IndisputableMonolith.Foundation.UniversalInstantiationFromDistinction
This module constructs logic realizations from a bare distinction on any carrier using the two-valued equality cost. Researchers tracing the forcing chain from distinction to physics cite it as the concrete bridge between the LogicRealization interface and the master theorem in RealityFromDistinction. The module defines eqCost together with distinctionInterpret and logicRealizationOfDistinction, then proves their elementary properties to instantiate the universal forcing objects.
claimThe two-valued equality cost $C(x,y) = 0$ if $x=y$ and $C(x,y)=1$ otherwise induces a canonical logic realization from any distinction $x eq y$ on an inhabited carrier $K$.
background
The module sits in the Foundation domain and imports the setting-independent interface of LogicRealization together with the UniversalForcing theorem. LogicRealization supplies the common object into which different Law-of-Logic settings map; its doc-comment states that the point is to create the common object into which continuous positive ratios, discrete propositions, and categorical settings can be mapped. UniversalForcing asserts that any two realizations have canonically equivalent forced arithmetic objects because those objects are initial Peano algebras.
The module introduces the two-valued equality cost as the basic distinction interpreter, along with the auxiliary maps distinctionInterpret and logicRealizationOfDistinction that turn a single inequality into an instance of the LogicRealization interface.
proof idea
This is a definition module. It defines eqCost as the indicator of inequality, proves the elementary facts eqCost_self, eqCost_symm and eqCost_ne_one, constructs distinctionInterpret and logicRealizationOfDistinction, and records the equivalence distinction_arithmetic_equiv_logicNat that identifies the induced arithmetic with the logicNat object.
why it matters in Recognition Science
The constructions supply the specific instantiation required by the master forcing-chain theorem in RealityFromDistinction, whose doc-comment describes the derivation of spacetime, the light cone, time as the canonical orbit, and the φ-derived constants from the bare proposition ∃ x y : K, x ≠ y. The module therefore closes the gap between the abstract UniversalForcing statement and the concrete distinction setting that begins the physical chain.
scope and limits
- Does not treat continuous or multi-valued cost functions.
- Does not derive physical constants, dimensions, or the light cone.
- Does not address multiple simultaneous distinctions or higher-order logic.
- Does not contain the full forcing chain to spacetime.
used by (1)
depends on (2)
declarations in this module (15)
-
def
eqCost -
theorem
eqCost_self -
theorem
eqCost_symm -
theorem
eqCost_ne_one -
def
distinctionInterpret -
def
distinctionStep -
theorem
distinctionInterpret_zero -
theorem
distinctionInterpret_step -
def
logicRealizationOfDistinction -
theorem
exists_logicRealization_of_distinction -
theorem
exists_named_logicRealization_of_distinction -
def
distinction_arithmetic_equiv_logicNat -
def
distinction_realizations_have_same_arithmetic -
structure
UniversalInstantiationCert -
theorem
universalInstantiationCert