IndisputableMonolith.Foundation.DiscreteLogicRealization
This module supplies the discrete boolean realization of the Law of Logic for the Universal Forcing theorem. It defines a cost function that is zero on equal elements and one on distinct elements, together with symmetry, identity-step, and arithmetic-invariance lemmas. Researchers citing the initial-Peano-algebra equivalence would reference this concrete model. The module is built from definitions and short lemmas that verify the realization meets the forcing requirements.
claimThe boolean realization equips the carrier with cost function $c(x,y)=0$ when $x=y$ and $c(x,y)=1$ otherwise, satisfying $c(x,y)=c(y,x)$, identity-step preservation, and arithmetic invariance under the initial Peano algebra structure.
background
The module imports UniversalForcing, whose doc-comment states that any two Law-of-Logic realizations have canonically equivalent forced arithmetic objects because those objects are initial Peano algebras. It introduces the boolean realization whose main object is the comparison cost (zero for equality, one for distinction) together with supporting properties such as self-symmetry and Peano-surface invariance. These definitions sit inside the Foundation domain and supply the discrete model required by the forcing chain.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the discrete realization into the Universal Forcing program and is imported by CategoricalLogicRealization, ModularLogicRealization, UniversalForcingAudit, and UniversalForcing.DiscreteRealization. It supplies the concrete boolean instance that demonstrates the theorem does not require faithful carrier embedding, closing one branch of the initial-Peano-algebra equivalence.
scope and limits
- Does not embed arithmetic faithfully into the carrier.
- Does not treat continuous or infinite realizations.
- Does not address higher-dimensional or non-boolean carriers.
- Does not discharge the full Universal Forcing theorem.