IndisputableMonolith.Information.NetworkTopologyFromSigma
Module fixing the scale-free network degree exponent at γ = 1 + φ from Recognition cost structure. Network theorists and RS information-layer auditors cite it for the predicted power-law tail. Definitions pin γ via φ, with short lemmas placing it in a numerical band above 2 and equating it to 2 + 1/φ; a certificate packages the claim.
claimThe scale-free degree exponent is $\gamma = 1 + \varphi$, equivalently $\gamma = 2 + \varphi^{-1}$, with $\varphi$ the golden ratio. The module records $\gamma > 2$ and a tight numerical band, and packages the prediction as a network-topology certificate.
background
Recognition Science forces the cost functional $J(x) = (x + x^{-1})/2 - 1$ (T5) and the self-similar fixed point $\varphi$ (T6). In the information layer, network topology is read off sigma-structure rather than postulated: the degree distribution of a recognition network is expected to be scale-free, $P(k) \propto k^{-\gamma}$, with a single dimensionless exponent fixed by $\varphi$.
The module imports RS constants (including the native tick $\tau_0$) and the cost layer. Sibling definitions introduce degreeExponent as the predicted $\gamma$, prove it equals $2 + \varphi^{-1}$ (hence $1 + \varphi$), place it strictly above 2 (so the mean degree converges), and bound its numerical value. A certificate type bundles the prediction for downstream use.
proof idea
Definition-first module. The exponent is defined directly from $\varphi$; algebraic identities of the golden ratio give $\gamma = 1 + \varphi = 2 + \varphi^{-1}$ in one or two rewrites. Strict inequality $\gamma > 2$ is immediate from $\varphi > 1$. A short numerical band lemma evaluates the constant. The certificate is a structure packing these facts, with a canonical inhabitant.
why it matters in Recognition Science
Places a concrete, falsifiable network prediction inside the RS information domain: scale-free degree tails must sit at $\gamma = 1 + \varphi \approx 2.618$, not at a free fit parameter. No downstream consumers are wired yet in the graph; the certificate is the intended hook for topology-from-sigma theorems and empirical checks against real recognition or citation networks. Ties to the forcing chain via $\varphi$ (T6) and the cost $J$ imported from the Cost module.
scope and limits
- Does not derive the power-law form $P(k) \propto k^{-\gamma}$ from sigma; it fixes only the exponent.
- Does not prove uniqueness of the topology beyond the degree exponent.
- Does not compare to empirical network data or fit residuals.
- Does not address directed, weighted, or temporal network extensions.
- Does not depend on eight-tick or $D=3$ forcing steps.