`Composition`

definition

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module: IndisputableMonolith.Foundation.CostAxioms
domain: Foundation
line: 68 · github
papers citing: none yet

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IndisputableMonolith.Foundation.CostAxioms on GitHub at line 68.

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formal source

  65
  66This is the d'Alembert functional equation in multiplicative form.
  67It forces F to be compatible with the multiplicative structure of ℝ₊. -/
  68class Composition (F : ℝ → ℝ) : Prop where
  69  dAlembert : ∀ x y : ℝ, 0 < x → 0 < y →
  70    F (x * y) + F (x / y) = 2 * F x * F y + 2 * F x + 2 * F y
  71
  72/-- **Axiom 3 (Calibration)**: The second derivative at the origin (in log coordinates) equals 1.
  73
  74Let G(t) = F(exp(t)). Then G''(0) = 1.
  75
  76This normalizes the "curvature" of the cost functional at unity,
  77ensuring a unique solution rather than a family. -/
  78class Calibration (F : ℝ → ℝ) : Prop where
  79  second_deriv_at_zero : deriv (deriv (fun t => F (exp t))) 0 = 1
  80
  81/-- The complete axiom bundle for a cost functional. -/
  82class CostFunctionalAxioms (F : ℝ → ℝ) extends
  83  Normalization F, Composition F, Calibration F : Prop
  84
  85/-! ## The Canonical Cost Functional J -/
  86
  87/-- The canonical cost functional:
  88  J(x) = ½(x + x⁻¹) - 1
  89
  90This is the **unique** solution to the three axioms (proven below). -/
  91noncomputable def J (x : ℝ) : ℝ := (x + x⁻¹) / 2 - 1
  92
  93/-- J equals the Cost.Jcost defined in the Cost module. -/
  94lemma J_eq_Jcost : J = Cost.Jcost := by
  95  ext x; rfl
  96
  97/-! ## Verification: J satisfies the axioms -/
  98

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