Constants Alpha Genesis Calibration Forcing Step Eq Sq

The step as a square

The declaration step_eq_sq establishes a small but load-bearing fact inside the Recognition Science framework. It says that for any self-similar dressing, a function that gives the survival fraction under a gap load, the value at a load of one is the square of the value at a load of one half. In symbols, if g is the survival function, then g(1) = g(1/2)². This is not an assumption; it is proved from the factorization premise, which says that the survival fraction for two independent loads multiplies together. Since a load of one can be split into two loads of one half, the survival at one must equal the survival at one half times itself.

That squaring fact is the first step in a chain that forces the step's value. The proof that g(1) = g(1/2)² immediately gives that the step is nonnegative, since a square is never negative. A separate argument shows the step cannot be zero, because that would contradict the self-similar balance equation g(1) = 1/(1 + g(1)). Together these imply the step is strictly positive. The balance equation then has exactly one admissible root, and that root is the reciprocal of the golden ratio, φ⁻¹. The step is derived, not calibrated: no unit convention and no normalization choice enters anywhere.

In Recognition Science, this matters because it removes a residual worry about the genesis of the fine-structure constant. The framework's forward α object, built from a channel budget and a spectral load, is obtained from every self-similar dressing. Since the dressing carries no calibration input, the α object inherits that property. The step equation is one of the structural facts, along with factorization and the self-similar balance, that force the form of the recognition measure itself.

What step_eq_sq does not claim is equally precise. It does not say that the survival function is differentiable, continuous, or even defined everywhere; it only concerns nonnegative loads. It does not assert that the step equals the golden ratio; that is the later theorem step_forced. And it says nothing about the measured value of the fine-structure constant. The declaration is a lemma about a single point, not a statement about the whole function or about experimental physics.

/-- The step is a square: `g(1) = g(1/2)²` (factorization). -/
theorem step_eq_sq : D.g 1 = D.g (1/2) ^ 2 := by
  have h := D.factorizes (1/2) (1/2) (by norm_num) (by norm_num)
  have h1 : (1/2 : ℝ) + 1/2 = 1 := by norm_num
  rw [h1] at h
  rw [h]
  ring

/-- The step is strictly positive (derived, not assumed). -/
theorem step_pos : 0 < D.g 1 :=
  lt_of_le_of_ne D.step_nonneg (Ne.symm D.step_ne_zero)

/-- **STEP FORCING.** The balance equation has exactly one admissible
root: `g(1) = φ⁻¹`. The reciprocal of the step satisfies the T6
self-similarity equation `r² = r + 1`, whose unique positive root is φ. -/
theorem step_forced : D.g 1 = 1 / Constants.phi := by
  set ρ := D.g 1 with hρdef
  have hpos : 0 < ρ := D.step_pos
  have hbal : ρ = 1 / (1 + ρ) := D.step_self_similar
  have hsum_pos : 0 < 1 + ρ := by linarith
  have hsum_ne : (1 + ρ) ≠ 0 := ne_of_gt hsum_pos
  -- ρ² + ρ − 1 = 0
  have hmul : ρ * (1 + ρ) = 1 := by
    calc ρ * (1 + ρ) = (1 / (1 + ρ)) * (1 + ρ) := by rw [← hbal]
    _ = 1 := by field_simp
  have hquad : ρ ^ 2 + ρ - 1 = 0 := by nlinarith [hmul]
  -- 1/φ satisfies the same quadratic
  have hφ : Constants.phi ^ 2 = Constants.phi + 1 := Constants.phi_sq_eq
  have hφpos : 0 < Constants.phi := Constants.phi_pos
  have hφne : Constants.phi ≠ 0 := ne_of_gt hφpos
  have hinv_pos : 0 < 1 / Constants.phi := by positivity
  have hinv_quad : (1 / Constants.phi) ^ 2 + (1 / Constants.phi) - 1 = 0 := by
    field_simp
    nlinarith [hφ]
  -- difference of the two quadratics factors: (ρ − 1/φ)(ρ + 1/φ + 1) = 0
  have hfactor : (ρ - 1 / Constants.phi) * (ρ + 1 / Constants.phi + 1) = 0 := by
    linear_combination hquad - hinv_quad
  have hsum2_pos : 0 < ρ + 1 / Constants.phi + 1 := by linarith
  rcases mul_eq_zero.mp hfactor with h | h
  · linarith [sub_eq_zero.mp h]
  · linarith

What this page does not claim

This does not claim that the survival function is differentiable or continuous. This does not claim that the step equals the golden ratio; that is the separate theorem step_forced. This does not claim any numerical agreement with the measured fine-structure constant.

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/AlphaGenesis/CalibrationForcing.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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