Spectral Methods

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 ✓