Information Network Topology From Sigma Network Topology Cert

The certificate

In the study of large networks, a scale-free network is one whose distribution of connections follows a power law: the fraction of nodes with k connections is proportional to k to the power of minus gamma. The exponent gamma is the number that describes how heavy the tail of the distribution is. Measured values for real networks, such as the internet or social graphs, typically fall between about 2.1 and 2.3.

The Recognition Science framework predicts a specific value for this exponent: gamma equals 1 plus the golden ratio phi, which is about 2.618. The framework derives this from its model of recognition cost: each new attachment to a network is treated as a decision with a forced cost, and the structure of that cost forces the exponent. This prediction is not a theorem about real networks; it is a hypothesis about them, and the certificate does not change that.

In Recognition Science, the machine-checked library of formal theorems contains a structure called NetworkTopologyCert. A structure in this context is a bundle of proved facts. The certificate packages three facts about the predicted exponent. First, it states that the exponent lies strictly between 2.61 and 2.63. Second, it states that the exponent is greater than 2, which is the defining condition for a scale-free network. Third, it states that the exponent equals 2 plus the reciprocal of phi, which is the same as 1 plus phi.

These three facts are proved in the framework's library, with no unproved assumptions. The certificate is a way to hold the prediction together: anyone can look at the single object and see the band, the scale-free condition, and the exact algebraic form. It is a compact record of what the framework claims, not a measurement of any actual network.

What the certificate does not claim is just as important. It does not say that real networks have this exponent; that is an empirical question, and the certificate contains no data. It also does not say that the derivation from recognition cost to the exponent is complete in the library; the derivation is described informally, not fully formalized. The certificate is a statement about the framework's internal prediction, not about the world.

/-- 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 certificate does not prove that any real network has the predicted exponent. The certificate does not contain a complete formal derivation from recognition cost to the exponent. The certificate does not claim that the measured gamma values are wrong.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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