Foundation Dalembert Wlogalpha One Cost Alpha One Eq Jcost

The rescaling identity

In the Recognition Science framework, a cost is a measure of the price of recognizing one state from another, a discrete record of events that reality keeps. The framework's central theorem proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 − 1. Before that theorem applies, however, the framework considers a family of candidate costs, each indexed by a positive parameter α.

For each α, the candidate cost is F_α(x) = (1/α²)(cosh(α ln x) − 1). This is not a new family of functions; it is the same canonical cost J, but with the multiplicative coordinate x rescaled by the power x^α. The declaration cost_alpha_one_eq_jcost establishes the simplest case: when α = 1, the candidate cost F₁(x) equals J(x) exactly for all positive x. In plain language, the parameter α does not introduce a structurally new cost; it merely reparametrizes the coordinate system.

The proof is a direct computation. The identity cosh(α ln x) − 1 = J(x^α) holds for x > 0, and setting α = 1 gives cosh(ln x) − 1 = J(x). The framework's machine-checked library of formal theorems verifies this chain of equalities. The broader statement, wlog_alpha_eq_one, packages four facts: the rescaling identity, the α = 1 recovery, the group automorphism property (xy)^α = x^αy^α, and the calibration invariance that the second derivative at zero equals 1 for every α.

What the declaration does not claim is that α itself is derived or forced by the framework's axioms. The parameter α is a free choice in the candidate family; the theorem only shows that every choice leads back to the same canonical cost under rescaling. The framework does not predict a specific value for α from its postulates. The uniqueness of J as the cost function is a separate theorem, proved under the five conditions, and this rescaling identity is a supporting lemma in that larger argument.

The practical consequence is that the framework's cost function is stable: no matter how one scales the coordinate system, the underlying structure is unchanged. This is what makes the later results, such as the golden ratio as the unique self-similar scaling, depend only on the canonical form J and not on any arbitrary choice of units or coordinates.

/-- 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]

/-- Core identity: cosh(α log x) − 1 = J(x^α) for x > 0.
    Proof uses x^α = exp(α log x), then Jcost ∘ exp = cosh − 1. -/
theorem cosh_log_eq_jcost_rpow (α x : ℝ) (hx : 0 < x) :
    cosh (α * log x) - 1 = Jcost (x ^ α) := by
  have h : x ^ α = exp (α * log x) := by
    rw [rpow_def_of_pos hx, mul_comm]
  rw [h, ← Jcost_exp_cosh]

/-- **WLOG α = 1**: Every calibrated cost F_α is the canonical cost J under
    coordinate rescaling. The parameter α does not introduce a structurally
    new cost function.

    Components:
    1. Rescaling identity: F_α(x) = (1/α²) · J(x^α)
    2. Recovery: F₁(x) = J(x)
    3. Group automorphism: (xy)^α = x^α · y^α
    4. Calibration invariance: G_α''(0) = 1 -/
theorem wlog_alpha_eq_one (α : ℝ) (hα : 0 < α) :
    (∀ x : ℝ, 0 < x → CostAlpha α x = (1 / α ^ 2) * Jcost (x ^ α))
    ∧ (∀ x : ℝ, 0 < x → CostAlpha 1 x = Jcost x)
    ∧ (∀ x y : ℝ, 0 < x → 0 < y → (x * y) ^ α = x ^ α * y ^ α)
    ∧ deriv (deriv (CostAlphaLog α)) 0 = 1 :=
  ⟨fun x hx => cost_alpha_rescaling α x hx,
   fun x hx => cost_alpha_one_eq_jcost x hx,
   fun _ _ hx hy => rpow_mul_hom' α hx hy,
   costAlphaLog_unit_curvature α hα.ne'⟩

/-- **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 parameter α is not derived from the framework's axioms; it is a free choice in the candidate family. The declaration does not prove the uniqueness of J; that is a separate theorem under the five conditions. The rescaling identity does not imply that all costs are equivalent; only that the α-parameterized family reduces to the canonical form.

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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