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What is this? A four-layer visualization of the chess-spectral encoding of the position. Each layer surfaces different structure; toggle them independently to dissect the field.
1. Hover overlay (top dropdown, Channel)
When you click a piece, each legal destination is tinted by the post-move signature on the chosen channel. Bright = more positional shake-up.
- A1 default
- Orbit projection — overall complexity. Best general intuition.
- STD4_X / Y / Z / W
- Coordinate-residuals along each of the 4 axes. Highlights moves that shift piece distribution along that one dimension. Signed.
- FIB_SYM_1 / 2 / 3
- Cross-piece interaction (rank-3 SVD basis). Slow — needs a full re-encode per destination.
- FA_PAWN_W / FA_PAWN_Y
- Pawn-antisymmetric channels, one per pawn axis (Oana & Chiru §3.10 Def. 11).
- FD_DIAG
- Diagonal-antisymmetric.
2. Board-signature cloud (Board signature dropdown)
Volumetric cloud over the entire 4D lattice. Per-cell color = channel intensity (viridis ramp by default), per-cell scale = magnitude. Refreshes only on real moves so it's free between moves.
- Log scale (log1p)
- Compress heavy-tailed channels via
sign(x)·log1p(|x|). Useful for orbit channels where one cell dominates.
- Signed (diverging) color mode
- Blue → white → red ramp; height tracks
|value| not value, so positive AND negative lobes get visible volume. Best for STD4_X/Y/Z/W (signed channels).
- Mark local maxima
- Bright spheres at every cell whose scalar strictly exceeds its 8 face-neighbors in 4D. The field's peaks.
- Slice axis + value
- Pin one lattice axis to one of 0..7 — collapses the 4D cloud to a 3D cross-section.
w=4 picks one column of 8 boards out of the 64.
- Threshold
- Hide cells below a normalized intensity. Sweeps "percentile shells": 0 = all, 0.95 = top 5%.
- Clip sphere
- Hide cells outside a 4D ball around the lattice center, the brightest peak, or a sticky pin set via console. Soft 10% falloff. Stacks with slice + threshold.
3. Streamline filaments (Streamline filaments checkbox)
Field lines through the active board-signature channel. Default seeding stride-tiles the lattice; topology mode replaces it with Morse–Smale extraction.
- Topology mode (Morse–Smale)
- Lines emanate from each critical point (max / min / saddle) along the Hessian eigenvectors, integrated in
sign(λ)·∇φ. Red = ascending manifold (toward peaks). Blue = descending manifold (toward pits). The visible lines ARE the topological 1-skeleton of the field. Fixes the "lines all converge to one spot" complaint definitively.
- Seeds / Steps sliders
- (Stride mode only) — number of seed cells and integration steps per line.
4. Camera control
- ⤺ Align view to flow
- Tween the camera so its view axis lies along the dominant Hessian eigenvector at the field's most prominent critical point. Streamlines tangent to the camera foreshorten to dots; perpendicular structure spreads out. Lets you "look down the flow" instead of fighting projection clutter (Hofmann/Rieck/Sadlo 2018).
What it all describes: the Morse–Smale complex of one chosen scalar field on the 8⁴ lattice. The cloud is the field; the spheres are its peaks; the red/blue filaments are the curves connecting peaks, pits, and saddles. Filtering tools let you isolate slices, percentile shells, or 4D balls. Together they give a verbal-only description even with aphantasia: "at every red endpoint a peak; at every blue endpoint a pit; where the colors cross, a saddle; the cloud is everywhere; the slicers and clipper let me hide everything except the slab / shell / ball I'm looking at."
4D CHESS