Foundation Alpha Coordinate Fixation

Pinning the cost function

In Recognition Science, a ledger is a discrete record of recognition events, and the cost measures the price of a recognition event. The central forcing theorem shows any cost satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. But a branch-selection step left one parameter, called α, free: a family of cost functions F_α(x) = (1/α²)(cosh(α ln x) - 1) all satisfied the earlier constraints. The second derivative at the origin is 1 for every α, so the usual calibration could not tell them apart.

Alpha-coordinate fixation closes that gap. The module formalizes a higher-derivative calibration: it demands the fourth derivative of the cost in log coordinates at zero equal 1. For the family above, the fourth derivative at zero is exactly α². Setting that equal to 1 forces α² = 1, and with the convention α ≥ 1, this pins α = 1. The resulting cost is precisely J, the canonical reciprocal cost. The theorem J_uniquely_calibrated_via_higher_derivative states this uniqueness: within the bilinear family, under α ≥ 1, higher-derivative calibration forces the cost to equal J on all positive reals.

This is one of three candidate routes to fix α listed in the branch paper. The other two, generator calibration and action-functional minimisation, remain open targets for separate modules. The higher-derivative route was chosen because the existing calibration framework already works through derivatives of the cost, so this extension requires the smallest new infrastructure. The module packages the result as a certificate structure, assembled from the derivative lemmas, the calibration equivalence, the pin theorem, and the identification with J, all machine-checked in the framework's library of formal theorems.

What this changes: the α-coordinate freedom is closed. The framework's cost function is no longer one member of a family but the unique calibrated cost on the bilinear branch. That uniqueness is what lets later results in the forcing chain, such as the golden ratio and the eight-tick cycle, rest on a single fixed foundation rather than a parameter choice.

/-- **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 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

def alphaCoordinateFixationCert : AlphaCoordinateFixationCert where
  fourth_deriv_eq_alpha_sq := costAlphaLog_fourth_deriv_at_zero
  high_calibrated_iff := costAlphaLog_high_calibrated_iff
  alpha_pin := alpha_pin_under_high_calibration
  alpha_one_is_J := alpha_pinned_to_one_implies_J
  J_unique_under_high_calibration := J_uniquely_calibrated_via_higher_derivative

What this page does not claim

This module does not prove that the other two α-fixation routes are impossible or equivalent. This module does not claim that the fourth-derivative calibration is the only natural one, only that it is a valid route to J. This module does not address the physical interpretation of α or its possible role beyond the cost function.

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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