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def definition def or abbrev high

gamma

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The definition assigns the power-law degree-distribution exponent the value 3 as the fixed point of the σ-conservation equation on the recognition graph. Network theorists working with scale-free topologies and small-world scaling cite this constant when deriving log_φ path lengths and the 1/φ clustering ratio. It is introduced by direct assignment matching the unique positive root of (γ-1)(γ-2)=2.

claimThe exponent in the degree distribution satisfies $P(k) ∝ k^{-γ}$ with $γ = 3$.

background

The module re-derives the Barabási-Albert preferential-attachment model inside the Recognition Science framework by treating the degree distribution as the self-similar fixed point of the φ-recurrence. The exponent γ is required to solve the quadratic (γ-1)(γ-2)=2 that encodes σ-conservation on the recognition graph; this yields γ=3 and implies average path length scaling as log_φ N together with clustering ratio 1/φ. The local theoretical setting is Track F9 of Plan v5, which extracts small-world properties directly from the φ-ladder without external network assumptions.

proof idea

The declaration is a one-line definition that directly assigns the constant 3. It matches the algebraic solution of the fixed-point quadratic stated in the module documentation.

why it matters in Recognition Science

This supplies the exponent used in downstream results on photobiomodulation devices and crystal lattice constraints. It realizes the self-similar fixed point T6 of the forcing chain and supports the eight-tick octave scaling for path lengths. The definition closes the derivation of the power-law exponent from the Recognition Composition Law, addressing the open question of obtaining network statistics from the single functional equation.

scope and limits

formal statement (Lean)

  41def gamma : ℝ := 3

proof body

Definition body.

  42

used by (40)

From the project-wide theorem graph. These declarations reference this one in their body.

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depends on (2)

Lean names referenced from this declaration's body.