gamma_fixed_point
plain-language theorem explainer
The σ-conservation fixed-point equation holds for the network degree exponent γ. Network theorists deriving small-world properties from preferential attachment models cite this when connecting Recognition Science to Barabási-Albert networks. The proof is a one-line term that unfolds the definition of γ as 3 and applies ring normalization to the arithmetic identity.
Claim. The power-law exponent $γ$ in the degree distribution satisfies $(γ-1)(γ-2)=2$.
background
In this module the degree distribution follows $P(k)∝k^{-γ}$ with γ set to 3 as the self-similar fixed point of the φ-recurrence on the recognition graph. The σ-conservation fixed-point equation is the algebraic relation $(γ-1)(γ-2)=2$. The module imports the Euler-Mascheroni constant definition but re-uses the identifier gamma for the network exponent; upstream structures such as PhiForcingDerived.of supply the J-cost background while LedgerFactorization.of calibrates the underlying multiplicative structure.
proof idea
Term-mode proof. It unfolds the definition of gamma (which is 3) and invokes the ring tactic to reduce the left-hand side to the constant 2.
why it matters
The result is invoked directly by networkScience_one_statement to bundle γ=3 with the fixed-point equation and the clusteringRatio_band bounds, and by smallWorldFromSigmaCert to certify the small-world properties. It supplies the concrete algebraic content for the claim that γ=3 is the unique positive solution of the σ-conservation equation, closing the derivation step in Track F9. The fixed-point relation parallels the self-similar fixed-point construction (T6) in the forcing chain.
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