Foundation Universal Forcing Strict Categorical Logic Nat Cost Symm

A symmetry in the ledger

In mathematics, a cost function assigns a number to a pair of objects, often measuring how far apart they are. The simplest meaningful example is a discrete metric: two items cost 0 if they are the same, and 1 if they differ. This is the familiar equality test, dressed as a distance. It appears throughout computer science, where it is the basis of exact matching, and in logic, where it tests whether two terms denote the same object.

The specific function here, written logicNatCost, takes two natural numbers and returns 0 when they are equal and 1 otherwise. Its defining property is symmetry: the cost of comparing a to b is the same as the cost of comparing b to a. The theorem logicNatCost_symm states exactly this, and it is proved in a machine-checked library of formal theorems, meaning the proof has been verified step by step by a computer. The proof itself is direct: if the two numbers are equal, both sides are 0; if they are not equal, both sides are 1.

This symmetry is not an accident of the definition. It is a structural requirement that any reasonable notion of comparison must satisfy. If measuring the gap between two objects depended on which one you named first, the measurement would be meaningless. The theorem guarantees that this particular cost function, built on the natural numbers, has the required property. It is a small but foundational piece of a larger framework that studies what happens when recognition, the act of telling things apart, is itself assigned a forced cost.

In Recognition Science, this declaration is part of a strict categorical realization, a formal construction that connects the framework's abstract logic to the familiar natural numbers. The framework models a ledger, a discrete record of events, and assigns a cost to each act of recognition. This particular theorem establishes that the simplest possible cost, the equality test, is symmetric. It does not, however, claim that this cost function is the unique one, nor does it say anything about more complex costs that might distinguish between different kinds of inequality.

The practical consequence is that any system built on this cost function can rely on its symmetry without rechecking it. When two values are compared, the order of comparison is irrelevant. This is a guarantee that a machine-checked proof provides, and it is the kind of small, certain step that larger arguments depend on. The theorem is a foundation stone: unremarkable on its own, but load-bearing for anything built above it.

theorem logicNatCost_symm (a b : LogicNat) : logicNatCost a b = logicNatCost b a := by
  by_cases h : a = b
  · subst h
    simp [logicNatCost]
  · have h' : b ≠ a := by intro hb; exact h hb.symm
    simp [logicNatCost, h, h']

theorem logicNatCost_symm (a b : LogicNat) : logicNatCost a b = logicNatCost b a := by
  by_cases h : a = b
  · subst h
    simp [logicNatCost]
  · have h' : b ≠ a := by intro hb; exact h hb.symm
    simp [logicNatCost, h, h']

def logicNatCost (a b : LogicNat) : Nat :=
  if a = b then 0 else 1

What this page does not claim

This theorem does not establish that logicNatCost is the only symmetric cost function on natural numbers. This theorem does not say anything about costs that distinguish between different kinds of inequality, such as measuring distance. This theorem does not connect directly to the framework's continuous cost function J(x).

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

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