Foundation Universal Forcing Order Realization Int Cost Symm

The symmetric ledger

In mathematics, a cost function assigns a number to each pair of objects, measuring how far apart they are. The simplest possible cost on integers is equality cost: 0 if the two numbers are the same, 1 if they differ. This is the discrete record, a ledger that charges a single unit for any mismatch and nothing for a match.

The declaration intCost_symm proves that this cost is symmetric: the cost from a to b equals the cost from b to a. Formally, intCost a b = intCost b a for all integers a and b. The proof is immediate from the definition, since equality itself is symmetric. If a equals b, both costs are 0; if they differ, both are 1.

This symmetry is one of the five plain conditions that force the framework's central cost function J(x) = (x + 1/x)/2 - 1. Reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity together pin down that unique form. The integer realization here is a lightweight model: it shows the symmetry condition holds in the simplest possible carrier, the integers with unit step.

In Recognition Science, this lemma is a building block, not a destination. It establishes that the equality cost on integers satisfies the symmetry requirement, nothing more. It does not derive the full cost function, does not involve the golden ratio, and does not touch the forcing chain that leads to three spatial dimensions. Those results live elsewhere in the framework's machine-checked library of formal theorems.

What the lemma changes for a reader: it shows that even the most basic recognition ledger already obeys the symmetry law. The framework's later theorems build on this foundation, but this declaration alone is a single verified step, not the whole staircase.

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

/-- Equality cost on integers. -/
def intCost (a b : ℤ) : Nat :=
  if a = b then 0 else 1

What this page does not claim

This lemma does not derive the full cost function J(x) = (x + 1/x)/2 - 1. It does not involve the golden ratio or the eight-tick recognition cycle. The declaration does not establish anything about physical space or dimensions.

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

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