and
+ * `tests/fixtures/mask_geometry.json` is what makes "they agree" a fact rather
+ * than a hope.
+ *
+ * ## The pipeline is fixed and its order is not configurable
+ *
+ * 1. `components` — which pieces of the mask survive the noise filter.
+ * 2. `filled` — the gaps in them narrower than a reach, closed.
+ * 3. `contour` — the boundary of what is left, traced, smoothed and reduced at
+ *    the floor.
+ * 4. `polygonAt` — that boundary, within a pixel tolerance somebody chose.
+ *
+ * The geometry branch happens after step 1: a polygon class takes steps 2–4 on
+ * the piece the prompt points at, a box class takes one extent over *every*
+ * surviving piece. **A box therefore does not depend on the tolerance.**
+ */
+
+import { DEFAULT_TOLERANCE, polygonAt } from "../geometry/simplify";
+import type { BboxGeometry, GeometryType, Point, PolygonGeometry } from "../types";
+import type { BinaryMask, Piece } from "./binaryMask";
+import { filled } from "./closing";
+import { components } from "./components";
+import { contour } from "./outline";
+import { bboxFrom } from "./runs";
+
+export type { BinaryMask } from "./binaryMask";
+
+/**
+ * One proposal: the geometry, and the contour it was reduced from.
+ *
+ * `contour` is empty for a box, because a box is not reduced from anything — it
+ * is the pieces' extent, and there is nothing a client could re-derive from a
+ * different setting.
+ */
+export interface ShapedGeometry {
+  readonly geometry: BboxGeometry | PolygonGeometry;
+  readonly contour: readonly Point[];
+}
+
+/**
+ * The kind an answer for that class will come back in.
+ *
+ * Polygon where a class admits it, because it is the more informative shape and
+ * a box can always be read off it; a box where that is all there is; nothing at
+ * all for a class that holds no shape. It reads a set, so the caller's ordering
+ * cannot change it.
+ */
+function targetKind(allowed: readonly GeometryType[]): "bbox" | "polygon" | null {
+  const kinds = new Set<GeometryType>(allowed);
+  if (kinds.has("polygon")) return "polygon";
+  if (kinds.has("bbox")) return "bbox";
+  return null;
+}
+
+/** A cropped piece's coordinates put back where the asset has them. */
+function shifted(points: readonly Point[], piece: Piece): Point[] {
+  return points.map(([x, y]): Point => [x + piece.x, y + piece.y]);
+}
+
+/** The piece's extent, in the asset's coordinates. */
+function boxed(piece: Piece): BboxGeometry | null {
+  const box = bboxFrom(piece.mask);
+  if (box === null) return null;
+  return { ...box, x: box.x + piece.x, y: box.y + piece.y };
+}
+
+/**
+ * One box over every piece, or `null` if none of them holds anything.
+ *
+ * The answer to a point prompt is *this object*, and a mask arriving in several
+ * pieces is nearly always one object seen around an occlusion — a railing across
+ * an animal, a post in front of a car. The largest piece alone would cut the
+ * object off at the occlusion, and a box per piece would annotate one thing
+ * twice.
+ */
+function unionOf(pieces: readonly Piece[]): BboxGeometry | null {
+  const boxes = pieces.map(boxed).filter((box): box is BboxGeometry => box !== null);
+  if (boxes.length === 0) return null;
+  let left = Infinity;
+  let top = Infinity;
+  let right = -Infinity;
+  let bottom = -Infinity;
+  for (const box of boxes) {
+    if (box.x < left) left = box.x;
+    if (box.y < top) top = box.y;
+    if (box.x + box.width > right) right = box.x + box.width;
+    if (box.y + box.height > bottom) bottom = box.y + box.height;
+  }
+  return { type: "bbox", x: left, y: top, width: right - left, height: bottom - top };
+}
+
+/**
+ * The whole pipeline: a mask and a class's geometries in, proposals out.
+ *
+ * `at` is the prompt's **positive** points, and only those — negatives say what
+ * the shape is not, and a piece is chosen before its shape is known.
+ *
+ * A list, though today it holds at most one: an empty list is how "nothing to
+ * propose" is already said, and accepting part of a plural proposal is work
+ * somebody is tracking.
+ */
+export function shapesFromMask(
+  mask: BinaryMask,
+  options: {
+    readonly allowed: readonly GeometryType[];
+    readonly tolerance?: number;
+    readonly at?: readonly Point[];
+  },
+): readonly ShapedGeometry[] {
+  if (mask.mask.length !== mask.width * mask.height) {
+    throw new Error(
+      `mask carries ${mask.mask.length} bytes for a ${mask.width}x${mask.height} extent`,
+    );
+  }
+  const { allowed, tolerance = DEFAULT_TOLERANCE, at = [] } = options;
+  const kind = targetKind(allowed);
+  if (kind === null) return [];
+
+  const pieces = components(mask, at);
+  if (pieces.length === 0) return [];
+
+  if (kind === "bbox") {
+    const box = unionOf(pieces);
+    return box === null ? [] : [{ geometry: box, contour: [] }];
+  }
+
+  const pointed = pieces[0]!;
+  const whole: Piece = { x: pointed.x, y: pointed.y, mask: filled(pointed.mask) };
+  const traced = shifted(contour(whole.mask), whole);
+  const points = polygonAt(traced, tolerance);
+  return points === null ? [] : [{ geometry: { type: "polygon", points }, contour: traced }];
+}
diff --git a/frontend/annotator/src/index.ts b/frontend/annotator/src/index.ts
index eaae5ca2..a0fe7e86 100644
--- a/frontend/annotator/src/index.ts
+++ b/frontend/annotator/src/index.ts
@@ -114,6 +114,12 @@ export {
   toleranceInAssetPixels,
   type Tolerances,
 } from "./core/geometry/tolerance";
+// Mask -> geometry: the kernel's pipeline, held to it by tests/fixtures/mask_geometry.json
+export {
+  shapesFromMask,
+  type BinaryMask,
+  type ShapedGeometry,
+} from "./core/mask/shapes";
 // Interaction — the state machine: states, events, effects, and the runner
 export {
   IDLE,
diff --git a/scripts/export_mask_geometry_fixtures.py b/scripts/export_mask_geometry_fixtures.py
new file mode 100644
index 00000000..8bb231af
--- /dev/null
+++ b/scripts/export_mask_geometry_fixtures.py
@@ -0,0 +1,287 @@
+"""Export mask-to-geometry golden cases to tests/fixtures/mask_geometry.json.
+
+`visionset.inference.masks` turns a segmenter's mask into this domain's
+geometry, and `@visionset/annotator` reproduces it so a host that runs a model
+in the browser gets the same shapes as one that runs it on the server. Those are
+two implementations of one pipeline, and the only thing that makes "they agree"
+a fact rather than a hope is a set of inputs both are held to.
+
+Two gates, sharing no toolchain, exactly like `simplification.json`:
+`tests/inference/test_mask_geometry_fixture.py` keeps this file matching the
+Python; `frontend/annotator/src/core/mask/maskGeometry.test.ts` keeps the
+TypeScript matching this file. The frontend CI job installs no Python and reads
+only what is committed.
+
+It carries **masks rather than contours**, because the mask is where these two
+implementations meet: `simplification.json` already owns everything from the
+contour onward, and what is unproven until this file exists is that the contour
+fed into `polygon_at` is the same contour.
+
+Masks travel as lit runs — `[y, first, last]`, inclusive — which both languages
+reconstruct into identical pixels and a reviewer can read.
+
+Written compact rather than indented, which is the one place this diverges from
+`simplification.json`. That file is 101 KB indented and fits; this one carries a
+mask, five pipeline stages and thirty-five shape answers per case, and indented it
+runs to 886 KB — past the tracked-file size ceiling in
+`tests/architecture/test_tracked_file_sizes.py`, which exists to keep media out of
+git. Nothing is lost: a golden fixture regenerated wholesale by a script is read
+through the tests that consume it, never line by line, and `sort_keys` keeps it
+deterministic either way.
+
+Usage: uv run python scripts/export_mask_geometry_fixtures.py
+"""
+
+from __future__ import annotations
+
+import json
+from pathlib import Path
+from typing import Any
+
+from visionset.inference.masks import (
+    CLOSING_REACH,
+    MAXIMUM_CLOSING_RADIUS,
+    MINIMUM_FRAGMENT_SHARE,
+    closing_radius,
+    components,
+    contour,
+    filled,
+    outline,
+    runs,
+    shapes_from,
+)
+from visionset.kernel.domain import DEFAULT_TOLERANCE, MINIMUM_TOLERANCE, GeometryType
+
+REPO_ROOT = Path(__file__).resolve().parent.parent
+OUTPUT_PATH = "tests/fixtures/mask_geometry.json"
+
+TOLERANCES: list[float] = [0.25, 0.5, 1.0, 2.0, 4.0, 8.0, 16.0]
+
+ALLOWED: list[list[GeometryType]] = [
+    [GeometryType.BBOX],
+    [GeometryType.POLYGON],
+    # Both orders, deliberately: the kind must come from the set and not from
+    # the caller's ordering, and two keys is how the fixture can say so.
+    [GeometryType.BBOX, GeometryType.POLYGON],
+    [GeometryType.POLYGON, GeometryType.BBOX],
+    # A class holding neither: the answer is nothing, and nothing is widened.
+    [GeometryType.POLYLINE],
+]
+
+Mask = list[list[int]]
+
+
+def blank(width: int, height: int) -> Mask:
+    return [[0] * width for _ in range(height)]
+
+
+def painted(width: int, height: int, lit: list[tuple[int, int, int]]) -> Mask:
+    mask = blank(width, height)
+    for y, first, last in lit:
+        for x in range(first, last + 1):
+            mask[y][x] = 1
+    return mask
+
+
+def rect(width: int, height: int, x0: int, y0: int, x1: int, y1: int) -> Mask:
+    return painted(width, height, [(y, x0, x1) for y in range(y0, y1 + 1)])
+
+
+def disc(radius: int) -> Mask:
+    size = 2 * radius + 8
+    centre = size // 2
+    return [
+        [1 if (x - centre) ** 2 + (y - centre) ** 2 <= radius * radius else 0 for x in range(size)]
+        for y in range(size)
+    ]
+
+
+def notched(depth: int, size: int = 64) -> Mask:
+    mask = rect(size, size, 0, 0, size - 1, size - 1)
+    for x in range(size - depth, size):
+        mask[size // 2][x] = 0
+    return mask
+
+
+def holed(hole: int, size: int = 64) -> Mask:
+    mask = rect(size, size, 0, 0, size - 1, size - 1)
+    low = size // 2 - hole // 2
+    for y in range(low, low + hole):
+        for x in range(low, low + hole):
+            mask[y][x] = 0
+    return mask
+
+
+def speckled() -> Mask:
+    mask = blank(10, 10)
+    mask[0][9] = 1  # a 1-px speck owning the topmost-leftmost pixel
+    for y in range(3, 9):
+        for x in range(1, 7):
+            mask[y][x] = 1  # the real object, 36 px
+    return mask
+
+
+def islands() -> Mask:
+    mask = blank(24, 12)
+    for y in range(2, 7):
+        for x in range(1, 6):
+            mask[y][x] = 1  # 5x5 = 25
+    for y in range(3, 7):
+        for x in range(14, 18):
+            mask[y][x] = 1  # 4x4 = 16
+    return mask
+
+
+def tied() -> Mask:
+    """Two five-pixel components rooted at run indices 3 and 8."""
+    mask = blank(30, 7)
+    for x in (0, 4, 8, 12, 16, 20, 24):
+        mask[0][x] = 1
+    mask[1][0] = 1
+    mask[1][2] = 1
+    for y in (1, 2, 3, 4):
+        mask[y][12] = 1
+    for y in (2, 3, 4, 5):
+        mask[y][2] = 1
+    return mask
+
+
+def halved() -> Mask:
+    """The half-pixel rounding case, and it takes two points to be one.
+
+    A *single* half-pixel click cannot tell the two roundings apart: the two
+    candidate pixels are either one run or 8-connected, so they are the same
+    piece, and a rounding that lands on nothing falls back to the piece the
+    click is nearly inside anyway. Both spellings answer the same.
+
+    So there are two points. ``A`` is 12 px, ``B`` is 4 px and far away. The
+    first point sits at ``y = 2.5``: Python reads row 2, which is ``A``, so both
+    labels are under the prompt and the larger wins. `Math.round` reads row 3,
+    which is empty, so only ``B`` is under it and ``B`` is the answer. Two
+    different pieces, and therefore two different shapes.
+    """
+    return painted(12, 8, [(1, 0, 5), (2, 0, 5), (1, 8, 9), (2, 8, 9)])
+
+
+def bayed() -> Mask:
+    """A 64x64 square with an 8x8 bite out of its right edge.
+
+    Wide in *both* directions, which is what makes it a bay rather than a notch:
+    a one-row bite closes at radius 1 however deep it is, because the dimension
+    the close has to bridge is its height, not its depth.
+    """
+    mask = rect(64, 64, 0, 0, 63, 63)
+    for y in range(28, 36):
+        for x in range(56, 64):
+            mask[y][x] = 0
+    return mask
+
+
+def diagonal() -> Mask:
+    """Two pixels touching only at a corner: one 8-connected piece, one ring."""
+    return painted(4, 4, [(1, 1, 1), (2, 2, 2)])
+
+
+def squared_distance_near_tie() -> Mask:
+    """Two one-pixel pieces whose squared distances differ below sqrt precision."""
+    return painted(64, 64, [(12, 11, 11), (26, 44, 44)])
+
+
+CASES: list[tuple[str, Mask, list[tuple[float, float]]]] = [
+    ("empty", blank(20, 20), []),
+    ("single-pixel", painted(5, 5, [(2, 3, 3)]), []),
+    ("rectangle", rect(40, 40, 6, 6, 30, 24), []),
+    ("disc", disc(20), []),
+    ("speck-and-object", speckled(), []),
+    ("two-islands", islands(), [(3.0, 4.0)]),
+    ("point-picks-the-smaller-island", islands(), [(15.0, 4.0)]),
+    ("points-across-components", islands(), [(15.0, 4.0), (3.0, 4.0)]),
+    ("equal-area-tie", tied(), [(12.0, 2.0), (2.0, 3.0)]),
+    ("equal-area-tie-reversed", tied(), [(2.0, 3.0), (12.0, 2.0)]),
+    ("click-outside", islands(), [(20.0, 4.0)]),
+    ("half-pixel-click", halved(), [(2.0, 2.5), (8.0, 1.0)]),
+    (
+        "squared-distance-near-tie",
+        squared_distance_near_tie(),
+        [(27.660028289416932, 18.62279046066009)],
+    ),
+    ("diagonal-touch", diagonal(), []),
+    ("narrow-notch", notched(2), []),
+    ("deep-one-row-notch", notched(40), []),
+    ("wide-bay", bayed(), []),
+    # The cap, and the only case that reaches it. Below about 98,000 lit pixels
+    # the reach never grows past six, so every other case here would pass a port
+    # that dropped the cap altogether rather than merely changing it.
+    ("capped-reach", rect(320, 320, 0, 0, 319, 319), []),
+    ("enclosed-hole", holed(2), []),
+    # A polygon at the floor and refused at the ceiling. 16 wide and one tall:
+    # a longer strip keeps a vertex at 16 px however thin it is.
+    ("thin", rect(20, 8, 2, 3, 17, 3), []),
+]
+
+
+def _runs(mask: Mask) -> list[list[int]]:
+    return [list(run) for run in runs(mask)]
+
+
+def _grid(mask: Mask) -> dict[str, Any]:
+    return {"width": len(mask[0]), "height": len(mask), "runs": _runs(mask)}
+
+
+def _shaped(shape: Any) -> dict[str, Any]:
+    return {
+        "geometry": json.loads(shape.geometry.model_dump_json()),
+        "contour": [list(point) for point in shape.contour],
+    }
+
+
+def _key(allowed: list[GeometryType]) -> str:
+    return ",".join(str(kind.value) for kind in allowed)
+
+
+def _case(name: str, mask: Mask, at: list[tuple[float, float]]) -> dict[str, Any]:
+    pieces = components(mask, at=at)
+    head = pieces[0] if pieces else None
+    whole = filled(head.mask) if head is not None else None
+    return {
+        "name": name,
+        "mask": _grid(mask),
+        "at": [list(point) for point in at],
+        "components": [{"x": piece.x, "y": piece.y, **_grid(list(piece.mask))} for piece in pieces],
+        "closing_radius": None if head is None else closing_radius(head.mask),
+        "filled": None if whole is None else _grid([list(row) for row in whole]),
+        "outline": None if whole is None else [list(point) for point in outline(whole)],
+        "contour": None if whole is None else [list(point) for point in contour(whole)],
+        "shapes": {
+            _key(allowed): {
+                str(tolerance): [
+                    _shaped(shape)
+                    for shape in shapes_from(mask, allowed=allowed, tolerance=tolerance, at=at)
+                ]
+                for tolerance in TOLERANCES
+            }
+            for allowed in ALLOWED
+        },
+    }
+
+
+def build_fixture() -> dict[str, Any]:
+    return {
+        "minimum_fragment_share": MINIMUM_FRAGMENT_SHARE,
+        "closing_reach": CLOSING_REACH,
+        "maximum_closing_radius": MAXIMUM_CLOSING_RADIUS,
+        "minimum_tolerance": MINIMUM_TOLERANCE,
+        "default_tolerance": DEFAULT_TOLERANCE,
+        "tolerances": TOLERANCES,
+        "cases": [_case(name, mask, at) for name, mask, at in CASES],
+    }
+
+
+def main() -> None:
+    out = REPO_ROOT / OUTPUT_PATH
+    out.write_text(json.dumps(build_fixture(), separators=(",", ":"), sort_keys=True) + "\n")
+    print(f"wrote {out}")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/scripts/verify_npm_packages.sh b/scripts/verify_npm_packages.sh
index 6ec94008..cce4d3cb 100755
--- a/scripts/verify_npm_packages.sh
+++ b/scripts/verify_npm_packages.sh
@@ -172,6 +172,7 @@ done
 echo "no dependency resolved back into the workspace"
 
 cat > esm-check.mjs <<'JS'
+import assert from "node:assert";
 import "@visionset/annotator";
 import "@visionset/ui-core";
 // The core entrypoint on its own: plain Node has no browser globals, so an
@@ -183,8 +184,18 @@ import "@visionset/media";
 import "@visionset/browser-inference";
 import { EFFICIENT_SAM_TI } from "@visionset/browser-inference";
 if (EFFICIENT_SAM_TI.maxPoints !== 6) throw new Error("core entry lost the model constants");
+
+// Mask -> geometry: a real call, not just a resolved symbol — proves the
+// kernel's pipeline still runs correctly from the packed tarball in plain Node.
+const { shapesFromMask } = await import("@visionset/annotator");
+const mask = { width: 3, height: 3, mask: Uint8Array.from([0, 0, 0, 0, 1, 1, 0, 1, 1]) };
+const shaped = shapesFromMask(mask, { allowed: ["bbox"] });
+assert.deepEqual(shaped, [{ geometry: { type: "bbox", x: 1, y: 1, width: 2, height: 2 }, contour: [] }]);
+const outlined = shapesFromMask(mask, { allowed: ["polygon"], tolerance: 1 });
+assert.equal(outlined[0].geometry.type, "polygon");
+
 console.log(
-  "Node ESM import OK: @visionset/annotator, @visionset/ui-core, @visionset/media (core), @visionset/browser-inference (core, EFFICIENT_SAM_TI)",
+  "Node ESM import OK: @visionset/annotator (core, shapesFromMask), @visionset/ui-core, @visionset/media (core), @visionset/browser-inference (core, EFFICIENT_SAM_TI)",
 );
 JS
 node esm-check.mjs
@@ -210,7 +221,7 @@ cat > tsconfig.json <<'JSON'
 JSON
 
 cat > ts-check.ts <<'TS'
-import "@visionset/annotator";
+import { shapesFromMask, type BinaryMask, type ShapedGeometry } from "@visionset/annotator";
 import "@visionset/ui-core";
 import "@visionset/media";
 import type { MediabunnyVideoMaterializer } from "@visionset/media/mediabunny";
@@ -233,7 +244,21 @@ export type {
   PixelImage, PointPrompt, PreparedImage, PromptableSegmentationRuntime, RawSegmentation,
 };
 export type CreateEfficientSamRuntime = typeof createEfficientSamRuntime;
+
+const packedMask: BinaryMask = { width: 1, height: 1, mask: new Uint8Array([1]) };
+const packedShaped: readonly ShapedGeometry[] = shapesFromMask(packedMask, { allowed: ["bbox"] });
+void packedShaped;
+
+type Equal<Left, Right> =
+  (<Value>() => Value extends Left ? 1 : 2) extends
+  (<Value>() => Value extends Right ? 1 : 2)
+    ? true
+    : false;
+type Assert<Condition extends true> = Condition;
+type _ShapesFromMaskReturnsReadonly = Assert<
+  Equal<ReturnType<typeof shapesFromMask>, readonly ShapedGeometry[]>
+>;
 TS
 
 pnpm exec tsc -p tsconfig.json
-echo "TypeScript consumer resolved @visionset/annotator, @visionset/ui-core, @visionset/media, @visionset/media/mediabunny, @visionset/browser-inference, @visionset/browser-inference/browser and the EfficientSAM model surface under nodenext OK"
+echo "TypeScript consumer resolved @visionset/annotator (shapesFromMask, BinaryMask, ShapedGeometry), @visionset/ui-core, @visionset/media, @visionset/media/mediabunny, @visionset/browser-inference, @visionset/browser-inference/browser and the EfficientSAM model surface under nodenext OK"
diff --git a/src/visionset/inference/masks.py b/src/visionset/inference/masks.py
index e23cbf9d..026f3078 100644
--- a/src/visionset/inference/masks.py
+++ b/src/visionset/inference/masks.py
@@ -266,12 +266,17 @@ def join(left: int, right: int) -> None:
     return [root(index) for index in range(len(found))]
 
 
-def _gap(point: Point, run: tuple[int, int, int]) -> float:
-    """Distance from a point to a run, which is a horizontal segment of pixels."""
+def _squared_gap(point: Point, run: tuple[int, int, int]) -> float:
+    """Squared distance from a point to a run, which is a horizontal segment of pixels.
+
+    The square root cannot change which run is nearest, and leaving it out keeps
+    a one-ulp libm rounding difference from turning a geometric ordering into a
+    reading-order tie.
+    """
     x, y = point
     row, first, last = run
     across = max(0.0, first - x, x - last)
-    return (across * across + (row - y) * (row - y)) ** 0.5
+    return across * across + (row - y) * (row - y)
 
 
 def _areas(found: Sequence[tuple[int, int, int]], labels: Sequence[int]) -> dict[int, int]:
@@ -296,8 +301,15 @@ def _pointed_at(
     rather than of what was clicked.
 
     Three cases, in order. A point inside a piece picks that piece; several
-    points inside several pieces pick the largest of them, because two positives
-    are a caller describing one object rather than proposing two. A point inside
+    points inside several pieces pick the largest of them, because two
+    positives are a caller describing one object rather than proposing two —
+    and two of the same size pick the lowest label, which is the piece whose
+    earliest run comes first. That last clause is not a new preference: it is
+    the ``(-size, label)`` order :func:`components` already applies to every
+    piece behind the head, written where it was missing. It has to be written
+    somewhere, because the alternative was ``max`` over a *set* of labels, and a
+    set's iteration order is an interpreter's hash-table layout rather than a
+    rule a second implementation could hold to. A point inside
     none of them — the model's mask need not cover the exact pixel clicked —
     picks the piece nearest the point, which is still an answer about where the
     user pointed. Negatives never select: they say what the shape is not, and a
@@ -315,9 +327,9 @@ def _pointed_at(
         if row == round(point[1]) and first <= round(point[0]) <= last
     }
     if under:
-        return max(under, key=lambda label: size[label])
+        return min(under, key=lambda label: (-size[label], label))
     nearest = min(
-        range(len(found)), key=lambda index: min(_gap(point, found[index]) for point in at)
+        range(len(found)), key=lambda index: min(_squared_gap(point, found[index]) for point in at)
     )
     return labels[nearest]
 
diff --git a/tests/fixtures/mask_geometry.json b/tests/fixtures/mask_geometry.json
new file mode 100644
index 00000000..b05bc7b0
--- /dev/null
+++ b/tests/fixtures/mask_geometry.json
@@ -0,0 +1 @@
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diff --git a/tests/inference/test_mask_geometry_fixture.py b/tests/inference/test_mask_geometry_fixture.py
new file mode 100644
index 00000000..ebb5c5ac
--- /dev/null
+++ b/tests/inference/test_mask_geometry_fixture.py
@@ -0,0 +1,191 @@
+"""The Python half of the mask-to-geometry parity gate.
+
+`tests/fixtures/mask_geometry.json` is a committed artifact and the only thing
+carrying this pipeline across the language boundary: the `frontend` CI job
+installs no Python and reads what is in the repository.
+
+So it needs two independent links. This is the first — the fixture is the
+application's own output, and this module fails when it drifts. The second is
+`frontend/annotator/src/core/mask/maskGeometry.test.ts`.
+
+The tests after the staleness check are about the fixture rather than the code:
+a golden fixture that never reaches a branch is not a gate, so each one names a
+branch and fails if no case exercises it.
+"""
+
+from __future__ import annotations
+
+import json
+from pathlib import Path
+from typing import Any
+
+import pytest
+from scripts.export_mask_geometry_fixtures import ALLOWED, OUTPUT_PATH, TOLERANCES, build_fixture
+
+from visionset.inference.masks import (
+    CLOSING_REACH,
+    MAXIMUM_CLOSING_RADIUS,
+    MINIMUM_FRAGMENT_SHARE,
+)
+from visionset.kernel.domain import DEFAULT_TOLERANCE, MINIMUM_TOLERANCE
+
+REPO_ROOT = Path(__file__).resolve().parents[2]
+
+
+def committed() -> dict[str, Any]:
+    payload: dict[str, Any] = json.loads((REPO_ROOT / OUTPUT_PATH).read_text())
+    return payload
+
+
+def named(name: str) -> dict[str, Any]:
+    return next(case for case in committed()["cases"] if case["name"] == name)
+
+
+def test_the_committed_fixture_matches_the_application() -> None:
+    assert committed() == build_fixture(), (
+        "tests/fixtures/mask_geometry.json is stale — run "
+        "`uv run python scripts/export_mask_geometry_fixtures.py` and commit the result."
+    )
+
+
+def test_the_fixture_carries_the_constants_the_port_needs() -> None:
+    """The TypeScript reads these rather than restating them, so they have to travel."""
+    payload = committed()
+    assert payload["minimum_fragment_share"] == MINIMUM_FRAGMENT_SHARE
+    assert payload["closing_reach"] == CLOSING_REACH
+    assert payload["maximum_closing_radius"] == MAXIMUM_CLOSING_RADIUS
+    assert payload["minimum_tolerance"] == MINIMUM_TOLERANCE
+    assert payload["default_tolerance"] == DEFAULT_TOLERANCE
+    assert payload["tolerances"] == TOLERANCES
+
+
+def test_every_case_covers_every_allowed_set_and_every_tolerance() -> None:
+    keys = {",".join(str(kind.value) for kind in allowed) for allowed in ALLOWED}
+    for case in committed()["cases"]:
+        assert set(case["shapes"]) == keys
+        for shapes in case["shapes"].values():
+            assert set(shapes) == {str(tolerance) for tolerance in TOLERANCES}
+
+
+def test_a_case_has_more_than_one_surviving_component() -> None:
+    assert [case for case in committed()["cases"] if len(case["components"]) > 1]
+
+
+def test_a_case_points_at_something_other_than_the_default_head() -> None:
+    """Without one, a port that took the first component would pass the whole gate."""
+    moved = [
+        case
+        for case in committed()["cases"]
+        if case["at"]
+        and len(case["components"]) > 1
+        and case["components"][0]["x"] > case["components"][1]["x"]
+    ]
+    assert moved, "no case proves the prompt moves the head"
+
+
+def test_a_case_has_two_pointed_components_of_equal_area() -> None:
+    """The tie `_pointed_at` settles by reading order — an interpreter cannot decide it."""
+    case = named("equal-area-tie")
+    assert case["components"][0]["x"] == 12
+    assert named("equal-area-tie-reversed")["components"][0]["x"] == 12
+
+
+def _head_runs(case: dict[str, Any]) -> list[list[int]]:
+    """The head piece's own runs, before the close touched it."""
+    return list(case["components"][0]["runs"])
+
+
+def test_a_case_is_changed_by_the_closing_and_another_is_not() -> None:
+    """Both halves, or the step is only half proved.
+
+    A fixture holding no closed gap would pass a port that skipped the close
+    entirely; one holding nothing the close leaves alone would pass a port that
+    closed everything, at any radius.
+    """
+    reaching = [case for case in committed()["cases"] if case["closing_radius"]]
+    changed = [case for case in reaching if case["filled"]["runs"] != _head_runs(case)]
+    unchanged = [case for case in reaching if case["filled"]["runs"] == _head_runs(case)]
+    assert changed, "no case is changed by the close"
+    assert unchanged, "no case with a reach is left alone by the close"
+
+
+def test_a_case_clicks_at_a_half_pixel() -> None:
+    """The coordinate where Python's rounding and JavaScript's disagree."""
+    halves = [
+        case
+        for case in committed()["cases"]
+        if any(value != int(value) for point in case["at"] for value in point)
+    ]
+    assert halves, "no case tells the two roundings apart"
+
+
+def test_a_case_touches_only_diagonally() -> None:
+    case = named("diagonal-touch")
+    assert len(case["components"]) == 1, "a diagonal touch is one 8-connected piece"
+    assert len(case["outline"]) == 8, "and one ring round both unit squares"
+
+
+def test_a_case_boxes_more_than_its_polygon_traces() -> None:
+    """The branch that is the whole point of keeping the two geometries apart."""
+    wider = []
+    for case in committed()["cases"]:
+        box = case["shapes"]["bbox"]["1.0"]
+        polygon = case["shapes"]["polygon"]["1.0"]
+        if not box or not polygon:
+            continue
+        span = max(point[0] for point in polygon[0]["geometry"]["points"]) - min(
+            point[0] for point in polygon[0]["geometry"]["points"]
+        )
+        if box[0]["geometry"]["width"] > span + 1:
+            wider.append(case)
+    assert wider, "no case shows a box spanning more than its polygon"
+
+
+def test_a_case_moves_its_vertex_count_across_the_ladder() -> None:
+    moving = [
+        case
+        for case in committed()["cases"]
+        if len(
+            {
+                len(shapes[0]["geometry"]["points"])
+                for shapes in case["shapes"]["polygon"].values()
+                if shapes and shapes[0]["geometry"]["type"] == "polygon"
+            }
+        )
+        >= 3
+    ]
+    assert moving, "no case tells the tolerances apart"
+
+
+def test_a_box_is_the_same_box_at_every_tolerance() -> None:
+    for case in committed()["cases"]:
+        answers = [shapes for shapes in case["shapes"]["bbox"].values()]
+        assert all(answer == answers[0] for answer in answers), case["name"]
+
+
+def test_the_two_orderings_of_one_allowed_set_agree() -> None:
+    for case in committed()["cases"]:
+        assert case["shapes"]["bbox,polygon"] == case["shapes"]["polygon,bbox"], case["name"]
+
+
+def test_a_class_admitting_neither_kind_is_offered_nothing() -> None:
+    for case in committed()["cases"]:
+        for shapes in case["shapes"]["polyline"].values():
+            assert shapes == []
+
+
+def test_a_case_proposes_nothing_at_all() -> None:
+    empty = [
+        case
+        for case in committed()["cases"]
+        if all(not shapes for shapes in case["shapes"]["polygon"].values())
+    ]
+    assert empty, "no case is below what a shape can be"
+
+
+@pytest.mark.parametrize("case", committed()["cases"], ids=lambda c: str(c["name"]))
+def test_a_mask_is_lit_runs_inside_its_own_frame(case: dict[str, Any]) -> None:
+    grid = case["mask"]
+    for y, first, last in grid["runs"]:
+        assert 0 <= y < grid["height"]
+        assert 0 <= first <= last < grid["width"]
diff --git a/tests/inference/test_masks.py b/tests/inference/test_masks.py
index 2869b4f7..370ff760 100644
--- a/tests/inference/test_masks.py
+++ b/tests/inference/test_masks.py
@@ -163,12 +163,68 @@ def test_a_click_just_off_the_piece_falls_back_to_the_nearest_one() -> None:
     assert (pieces[0].x, pieces[0].y) == (1, 3)
 
 
+def test_nearest_piece_compares_squared_distances_without_a_square_root_tie() -> None:
+    """The smaller geometric distance wins even when its square roots round equal.
+
+    These two one-pixel pieces are separate survivors. Their squared distances
+    from the in-frame point differ by one ulp, while their square roots can
+    round equal on a platform. Comparing the squared values keeps the answer
+    the actual nearer piece rather than letting a libm tie fall back to reading
+    order.
+    """
+    mask = [[0] * 64 for _ in range(64)]
+    mask[12][11] = 1
+    mask[26][44] = 1
+
+    pieces = components(mask, at=[(27.660028289416932, 18.62279046066009)])
+
+    assert (pieces[0].x, pieces[0].y) == (44, 26)
+
+
 def test_several_points_spanning_pieces_prefer_the_largest_of_them() -> None:
     """Two positives can straddle two pieces; the bigger one is the better answer."""
     pieces = components(speckled(), at=[(9.0, 0.0), (3.0, 5.0)])
     assert (pieces[0].x, pieces[0].y) == (1, 3), "the 36-px object beats the 1-px speck"
 
 
+def tied() -> list[list[int]]:
+    """Two five-pixel components, rooted at run indices 3 and 8.
+
+    The row of specks is what puts the roots at those indices, and 3 and 8 are
+    what make the tie visible: a set holding them iterates ``[8, 3]``, because
+    ``8 & 7`` is 0 and lands in an earlier bucket, so ``max`` over the set used
+    to answer with the *later* component.
+    """
+    mask = [[0] * 30 for _ in range(7)]
+    for x in (0, 4, 8, 12, 16, 20, 24):
+        mask[0][x] = 1  # run indices 0..6
+    mask[1][0] = 1  # run index 7, joins the component rooted at 0
+    mask[1][2] = 1  # run index 8, a new component
+    for y in (1, 2, 3, 4):
+        mask[y][12] = 1  # the component rooted at run index 3
+    for y in (2, 3, 4, 5):
+        mask[y][2] = 1  # the component rooted at run index 8
+    return mask
+
+
+def test_two_equally_large_pointed_pieces_are_settled_by_reading_order() -> None:
+    """The tie the module never stated, and the one a port cannot guess.
+
+    A set's iteration order is an interpreter's hash-table layout, not a rule, so
+    the answer here is fixed the way `components` already fixes it for every
+    other piece: biggest first, ties by the lowest label — the piece whose
+    earliest run comes first.
+    """
+    pieces = components(tied(), at=[(12.0, 2.0), (2.0, 3.0)])
+    assert pieces[0].x == 12, "the earlier component leads when the areas are equal"
+
+
+def test_the_tie_is_broken_by_reading_order_whichever_way_the_points_arrive() -> None:
+    """Point order is not a tie-break either — only the pieces decide."""
+    reversed_points = components(tied(), at=[(2.0, 3.0), (12.0, 2.0)])
+    assert reversed_points[0].x == 12
+
+
 def test_without_a_point_the_noise_filter_still_answers_with_the_object() -> None:
     """Nothing outside a point prompt has an opinion — but the speck is gone first.