Foundation Discrete Logic Realization Bool Has Identity Step

A Boolean shadow of arithmetic

In classical logic, a Boolean value is one of two states: true or false. A Boolean comparison cost is a rule that charges zero when two values are equal and one when they differ. This is the simplest possible discrete ledger: a record of whether two propositions agree. The Recognition Science framework studies what structure is forced when reality keeps such a ledger of recognition events, and this two-value case is its first non-continuous test case.

The framework's library, a machine-checked collection of formal theorems, defines a Boolean realization of its core logic. The declaration bool_hasIdentityStep establishes that this two-value structure has a non-trivial identity-step shadow. In plain terms, the Boolean carrier, despite having only two elements, contains a distinct step from identity to its negation, and this step is not trivial. The theorem is proved by showing the realization is non-trivial, which is a precondition for having such a step.

The deeper result is that this Boolean realization forces the same arithmetic as every other realization in the framework. The library proves that the arithmetic derived from the Boolean carrier is equivalent to the arithmetic derived from any other realization, and that this forced arithmetic has the Peano surface, meaning it satisfies the standard axioms for natural numbers. This is a structural invariance: the two-value logic circuit, with no numbers in it, still carries the same arithmetic skeleton as any other structure in the framework.

What this does not claim is that Boolean logic itself is arithmetic. The theorem shows that the framework's forced arithmetic is invariant across realizations, including the Boolean one. It does not say that classical Boolean algebra contains numbers, nor that the framework derives arithmetic from logic alone. The Peano surface is a property of the forced arithmetic, not a claim that the Boolean carrier itself is a model of arithmetic.

The consequence is a sharpening of the framework's central claim: the arithmetic that emerges from the logic of recognition does not depend on the carrier chosen to realize it. Whether the carrier is continuous or as simple as a two-value switch, the same arithmetic structure is forced. This is what makes the Boolean case a meaningful test, not a trivial one.

/-- The discrete realization has a non-trivial identity-step shadow. -/
theorem bool_hasIdentityStep : boolRealization.hasIdentityStep :=
  LogicRealization.hasIdentityStep_of_nontrivial boolRealization

/-- 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

Classical Boolean algebra contains numbers. The framework derives arithmetic from logic alone without any additional structure. The Boolean carrier itself is a model of Peano arithmetic.

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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