--- name: spectral-methods-1dab3839b534 description: Fourier-Laplacian eigenmodes for frequency domain analysis of graph and signal structures version: 1.0.0 --- # 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: 1. **Graph Laplacian**: L = D - A, normalized Laplacian L_norm = I - D^(-1/2)AD^(-1/2) 2. **Eigenvalue computation**: Power iteration and shifted inverse iteration 3. **Spectral clustering**: Fiedler vector bisection for graph partitioning 4. **Discrete Fourier Transform**: DFT with band power analysis (alpha/beta/gamma) 5. **Heat kernel**: Tr(exp(-tL)) diffusion trace across time scales 6. **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: ```clojure (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: ```clojure (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: ```clojure (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: ```clojure (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: ```clojure (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: ```clojure (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 ```sql 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 ✓ ``` --- Source attribution: plurigrid Original project: https://github.com/plurigrid/asi/blob/293ce9f76e3abcd5ba18db284688bc0d9ac6691d/skills/spectral-methods/SKILL.md License: MIT License information: https://opensource.org/license/mit Original upstream instructions; downloaded from Mega AI Hub. ## Upstream license notice MIT License Copyright (c) 2025 SkillCreator.ai Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. 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