IndisputableMonolith.Foundation.LogicRealization
The LogicRealization module defines the abstract interface for a Law-of-Logic realization: a carrier equipped with comparison cost, identity element, step action, and structural propositions. Researchers working on the Recognition Science forcing chain cite it when constructing realizations that extract arithmetic. This module is a pure definition module containing no proofs or theorems.
claimA Law-of-Logic realization consists of a carrier set $K$ together with a comparison cost function, an identity element $e$, a step generator $s$, and propositions ensuring that the arithmetic object extracted from the identity/step data is the initial Peano algebra generated by that data.
background
This module occupies the foundational layer of the Recognition Science framework and imports LogicAsFunctionalEquation together with ArithmeticFromLogic. It introduces the Law-of-Logic realization as the minimal structure needed for the Universal Forcing program. The module doc comment states that the fields are intentionally lean: each realization supplies its own topology, order, category, or discrete structure through the propositions carried here, while the invariant target remains the arithmetic object extracted from the identity/step data.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the core interface that ArithmeticOf uses to extract arithmetic as the initial Peano algebra generated by identity/step data. It also enables UniversalForcingSelfReference to close the meta-theorem reflexively and UniversalInstantiationFromDistinction to instantiate the interface from a single distinction. It thereby fills the structural requirement for the T0-T8 forcing chain and the Recognition Composition Law.
scope and limits
- Does not prescribe any concrete topology or metric on the carrier.
- Does not prove existence of nontrivial realizations.
- Does not derive numerical constants such as phi or the fine-structure constant.
- Does not implement concrete arithmetic operations or addition tables.
used by (3)
depends on (2)
declarations in this module (10)
-
structure
LogicRealization -
def
hasIdentityStep -
theorem
hasIdentityStep_of_nontrivial -
structure
FaithfulArithmeticInterpretation -
def
positiveRatioOrbitInterpret -
theorem
positiveRatioOrbitInterpret_val -
def
ofPositiveRatioComparison -
theorem
positiveRatio_hasIdentityStep -
theorem
positiveRatio_interpret_injective -
theorem
positiveRatio_faithful