1. Plain English
The declaration states that if you take the mathematical model of the Internet's k-core spectral gap at level $k+1$ and divide it by the gap at level $k$, the ratio is always exactly $1/\varphi$ (the inverse of the golden ratio). In other words, as you move one step deeper into the network's core, the spectral gap decreases by a constant factor of $\varphi^{-1}$.
2. Why it matters in Recognition Science
In Recognition Science, the appearance of the golden ratio in the Internet's routing graph serves as an empirical HYPOTHESIS. RS posits that self-organizing ledgers (like the Autonomous System graph) optimize the unique $J$-cost function. Because self-similarity under $J$-cost canonically forces $\varphi$, the framework predicts that the macroscopic topological properties of the Internet (such as its spectral gap at various k-core depths) must scale as a $\varphi$-decay ladder.
3. Reading the formal statement
theorem spectralGapRatio (k : ℕ) :
spectralGap (k + 1) / spectralGap k = phi⁻¹
k : ℕ: For any natural number $k$ (representing the k-core level).spectralGap (k + 1) / spectralGap k: The ratio of the spectral gap at the next core level to the current one.= phi⁻¹: The result is strictly equal to $\varphi^{-1}$.
4. Visible dependencies and certificates
The theorem depends on the MODEL of the spectral gap, defined in spectralGap as $( \varphi^k )^{-1}$.
The proof is bundled into a mathematical certificate, internetSpectralGapCert, which verifies two properties simultaneously for the whole ladder: the spectral gap is always strictly positive, and the ratio between adjacent steps is exactly $\varphi^{-1}$.
5. What this declaration does not prove
This Lean theorem does not prove that the physical Internet actually has these spectral gaps. It proves a mathematical THEOREM about the idealized MODEL of the gap. Proving that real-world routers form a graph matching this spectrum from first principles is not in this file; rather, the scaling is presented as an empirical prediction of the framework whose target exact values are locked in formally.