Information Network Topology From Sigma Degree Exponent Val Band

The predicted exponent band

In the study of complex networks, a scale-free network is one whose degree distribution, the count of connections per node, follows a power law: P(k) ∝ k^(-γ). The exponent γ typically falls between 2 and 3 for real-world networks like the internet, social graphs, and metabolic pathways. A specific theorem in the Recognition Science framework's machine-checked library of formal theorems establishes that its predicted value for this exponent lies in a narrow band: γ is strictly greater than 2.61 and strictly less than 2.63. The theorem, named degreeExponent_val_band, is a formal proof that the defined quantity 1 + φ, where φ is the golden ratio, satisfies this inequality.

The number itself is easy to compute. The golden ratio φ ≈ 1.618, so the predicted exponent γ = 1 + φ ≈ 2.618. This sits squarely in the band (2.61, 2.63). A second theorem in the same file proves the exponent is greater than 2, which is the defining condition for a scale-free network. A third proves the equivalent form γ = 2 + 1/φ, highlighting the relationship to the golden ratio's reciprocal. These are all formal consequences of the definition, not empirical measurements.

The framework's derivation, stated in the accompanying documentation, models each attachment step in network growth as a recognition cost decision. In this account, a ledger, a discrete record of events, tracks these decisions, and a forced cost function drives the exponent to 2 + J(φ)/J(φ) = 2 + 1/φ. This derivation is a hypothesis about how real networks form, not a proven theorem. The measured γ for the internet and other scale-free networks is often cited around 2.1 to 2.3, a range that does not overlap with the predicted band of 2.61 to 2.63.

What the theorem itself establishes is narrow and precise: a defined real number falls within a specific interval. It does not claim that real networks have this exponent, nor that the derivation from recognition costs is physically validated. The gap between the predicted 2.618 and the commonly measured 2.1 to 2.3 remains an open empirical question. The framework's contribution here is a clean, checkable mathematical statement about the golden ratio, not a settled fact about the internet's structure.

/-- 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]⟩

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

/-- 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 that any real-world network has a degree exponent in the band (2.61, 2.63). The derivation from recognition costs is a hypothesis, not a proved theorem about physical network formation. The theorem does not address the measured range of γ ≈ 2.1 to 2.3 often reported for scale-free networks.

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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