IndisputableMonolith.Foundation.MultiplicativeRecognizerL4
The module defines multiplicative recognizers on the positive reals equipped with a continuous comparison operator that satisfies the Law of Logic. It pairs geometric recognizer data with cost-functional data to enable automatic derivation of L4. Researchers building the Recognition Science forcing chain cite it when extending the recognizer-induced logic construction to the multiplicative setting. The module imports three upstream foundation modules and exposes sibling definitions that feed the root IndisputableMonolith.
claimA multiplicative recognizer consists of a recognizer $r : C → ℝ_{>0}$ together with a continuous comparison operator on its event space that satisfies the Law of Logic, paired with the cost-functional data required to derive (L4) automatically.
background
The module sits inside the Foundation layer and imports LogicAsFunctionalEquation, RecognizerInducesLogic, and PrimitiveDistinction. RecognizerInducesLogic states that any recognizer mapping configurations to events automatically generates a Law-of-Logic realization on its event space. The supplied DOC_COMMENT defines the multiplicative case: the event space is restricted to the positive reals and the comparison operator is required to be continuous. Sibling declarations introduce the concrete structures MultiplicativeRecognizer, cost, MultiplicativeL4, and the certificates L4DerivableCert that witness automatic L4 satisfaction.
proof idea
This is a definition module, no proofs. It assembles the multiplicative recognizer structure by importing the three upstream modules and declaring the paired geometric-plus-cost data objects together with the L4-derivability certificates.
why it matters in Recognition Science
The module supplies the multiplicative case required by the master forcing-chain theorem exposed in the root IndisputableMonolith module. It fills the multiplicative-recognizer slot in the T0-T8 chain and the Recognition Geometry-Logic unification described in the companion paper RS_Recognition_Geometry_Logic_Unification.tex. Downstream results cite it when constructing the full IndisputableMonolith surface that includes the T-1 absolute-floor and foundation-repair theorems.
scope and limits
- Does not treat non-multiplicative event spaces.
- Does not derive the T5 J-uniqueness or T8 D=3 results.
- Does not contain numerical checks of the alpha band or phi-ladder masses.
- Does not address the IntegrationGap or eta_B worked examples.
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