internetSpectralGapCert
plain-language theorem explainer
The definition assembles a certificate for the k-core spectral gap of the Internet AS graph under phi-ladder decay. Network theorists applying Recognition Science to self-similar topologies would cite it to confirm that consecutive gaps satisfy the exact ratio phi inverse. It is a direct record constructor that supplies the positivity theorem from the local spectralGap_pos and the ratio from the sibling spectralGapRatio.
Claim. The Internet spectral gap certificate is the structure asserting that for every natural number $k$ the k-core spectral gap satisfies $0 <$ spectralGap $k$ and that the ratio of consecutive gaps obeys spectralGap$(k+1)/$spectralGap $k = phi^{-1}$.
background
In this module the k-core spectral gap of the Internet AS-level graph is placed on the phi-decay ladder, with the prediction that the gap at level k+1 is exactly the gap at level k scaled by phi inverse. The structure InternetSpectralGapCert packages two properties: positivity of the gap for all k, and the self-similar ratio condition. The function spectralGap k denotes the second eigenvalue of the normalized Laplacian on the k-core subgraph.
proof idea
The definition is a one-line record constructor. It assigns gap_pos to the local theorem spectralGap_pos (which proves positivity via inv_pos.mpr on phi powers) and phi_inv_ratio to the sibling spectralGapRatio.
why it matters
This certificate is consumed by the parent definition internetSpectralGapCert in the InternetSpectralGap module, which augments it with strict decrease and as-core positivity. It realizes the RS prediction that the spectral gap decays geometrically with ratio phi inverse on the phi-ladder, consistent with the self-similar fixed point T6 and the eight-tick octave T7. No open questions are flagged.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.