Foundation Discrete Logic Realization Bool Cost

The Boolean cost

A Boolean value is one of two states, usually written true and false, or 1 and 0. The declaration boolCost defines a cost, a number assigned to comparing two such values: it returns 0 when the two values are equal, and 1 when they differ. This is the simplest possible discrete ledger, a record of whether two propositions agree or disagree. The framework proves two elementary facts about this cost: comparing a value with itself always costs 0, and the cost is symmetric, meaning comparing p with q costs the same as comparing q with p.

The declaration also builds a Boolean realization, a model of the framework's logic on this two-symbol carrier. The model interprets the free step orbit, the sequence of repeated applications of a logical operation, by parity: an even number of steps maps to false, an odd number to true. This gives the model a nontrivial identity-step shadow, a technical condition that the identity operation is not the same as a single step. The framework's library then proves that this Boolean realization carries the same forced arithmetic as every other realization, and that this arithmetic has the Peano surface, meaning it satisfies the axioms of natural number arithmetic.

In Recognition Science, the cost of recognition is forced, not chosen. The Boolean cost is the first non-continuous test case for this forcing: it shows that even a discrete, two-symbol carrier obeys the same universal arithmetic structure as the continuous cost function that the framework derives elsewhere. The result is that the framework's arithmetic is not an accident of a particular continuous model; it is invariant across realizations, including this minimal discrete one.

What the declaration does not claim is equally important. It does not define the continuous cost function J(x) = (x + 1/x)/2 - 1, nor does it derive the golden ratio, the eight-tick cycle, or any of the framework's larger constants. It establishes only the Boolean comparison cost and the fact that the Boolean realization shares the forced arithmetic. It does not claim that Boolean logic itself is the fundamental carrier of the framework; it is one test case among others. The declaration is a definition plus a small set of theorems about that definition, not a claim about the physical world.

/-- Boolean comparison cost: zero for equality, one for distinction. -/
def boolCost (p q : Bool) : Nat :=
  if p = q then 0 else 1

@[simp] theorem boolCost_self (p : Bool) : boolCost p p = 0 := by
  simp [boolCost]

theorem boolCost_symm (p q : Bool) : boolCost p q = boolCost q p := by
  by_cases h : p = q
  · subst h
    simp [boolCost]
  · have h' : q ≠ p := by intro hqp; exact h hqp.symm
    simp [boolCost, h, h']

/-- Boolean realization has the same forced arithmetic as every realization. -/
noncomputable def bool_arithmetic_invariant (R : LogicRealization.{0, 0}) :
    (UniversalForcing.arithmeticOf boolRealization).peano.carrier
      ≃ (UniversalForcing.arithmeticOf R).peano.carrier :=
  ArithmeticOf.equivOfInitial
    (UniversalForcing.arithmeticOf boolRealization) (UniversalForcing.arithmeticOf R)

/-- The Boolean realization's forced arithmetic has the Peano surface. -/
theorem bool_peano_surface :
    ArithmeticOf.PeanoSurface (UniversalForcing.arithmeticOf boolRealization) :=
  UniversalForcing.peano_surface boolRealization

What this page does not claim

It does not define or derive the continuous cost function J(x) = (x + 1/x)/2 - 1. It does not derive the golden ratio, the eight-tick cycle, or any of the framework's larger constants. It does not claim that Boolean logic is the fundamental carrier of the framework.

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

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