Cost F Eq J On Pos Of Averaging

The forced cost function

In mathematics, a cost function assigns a number to a ratio, measuring how far a quantity has moved from a baseline. The function J(x) = (x + 1/x)/2 - 1 does this in a particular way: it is zero when x = 1, grows as x moves away from 1, and treats x and 1/x as equally costly, so that doubling and halving incur the same penalty. This J is not an arbitrary choice. It is the unique function satisfying five plain conditions: reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity. The theorem T5_cost_uniqueness_on_pos in the machine-checked library proves that any function meeting those conditions must equal J on all positive real numbers.

The key to the proof is a change of variables. Writing x = e^t, the cost J becomes cosh(t) - 1, a standard hyperbolic function. The composition law, which relates the cost of a product to the costs of its factors, becomes a functional equation on the real line. The theorem shows that the only continuous solution is the hyperbolic cosine form. This is the classical d'Alembert equation, studied by Jean le Rond d'Alembert in the 1740s, and its regularity condition matters: without continuity, pathological solutions exist, but with it, the solution is unique. The framework's library formalizes this argument in a machine-checked proof, establishing the uniqueness theorem as a fact about the real numbers, not a conjecture.

What the declaration does not claim is also important. It does not claim that J is the only possible cost function in every conceivable setting; it claims uniqueness only under the five stated conditions. It does not claim that J is derived from physical measurements or that it explains why nature uses this cost; it is a mathematical theorem about a functional equation. The theorem does not assert that the cost of a product is simply the sum of costs, only that it obeys the specific composition law stated. Finally, the theorem is about positive real numbers; it says nothing about zero or negative inputs, where the expression (x + 1/x)/2 - 1 is not defined or not meaningful.

In Recognition Science, this theorem is the first step in a chain. The framework models recognition events as a ledger, a discrete record of comparisons, and the cost of a recognition is forced by the five conditions. The uniqueness of J is the foundation: it shows that any system that maintains such a ledger must use this specific cost function, not an arbitrary one. From this, the framework derives the golden ratio, an eight-tick cycle, and three spatial dimensions. The theorem itself is a piece of pure mathematics, but it is the load-bearing wall for the framework's physical claims.

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

/-- 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

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

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 theorem does not claim J is the only cost function in every conceivable setting, only under the five stated conditions. The theorem does not claim J is derived from physical measurements or explains why nature uses this cost. The theorem says nothing about zero or negative inputs, where the expression (x + 1/x)/2 - 1 is not defined or not meaningful.

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)

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.

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