Module Guide: InternetSpectralGapFromPhiLadder
Purpose
This module formalizes a specific Recognition Science (RS) HYPOTHESIS regarding macroscopic network topology: the spectral gap $\lambda_2(k)$ of the $k$-core of the Internet's Autonomous System (AS) graph decays according to the powers of the golden ratio $\phi$. It establishes the theoretical exact values on the "phi-decay ladder" against which empirical network measurements can be compared.
Main Declarations
The formalization is tight and relies on the RS golden ratio $\phi$:
- spectralGap: A MODEL definition setting the ideal spectral gap at depth $k$ to $\phi^{-k}$.
- spectralGapRatio: A THEOREM proving that the ratio of the gap at $k+1$ to the gap at $k$ is exactly $\phi^{-1}$.
- spectralGap_k2_val: A THEOREM computing the specific predicted gap for the 2-core as $\phi^{-2} \approx 0.382$.
- internetSpectralGapCert: The module's master certificate, formally verifying that the spectral gap is always positive and respects the $\phi^{-1}$ decay ratio for all $k$.
Fit into the Recognition Science Chain
In the RS framework, $\phi$ is uniquely forced by self-similarity in a discrete ledger with $J$-cost (proved in phi_equation). While fundamental constants like $c$ and $\hbar$ are derived as ratios of RS-native quantities at the microscopic level, this module projects the same invariant scale-free properties outward to human-engineered complexity. It acts as an empirical falsifier (F5 Depth): if the topological scaling of the Internet drastically violated the $\phi$-ladder, it would challenge the universality of the forcing chain.
What Remains Outside
- Empirical Data: The module defines the theoretical ladder. Lean does not contain CAIDA datasets or BGP routing snapshots to empirically measure the true $\lambda_2(k)$.
- Topological Forcing: The supplied module posits the prediction as a MODEL. The rigorous formal proof mapping the structural growth of the Internet graph back to the universal $J$-cost functional equation (proving it must decay exactly this way, rather than just postulating the ladder) is currently outside this module.