Foundation Alpha Coordinate Fixation Deriv Deriv Deriv Cost Alpha Log Eq

The third derivative

In calculus, the third derivative of a function measures how the curvature itself is changing. For the family of functions CostAlphaLog (a cost, a measure of recognition effort, written in log coordinates), the declaration deriv_deriv_deriv_costAlphaLog_eq states a precise formula: the third derivative equals α * sinh(α * t). Here α is a positive parameter, and t is the log-coordinate variable. The formula is a lemma, proved in the machine-checked library of formal theorems, and it is a stepping stone to a larger result.

The larger result concerns a one-parameter family of cost functions, written as F_α(x) = (1/α²)(cosh(α ln x) - 1). The second derivative of this family at zero is always 1, so it cannot tell the functions apart. The fourth derivative at zero, however, equals α², which does distinguish them. A calibration condition, that the fourth derivative at zero equals 1, therefore forces α² = 1. Combined with the convention α ≥ 1, this pins α = 1. The cost then becomes exactly J(x) = (1/2)(x + 1/x) - 1, the canonical reciprocal cost.

The third-derivative formula itself does not force α = 1. At zero, the third derivative is 0 for every α, so it is blind to the parameter. The declaration merely supplies the derivative computation that the fourth-derivative theorem builds upon. It is a lemma in a chain, not the headline result.

In Recognition Science, this chain of results closes a freedom in the framework. The branch-selection theorem reduces the possible cost functions to the one-parameter family above. The higher-derivative calibration then selects α = 1, making J the unique calibrated cost on that branch. This is one of three candidate routes to fixing α; the other two remain open as targets for future modules.

private lemma deriv_deriv_deriv_costAlphaLog_eq (α : ℝ) (hα : α ≠ 0) :
    deriv (deriv (deriv (CostAlphaLog α))) = fun t => α * sinh (α * t) :=
  funext fun t => (hasDerivAt_costAlphaLog_third α hα t).deriv

/-- **The fourth derivative of `CostAlphaLog α` at zero is `α²`.**

This is the calibration invariant that distinguishes different `α` values
within the bilinear family: the second derivative `G_α''(0) = 1` is
constant, but the fourth derivative `G_α^(4)(0) = α²` separates them. -/
theorem costAlphaLog_fourth_deriv_at_zero (α : ℝ) (hα : α ≠ 0) :
    deriv (deriv (deriv (deriv (CostAlphaLog α)))) 0 = α ^ 2 := by
  have := (hasDerivAt_costAlphaLog_fourth α hα 0).deriv
  rw [this]
  simp [mul_zero, cosh_zero]

/-- **The α-pin theorem.** Within the bilinear `α`-family with the
rigidity-paper convention `α ≥ 1`, higher-derivative calibration forces
`α = 1`. -/
theorem alpha_pin_under_high_calibration
    (α : ℝ) (h_pos : 1 ≤ α)
    (h_calib : IsHighCalibratedLog (CostAlphaLog α)) :
    α = 1 := by
  have hα_ne : α ≠ 0 := by linarith
  have hα_sq : α ^ 2 = 1 :=
    (costAlphaLog_high_calibrated_iff α hα_ne).mp h_calib
  -- α ≥ 1 and α² = 1 forces α = 1.
  nlinarith

/-- The `α = 1` bilinear cost is exactly `Jcost`. -/
theorem alpha_pinned_to_one_implies_J (x : ℝ) (hx : 0 < x) :
    CostAlpha 1 x = Jcost x :=
  cost_alpha_one_eq_jcost x hx

What this page does not claim

The third derivative formula alone does not force α = 1. The higher-derivative calibration is not the only route to fixing α; two alternatives remain open. The declaration does not derive the fine-structure constant α from first principles.

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/AlphaCoordinateFixation.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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