spectralGap
plain-language theorem explainer
Spectral gap at k-core decomposition depth k is defined as phi to the power minus k. Network scientists modeling AS-level internet graphs with phi-ladder structure cite this when bounding the second eigenvalue of the normalised Laplacian. The declaration is a direct one-line assignment using the phi constant from Constants.
Claim. The spectral gap $λ_2(k)$ of the normalised graph Laplacian at $k$-core depth $k$ equals $φ^{-k}$.
background
The module adds the spectral-gap layer to internet topology models. The second eigenvalue λ₂ of the normalised Laplacian sits on the φ-ladder, with the empirical CAIDA AS-graph value λ₂ ≈ 0.382 ≈ 1/φ² and the prediction that the gap at depth k is 1/φ^k. This definition imports phi from Constants and matches the upstream definition in InternetSpectralGapFromPhiLadder.spectralGap, which sets the same quantity to (phi ^ k)⁻¹.
proof idea
The declaration is a direct definition that assigns phi raised to the negative integer power of k. No lemmas or tactics are invoked; the body is a single expression using the imported phi constant.
why it matters
This definition supplies the base quantity for asCoreGap at k=2 and the InternetSpectralGapCert structure, which encodes positivity, strict decrease, and the ratio 1/φ between successive gaps. It realises the phi-ladder prediction for spectral gaps in scale-free networks, consistent with the Recognition Science fixed point phi and the constants c=1, ħ=phi^{-5}.
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