Rubik's Cube · mimo-v2.6-pro

Let me build the app. I'll write index.html with three.min.js script tag.

Key design: 27 cubies (or 26 visible), each a BoxGeometry with 6 materials (colored stickers on faces), plus visible cubie boundaries (black edges / slightly smaller sticker planes). Simpler: each cubie is a rounded-ish box with dark material, and stickers as slightly offset planes on outer faces with colors.

Implementation:
- cubies: THREE.Mesh of BoxGeometry(0.96) with dark material, plus sticker planes on outer faces at positions where coordinate is ±1.
- Group by layer for rotation: when a move is applied, gather cubies in that layer into a pivot THREE.Group, animate rotation around axis, on complete, apply rotation to cubie positions (snap), reset pivot.

State preservation: actual object transforms, so stickers move with cubies.

Camera controls: custom orbit with spherical coords, pointer drag on background (pointerdown on canvas not on a face? Requirement: "Dragging the background orbits the camera without turning a face." So no face turning via drag at all — dragging orbits. That's simplest: drag = orbit always. Fine.

Wheel zoom bounded: radius between e.g. 7 and 20.

Reset: restore solved cube (reset each cubie position/rotation) and camera to initial.

Scramble: queue ~20 random moves.

Move queue: moves array processed one at a time; animation ~250ms; buttons disabled while animating? Requirement says "queue or disable further moves until it completes so state is not corrupted" — I'll queue them (allow queuing) but disable buttons during scrambling maybe. Simpler: queue.

Face turning correctness: axis and direction. Define moves:
- U: rotate top layer (y=1) around +Y by -90° (clockwise looking from +Y down). Standard: U is clockwise viewed from top. Rotation about +Y by -90° (right-hand rule: positive rotation about Y is counterclockwise viewed from +Y). Clockwise = -90°. Yes.
- D: layer y=-1, clockwise viewed from bottom (-Y): rotation about +Y... viewed from -Y, clockwise is +90° about +Y? Let's think: rotation about axis a by angle θ, viewed from the tip of a looking toward origin, positive θ is counterclockwise. For D viewed from -Y looking toward +Y, clockwise means counterclockwise viewed from +Y... hmm. Viewed from -Y is the opposite orientation, so clockwise-from-below = counterclockwise-from-above = +90° about +Y... wait. Viewed from +Y: positive angle = CCW. D clockwise as viewed from -Y direction = CCW as viewed from +Y = +90° about Y. Hmm, let me double check: if you look from below (from -Y up), a rotation that appears clockwise there appears counterclockwise from above. So D = +90° about +Y axis. But conventionally D = rotation of layer y=-1 about Y by +90°? Let's just verify with a simple test later maybe. Actually easier: define each move as (axis, layer, angle) where angle for face X is -90° for clockwise as viewed from the positive side of that axis, and +90° for negative-side faces (D, L, B) — with angle about the positive axis. So:
  - U: axis Y, layer +1, angle -π/2
  - D: axis Y, layer -1, angle +π/2
  - R: axis X, layer +1, angle -π/2
  - L: axis X, layer -1, angle +π/2
  - F: axis Z, layer +1, angle -π/2
  - B: axis Z, layer -1, angle +π/2
Inverse: negate angle.

Correctness of clockwise convention doesn't really matter for the test (turn and inverse returns to start), as long as consistent. Good.

Animation: pivot group added to scene; move cubies whose rounded position component == layer into pivot via pivot.attach(cubie) (three.min.js r1xx has Object3D.attach? Yes since r103 or so). Safer to do manually: reparent preserving world transform. But pivot rotation is about origin and cubies are at integer positions, so attaching is easy: just add cubie to pivot (positions are local, and pivot is at origin with identity rotation, so local coords unchanged). On completion, set pivot rotation to target, then recompute each cubie's world position/quaternion and move back to root group... Since pivot is child of scene or cubeGroup at identity, cubie.position after rotation is rotated. Simplest: after animation completes, for each cubie: cubie.position = round(rotate(position by angle about axis)), quaternion = rotationQuat * quaternion; then move back to root group with root.attach or just root.add(cubie) and set position/quaternion directly (since root is at origin identity... but cubeGroup might be identity). I'll keep the root group at identity transform always (only camera moves).

Actually careful: while in pivot, cubie quaternion is pivot.quaternion * localQuaternion. At completion, compute world quat = pivot.quaternion * cubie.quaternion, world pos = pivot.quaternion applied to cubie.position. Then root.add(cubie) and set those, with snapping positions to integers and quaternion components to nearest values (normalize, and round tiny errors). Good.

Sticker colors: standard: U white, D yellow, F green, B blue, R red, L orange. Six distinct.

Cubie boundaries: use dark cubie body with stickers inset (sticker size 0.88 with gap) and also maybe EdgesGeometry lines. Dark body with stickers gives visible boundaries.

Lighting: ambient + directional. Materials: MeshStandardMaterial or MeshPhongMaterial (three.min.js includes both). Use MeshPhongMaterial for reliability? MeshStandardMaterial works fine. I'll use MeshLambertMaterial or Phong.

Also add subtle bevel? Keep simple: BoxGeometry(0.98,0.98,0.98) dark with stickers planes at ±0.501.

Let me check three.min.js version first.

UI: buttons U U' D D' L L' R R' F F' B B', Scramble, Reset. Status text showing move count/queue.

Camera: perspective, position via spherical (radius, theta, phi), lookAt origin. Initial radius ~9.5, phi ~ 60°... Keep cube fully visible: fov 45, radius 9.5, cube half-diagonal ~2.6, fine.

Drag orbit: pointer events on renderer.domElement; update theta -= dx*0.006, phi -= dy*0.006, clamp phi to [0.15, π-0.15].

Wheel: radius *= (1 + delta*0.001), clamp [6, 20]. preventDefault.

Also maybe click on a face to turn? Not required; skip to avoid conflicting with orbit.

Let me write the code, then run browser checks.

Animation loop uses requestAnimationFrame with clock delta; move speed: duration 220ms, ease with easeInOutQuad/cubic.

Scramble: generate 20 random moves from the 12 moves, queue them with faster animation maybe (150ms). Reset: clear queue, snap any in-flight animation to complete? Simpler: reset sets a flag; if animating, finish instantly then reset. I'll implement: reset() → clear queue, if current animating, instantly complete it (apply final state), then reset all cubies to solved positions & quaternions, reset camera.

Also disable Scramble/Reset? No, allow them anytime (reset handles in-flight).

Keyboard shortcuts optional.

Now write the file. Then run browser checks.

For the verify checks: turn F then F' and compare sticker arrangement — I can verify visually via screenshots and also programmatically by reading cube state? I can expose a debug function like window.__cubeState() returning each cubie's position and quaternion, and check solved state after F,F'. Also do camera orbit drag and reset.

Let me write it.

Details for stickers: for each cubie at (x,y,z) in {-1,0,1}^3, create body mesh, then for each face where coordinate is ±1, add sticker plane:
- +X: position (0.501,0,0), rotation.y = π/2
- -X: rotation.y = -π/2
- +Y: rotation.x = -π/2
- -Y: rotation.x = π/2
- +Z: rotation.x = 0 (facing +z)
- -Z: rotation.y = π
Colors mapped by face.

Sticker geometry PlaneGeometry(0.86,0.86). Maybe use slightly rounded look with a border — the dark body shows through as borders. Good.

Better visual: add edges lines with black color for cubie boundaries — dark body already does it.

Materials: create one material per color, shared.

Let's also add subtle shine: MeshPhongMaterial shininess 30.

Check three.min.js global build version — read first lines.

Three r128 (REVISION "128"). Good — has MeshPhongMaterial, PlaneGeometry (in r128, PlaneGeometry exists as alias? In r125+, Geometry was removed and PlaneBufferGeometry renamed... In r128, `THREE.PlaneGeometry` exists (buffer-based). Yes, since r125 PlaneGeometry = PlaneBufferGeometry). BoxGeometry fine.

Note r128 uses `outputEncoding` etc. Fine.

Now write index.html.

Note: sticker.translateZ(0.012) after setting rotation — translateZ moves along local +Z which is the sticker's outward normal. Good. But sticker is child of group g which is at integer position; the sticker offset is in local coords — fine.

Wait: sticker position set to ±0.487 then translateZ(0.012) — for +X face, rotation.y=π/2 means local +Z points along +X. So translateZ moves to x = 0.487+0.012 = 0.499, just above body half-width 0.48. Good. For -X: rotation.y=-π/2 → local +Z points along -X, translateZ moves to -0.499. Good. For +Y: rotation.x = -π/2 → +Z points to +Y. Good. -Y: rotation.x = π/2 → +Z points to -Y. Good. -Z: rotation.y = π → +Z points -Z. Good.

Edge case: in resetCube, finishMove() increments moveCount then we set moveCount = 0 after — fine.

Potential problem: finishMove uses snapQuaternion which snaps Euler to multiples of 90° — fine since all rotations are 90° multiples relative to world axes. But Euler decomposition of a quaternion that's product of 90° axis rotations is still a 90°-multiple Euler triple in XYZ order? Any rotation in the cube group (the 24 orientations) maps axes to axes, and any such rotation can be expressed as XYZ Euler with multiples of 90°? Hmm — is that always true? The rotation group of the cube (octahedral group, 24 elements) — each maps x,y,z axes to ± axes. XYZ Euler angles for such rotations: R = Rx(a)Ry(b)Rz(c) with a,b,c ∈ {0, ±90, 180}. Is every cube-symmetry rotation representable this way? The set of matrices with entries in {0,±1} and determinant +1. XYZ Euler with angles in {0,90,180,-90} gives exactly matrices of that form? Let's think: Rz(c) with c multiple of 90 gives a signed permutation matrix. Ry(b), Rx(a) too. Product of three such is a signed permutation matrix with det 1. Conversely, any signed permutation matrix with det 1 has Euler XYZ decomposition with angles that are multiples of 90? Three.js Euler.setFromRotationMatrix for XYZ: it computes via standard formulas and clamps. For rotation matrices with entries 0/±1, sy = m13... For gimbal lock case sy = -1 handled. In gimbal lock cases, it sets a = atan2(-m21, m11), b = ±π/2, c = 0. Is atan2 result a multiple of 90 for these matrices? m21, m11 ∈ {0,±1}, so atan2 gives multiples of 90 or 45? atan2(±1, ±1) = ±135° — that would be a problem! E.g. matrix from rotation about Y by 90 then... Let's test: rotation = Ry(90)*Rx(90)? Its matrix: does it have gimbal? sy = m13 (for XYZ order, three.js uses sy = sqrt(m13^2+m33^2)... let me recall three.js code:

For XYZ:
```
m11..m33
if (order === 'XYZ') {
  this._y = Math.asin(clamp(m13, -1, 1));
  if (Math.abs(m13) < 0.9999999) {
    this._x = Math.atan2(-m23, m33);
    this._z = Math.atan2(-m12, m11);
  } else {
    this._x = Math.atan2(m32, m22);
    this._z = 0;
  }
}
```
So non-gimbal case: atan2(-m23, m33) where entries are 0/±1 → atan2 of (0, ±1) = 0 or π; (±1, 0) = ±π/2. Never both nonzero magnitude 1 simultaneously? m23 and m33 both nonzero ±1 possible? In a signed permutation matrix each row/column has exactly one nonzero. So row 2: (m21,m22,m23) has one nonzero. m23 nonzero → m22 = 0. m33 is in row 3, could be nonzero along with m23. So atan2(-m23, m33) can be atan2(±1, ±1) = ±135°/±45°. Hmm, that would be wrong-ish? But wait: in the non-gimbal case |m13| < 1, so m13 = 0. Then column 3's nonzero entry is m23 or m33. If m23 nonzero then m33 = 0 → atan2(±1, 0) = ±90°. If m33 nonzero, m23 = 0 → atan2(0, ±1) = 0 or π. So fine! Similarly for z: atan2(-m12, m11): m13=0 means row 1 nonzero entry is m11 or m12. If m12 ≠ 0 then m11 = 0 → ±90. Else m12 = 0 → 0 or π. Good.

Gimbal case |m13| = 1: then row 1 has m13 = ±1, so m11 = m12 = 0, and column 3's other entries 0. x = atan2(m32, m22): row 2 and row 3 nonzeros in columns 1 or 2. m22 and m32 both nonzero? Column 2's nonzero entry is one of m12(=0)... wait m13=±1 uses column 3 and row 1. Remaining entries: rows 2,3 × columns 1,2 form a 2×2 signed permutation. So exactly one nonzero in each row and column among {m21,m22},{m31,m32}. So m22 and m32 can't both be nonzero. So atan2(m32,m22) is 0/π/±90. Good. So snapping works.

But the actual quaternion accumulates tiny float error; setFromQuaternion uses quaternion-to-matrix conversion then Euler, and the values are near 0/±1, so atan2 lands near multiples of 90 — Math.round(_e.x/HALF)*HALF snaps correctly. Good.

Also `q.setFromEuler(new THREE.Euler(...))` — fine.

Alternative simpler: instead of snapQuaternion, just round quaternion components to nearest of exact set... Euler approach is fine.

Now let's run the browser checks.

One issue: the reveal of reset button restore camera — fine.

Let me load in browser.

Cube renders nicely, solved, fully visible. But note the panel occludes part of the view? Cube is fully visible. Good. The colors look a bit washed/saturated but fine. The stickers look big without much gap — ok, boundaries visible via dark gaps.

Now verify moves: click F then F', check state. Use browser click on button at (211, 180) for F, then F' at (267,180). Then check window.__cube state.

Actually I can evaluate JS? The browser tool doesn't have an eval action; but "inspect" returns console. I can use click on buttons and then inspect. To check state programmatically I could use console output... Hmm. Maybe I can add a temporary approach: use browser click + screenshot to visually check. But better: I can run node with a headless check? No puppeteer probably. Let me just use screenshots and DOM inspection.

Actually the "inspect" action returns DOM and console. I could add a small debug line in the DOM... The status shows move count and queue. For solved check, I could make the toast show "solved" — but easier: run a check in browser via clicking, and read __cube.isSolved through... maybe browser "type" into something? No eval.

Alternative: use node with three.min.js to simulate? Not needed — but I could verify move/inverse logic mathematically in node by requiring three.min.js (it's UMD, exports). Let's do that: replicate the move logic in node and verify F then F' returns state. That's a decent correctness check. But the real rendering check in the browser matters more.

Simpler approach for browser verification: temporarily expose state in DOM? I could make the status bar show a "solved" indicator permanently — actually a nice feature: show "Solved" / "Mixed" status. That both helps user and gives me DOM-visible state for verification. Let me add that: a state indicator in #status that updates after each move (using isSolved()).

Let me add: in updateStatus, compute solved and set a #solvedVal text. But isSolved defined later... define earlier or call lazily. I'll restructure: move `cubies`-based isSolved function to a plain function used in updateStatus.

Let me implement: add third span "State: <span id=stateVal>Solved</span>".

Also verify orbit drag and reset visually via screenshots.

Edit index.html.

Also updateStatus is called in animate loop after finishMove — fine. But updateStatus is only called on moves; during in-flight animation the state indicator may lag by one frame — acceptable, it's called in finishMove.

Also isSolved for a rotated face mid-animation would say Mixed — fine.

Reload and test.

The UMD needs `globalThis` handling — it sets `t.THREE` where t = globalThis... Actually the UMD assigns to `globalThis.THREE` when `exports` is undefined — in node, `module.exports` exists so it takes the CommonJS branch and exports into `exports`, but `require` returns module.exports which the factory populates... The factory is called with `e(exports)` so `require('./three.min.js')` should return the THREE object directly. Let me fix: `const THREE = require('./three.min.js')`.

Meanwhile, DOM shows "State: Solved". Now test F then F' in browser.

The F turn animated and completed: yellow row now on top front (D face stickers moved up to F top row) and white stickers on the right column of F face (from U face). That's the correct result of an F clockwise turn viewed from front... Let's verify: F clockwise (viewed from +Z front): the front face's adjacent: U's front row → R's left column... wait. F clockwise: U front row goes to L? Let's think: F clockwise viewed from front: top row of front stays; the up face's front row moves to the right face's left column; right face's left column moves to down face's front row... hmm.

Observed: yellow (D stickers) appear on the top row of the F face — meaning the down face's front row rotated to the front face's top row? For F clockwise viewed from front (looking along -Z): the U face's front row goes to R face's left column, R's left column goes to D's front row, D's front row goes to L's right column, L's right column goes to U's front row. Hmm, then the F face's top row would be from... wait no, the front face's own stickers rotate within the face. But the observed top row of the F face is yellow — meaning D stickers moved to the front-top. That means the rotation is counter-clockwise viewed from front (D→F top? Let's redo).

Actually careful: the visible top face in the screenshot is U (white) at the top-back area and the front-top row shows yellow. Wait, the yellow strips appear at the boundary between the top face and the front face. Looking at the image: white face is at top (back-right top), green face front-left, red face right. The yellow stickers appear at the top row of the green (F) face — the row nearest the top.

Hmm, but they could also be the front row of the U face viewed... In this camera view (from +X+Y+Z region roughly, theta 0.72, phi 1.12 → camera at (r sp sin θ, r cp, r sp cos θ) = (10.4*0.9*0.66, 10.4*0.41, 10.4*0.9*0.75) ≈ (6.2, 4.3, 7.0). So we see F (+Z) on the left, R (+X) on the right, U on top. The yellow strips are in the top row of the front-left (green) face, i.e., the F face top row.

Wait — but that row is where U's front row meets F. After an F clockwise turn, F's own stickers rotate within the face; the row of the F face adjacent to U would come from L's right column, and U's front row would move onto R's left column. Observed: F face's top row is yellow (D color!) and R face's left column is white (U color!).

Hmm, that doesn't match "F clockwise". Let's think about what F clockwise does:
- F clockwise viewed from +Z (looking toward origin, i.e., -Z direction): rotation about +Z axis by -90° (as I defined, since positive about +Z is CCW viewed from +Z).
- Rotation by -90° about Z maps: +X → -Y (i.e., R side goes to D side), +Y → +X (U goes to R), -X → +Y (L goes to U), -Y → -X (D goes to L).

So U's front row → R's left column (white on R left column ✓ matches screenshot: white column on the left edge of the red face). R's left column → D's front row. D's front row (yellow) → L's right column. L's right column → U's front row.

So the F face's own top row should be L's stickers (orange), not yellow. But the screenshot shows yellow on the top row of the green face... Unless what I'm seeing as "green face" is actually showing the F face top row as yellow because the F face's top row received from... hmm.

Wait, maybe I'm misreading which face is which. Perhaps the camera at +X,+Y,+Z sees F on the left. The F face top row: after F turn, F's stickers rotate within the plane: F top row comes from F left column... no wait, F's stickers rotate with the face: the F face's top row after clockwise rotation comes from F's left column. F's left column was green. So F's top row should be green still (all green face is green). Hmm, but the F face is entirely green stickers so rotating within the face keeps it green.

So yellow appearing on the top row of the front face is impossible for F turn... unless the yellow is actually on the U face's front row (the row of the top face nearest to front). Looking again at the image: the yellow strips appear as three tiles at the boundary between the top white region and the green face. The top face shows white tiles in the back 2 rows, and yellow in the front row of the TOP face. That's consistent with D's front row → ... no. For F clockwise, U's front row goes to R. So U's front row should now contain L's right column (orange), not yellow.

Hmm. So maybe my rotation is F counter-clockwise? Let's recompute: which face's stickers land on U's front row under rotation by -90° about Z? The inverse: U front row receives from the face that maps onto +Y. Rotation by -90° about Z maps direction -X → ... rotation by angle θ=-90° about Z: v' = R v. R(-90°) maps +X → -Y, -Y → -X, -X → +Y, +Y → +X. So the stickers originally at -X (L face right column) move to +Y (U face front row). So U front row = orange (L). And D front row receives from +X (R's left column)... wait: R's left column at +X moves to -Y (D front row) ✓. And L right column receives from -Y (D front row, yellow) → yellow at L's right column.

But the screenshot shows yellow on the U face front row region and white on R's left column. White on R left column: from +Y (U front row) → +X (R left column) ✓ correct. So both observations... yellow at U front row would be wrong for -90° and right for +90°.

Hmm wait — maybe I'm misreading the image. The yellow tiles might be at the top row of the F face. In the image, the green face occupies the left. The yellow tiles: three tiles at approximately y≈255-320, x≈375-680. The green face's top row would be right below the boundary with the top face. Hmm.

Actually with the perspective, the boundary between U and F faces runs diagonally. The yellow tiles at x 375-680, y 255-320 form a parallelogram band that slants down-right — matching the F face's top row orientation? The F face's top row boundary would slant down-to-the-right? Let's see: F face is on the left, its top edge runs from upper-left (x≈375,y≈255) down to... in the first screenshot, the boundary between the white top face and green face runs from (375,255) to (540,205)-ish going up-right? Actually in the solved screenshot, the U/F boundary runs from (375, 255) up-right to (540, 207)? Look at the first screenshot: the green face top edge goes from around (375,255) to (610,320)? Hmm hard.

Let me just check deterministically: compare screenshots. In the solved image, the front (green) face occupies the region with corners approx: leftmost (375,255) top, going right-down to (610,320)? and the white face occupies the top region. In the post-F image, the yellow band is between the white top tiles and the green front tiles.

Rather than squinting, better: verify state numerically in the browser. I can't eval directly... but the DOM shows State: Mixed, and after F then F' it must show Solved (which is the key verification). Additionally I can add a temporary debug text? Actually I could verify the turn direction correctness by a DOM-visible check... but honestly the direction convention (clockwise vs counter-clockwise) doesn't affect required behavior as long as move/inverse are consistent and buttons are labeled. Still, labeling "F" as clockwise from the front is a nice-to-have; if it's inverted, the labels would be wrong.

Let me determine using node: compute where the yellow (D) stickers go under my F move. From my analysis: D front row → L right column. So no yellow should be on the top. But the screenshot shows yellow near the top. Unless the band I see is the L-ward... no.

Hmm wait — maybe the yellow band is on the TOP face's front row and it came from D? For that, the rotation would need D → U directly, which happens around... that would be a rotation about the X axis? No — D front row to U front row isn't a single face turn... Actually a rotation about the Z axis maps D-front-row stickers to L or R columns only, never to U front row. Yellow at U front row is impossible with a Z-axis turn! Unless the yellow band is on the F face top row and it came from D? D's front row under Z rotation goes to L's right column, not F. F's face stickers under Z rotation stay on F. So yellow cannot appear on F's top row either!

Hmm, so yellow can't be at either. Unless... the yellow is at the top row of the FRONT face and the rotation was about X? No, we only did F.

Wait, maybe I misidentified the faces: maybe the top face visible is not U? Let's reconsider: after F turn, the U face's front row should be orange (from L). In the screenshot, the top face's front row... the yellow band. Hmm, and where's orange? Not visible in the screenshot at all! Orange should be somewhere visible: L's right column became yellow (from D). And U's front row orange. But the screenshot shows no orange.

Unless the yellow band IS the U face's front row and it's actually... yellow = D color = 0xffd21c. The band color is #ffd21c-ish yellow.

Hold on. Maybe my layer selection is wrong: `Math.round(c.group.position.dot(axisVec)) === m.layer` — for F, axisVec = Z, layer 1. Fine.

Hmm, maybe the animation applied -90° about Z but the pivot children keep world transforms... `pivot.add(c.group)` — since pivot is a new Group added to scene with identity transform, and c.group's parent changes from cubeGroup (identity) to pivot (identity) — position preserved. Fine.

Then finishMove: q = setFromAxisAngle(Z, -90°). p = position.applyQuaternion(q). For a cubie at (0,1,1) (top-front-center, part of both U and F layers... but only z=1 layer selected): p = rotate (0,1,1) by -90 about Z: (x,y) → (x cosθ + ... ) let's compute: rotation by θ=-90° about Z: (x,y) → (x cosθ - y sinθ, x sinθ + y cosθ) = (0*0 - 1*(-1), 0*(-1)+1*0) = (1, 0). So (0,1,1) → (1,0,1). That cubie's stickers: it had U (top, +Y) and F (+Z). After rotation, it sits at (1,0,1) — the R/F edge. Its +Y sticker now points... rotate the sticker normal +Y by -90° about Z → (0,1,0)→(1,0,0)*? compute: (x,y)=(0,1) → (0*0 - 1*(-1), 0*(-1)+1*0) = (1, 0) → +X. So the white sticker faces +X (right) ✓ matches screenshot (white column on red face's left edge). 

And the cubie that ends at U's front-center (0,1,1) came from (-1,1,0)? Inverse rotation of (0,1,1) by -90 = forward by +90: (x,y)→(x cos90 - y sin90, x sin90 + y cos90) = (-y, x) = (-1, 0)... with z=1? No: the z coordinate is unchanged under Z rotation. So a cubie at z=1 layer maps to z=1. Positions: (0,1,1) is in the z=1 layer, and it's the U-face front-center. Its stickers: +Y and +Z. After rotation by -90° about Z: position (1,0,1), stickers normal +Y→+X and +Z stays +Z. So the white sticker went to the R face ✓.

But which cubie lands at (0,1,1)? Inverse rotation: apply +90° to (0,1,1): (x,y) → (-y, x) = (-1, 0) so the cubie from (-1, 0, 1)?? That gives (−1,0,1) → rotate by +90: (x,y)→(−y,x)=(0,−1)?? Hmm let me redo. Rotation by +90° about Z maps (x,y) → (x cos90 − y sin90, x sin90 + y cos90) = (−y, x). So (-1,0,1) → (0,-1,1)? That's not (0,1,1). Which cubie maps to (0,1,1) under −90°? Its inverse image under −90 is applying +90 to (0,1,1): (−1, 0, 1). Wait: applying +90 to (0,1,1): (x,y)=(0,1) → (−y, x) = (−1, 0), z=1 → (-1, 0, 1). Yes so the cubie from the L-F edge (−1,0,1) moves to (0,1,1) = U-F edge. That cubie's stickers: −X (orange) and +Z (green). After rotation, its −X normal → rotate (−1,0) by −90: (x,y)=(−1,0) → (x cosθ − y sinθ, x sinθ + y cosθ) with θ=−90: (−1*0 − 0*(−1), −1*(−1) + 0*0) = (0, 1) = +Y ✓. So orange sticker appears on the U face front row!

So the U face front row should be ORANGE, and the F face's top row is green (unchanged, since F's own stickers stay on F... wait no! The F face's top row of stickers: which cubies? (x,1,1) for x=-1..1. Those cubies' +Z stickers stay +Z. So F's top row remains green.

So where does yellow come from?! D's front row (y=-1, z=1) cubies: (x,-1,1) → rotate (x,-1) by -90: (x*0 - (-1)(-1), x*(-1) + (-1)*0) = (-1, -x). So (x,-1,1) → (-1,-x,1): for x=-1 → (-1,1,1) (the L-U-F corner) — that's the L face's top-right... its −X sticker goes to... normal (−1,0) rotated by −90 = (0,1) = +Y. So the yellow sticker from D's front-left corner goes to the U face at position (-1,1,1) — the U face's front-left corner! So yellow DOES appear on the U face front row (its left portion). Hmm interesting. So U front row gets: from x=-1 → yellow at (-1,1,1); from x=0 → cubie (0,-1,1) → (-1,0,1)?? wait that maps to L face. Let me redo more carefully:

Map under -90° about Z: (x,y,z) → (y, -x, z)? Check: rotation by θ=−90: (x,y)→(x cosθ − y sinθ, x sinθ + y cosθ) = (x*0 − y*(−1), x*(−1) + y*0) = (y, −x). Yes: (x,y,z) → (y, −x, z).

So D front row (x,−1,1) → (−1, −x, 1): x=−1 → (−1,1,1) ✓ yellow on U face front-left corner (its −X sticker normal (−1,0,0) → (0,1,0) = +Y ✓ so yellow shows on U face). x=0 → (−1,0,1): yellow sticker on L face (normal −X → +Y? wait that cubie is (0,-1,1) with stickers −Y and +Z... hmm its D sticker normal (0,−1,0) → (−1, 0, 0) = −X → shows on L face ✓). x=1 → (−1,−1,1): the D-F-right corner cubie has stickers −Y (yellow), +Z (green), +X (red). Under rotation: −Y normal (0,−1,0)→(−1,0,0)=−X; +Z stays; +X normal (1,0,0)→(0,−1,0)=−Y. So it lands at (−1,−1,1) with yellow on L face, green on F, red on D face.

So on the U face front row: x=−1 gets yellow (from D's front-left). x=0: which cubie maps to (0,1,1)? Inverse: (0,1,1) → preimage under (x,y)→(y,−x) is (x,y)→(−y,x): preimage of (0,1) = (−1, 0) → cubie (−1,0,1), the L-F edge, whose stickers are −X (orange) and +Z (green). Its −X normal → +Y, so orange on U face. x=1: preimage of (1,1) = (−1,1) → cubie (−1,1,1) = L-U-F corner with stickers −X(orange), +Y(white), +Z(green). Its −X normal (−1,0,0) → (0,1,0) = +Y (orange on U), and its +Y normal (0,1,0) → (1,0,0) = +X (white on R).

So the U face front row after F = [yellow, orange, orange] left→right. And the screenshot shows yellow tiles in that row! Partially — the visible yellow band was 3 tiles... Hmm, at x≈375-680 there appear to be 3 yellow tiles. But my analysis says only the leftmost of the U front row is yellow and the rest orange.

Hmm wait, but which row of U is "front"? The row nearest the viewer. In the screenshot, the yellow tiles form the band just above the green face. If the U front row is [yellow, orange, orange], we'd see one yellow tile at the left and two orange.

Unless my analysis of the visible band is wrong and the band shows something else. Let me look at the image again... The band: three tiles, all yellow-ish, at the top of the green region. Hmm.

Hold on — maybe the image shows the F face's top row as it appears AFTER the turn, and what I think is the green face is actually the front face which received the D stickers... no, F face's own stickers stay.

Hmm, wait. Actually, maybe the visible "green face" is not F but... Let's consider the camera position: (6.2, 4.3, 7.0) — positive X, Y, Z. So we see +X (R, red), +Y (U, white), +Z (F, green). In the screenshot the LEFT face is green = F and the RIGHT face is red = R ✓.

Now, the top face (white) shows 3 rows: back rows white and the front row (adjacent to green) as yellow. But my analysis says U front row = [yellow, orange, orange]. So only 1/3 yellow.

Unless the rotation direction is the opposite (F' instead of F): map (x,y,z)→(−y,x,z). Then U front row preimages: (0,1,1) → preimage under (x,y)→(−y,x): preimage of (0,1) is (1,0)?? Let's do: forward map f(x,y)=(−y,x). f(1,0)=(0,1) ✓. So the cubie from (1,0,1) (R-F edge, stickers +X red, +Z green) → (0,1,1): red on U face. Hmm no yellow on U either.

So neither direction puts 3 yellow tiles on the U front row. Something else: maybe the yellow band is not on the U face, but rather on the F face's top row. Can the F top row become yellow? F top row cubies are (x,1,1). Their +Z stickers stay +Z under Z-rotation. So the F face top row = the +Z stickers of cubies at (x,1,1) after rotation, which came from preimages. Under (x,y)→(y,−x): preimage of (x,1) is (−1, x)... wait preimage g(x,y)=(−y,x). g(x,1) = (−1, x). So cubies from (−1,x,1): x=−1 → (−1,−1,1) [stickers −Y yellow, +Z green, +X red] — its +Z sticker (green) shows on F. Hmm all green.

I'm confusing myself. Let me just simulate in node with the actual cube sticker model and print the top face and front face colors after the F move. That's deterministic and fast. Let me write a small node script replicating cubies with stickers and print ASCII faces.

Actually simpler: in the browser I can get a screenshot but not eval. But node simulation using the same math gives ground truth. Let me do that: simulate and print faces, then compare with the screenshot.

Hmm — the node sim shows U front row = L (orange) and R left column = U (white) — matching my analysis (F clockwise). But the screenshot showed a yellow band. Wait, my sim says U front row = "L L L" (orange). But the screenshot showed yellow. Contradiction!

Hmm — unless the screenshot's yellow band is on the U face's front row and it's actually... Let me look at the screenshot once more: the band is yellow-ish. Hmm, but maybe the colors: L = 0xff8b1f orange, D = 0xffd21c yellow. In the screenshot the band looks yellow (#ffd21c-ish bright yellow).

Wait — maybe the visible band is not U's front row but the F face's top row, and the sim says F's top row is F (green)... but the screenshot shows green face with the top row yellow.

Hmm, discrepancy. Unless... the screenshot was taken mid-animation? No, Moves: 1, Queue: 0.

Or maybe my reading of the image is off and the band tiles are the U face's front row (which per sim should be orange) — showing as yellow would be wrong. OR the band is at the boundary and it's actually the top row of the F face which per sim should be green.

Something's off. Possibility: the browser's actual rotation direction differs from my sim? Both use -π/2 about Z... In the app, `startMove` uses `m.angle` and the pivot rotation via setRotationFromAxisAngle during animation, then finishMove uses q from same angle. Same as sim.

Wait, maybe the screenshot shows the D face!? If the camera... no.

Hmm, let me reconsider: maybe the yellow band tiles are actually on the F face's TOP row and my sim is wrong about which cubies are in the z=1 layer... Actually hold on: maybe the screenshot's yellow comes from the fact that the visible band is the U face's front row but with F turn the D stickers... my sim says D front row → L face. So the D stickers move to the L face (not visible from this camera). And the visible U front row = orange.

So the screenshot should show orange band, not yellow. Unless the band tiles in the screenshot are... I described "yellow" but they might be orange! #ff8b1f vs #ffd21c — in the rendered screenshot the band looks like a bright yellow-orange. I can't distinguish reliably from my reading. Actually in the image, the tiles look distinctly yellow compared with... there's no orange reference in the scene. Hmm.

Given time pressure, let me do a quick pixel check: take a screenshot and... I can't read pixel values directly (screenshots come as images to me). I could use the browser screenshot and visually compare colors — the legend swatch for D is #ffd21c and L is #ff8b1f.

Better approach: check via DOM. I can't eval JS... but I can use the keyboard? No.

Alternative quick check: click F then F' and confirm "State: Solved" returns — that's the key requirement. And the color arrangement check: since the sim (exact same math) proves correctness, and the app's rendering is just applying the same quaternion, the visual must match the sim. The band being yellow/orange is my misreading — note that the F turn in the app might actually be showing U front row = orange which could look yellowish in the render... Actually orange #ff8b1f is clearly orange vs yellow #ffd21c.

Hmm, hold on. Let me re-examine: maybe the band is on the U face's front row AND the app applied the rotation in the +90 direction (i.e., F') — then U front row would get from R's left column... earlier I computed for +90 (F'): map (x,y)→(−y,x). U front row (0,1,1): preimage under f(x,y)=(−y,x) of (x,1)... f(a,b)=(−b,a)=(x,1) → a=1, b=−x → cubies (1,−x,1): for x=−1 → (1,1,1) corner (stickers +X red, +Y white, +Z green): its +Y normal (0,1,0) → f(0,1)=(−1,0) = −X... so white goes to L, and red (+X normal (1,0) → (0,1) = +Y) → red on U. So U front row would be red, red, red. Not yellow either.

So neither direction gives yellow on U front row. And F top row stays green always. So where can yellow appear near the top? Only if... hmm! What if the band tiles I see are on the U face but they are at the row adjacent to the F face, and the yellow comes from D via a different mechanism — impossible per group theory.

Unless the app's move applied to the WRONG layer (e.g., rotated the y=-1 layer about Z? no...). OR the pivot parenting broke and cubies moved incorrectly! `pivot.add(c.group)` preserves local position because both parent transforms are identity — true. But wait: `pivot.setRotationFromAxisAngle` during animation — fine.

Hmm, but there's a subtle bug: `finishMove` computes `g.position` and `g.quaternion` as LOCAL to pivot (identity rotation at finish since we set pivot.rotation to 0 first!). Look at finishMove: `mv.pivot.rotation.set(0,0,0);` then computes p = g.position.applyQuaternion(q). But g.position is the local position within the pivot — which was set before the pivot rotated, so it's the ORIGINAL position ✓. And g.quaternion is the local quaternion ✓. So we apply q to those — correct.

OK so the app is right. But then what's yellow in the screenshot? Let me look at the image once more mentally... The band of 3 tiles: colors at y≈255-320. Actually — could those be the D face?! No, D is at the bottom, hidden.

Hmm, could the app have applied "F" but the visible band is the U face front row showing L(orange) rendered as... The rendered orange at 0xff8b1f under white light would look orange-yellow. The red face is 0xe33b3b clearly red. The green clearly green. The band could plausibly be described as "yellow-orange".

I think the fastest resolution: take a fresh screenshot with better comparison — turn D (bottom face, not visible from above) — no. Alternative: click F' (inverse) and check "State: Solved" appears — that's the required verification. And to check sticker arrangement matches start, I can compare screenshots visually (F then F' returns to the exact first image).

Also to disambiguate colors: click L' which would bring D (yellow) stickers to the top face per theory — then compare band colors. Eh, time.

Better idea for ground truth in-browser: I can't eval... but I CAN type into something? There's no input. Hmm, the `browser` tool has "type" action which types text — into focused element. Not helpful.

Alternative ground truth: render the app and read pixels via a screenshot — I can't read pixels programmatically.

Given the node sim uses the identical math and proves: after F, U front row = orange (L), R left column = white (U), D front row → L. The screenshot shows white column on the red face's left edge ✓ (that part matches the sim exactly: "U R R" rows). So the app matches the sim. Therefore the yellow-looking band must be the U front row rendered as L=orange #ff8b1f which in a JPEG-ish screenshot could appear yellowish... but wait, the sim says U front row = L L L (orange) — consistent with three orange tiles! The screenshot shows exactly three tiles in that band! So the band = U face front row = 3 orange tiles, and I misjudged the hue. 

Wait but earlier I thought the band is at the top of the green face (F's top row). U's front row IS adjacent to F's top row — the boundary band right above the green face IS the U front row when viewed from above-front. Yes! The band between the white back rows and the green face is the U face's front row. So it's orange (from L) ✓ matches sim. And white column on R's left edge ✓ matches sim. 

So the implementation is correct and matches standard F clockwise. Now quickly verify F then F' in browser → State: Solved, and orbit + reset. Let me do these checks quickly.

Click F' at (267,180).

F then F' → State: Solved, and the render matches the initial solved screenshot exactly (white top, green front, red right). Move/inverse verified visually.

Now: orbit drag (background), check camera changed and cube still visible, then Reset restores framing. Also test Scramble then Reset quickly.

Drag from (950,400) to (750,300).

Orbit works — camera moved to show R (red), B (blue), D (yellow) — solved colors correct from behind (red left, blue right, yellow bottom). No face turned (State: Solved, Moves: 2 unchanged) ✓.

Now test scramble, then reset restores framing & solved.

Scramble is animating and queueing (Moves3, Queue 19, State: Mixed) — stickers are being permuted properly. Now click Reset and confirm solved + camera restored to initial framing.