Foundation Alpha Coordinate Fixation Cost Alpha Log Fourth Deriv At Zero

The fourth derivative at zero

The fourth derivative of a function measures how sharply its curvature itself changes. For the family of functions F_α(x) = (1/α²)(cosh(α ln x) − 1), each member is a candidate cost function in a framework where recognition events carry a forced price. The parameter α stretches the log-coordinate scale. The second derivative of every member at the point x = 1 equals 1, so that standard calibration cannot distinguish one α from another. The fourth derivative at the same point, however, equals α², a value that changes with α. The theorem costAlphaLog_fourth_deriv_at_zero states this fact exactly: the fourth derivative of the α-cost at zero is α².

This single identity becomes a selection rule. The framework's ledger, a discrete record of recognition events, requires a cost function that satisfies a calibration condition. The second-derivative condition leaves a one-parameter family of possibilities. The fourth-derivative condition, G⁽⁴⁾(0) = 1, combined with the convention α ≥ 1, forces α² = 1 and hence α = 1. The theorem alpha_pin_under_high_calibration proves this pinning. With α fixed at 1, the cost function reduces exactly to J(x) = (1/2)(x + x⁻¹) − 1, the canonical reciprocal cost that anchors the framework. The declaration J_uniquely_calibrated_via_higher_derivative states this uniqueness: within the bilinear family, under the higher-derivative calibration, the cost on positive reals is exactly J.

What this does not claim matters as much as what it proves. The theorem does not say that the fourth-derivative condition is the only way to fix α. The branch paper lists three candidate fixations; the other two, generator calibration and action-functional minimisation, remain open targets for separate formalisation. The theorem also does not claim that the fourth derivative of every cost function equals α²; it applies specifically to the bilinear family defined above. Finally, the theorem does not assert that the fourth-derivative condition is physically necessary. It is a definitional choice, a calibration convention, not a derived physical law. The framework's library proves that if one adopts this convention, then α = 1 and J is the unique cost; it does not prove that nature must adopt it.

The consequence is a closed coordinate freedom. The branch-selection chain now reads: from the general cost family, branch selection yields the bilinear α-family; second-derivative calibration is blind to α; fourth-derivative calibration pins α = 1; and the cost is J. The α-coordinate freedom is closed, and J is the unique calibrated cost on the bilinear branch under this combined encoding. A reader can now see exactly which calibration convention selects the canonical cost, and which alternatives remain open.

/-- **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

/-- **The full uniqueness theorem.** Within the bilinear `α`-family,
under the convention `α ≥ 1`, higher-derivative calibration forces
`α = 1`, and the cost on positive reals is exactly `Jcost`. -/
theorem J_uniquely_calibrated_via_higher_derivative
    (α : ℝ) (h_pos : 1 ≤ α)
    (h_calib : IsHighCalibratedLog (CostAlphaLog α)) :
    ∀ x : ℝ, 0 < x → CostAlpha α x = Jcost x := by
  intro x hx
  have hα_eq : α = 1 := alpha_pin_under_high_calibration α h_pos h_calib
  rw [hα_eq]
  exact cost_alpha_one_eq_jcost x hx

What this page does not claim

The fourth derivative of every cost function equals α²; this holds only for the bilinear family CostAlphaLog. The fourth-derivative condition is the only way to pin α; two other candidate fixations remain open. The fourth-derivative calibration is physically necessary; it is a definitional choice within the framework.

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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