IndisputableMonolith.Foundation.UniversalInstantiationFromDistinction
This module defines the two-valued equality cost and constructs logic realizations directly from distinctions on a carrier set. Researchers building the forcing chain from bare distinction to arithmetic objects would cite these definitions to instantiate the universal forcing interface. The module consists of targeted definitions and elementary lemmas that establish the cost function and the induced realizations.
claimDefine the two-valued equality cost by $\mathrm{eqCost}(x,y)=0$ when $x=y$ and $\mathrm{eqCost}(x,y)=1$ otherwise. The module then constructs $\mathrm{distinctionInterpret}$ and $\mathrm{logicRealizationOfDistinction}$ that map any distinction on an inhabited carrier to a logic realization compatible with the universal forcing theorem.
background
The module sits in the Foundation domain and imports LogicRealization, which supplies a setting-independent interface for the Universal Forcing program so that different Law-of-Logic settings can map to common arithmetic objects. It also imports UniversalForcing, whose doc-comment records the first formal statement that any two realizations have canonically equivalent forced arithmetic objects because those objects are initial Peano algebras. The central definition introduced is eqCost, described as the two-valued equality cost that returns zero on equal inputs and one on distinct inputs, together with the auxiliary maps distinctionInterpret and logicRealizationOfDistinction.
proof idea
This is a definition module. It introduces eqCost together with the supporting lemmas eqCost_self, eqCost_symm, eqCost_ne_one, distinctionInterpret, distinctionStep, distinctionInterpret_zero, distinctionInterpret_step, logicRealizationOfDistinction, exists_logicRealization_of_distinction, exists_named_logicRealization_of_distinction and distinction_arithmetic_equiv_logicNat, each establishing the required algebraic or existence properties of the cost and the induced realization.
why it matters in Recognition Science
These constructions supply the concrete distinction-based instantiation required by the master forcing-chain theorem in RealityFromDistinction, whose doc-comment states that the module derives the entire chain to spacetime, the light cone, time as the canonical orbit, and the φ-derived physical constants from the bare proposition ∃ x y : K, x ≠ y on any inhabited carrier K.
scope and limits
- Does not derive the full forcing chain to spacetime or constants.
- Does not treat continuous positive ratios or categorical settings.
- Does not establish uniqueness of the induced realization beyond the listed lemmas.
- Does not address the eight-tick octave or spatial dimension D=3.
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