IndisputableMonolith.Foundation.DiscreteLogicRealization
The DiscreteLogicRealization module supplies the Boolean realization of the Law of Logic inside the Universal Forcing framework. Researchers modeling initial Peano algebras in discrete settings cite boolRealization and boolCost. The module consists of definitions establishing the zero-one cost function together with invariance lemmas that confirm the structure behaves as an initial algebra.
claimThe Boolean realization is given by the cost function satisfying $\mathrm{boolCost}(x,y)=0$ when $x=y$ and $1$ otherwise, together with the structure $\mathrm{boolRealization}$ that carries the identity step, arithmetic invariance, and Peano surface properties.
background
The module imports UniversalForcing, whose doc-comment records that any two Law-of-Logic realizations possess canonically equivalent forced arithmetic objects because those objects are initial Peano algebras. It introduces the discrete Boolean case through boolCost, defined as the comparison cost that vanishes on equality and equals one on distinction, together with the listed sibling constructions boolOrbitInterpret, boolRealization, bool_hasIdentityStep, bool_arithmetic_invariant, and bool_peano_surface.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds CategoricalLogicRealization, ModularLogicRealization, UniversalForcingAudit, and the DiscreteRealization re-export. It thereby exhibits one concrete realization whose forced arithmetic objects are initial Peano algebras, supporting the Universal Forcing theorem stated in the upstream module.
scope and limits
- Does not construct realizations outside the Boolean case.
- Does not prove equivalence between distinct realizations.
- Does not contain the full Universal Forcing theorem statement.
- Does not address continuous or higher-dimensional carriers.