networkScience_one_statement
plain-language theorem explainer
The declaration asserts that the power-law exponent γ equals 3, satisfies the fixed-point equation (γ-1)(γ-2)=2, and that the clustering ratio lies between 0.617 and 0.622. Network scientists studying scale-free graphs would cite this when linking recognition-graph recurrences to observed small-world statistics. The proof is a term-mode conjunction that applies reflexivity on the definition of γ together with the fixed-point lemma and the two conjuncts of the clustering band theorem.
Claim. Let γ denote the predicted power-law degree-distribution exponent and let C denote the predicted clustering ratio. Then γ = 3, (γ − 1)(γ − 2) = 2, and 0.617 < C < 0.622.
background
The module re-derives power-law degree distributions from φ-recurrence on the recognition graph. γ is defined as the constant 3. The clustering ratio is defined as 1/φ. This one-statement theorem bundles the value of γ, verification that it solves the σ-conservation fixed-point equation, and numerical bounds on the clustering ratio. It forms part of the claim that average path length scales as log_φ N for small-world networks. Upstream, the gamma_fixed_point lemma establishes that γ = 3 solves (γ − 1)(γ − 2) = 2 with γ > 2, while clusteringRatio_band proves the interval bounds using the inequalities phi > 1.618 and phi < 1.62.
proof idea
The proof is a term-mode constructor that builds the conjunction by reflexivity on the definition of gamma, followed by direct application of the gamma_fixed_point theorem for the algebraic identity, and the two projections of the clusteringRatio_band theorem for the strict inequalities.
why it matters
This theorem consolidates the core predictions for scale-free networks under Recognition Science, with γ = 3 as the self-similar fixed point of the σ-conservation equation and clustering ratio 1/φ. It supports the small-world property in the module and aligns with the φ-ladder and T6 fixed-point structure. The result fills Track F9 by re-deriving the Barabási-Albert exponent from φ-recurrence. The module falsifier is real networks whose best-fit exponent falls outside [2.5, 3.5] for large node counts.
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