Foundation Discrete Logic Realization Bool Peano Surface

The Boolean carrier

A Boolean algebra is the simplest possible discrete logic: two values, true and false, with the usual operations of AND, OR, and NOT. In the Recognition Science framework, this carrier is a test case for a central claim: that the arithmetic structure of natural numbers emerges from any recognition process, regardless of the underlying material. The framework's ledger, a discrete record of recognition events, can be built on this two-valued carrier, and the framework's machine-checked library of formal theorems proves that the resulting arithmetic is the same as for any other realization.

The specific declaration bool_peano_surface establishes that the arithmetic forced by this Boolean realization has the Peano surface: it satisfies the axioms of natural number arithmetic, including zero, successor, and induction. The proof is not a hand-waving argument but a formal derivation in the framework's library, which means the claim is verified by a computer-checked proof. This is the first non-continuous test case for the framework's Universal Forcing principle, which states that the structure of arithmetic is not an assumption but a consequence of the recognition process itself.

The Boolean cost function is stark: it assigns zero cost when two propositions are equal and one when they differ. This is the simplest possible distinction, yet the framework proves it still yields the full Peano surface. The theorem also shows the Boolean realization has a non-trivial identity-step shadow, meaning even the most minimal recognition structure contains the seeds of arithmetic structure. This result matters because it demonstrates the forcing principle does not depend on the richness of the carrier; a two-valued logic is enough to generate the same arithmetic as any continuous or more complex realization.

In Recognition Science, this is a milestone because it shows the framework's core claim holds across different mathematical settings. The theorem is tagged THEOREM in the framework's library, meaning it is proved in the machine-checked system with no unverified assumptions beyond the standard logical axioms. The declaration does not claim that Boolean algebra itself is the foundation of arithmetic in a philosophical sense, nor does it claim that the Boolean realization is the only possible discrete carrier. It establishes a specific formal equivalence: the arithmetic forced by the Boolean realization is isomorphic to the arithmetic forced by any other realization, including continuous ones.

What the declaration does not claim is equally important. It does not claim that the Boolean carrier is the actual physical substrate of the universe, nor that natural numbers are somehow made of true and false values. It does not claim that the Peano surface is the only arithmetic structure possible, only that this particular realization produces it. The theorem is a structural result about the framework's own construction, not a statement about the empirical world. It is a proof that the framework's forcing principle is not an artifact of a particular carrier, but a general property of the recognition process itself.

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

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

/-- 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)

What this page does not claim

The Boolean carrier is the actual physical substrate of the universe. Natural numbers are made of true and false values in any empirical sense. The Peano surface is the only arithmetic structure the framework can produce.

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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