IndisputableMonolith.Foundation.LogicRealization
A Law-of-Logic realization is a carrier with comparison cost, identity element, and step generator under the structural laws required by Universal Forcing. The invariant is the arithmetic object extracted from identity/step data, not the ambient carrier. Foundation authors cite this interface when building forced Peano arithmetic or closing the T-1–T8 spine. The module assembles the structure and elementary faithfulness lemmas for positive-ratio interpretations.
claimA Law-of-Logic realization is a carrier $K$ with comparison cost, identity $e$, and step/generator $s$, plus the structural propositions needed so that the arithmetic extracted from $(e,s)$ is the initial Peano algebra of Universal Forcing. Related notions include identity-step existence, faithful arithmetic interpretation, and positive-ratio orbit interpretation.
background
Recognition Science derives physics from one functional equation through the Universal Forcing program. Upstream, LogicAsFunctionalEquation supplies the cost-theoretic reading of logical comparison; ArithmeticFromLogic extracts arithmetic from identity and step data.
A LogicRealization is deliberately thin. Topology, order, category, or discreteness are not hard-wired into the carrier; they appear as propositions the realization carries. The target invariant is the arithmetic object generated by identity and step, not the ambient set.
Sibling definitions include hasIdentityStep, FaithfulArithmeticInterpretation, and positive-ratio orbit interpretation (with injectivity and faithfulness lemmas). Nontrivial carriers yield identity steps; comparison data on positive ratios yields a concrete realization.
proof idea
This is a definition-and-interface module, not a deep forcing proof. It packages the LogicRealization structure and the predicates that witness identity/step data and faithful arithmetic interpretation. Short lemmas show nontrivial carriers admit identity steps, that positive-ratio orbit maps are injective and faithful, and that positive-ratio comparison data produces a realization. Downstream modules supply the initiality and forcing arguments.
why it matters in Recognition Science
ArithmeticOf imports this interface to extract the initial Peano algebra from any realization; initial objects are unique up to unique isomorphism, which is the mechanism of Universal Forcing. TMinus1ToT8Bridge and UnifiedForcingChain sit on the public forcing spine that begins from cost foundation and Law-of-Logic data (T-1 distinguishability through T8 dimension). UniversalForcingSelfReference shows the meta-theorem itself fits the LogicRealization shape, giving reflexive closure. UniversalInstantiationFromDistinction builds a realization from a bare distinction $x\neq y$, so a single distinction instantiates the interface rather than sitting beside a pre-existing certificate.
scope and limits
- Does not prove T0–T8 forcing; that lives in UnifiedForcingChain and the bridge modules.
- Does not fix carrier topology, order, or category; those are propositional parameters.
- Does not construct physical constants or the mass ladder; only the logical/arithmetic interface.
- Does not assert every set is a realization without identity/step and cost data.
- Does not discharge initiality uniqueness; ArithmeticOf owns that argument.
used by (5)
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