Cost Uniqueness Jcost Is Reciprocal

The reciprocal cost

The cost of recognizing one thing in terms of another is a number that measures how hard the recognition is. In the Recognition Science framework, that cost is not chosen freely; it is forced by a small set of plain conditions. The first of those conditions is reciprocal symmetry: recognizing x in terms of y costs exactly as much as recognizing y in terms of x. In symbols, if the cost is a function F of the ratio x/y, then F(r) = F(1/r) for every positive ratio r. This is the property that the declaration Jcost_is_reciprocal establishes for the framework's specific cost function, Jcost.

The function Jcost itself is defined by the formula J(x) = (x + 1/x)/2 - 1. It is the unique function, proved in a machine-checked library of formal theorems, that satisfies reciprocal symmetry together with four other conditions: zero cost at unity, a forced composition law, a calibration condition, and continuity. The reciprocal property is the simplest of these, and it is the one that makes the cost symmetric under exchanging the roles of the two things being compared. The theorem Jcost_is_reciprocal states exactly that Jcost(r) = Jcost(1/r) for every positive r, and it does so by citing the earlier lemma Jcost_symm.

What the declaration does not claim is any part of the uniqueness story. It does not say that Jcost is the only function with reciprocal symmetry; many functions have that property. It does not say that the other conditions hold for Jcost, though they do, each proved separately. It does not say anything about what the cost means physically, or about the golden ratio, the eight-tick cycle, or three spatial dimensions that the framework derives elsewhere. The declaration is a single, narrow fact: the reciprocal symmetry of Jcost, nothing more.

In Recognition Science, this symmetry is the first rung of a ladder. Once reciprocal symmetry is in place, the other conditions can be added one by one, and the full uniqueness theorem follows: any cost function satisfying all five conditions must equal Jcost on the positive reals. That theorem is the framework's central result, and the reciprocal property is its foundation. The declaration Jcost_is_reciprocal is the formal record that the foundation is sound.

/-- `Jcost` satisfies reciprocal symmetry in the theorem-surface format. -/
theorem Jcost_is_reciprocal : FunctionalEquation.IsReciprocalCost Jcost :=
  fun x hx => Jcost_symm hx

/-- Main uniqueness statement on ℝ₊: any admissible cost equals Jcost on (0,∞). -/
theorem unique_cost_on_pos (F : ℝ → ℝ) (hF : UniqueCostAxioms F) :
  ∀ {x : ℝ}, 0 < x → F x = Jcost x :=
  T5_uniqueness_complete F hF.symmetric hF.unit hF.convex hF.calibrated hF.continuousOn_pos hF.coshAdd
    hF.dAlembert_smooth hF.dAlembert_toODE hF.ode_cont hF.ode_diff hF.ode_bootstrap

What this page does not claim

This declaration does not prove uniqueness; it only establishes one of the five conditions. This declaration does not define the cost function Jcost; that is done elsewhere. This declaration does not connect the cost to any physical quantity or derived constant.

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

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