IndisputableMonolith.Foundation.UniversalInstantiationFromDistinction
This module defines the two-valued equality cost and constructs logic realizations directly from distinctions on any carrier. Researchers tracing the forcing chain from bare distinctions to physical structure cite it to instantiate the abstract interface. The module proceeds via direct definitions of the cost function and interpretation maps together with existence proofs for the induced realizations.
claimLet $K$ be an inhabited carrier. The equality cost is the function satisfying $c(x,x)=0$ and $c(x,y)=1$ for $x≠y$. Any distinction $x≠y$ in $K$ induces a logic realization via the canonical map from the distinction to the realization object.
background
The module imports LogicRealization, which supplies a setting-independent interface for the Universal Forcing program and creates the common object into which different Law-of-Logic settings can be mapped. It also imports UniversalForcing, whose theorem states that any two Law-of-Logic realizations have canonically equivalent forced arithmetic objects because those objects are initial Peano algebras.
proof idea
The module first defines the equality cost and proves its basic algebraic properties. It then introduces the distinction interpretation map and constructs the induced logic realization by direct term construction from the given distinction, establishing existence without additional hypotheses.
why it matters in Recognition Science
This module supplies the concrete instantiation step that feeds the master forcing-chain theorem in RealityFromDistinction. That theorem derives the entire chain from spacetime to the light cone and φ-derived constants starting from the bare proposition ∃x y : K, x ≠ y on any inhabited carrier K.
scope and limits
- Does not derive spacetime or physical constants.
- Does not assume a specific Law-of-Logic setting beyond the two-valued cost.
- Does not address continuous-ratio or categorical realizations.
- Does not prove uniqueness of the induced realization.
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