asCoreGap
The declaration defines the AS-level spectral gap at core depth 2 as the value of the phi-ladder function at k=2, yielding φ^{-2} ≈ 0.382. Network theorists working on scale-free Internet graphs would cite this when matching the CAIDA AS-graph observation λ₂ ≈ 0.382 to the Recognition Science prediction. It is a direct one-line instantiation of the general spectralGap definition at the empirically relevant depth.
claimDefine the AS-level core spectral gap by $λ_2(2) := φ^{-2}$, where $φ$ is the golden ratio and the spectral gap at k-core depth $k$ is given by $φ^{-k}$.
background
The module develops the spectral gap layer for Internet topology on the phi-ladder. The spectral gap at k-core decomposition depth k is defined as φ^{-k}. This follows from the phi-ladder construction in NetworkScience.InternetSpectralGapFromPhiLadder, where spectralGap(k) := (phi^k)^{-1}. The local setting predicts that successive k-core gaps decrease by the factor 1/φ, matching the structural self-similarity of the AS graph.
proof idea
One-line definition that applies the spectralGap function to the argument 2.
why it matters in Recognition Science
This definition supplies the concrete value for the AS-level gap used in the InternetSpectralGapCert structure and the positivity theorem asCoreGap_pos. It instantiates the general prediction λ₂(k) = 1/φ^k at the observed depth k=2, where the CAIDA measurement ≈0.382 aligns with φ^{-2}. The placement closes the empirical anchor for the spectral gap layer in the Recognition framework.
scope and limits
- Does not prove the empirical match between CAIDA data and φ^{-2}.
- Does not derive the phi-ladder form from the J-cost or forcing chain.
- Does not compute the spectral gap for arbitrary graphs or higher k.
formal statement (Lean)
52def asCoreGap : ℝ := spectralGap 2
proof body
Definition body.
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