gamma_pos
The theorem establishes positivity of the power-law degree-distribution exponent set to 3. Network scientists working on scale-free graphs and small-world properties would cite it to anchor the Barabási-Albert exponent inside the Recognition Science derivation. The proof is a one-line term that unfolds the constant definition and applies norm_num.
claim$0 < 3$, where $3$ is the power-law exponent in the degree distribution $P(k) ∝ k^{-3}$ arising as the fixed point of the σ-conservation equation.
background
In the NetworkScience.SmallWorldFromSigma module the power-law exponent is introduced as the constant 3, obtained as the unique positive solution of the fixed-point equation (γ − 1)(γ − 2) = 2 that encodes σ-conservation on the recognition graph. The module derives the Barabási-Albert preferential-attachment model from φ-recurrence, yielding both this exponent and the small-world scaling of average path length as log_φ N together with a clustering ratio of 1/φ. Upstream results supply the local definition of the exponent as 3 and the positivity theorem for the Euler-Mascheroni constant, although the network exponent remains independent of that constant.
proof idea
The proof is a one-line wrapper that unfolds the definition of the exponent to the numeral 3 and invokes norm_num to confirm the strict inequality over the reals.
why it matters in Recognition Science
The result supplies the elementary positivity fact required for the small-world certification SmallWorldFromSigmaCert and for the Euler-Mascheroni bounds that appear in the Constants module. It fills the network-science step of the φ-recurrence track (F9), confirming that the self-similar fixed point at T6 produces a positive exponent compatible with the eight-tick octave and D = 3. The open question it touches is the full Recognition Science derivation of the Euler-Mascheroni constant itself, which remains blocked on the Riemann hypothesis.
scope and limits
- Does not derive the exponent value from the φ-recurrence or σ-conservation equation.
- Does not relate the network exponent to the Euler-Mascheroni constant.
- Does not prove uniqueness of the fixed-point solution.
- Does not address clustering coefficients or path-length scaling.
formal statement (Lean)
43theorem gamma_pos : 0 < gamma := by unfold gamma; norm_num
proof body
Term-mode proof.
44
45/-- The σ-conservation fixed-point equation: `(γ - 1)(γ - 2) = 2`. -/