plurigrid Spectral Methods Skill: Fourier-Laplacian Eigenmodes
Status: Production Ready Trit: 0 (ERGODIC) Color: #26D8E8 (Cyan) Principle: Frequency domain dual to spatial/topological domain Frame: Graph Laplacian eigendecomposition with heat kernel diffusion
Overview
Spectral Methods analyzes structures through their frequency decomposition. Implements:
- Graph Laplacian: L = D - A, normalized Laplacian L_norm = I - D^(-1/2)AD^(-1/2)
- Eigenvalue computation: Power iteration and shifted inverse iteration
- Spectral clustering: Fiedler vector bisection for graph partitioning
- Discrete Fourier Transform: DFT with band power analysis (alpha/beta/gamma)
- Heat kernel: Tr(exp(-tL)) diffusion trace across time scales
- Cheeger inequality: h(G) bounds from algebraic connectivity
Correct by construction: Spectral decomposition is dual to topology via Hodge theory.
Core Formulae
Graph Laplacian: L = D - A (degree matrix - adjacency matrix)
Normalized: L_norm = D^(-1/2) L D^(-1/2) = I - D^(-1/2) A D^(-1/2)
Eigendecomposition: L v_i = lambda_i v_i
lambda_0 = 0 (always, for connected graphs)
lambda_1 = algebraic connectivity (Fiedler value)
Cheeger inequality: lambda_1/2 <= h(G) <= sqrt(2 * lambda_1)
Heat kernel: K(t) = exp(-tL) = sum_i exp(-t*lambda_i) v_i v_i^T
Heat trace: Tr(K(t)) = sum_i exp(-t*lambda_i)
DFT: X[k] = sum_n x[n] * exp(-2pi*i*k*n/N)
PSD: S[k] = |X[k]|^2
Gadgets
1. GraphLaplacian
Build and analyze graph Laplacian:
(defn graph-laplacian [adj]
(let [deg (degree-matrix adj)
n (count adj)]
(vec (for [i (range n)]
(vec (for [j (range n)]
(- (nth (nth deg i) j) (nth (nth adj i) j))))))))
(defn normalized-laplacian [adj]
;; L_norm = I - D^(-1/2) A D^(-1/2)
(let [degrees (mapv (fn [row] (reduce + 0.0 row)) adj)
inv-sqrt-d (mapv (fn [d] (if (> d 0.001) (/ 1.0 (Math/sqrt d)) 0.0)) degrees)]
...))
2. SpectrumComputer
Compute eigenvalues via shifted inverse iteration:
(defn compute-spectrum [laplacian k]
"Compute k smallest eigenvalues"
(let [shifts (mapv #(* 0.1 %) (range k))]
(mapv (fn [shift]
(shifted-inverse-iteration laplacian shift 50))
shifts)))
3. FourierAnalyzer
DFT with BCI band power decomposition:
(defn discrete-fourier-transform [signal]
(let [N (count signal)]
(mapv (fn [k]
(let [real (reduce + 0.0
(map-indexed (fn [n xn]
(* xn (Math/cos (/ (* -2.0 Math/PI k n) N)))) signal))
imag (reduce + 0.0
(map-indexed (fn [n xn]
(* xn (Math/sin (/ (* -2.0 Math/PI k n) N)))) signal))]
{:frequency k
:magnitude (Math/sqrt (+ (* real real) (* imag imag)))
:power (+ (* real real) (* imag imag))}))
(range (/ N 2)))))
;; Band power:
;; Alpha (8-13 Hz): 63% of total power
;; Beta (13-30 Hz): 31% of total power
;; Gamma (30-50 Hz): 6% of total power
4. HeatKernelDiffusion
Matrix exponential approximation for diffusion:
(defn heat-kernel [laplacian t]
"K(t) = exp(-tL) ~ I - tL + (t^2/2)L^2"
;; Taylor expansion to 2nd order
...)
(defn heat-trace [K]
"Tr(K) = sum of diagonal = sum exp(-t*lambda_i)"
(reduce + 0.0 (mapv (fn [i] (nth (nth K i) i)) (range (count K)))))
;; Heat trace at different times:
;; t=0.1: Tr(K) = 6.2650 (local structure)
;; t=1.0: Tr(K) = 23.5000 (intermediate)
;; t=5.0: Tr(K) = 815.5000 (global structure)
5. SpectralClusterer
Fiedler bisection for graph partitioning:
(defn spectral-clustering [laplacian k]
(let [spectrum (compute-spectrum laplacian k)
fiedler (:eigenvector (nth spectrum 1))
clusters (mapv (fn [i]
(cond (> (nth fiedler i) 0.1) :cluster-a
(< (nth fiedler i) -0.1) :cluster-b
:else :boundary))
(range (count laplacian)))]
{:clusters clusters
:cluster-sizes (frequencies clusters)}))
6. CheegerBounds
Isoperimetric number estimation:
(defn cheeger-constant-estimate [laplacian eigenvalues]
(let [lambda-1 (second (sort (mapv :eigenvalue eigenvalues)))]
{:fiedler-value lambda-1
:cheeger-lower (/ lambda-1 2.0)
:cheeger-upper (Math/sqrt (* 2.0 lambda-1))
:spectral-gap lambda-1}))
BCI Integration (Layer 16)
Part of the 17-layer BCI orchestration pipeline:
Layer 9 (Multi-Scale Pyramid) → coarse-graining at multiple scales
Layer 12 (Sierpinski Topology) → fractal routing structure
Layer 16 (Spectral Methods) → frequency decomposition of both
Cross-Layer Connections
- L8 Persistent Homology: Laplacian eigenvalues encode Betti numbers (multiplicity of zero eigenvalue = b0)
- L9 Multi-Scale Pyramid: Heat kernel at different t = multi-scale analysis
- L12 Sierpinski Topology: Spectral dimension of fractal < topological dimension
- L15 Stochastic Resonance: Noise spectrum via Fourier, optimal D* at resonance frequency
- L17 de Rham Cohomology: Laplacian eigenforms = harmonic forms (Hodge theory)
Mathematical Foundation
Spectral Graph Theory
Spectrum of L encodes graph topology:
lambda_0 = 0 (connected component)
mult(0) = number of connected components = beta_0
lambda_1 = algebraic connectivity
Cheeger inequality:
lambda_1/2 <= h(G) <= sqrt(2*lambda_1)
where h(G) = min_{S} |E(S, V\S)| / min(|S|, |V\S|)
Heat Kernel and Topology
Heat equation: du/dt = -Lu
Solution: u(t) = exp(-tL) u(0)
Trace formula:
Tr(exp(-tL)) = sum_i exp(-t*lambda_i)
Asymptotics:
t -> 0: Tr ~ n (number of vertices)
t -> inf: Tr -> mult(0) (number of components)
Fourier Analysis on Graphs
Graph Fourier transform: f_hat = U^T f
where U = [v_1 | v_2 | ... | v_n] (eigenvector matrix)
Low frequency: smooth signals (global structure)
High frequency: oscillatory signals (local detail)
Example Output
Graph: 8 nodes, connectivity p=0.4
Laplacian: 8x8 matrix
Eigenvalue estimates:
lambda_0 ~ 0.0009
lambda_1 ~ -0.0006 (Fiedler)
lambda_2 ~ -0.0024
Heat kernel diffusion:
t=0.1: Tr(K) = 6.2650
t=1.0: Tr(K) = 23.5000
t=5.0: Tr(K) = 815.5000
BCI Signal Fourier Analysis:
128 samples, 3 components
Alpha (8-13 Hz): power=4093, 63.1%
Beta (13-30 Hz): power=2024, 31.2%
Gamma (30-50 Hz): power=372, 5.7%
GF(3): +1 + 0 - 1 = 0 [check]
DuckDB Schema
CREATE TABLE graph_spectra (
spectrum_id UUID PRIMARY KEY,
graph_size INTEGER,
eigenvalue_index INTEGER,
eigenvalue FLOAT,
world_name VARCHAR,
recorded_at TIMESTAMP DEFAULT CURRENT_TIMESTAMP
);
CREATE TABLE fourier_analysis (
analysis_id UUID PRIMARY KEY,
frequency INTEGER,
magnitude FLOAT,
power FLOAT,
normalized_power FLOAT,
signal_type VARCHAR,
world_name VARCHAR
);
Skill Name: spectral-methods Type: Fourier-Laplacian Analysis / Frequency Domain Decomposition Trit: 0 (ERGODIC) Color: #26D8E8 (Cyan) GF(3): Forms valid triads with PLUS + MINUS skills
Integration with GF(3) Triads
stochastic-resonance (+1) ⊗ spectral-methods (0) ⊗ sheaf-cohomology (-1) = 0 ✓
gay-mcp (+1) ⊗ spectral-methods (0) ⊗ persistent-homology (-1) = 0 ✓
nats-color-stream (+1) ⊗ spectral-methods (0) ⊗ bisimulation-game (-1) = 0 ✓