Foundation Dalembert Wlogalpha One Deriv Cost Alpha Log Eq

The rescaling theorem

The declaration deriv_costAlphaLog_eq states the derivative of the α-parameterized cost function in log coordinates: it equals sinh(α·t)/α. Here α is a positive scaling parameter and t is the logarithmic coordinate. The cost function itself, written G_α(t) = (1/α²)(cosh(α·t) − 1), measures the price of a recognition event in the framework's ledger, a discrete record of events where each entry carries a forced cost.

This derivative is a stepping stone to a larger result. The framework's machine-checked library of formal theorems proves that every calibrated cost function F_α(x) = (1/α²)(cosh(α·ln x) − 1) can be rewritten as (1/α²)·J(x^α), where J(x) = (x + x⁻¹)/2 − 1 is the canonical cost. The map x ↦ x^α is a group automorphism of the positive reals under multiplication, so α merely reparametrizes the coordinate. Setting α = 1 recovers J exactly. The unit-curvature condition G_α''(0) = 1 holds for every α ≠ 0, so calibration does not single out any particular α.

The theorem wlog_alpha_eq_one packages these facts: for any positive α, the rescaled cost equals (1/α²)·J(x^α), α = 1 gives J, the rescaling respects multiplication, and the second derivative at zero is always 1. The conclusion is that α introduces no structurally new cost function. Without loss of generality, one may assume α = 1.

What this does not claim: it does not say that α is physically meaningless, only that it is a coordinate choice. It does not assert that the canonical cost J is the only possible cost function; that is a separate theorem requiring the five axioms. It does not claim that the derivative formula holds for α = 0, which is excluded by the hypothesis. The declaration is a technical lemma, not a standalone physical law.

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

/-- **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

/-- Setting α = 1 gives F₁(x) = J(x) for x > 0. -/
theorem cost_alpha_one_eq_jcost (x : ℝ) (hx : 0 < x) :
    CostAlpha 1 x = Jcost x := by
  rw [cost_alpha_rescaling 1 x hx]
  simp [rpow_one]

/-- **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]

What this page does not claim

The declaration does not prove that α is physically meaningless, only that it is a coordinate choice. It does not assert that J is the only possible cost function; that requires the five axioms. It does not claim the derivative formula holds for α = 0.

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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