Cost Agrees On Exp Of Symm Unit

The exponential agreement

The function J(x) = (x + 1/x)/2 - 1 is a cost function: it measures the price of a recognition event, a discrete record of a comparison, in units where the cost of perfect agreement is zero. The declaration agrees_on_exp_of_symm_unit establishes a precise fact about this cost function when its input is written as an exponential. For any real number t, the cost of the exponential exp(t) equals the hyperbolic cosine of t minus one: J(exp(t)) = cosh(t) - 1. This identity is not a new assumption; it is a direct consequence of the cost function's definition, and the machine-checked library of formal theorems proves it as a lemma.

The identity matters because it connects the cost function to the classical hyperbolic geometry of the exponential curve. The hyperbolic cosine measures the average of exp(t) and exp(-t), so the identity says the cost of a ratio x = exp(t) is the average of the ratio and its reciprocal, minus one. This is the same as the squared-ratio form J(x) = (x-1)^2/(2x), which shows the cost is always nonnegative and vanishes only at x = 1. The exponential agreement also reveals the cost function's symmetry under reciprocals: J(x) = J(1/x), a property that the framework treats as a fundamental requirement for any recognition cost.

In Recognition Science, the framework models recognition events as discrete records and forces the cost function to satisfy five plain conditions: reciprocal symmetry, zero cost at unity, a composition law, calibration, and continuity. The uniqueness theorem T5_cost_uniqueness_on_pos states that any function satisfying these conditions must equal J(x) on positive inputs. The exponential agreement is a building block in that proof: it shows that any candidate function agreeing with J on the exponential curve must agree everywhere, because every positive real number is an exponential of some real t.

The declaration does not claim that the exponential agreement alone forces the cost function's uniqueness. The full uniqueness theorem requires the other conditions, especially the composition law, which the framework proves separately. The exponential agreement also does not claim that the cost function is the only possible function on the exponential curve; it merely states the identity for the specific J defined by the framework. It does not assert anything about negative inputs, which the framework handles separately through the reciprocal symmetry.

/-- J in terms of cosh: J(exp(t)) = cosh(t) - 1 -/
lemma Jcost_exp_cosh (t : ℝ) : Jcost (Real.exp t) = Real.cosh t - 1 :=
  Jlog_as_cosh t

/-- J(x) ≥ 0 for positive x (AM-GM inequality) -/
lemma Jcost_nonneg {x : ℝ} (hx : 0 < x) : 0 ≤ Jcost x := by
  have hx0 : x ≠ 0 := hx.ne'
  rw [Jcost_eq_sq hx0]
  positivity

/-- From J(x) = 0 and x > 0, conclude x = 1. -/
lemma Jcost_zero_iff_one {x : ℝ} (hx : 0 < x) (h : Jcost x = 0) : x = 1 :=
  (Jcost_eq_zero_iff x hx).mp h

lemma Jcost_symm {x : ℝ} (hx : 0 < x) : Jcost x = Jcost x⁻¹ := by
  have hx0 : x ≠ 0 := ne_of_gt hx
  rw [Jcost_eq_sq hx0, Jcost_eq_sq (inv_ne_zero hx0)]
  field_simp [hx0]
  ring

theorem T5_cost_uniqueness_on_pos {F : ℝ → ℝ} [JensenSketch F] :
  ∀ {x : ℝ}, 0 < x → F x = Jcost x :=
by
  intro x hx
  have hAgree : AgreesOnExp F := by
    intro t
    exact le_antisymm (JensenSketch.axis_upper (F:=F) t) (JensenSketch.axis_lower (F:=F) t)
  exact (agree_on_exp_extends (F:=F) hAgree) hx

What this page does not claim

The exponential agreement alone does not force the cost function's uniqueness; the full theorem needs the other conditions. The declaration does not define the cost function; it states a property of the already-defined J. The identity does not extend to negative inputs without invoking the reciprocal symmetry.

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/Cost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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