IndisputableMonolith.Foundation.MultiplicativeRecognizerL4
The module introduces multiplicative recognizers on positive reals equipped with continuous comparison operators that satisfy the Law of Logic. Researchers tracing the Recognition Science forcing chain would cite it when constructing the multiplicative case for L4 derivation. It pairs geometric recognizer data with cost-functional definitions to support automatic satisfaction of the L4 polynomial condition.
claimA multiplicative recognizer consists of a map $r : C → ℝ_{>0}$ together with a continuous comparison operator satisfying the Law of Logic, augmented by cost-functional data that automatically yields the L4 polynomial condition.
background
This module operates in the foundation layer of Recognition Science, specializing the general recognizer to the multiplicative event space of positive reals. It equips this space with a continuous comparison operator obeying the Law of Logic. The upstream RecognizerInducesLogic module states that any recognizer mapping configurations to events automatically generates a Law-of-Logic realization on its event space, while LogicAsFunctionalEquation and PrimitiveDistinction supply the underlying functional equations and distinction primitives.
proof idea
This is a definition module that introduces the multiplicative recognizer structure and the cost function. It then establishes lemmas verifying L4 satisfaction for the multiplicative case through direct verification using the cost data, relying on the unification supplied by the imported modules.
why it matters in Recognition Science
This module supplies the multiplicative recognizer foundation imported by the root IndisputableMonolith module to assemble the master forcing-chain theorem. It advances the unification of Recognition Geometry with the Law of Logic from the companion paper RS_Recognition_Geometry_Logic_Unification.tex and supports L4 derivation within the T0-T8 chain.
scope and limits
- Does not treat non-positive or non-multiplicative event spaces.
- Does not prove the complete T0 to T8 forcing chain.
- Does not compute explicit values for physical constants such as alpha or G.
- Does not address discrete or non-continuous comparison operators.
used by (1)
depends on (3)
declarations in this module (13)
-
structure
MultiplicativeRecognizer -
def
cost -
theorem
cost_def -
def
MultiplicativeL4 -
def
MultiplicativeL4Polynomial -
theorem
multiplicativeRecognizer_satisfies_L4_polynomial -
theorem
multiplicativeRecognizer_satisfies_L4 -
theorem
L4_derivable_on_multiplicative_event_space -
theorem
multiplicative_identity -
theorem
multiplicative_reciprocal_symmetry -
structure
L4DerivableCert -
def
l4DerivableCert -
theorem
l4DerivableCert_inhabited