Foundation Alpha Coordinate Fixation Alpha Pinned To One Implies J

Pinning the cost function

A cost function assigns a price to every possible comparison between two quantities. In the Recognition Science framework, the cost of a recognition event is forced, not chosen, and the central object is the function J(x) = (1/2)(x + 1/x) − 1. The declaration alpha_pinned_to_one_implies_J is the final step in a chain that isolates this J from a whole family of candidate functions, using a condition on the fourth derivative.

The story begins with a family of candidate cost functions written in log coordinates: F_α(x) = (1/α²)(cosh(α ln x) − 1), with α ≥ 1. Each choice of the parameter α gives a different cost function, and the framework needs a principle to pick one. The usual calibration condition, which fixes the second derivative at zero to be 1, is blind to α: G''(0) = 1 holds for every α. The framework's library proves that the fourth derivative at zero is α², so the condition G^(4)(0) = 1 forces α² = 1, and with α ≥ 1 this pins α = 1 exactly.

Once α is pinned to 1, the cost function F_1(x) is exactly J(x) = (1/2)(x + 1/x) − 1. The theorem alpha_pinned_to_one_implies_J states this equivalence for all positive x. The full uniqueness theorem, J_uniquely_calibrated_via_higher_derivative, extends this: within the bilinear family, under the convention α ≥ 1, the higher-derivative calibration forces α = 1, and the cost on positive reals is exactly Jcost. This closes the α-coordinate freedom that the second-derivative calibration left open.

This is one of three candidate routes to pin α, listed in the branch paper's section 5. The other two, generator calibration and action-functional minimisation, remain open targets for separate modules. This route was chosen because the existing IsCalibrated framework already routes through derivative-of-G calibration, so the extension has the smallest infrastructure surface. The machine-checked library of formal theorems verifies every derivative computation and the final equivalence.

What this does not claim: it does not claim that the higher-derivative calibration is the only way to reach J, nor that the other two candidate fixations are impossible. It does not claim that the α-family itself is derived from first principles; the bilinear branch is produced by a branch-selection theorem, and the convention α ≥ 1 is a paper choice. The theorem operates within the bilinear family only, and the cost is identified on positive reals.

/-- **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-cost is high-calibrated iff `α² = 1`. -/
theorem costAlphaLog_high_calibrated_iff (α : ℝ) (hα : α ≠ 0) :
    IsHighCalibratedLog (CostAlphaLog α) ↔ α ^ 2 = 1 := by
  unfold IsHighCalibratedLog
  rw [costAlphaLog_fourth_deriv_at_zero α hα]

/-- **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 higher-derivative calibration is not claimed to be the only route to J; two other candidate fixations remain open. The bilinear α-family itself is not derived from first principles; it is produced by a branch-selection theorem. The convention α ≥ 1 is a paper choice, not a theorem consequence.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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