J

formal source

  60
  61/-- The J-cost function: the unique cost satisfying the Recognition Composition Law.
  62    J(x) = ½(x + x⁻¹) − 1 -/
  63noncomputable def J (x : ℝ) : ℝ := Jcost x
  64
  65/-- **Normalization**: The multiplicative identity has zero cost. -/
  66theorem J_at_one : J 1 = 0 := Jcost_unit0
  67
  68/-- **Reciprocal symmetry**: Cost is invariant under inversion.
  69    This is the algebraic encoding of "double-entry": every ratio x
  70    and its reciprocal 1/x carry the same cost. -/
  71theorem J_reciprocal (x : ℝ) (hx : 0 < x) : J x = J x⁻¹ :=
  72  Jcost_symm hx
  73
  74/-- **Non-negativity**: All costs are non-negative on ℝ₊. -/
  75theorem J_nonneg (x : ℝ) (hx : 0 < x) : 0 ≤ J x :=
  76  Jcost_nonneg hx
  77
  78/-- **Defect characterization**: J(x) = (x − 1)²/(2x) for x ≠ 0. -/
  79theorem J_defect_form (x : ℝ) (hx : x ≠ 0) : J x = (x - 1) ^ 2 / (2 * x) :=
  80  Jcost_eq_sq hx
  81
  82/-! ## §2. The Recognition Composition Law (RCL) -/
  83
  84/-- The **Recognition Composition Law**: the ONE primitive of Recognition Science.
  85
  86    J(xy) + J(x/y) = 2·J(x)·J(y) + 2·J(x) + 2·J(y)
  87
  88    In the log-coordinate form (t = ln x, u = ln y), this becomes:
  89    G(t+u) + G(t−u) = 2·G(t)·G(u) + 2·(G(t) + G(u))
  90
  91    which is a calibrated multiplicative form of the d'Alembert functional equation. -/
  92def SatisfiesRCL (F : ℝ → ℝ) : Prop :=
  93  ∀ x y : ℝ, 0 < x → 0 < y →