Foundation Dalembert Ultimate Rcl Is Inevitable

The unavoidable combiner

In mathematics, a functional equation is an equation where the unknown is a function rather than a number. A classic example is the equation f(x+y) = f(x) f(y), whose continuous solutions are the exponential functions. The declaration rcl_is_inevitable concerns a different functional equation, one that arises when a function is used to measure the cost of a comparison.

Consider a cost function F that assigns a nonnegative number to any positive ratio x, where F(x) is meant to represent the cost of the deviation when one quantity is x times another. The framework of Recognition Science starts with three plain requirements. Symmetry says F(x) = F(1/x): comparing x to 1 costs the same as comparing 1 to x. Normalization says F(1) = 0: no deviation, no cost. Consistency says that F(xy) + F(x/y) can be expressed in terms of F(x) and F(y) through some fixed combiner function P. These are not arbitrary postulates; each is the definition of what the corresponding word means.

The theorem rcl_is_inevitable proves that if F is the specific cost function J(x) = (x + 1/x)/2 - 1, and if P is any combiner that satisfies the consistency equation for all positive x and y, then P must equal 2uv + 2u + 2v for all nonnegative u and v. In other words, once you fix the cost function J, there is exactly one way to combine costs that is consistent with the symmetry and normalization requirements. The proof is machine-checked: it is a formal theorem in the framework's library of verified mathematics, with no gaps in its logical steps.

The declaration does not claim that J is the only possible cost function, nor does it claim that the consistency equation has no other solutions. It assumes J as given and derives the unique form of the combiner P that works with it. The broader claim, that any smooth symmetric normalized cost function with multiplicative consistency must be J, is a separate theorem in the same module. The philosophical point attached to the declaration is that the combiner is not a free choice: within this framework, it is forced by the structure of comparison itself.

/-- The RCL is not a choice. It's what "comparison" IS.

    Just as the Pythagorean theorem is not a choice in Euclidean geometry
    (it follows from the axioms), the RCL is not a choice in comparison theory
    (it follows from symmetry + normalization + consistency).

    But unlike Euclidean geometry (where non-Euclidean alternatives exist),
    there is NO alternative to the RCL. Any symmetric, normalized, consistent
    cost function is J, and its combiner is the RCL.

    This is the deepest sense in which Recognition Science is "inevitable."
-/
theorem rcl_is_inevitable :
    ∀ P : ℝ → ℝ → ℝ,
    (∀ x y, 0 < x → 0 < y →
      Cost.Jcost (x*y) + Cost.Jcost (x/y) = P (Cost.Jcost x) (Cost.Jcost y)) →
    ∀ u v, 0 ≤ u → 0 ≤ v →
      P u v = 2*u*v + 2*u + 2*v :=
  rcl_unconditional

What this page does not claim

The declaration does not prove that J is the only possible cost function. The declaration does not claim that the consistency equation has no other solutions for different cost functions.

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

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