Constants Ilg Alpha Locked Pos

The locked alpha constant

The fine-structure constant α is a dimensionless number, about 1/137.036, that sets the strength of electromagnetic interactions between charged particles. In the Recognition Science framework, a related quantity is defined as alpha_locked, a number fixed by the golden ratio φ: α_locked = (1 − 1/φ)/2. Since φ ≈ 1.618, this evaluates to roughly 0.191. The framework's machine-checked library of formal theorems proves two facts about this number: it is positive, and it is less than one. That is all the lemma alpha_locked_pos establishes, and it is a narrow, precise claim.

The positive bound is proved from the positivity of φ and the algebraic structure of the definition. The less-than-one bound follows from φ being greater than one. These are internal consistency checks, not empirical claims. They guarantee that the framework's constant sits in the interval (0,1), which matters because the framework uses this number in further derivations; a number outside that range would break later steps. The proof is formal and machine-checked, meaning the reasoning is verified step by step by a computer, with no gaps in the logic.

What the lemma does not claim is any connection to the measured fine-structure constant. The value 0.191 is far from 1/137.036 ≈ 0.007297. The framework has a separate, much more involved expression that lands within a few parts per million of the measured value, but that expression is not this lemma, and its seed is an identification, not a derived coupling. The exact derivation of α from first principles remains open. The lemma alpha_locked_pos is a small, correct piece of mathematics, and it should not be mistaken for a physical result.

For a reader, the practical takeaway is simple: the framework defines a number, proves it lies between zero and one, and stops there. Any claim that this number is the fine-structure constant, or that its positivity proves anything about electromagnetism, goes beyond what the lemma supports. The lemma is a building block, not a headline.

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

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 lemma does not claim alpha_locked equals the measured fine-structure constant. The lemma does not claim any physical significance for the number 0.191. The lemma does not claim the exact fine-structure constant is derived.

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.

Derived articles

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