Constants Ilg

The ILG constants

The constants ilg module is a small machine-checked library of formal theorems that defines two real numbers and proves basic facts about them. The first, called Clag, is defined as 1 divided by the golden ratio raised to the fifth power. The golden ratio is the number φ ≈ 1.618, the classical solution to r² = r + 1. So Clag is about 0.09017. The second number, alpha_locked, is defined as (1 − 1/φ)/2, which evaluates to about 0.19098.

In the Recognition Science framework, these are not free parameters chosen to fit data. They emerge from a forcing chain: a sequence of proved theorems starting from a single cost function J(x) = (x + 1/x)/2 − 1. That cost function is forced by five plain conditions, and from it the framework derives φ as the unique self-similar scaling, then an eight-tick recognition cycle, then 2³, then three spatial dimensions. The constants Clag and alpha_locked sit at the end of that chain, expressed in native units where hbar = φ⁻⁵ and G = φ⁵/π.

What the module itself establishes is narrow but clean. It proves that Clag is positive, that alpha_locked is positive, and that alpha_locked is less than 1. These are the kind of sanity checks a formal library needs before any physical interpretation: the numbers live in the expected ranges. The module does not prove that these constants match any measurement; that comparison is an empirical check outside the formal theorems.

In plain language, the module pins down two numbers that the framework claims are structural, not chosen. Clag plays the role of a quantum scale, the inverse of the fifth power of φ. Alpha_locked is a locked coupling-like value, half of one minus the reciprocal of φ. Both are positive and below one, which is what a scale and a coupling should be.

The payoff for a reader is seeing how far the forcing chain reaches: from a single cost function to specific numerical constants, all in a machine-checked library with no hidden axioms. The classical golden ratio, known since Euclid for its role in the pentagon and Fibonacci numbers, here reappears as the root of physical scales. The module is a small but load-bearing piece of that larger derivation.

@[simp] noncomputable def Clag : ℝ := 1 / (phi ^ (5 : Nat))

@[simp] noncomputable def alpha_locked : ℝ := (1 - 1 / phi) / 2

lemma Clag_pos : 0 < Clag := by
  have hφ : 0 < phi := phi_pos
  have hpow : 0 < phi ^ (5 : Nat) := pow_pos hφ 5
  simpa [Clag, one_div] using inv_pos.mpr hpow

lemma alpha_locked_pos : 0 < alpha_locked := by
  dsimp [alpha_locked]
  have hφ : 0 < phi := phi_pos
  have hφ_gt_1 : 1 < phi := one_lt_phi
  -- 1/φ < 1 because φ > 1
  have hinv_lt_one : 1 / phi < 1 := by
    rw [div_lt_one hφ]
    exact hφ_gt_1
  have hsub : 0 < 1 - 1 / phi := by
    linarith
  have hdiv : 0 < (1 - 1 / phi) / 2 := by
    apply div_pos hsub
    exact zero_lt_two
  exact hdiv

lemma alpha_locked_lt_one : alpha_locked < 1 := by
  dsimp [alpha_locked]
  have hφ_pos : 0 < phi := phi_pos
  have hφ : 1 < phi := one_lt_phi
  -- We need to show: (1 - 1/φ) / 2 < 1
  -- Since φ > 1, we have 0 < 1/φ < 1, so 0 < 1 - 1/φ < 1, so (1 - 1/φ)/2 < 1/2 < 1
  have hinv_pos : 0 < 1 / phi := div_pos one_pos hφ_pos
  have hinv_lt_one : 1 / phi < 1 := by
    rw [div_lt_one hφ_pos]
    exact hφ
  have hsub_lt : 1 - 1 / phi < 1 := by
    have : 0 < 1 / phi := hinv_pos
    linarith
  have hdiv_lt : (1 - 1 / phi) / 2 < 1 / 2 := by
    apply div_lt_div_of_pos_right hsub_lt
    exact zero_lt_two
  have half_lt_one : (1 : ℝ) / 2 < 1 := by norm_num
  linarith

What this page does not claim

The module does not prove these constants match any measured physical value. The module does not derive the fine-structure constant alpha; alpha_locked is a different, framework-internal number. The module does not establish the physical recognition-to-linking bridge for three dimensions.

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/Constants/ILG.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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