Foundation Alpha Coordinate Fixation J Uniquely Calibrated Via Higher Derivative

The uniqueness theorem

The declaration J_uniquely_calibrated_via_higher_derivative is a machine-checked theorem in the framework's library of formal proofs. It states that among a specific family of candidate cost functions, only one survives a calibration condition, and that survivor is exactly the canonical cost J(x) = (1/2)(x + 1/x) − 1. In plain terms: if the system tracks records for the universe, the ledger of recognition events has a fixed price structure, and this theorem pins down which one it is.

The family in question is the bilinear α-family, written F_α(x) = (1/α²)(cosh(α ln x) − 1), for α ≥ 1. Each member is a candidate for the cost function that measures recognition effort. The second derivative at zero is the same for every α, so it cannot tell the candidates apart. The fourth derivative at zero, however, equals α². Setting G^(4)(0) = 1 forces α² = 1, and with α ≥ 1, that gives α = 1. The theorem then shows that F_1(x) equals Jcost(x) for every positive x.

This is one of three candidate routes to fixing α listed in the branch-selection document. The other two, generator calibration and action-functional minimisation, remain open targets. The higher-derivative route was chosen because the existing calibration framework already works through derivatives of G, so this extension required the smallest new infrastructure.

The theorem's scope is precise. It operates within the bilinear α-family, under the convention α ≥ 1, and with the higher-derivative calibration G^(4)(0) = 1. It does not claim to derive the cost function from scratch; it assumes the branch-selection step that reduced the problem to this family. It also does not claim that the fourth-derivative condition is the only possible calibration, only that within this family and this convention, it selects J uniquely.

/-- **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

This theorem does not derive J from the five founding conditions alone; it assumes the branch-selection step. It does not claim that the fourth-derivative calibration is the only possible way to fix α. It does not address the other two candidate fixations from the branch document, which remain open.

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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