asCoreGap_pos
plain-language theorem explainer
The theorem establishes positivity of the spectral gap for the 2-core of the AS-level Internet graph. Network theorists modeling scale-free topologies on the phi-ladder would cite this when certifying the Cheeger bound for the observed CAIDA structure. The proof is a direct one-line instantiation of the general spectral-gap positivity result at depth k=2.
Claim. $0 < λ_2(2)$, where $λ_2(k)$ denotes the second eigenvalue of the normalized Laplacian on the $k$-core of the AS-level topology.
background
The module treats Internet topology via the phi-ladder, where the spectral gap of the $k$-core equals $1/φ^k$ and successive cores differ by the ratio $1/φ$. The definition asCoreGap specializes the general spectralGap function to the $k=2$ case that matches the empirical CAIDA AS-graph value ≈0.382≈1/φ². The upstream theorem spectralGap_pos asserts 0 < spectralGap k for every natural number k, with one proof using zpow_pos on Constants.phi_pos and the other using inv_pos.mpr of pow_pos phi_pos k.
proof idea
The proof is a one-line wrapper that applies the general spectralGap_pos theorem at the concrete argument 2.
why it matters
This supplies the as_core_pos field inside the InternetSpectralGapCert record that aggregates gap positivity, strict decrease, and ratio properties. It closes the formal certificate for the structural prediction that k-core gaps lie on the phi-ladder, consistent with the Recognition Science T6 fixed-point and T7 eight-tick octave. The result enables axiom-free verification of the observed spectral gap without further scaffolding.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.