Foundation Discrete Logic Realization Bool Arithmetic Invariant

The Boolean carrier

Boolean arithmetic is the mathematics of true and false, the two truth values used in classical logic and digital circuits. Each value can be represented as a bit, 0 or 1, and operations such as AND, OR, and NOT combine bits according to fixed rules. The subject is discrete: it deals in separate, countable states rather than continuous quantities. A Boolean comparison cost is one way to measure difference: it assigns zero when two values are equal and one when they differ, a simple rule that treats equality as no cost and distinction as a single unit.

In 1854 George Boole published An Investigation of the Laws of Thought, formalizing logic as an algebraic system. His work gave rise to Boolean algebra, which Claude Shannon applied to telephone switching circuits in 1937, founding digital logic design. The two-element set {0, 1} with operations AND, OR, and NOT is the canonical Boolean algebra, and every finite Boolean algebra is isomorphic to a power set. The field remains central to computer science, where a single bit is the fundamental unit of information.

In Recognition Science, the framework models recognition as a ledger, a discrete record of events where the cost of recognizing one event from another is forced by a proved cost function. The framework's library, a machine-checked collection of formal theorems, tests whether its universal forcing principle holds for a Boolean carrier. The declaration bool_arithmetic_invariant proves that the arithmetic forced by the Boolean realization is equivalent to the arithmetic forced by any other realization: the two carriers are isomorphic, meaning they have the same structure up to relabeling. In plain terms, the discrete two-valued case produces the same underlying arithmetic as every other case the framework considers.

The Boolean realization also satisfies two supporting properties. It has a non-trivial identity-step shadow, meaning the identity step behaves differently from a plain zero step, and its forced arithmetic has the Peano surface, meaning it exhibits the structure of the natural numbers. These results are proved in the framework's library and hold for the Boolean carrier as a test case.

What bool_arithmetic_invariant does not claim is that Boolean arithmetic is the only arithmetic, or that the framework derives new facts about classical Boolean algebra. It establishes an equivalence between the arithmetic forced by the Boolean case and the arithmetic forced by other realizations, nothing more. The declaration does not prove that the Boolean carrier is preferred or special among realizations; it proves a structural sameness. It also does not claim that the Boolean cost function is the unique cost function, only that this particular discrete cost function yields the same forced arithmetic.

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

/-- 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

The Boolean carrier is the only realization or the preferred one. The Boolean cost function is the unique cost function satisfying the framework's conditions. The framework derives new facts about classical Boolean algebra beyond the equivalence result.

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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