gamma_fixed_point
plain-language theorem explainer
The declaration verifies that the network degree exponent gamma satisfies the quadratic fixed-point equation (gamma - 1)(gamma - 2) = 2. Recognition Science researchers deriving small-world properties from phi-recurrence on the recognition graph cite it to anchor the Barabasi-Albert exponent at 3. The proof is a one-line term that unfolds the definition of gamma and reduces the identity via the ring tactic.
Claim. The network degree exponent $gamma$ satisfies the sigma-conservation fixed-point equation $(gamma - 1)(gamma - 2) = 2$.
background
The module re-derives the Barabasi-Albert preferential-attachment model from the phi-recurrence on the recognition graph, producing the power-law degree distribution $P(k) propto k^{-gamma}$ with gamma set to 3. The sigma-conservation fixed-point equation is the algebraic relation that gamma must obey for self-similarity. Upstream results include the J-cost structure from PhiForcingDerived.of and the ledger factorization from DAlembert.LedgerFactorization.of, which calibrate the recognition costs that generate the recurrence; the module also imports the Euler-Mascheroni constant but treats gamma as a distinct network exponent.
proof idea
The proof is a term-mode one-liner. It unfolds the definition of gamma and applies the ring tactic to reduce the resulting arithmetic identity to reflexivity.
why it matters
This result is invoked directly in networkScience_one_statement to bundle gamma = 3 with the fixed-point equation and the clustering-ratio band, and again in smallWorldFromSigmaCert to certify the small-world properties. It fills the fixed-point step in the phi-recurrence derivation of scale-free networks, linking to the self-similar fixed point (T6) and the eight-tick octave. It touches the open empirical question of whether real networks with more than 10^5 nodes exhibit best-fit exponents strictly inside [2.5, 3.5].
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