Information Network Topology From Sigma Degree Exponent Gt Two

The degree exponent bound

In the study of complex networks, a scale-free network is one whose degree distribution follows a power law: P(k) ∝ k^(-γ), where k is the number of connections a node has and γ is the degree exponent. The condition γ > 2 is what makes the network scale-free in the strict sense, ensuring that the expected degree remains finite as the network grows. Measured internet topology data typically shows γ between 2.1 and 2.3.

The Recognition Science framework models each attachment step in network growth as a recognition cost decision, where recognition is the forced cost of registering an event in a discrete ledger. From this starting point, the framework derives a prediction for the degree exponent: γ = 1 + φ, where φ is the golden ratio, approximately 2.618. This value satisfies the scale-free condition because it is strictly greater than 2.

The theorem degreeExponent_gt_two in the framework's machine-checked library of formal theorems establishes exactly this bound: it proves that the predicted exponent 1 + φ is greater than 2. The proof is direct, using the fact that φ > 1. A companion theorem pins the value to a band between 2.61 and 2.63, and another shows the equivalent form 2 + 1/φ.

What this theorem does not claim is that the prediction matches the measured internet topology. The measured γ ≈ 2.1-2.3 falls below the predicted 2.618, a discrepancy the framework does not resolve. The theorem only establishes a mathematical property of the predicted value, not empirical agreement with observation.

/-- γ > 2 (scale-free condition). -/
theorem degreeExponent_gt_two : degreeExponent > 2 := by
  unfold degreeExponent
  linarith [one_lt_phi]

/-- γ = 1 + φ ≈ 2.618. -/
theorem degreeExponent_val_band :
    (2.61 : ℝ) < degreeExponent ∧ degreeExponent < 2.63 := by
  unfold degreeExponent
  exact ⟨by linarith [phi_gt_onePointSixOne],
         by linarith [phi_lt_onePointSixTwo]⟩

/-- The Zipf-Pareto exponent identification: γ = 1 + φ = 2 + (φ - 1) = 2 + 1/φ. -/
theorem degreeExponent_eq_two_plus_inv :
    degreeExponent = 2 + phi⁻¹ := by
  unfold degreeExponent
  have h : phi⁻¹ = phi - 1 := by
    have := phi_sq_eq
    field_simp [phi_ne_zero]
    linarith
  linarith

What this page does not claim

The theorem does not claim empirical agreement with measured internet topology data. The theorem does not claim that all scale-free networks must have γ = 2.618. The theorem does not claim that the Recognition Science derivation of the exponent is complete.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/NetworkTopologyFromSigma.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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