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docs(perturbation-sim): VISUALIZE.md fig 2 — Morton-tile pyramid + 4-theorem mapping
Adds the second figure spec: the stacked 2-bit×2-bit (4×4) Morton/z-order pyramid for spatial perturbation, with each theorem mapped to its pyramid structure — Weyl/Δλ₂ at the coarse top, Davis-Kahan/sinθ_DK at the reorienting partition, Cheeger/Δφ on the inter-tile seam (the exchange rate), infight at the fine bottom, Kron on the 4→1 coarsen arrow. Two algebras labelled (Walsh/XOR sign axis, EWA-splat magnitude axis); the bipolar-phase = Walsh-Hadamard identity; and the geography-trap + screen-not-exact + SIMD-WHT honesty strip.
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‎crates/perturbation-sim/VISUALIZE.md‎

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@@ -53,6 +53,69 @@ Cheeger duality — not for decoration.
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> red/orange cascade). Desaturate the glow ~20 %; keep all the same labels,
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> symbols, and the honesty disclaimer. Higher drama, identical rigor.
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## Figure 2 — Morton-tile pyramid: spatial perturbation & where the 4 theorems map
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Explains the stacked 2-bit×2-bit (4×4) Morton/z-order pyramid and maps each
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theorem to the pyramid structure it lives in. Faithful to the OGAR two-algebra
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rule (sign = Walsh/XOR, magnitude = EWA splat) and the geography/screen caveats.
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### Theorem → pyramid-structure mapping (the honest one-liner each)
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| Theorem (factor) | Lives at | Meaning |
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|---|---|---|
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| **Weyl** (`Δλ₂`) | **top** (coarse/global) | whole-graph field eigenvalue λ₂ (Raumgewinn) |
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| **Davis–Kahan** (`sinθ_DK`) | **mid** — the partition reorienting between levels | Fiedler-vector rotation / subspace stability |
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| **Cheeger** (`Δφ`) | **the inter-tile seam** | conductance of the cut = the coarse↔fine exchange rate `μ₂/2 ≤ h ≤ √(2μ₂)` |
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| **infight** | **bottom** (fine tiles) | local collapse (cascade trips) |
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| *(Kron reduction)* | **the 4→1 coarsen arrow** | Schur complement: 4 fine tiles → 1 coarse super-node (basin tiering) |
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Two algebras (axes of the pyramid): **sign = Walsh/XOR** (`vsa_bind`, the *scale*
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axis — coarse coeff = field, fine coeff = infight); **magnitude = EWA Gaussian
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splat** (`vsa_bundle`, the anisotropic footprint, anti-aliases the Z-seam).
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`perturbation = Σ_L sign(addr,L)·magnitude(addr,L)` (bipolar-phase pyramid =
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Walsh–Hadamard on the address tree).
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### Honesty (must appear on the figure)
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Tiles ride the **electrical** embedding (effective-resistance / spectral coords),
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**NOT geography**. Walsh basis = graph eigenbasis **exactly only on hypercubes**
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→ a fast O(n log n) **screen**; the exact eigensolve certifies the flagged tiles.
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SIMD WHT via `ndarray::simd::wht_f32`.
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### Prompt — Morton-pyramid figure
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> Clean light/white scientific figure, 16:9, titled "Stacked Morton-Tile Pyramid
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> — Spatial Perturbation & the Four Theorems" (thin teal/navy linework,
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> perceptually-uniform inferno accents, monospace numbers).
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>
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> **Center-left — the pyramid:** a vertical stack of 4 receding plates, bottom
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> (fine) → top (coarse), each a 2-bit×2-bit (4×4) Morton/z-order quadtree level.
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> L0 (bottom): 16×16 cells with a faint Z-order (Morton) curve threading them, a
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> bright red perturbation spike in one central cell. L1, L2: coarser 8×8 then 4×4
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> grids; the perturbation rises as a widening anisotropic glow cone. L3 (top): a
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> single tile (whole-network summary). Upward "coarsen 4→1" arrows between levels.
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>
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> **Map the four theorems (callout labels + leader lines):** L3 top →
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> `Weyl — Δλ₂` "algebraic connectivity λ₂ (global field / Raumgewinn)"; mid
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> partition boundary → `Davis–Kahan — sinθ_DK` "Fiedler-vector rotation / subspace
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> stability"; inter-tile seam → `Cheeger — Δφ` "conductance of the cut; exchange
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> rate μ₂/2 ≤ h ≤ √(2μ₂)"; L0 bottom → `infight` "local collapse (cascade trips)";
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> on the 4→1 arrow → `Kron reduction (Schur complement)` "4 fine tiles → 1 coarse
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> super-node".
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>
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> **Right column — two algebras:** Sign side: a small bipolar ±1 (black/white)
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> Walsh pattern, "Walsh / XOR (vsa_bind) — SCALE axis: coarse coeff = field/
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> Raumgewinn, fine coeff = infight". Magnitude side: an anisotropic Gaussian
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> ellipse over a 4×4 tile straddling a Z-seam, "EWA Gaussian splat (vsa_bundle) —
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> anti-aliases the Morton seam; Σ-anisotropy = spread direction ≈ cut normal". A
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> one-line equation between them: perturbation = Σ_L sign(addr,L)·magnitude(addr,L)
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> ("bipolar-phase pyramid = Walsh–Hadamard on the address tree").
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>
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> **Bottom honesty strip (visible):** "Tiles ride the electrical embedding
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> (effective-resistance / spectral coords) — NOT geography. Walsh basis = graph
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> eigenbasis exactly only on hypercubes → a fast O(n log n) screen; the exact
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> eigensolve certifies the flagged tiles. SIMD WHT via ndarray::simd::wht_f32."
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>
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> Legible labels, no photorealism; the rising red→orange perturbation cone and
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> the four theorem-callouts are the dominant motifs.
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## Future iterations
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- Map the glow to the actual `node_field` values from a real `simulate_outage`
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run (export `(lon, lat, |Δθ|)` and render to scale), not an artistic gradient.

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