Foundation Dalembert Wlogalpha One Has Deriv At Sinh Div Alpha

The rescaling lemma

The declaration hasDerivAt_sinhDivAlpha is a lemma in the machine-checked library of formal theorems. It states that the derivative of the function s ↦ sinh(αs)/α is cosh(αt) at the point t, for any real α ≠ 0. In plain words, it computes how fast the hyperbolic sine of a scaled input changes as the input varies.

This lemma is a technical step in a larger proof about cost functions. In the framework, a cost function assigns a number to a recognition event, measuring how expensive it is for reality to keep a discrete record of that event. The family of calibrated costs is F_α(x) = (1/α²)(cosh(α ln x) − 1), with α ≥ 1. The lemma shows that the second derivative of the log-coordinate version of F_α at zero equals 1 for every α ≠ 0. This is the calibration invariance property: the unit-curvature condition does not depend on α.

The consequence is the WLOG α = 1 theorem. Since F_α(x) = (1/α²)J(x^α), where J(x) = (x + x⁻¹)/2 − 1, and the map x ↦ x^α is a group automorphism of the positive reals under multiplication, α merely reparametrizes the coordinate. Setting α = 1 recovers J exactly. The lemma is the derivative calculation that makes the calibration check go through for all α at once.

What the lemma does not claim is that α itself is derived or forced by the framework. It is a parameter in a family of cost functions, and the lemma shows that the family collapses to a single structural cost J under rescaling. The lemma does not prove that α must be 1; it proves that assuming α = 1 loses no generality. It also does not claim anything about the physical value of α or about the fine-structure constant, which remains an open identification.

private lemma hasDerivAt_sinhDivAlpha (α : ℝ) (hα : α ≠ 0) (t : ℝ) :
    HasDerivAt (fun s => sinh (α * s) / α) (cosh (α * t)) t := by
  have h1 : HasDerivAt (fun s => sinh (α * s)) (cosh (α * t) * α) t :=
    (hasDerivAt_sinh (α * t)).comp t (hasDerivAt_alpha_mul α t)
  convert h1.div_const α using 1
  field_simp

/-- **Calibration Invariance**: G_α''(0) = 1 for every α ≠ 0.
    The unit-curvature condition is independent of the rescaling parameter. -/
theorem costAlphaLog_unit_curvature (α : ℝ) (hα : α ≠ 0) :
    deriv (deriv (CostAlphaLog α)) 0 = 1 := by
  rw [deriv_costAlphaLog_eq α hα, (hasDerivAt_sinhDivAlpha α hα 0).deriv,
      mul_zero, cosh_zero]

/-- **Rescaling Identity**: F_α(x) = (1/α²) · J(x^α). -/
theorem cost_alpha_rescaling (α x : ℝ) (hx : 0 < x) :
    CostAlpha α x = (1 / α ^ 2) * Jcost (x ^ α) := by
  unfold CostAlpha CostAlphaLog
  congr 1
  exact cosh_log_eq_jcost_rpow α x hx

What this page does not claim

The lemma does not prove that α must equal 1; it proves that assuming α = 1 loses no generality. The lemma does not derive the fine-structure constant or any physical coupling constant. The lemma does not establish the full uniqueness theorem for J; it is a component of that proof.

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

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Derived articles

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