Cost Jcost Logic Jcost L Unit0

The unit point

The function at the center of Recognition Science's cost framework is J, defined for positive numbers by the formula J(x) = (x + 1/x)/2 - 1. It measures the price of recognizing one quantity in terms of another. The declaration JcostL_unit0 establishes the most basic property of this function: when the two quantities are the same, when x equals 1, the cost is exactly 0. This is the anchor point from which the entire cost structure hangs.

The formula itself has a clear shape. For any positive x, the term (x + 1/x) is always at least 2, so J(x) is always at least 0. The minimum occurs precisely at x = 1, where the cost vanishes. This is not an accident of the formula; it is a forced consequence of the framework's five founding conditions, which include the requirement that recognizing something as itself costs nothing. The declaration JcostL_unit0 is the machine-checked proof of this fact within the framework's logical system.

In Recognition Science, this unit point is not merely a convenience. It is the reference against which all other costs are measured. The framework models recognition as a discrete record of events, a ledger, and the cost function assigns a price to each comparison. The zero at unity is what makes the whole scale meaningful: every other cost is a positive deviation from this baseline. Without it, the numbers would float without a fixed origin.

What the declaration does not claim is equally important. It does not assert that the formula for J is the only possible cost function; that uniqueness is a separate, deeper theorem. It does not imply anything about what happens at x = 0, where the formula is undefined. And it does not, by itself, justify any physical interpretation of the cost. The unit point is a logical fact about a defined function, not a statement about the world.

theorem JcostL_unit0 : JcostL (fromReal 1) = fromReal 0 := by
  rw [eq_iff_toReal_eq, toReal_JcostL, toReal_fromReal, toReal_fromReal]
  exact Jcost_unit0

/-- Canonical reciprocal cost on recovered reals. -/
def JcostL (x : LogicReal) : LogicReal :=
  (x + x⁻¹) / fromReal 2 - fromReal 1

/-- Recognition Composition Law on recovered reals for a cost function. -/
def SatisfiesCompositionLawL (F : LogicReal → LogicReal) : Prop :=
  ∀ x y : LogicReal, (0 : LogicReal) < x → (0 : LogicReal) < y →
    F (x * y) + F (x / y)
      = fromReal 2 * F x * F y + fromReal 2 * F x + fromReal 2 * F y

What this page does not claim

The declaration does not prove that J is the unique cost function satisfying the framework's conditions. The declaration says nothing about the behavior of J at x = 0, where the formula is undefined. The unit point is a logical fact about a defined function, not an empirical claim about the world.

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/JcostLogic.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.

Derived articles

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