IndisputableMonolith.Foundation.LogicAsFunctionalEquation.OperativeDomain
OperativeDomain defines the operative-domain structure as finite logical comparison on continuous positive ratios. It bridges the sharpened finite theorem to the continuous setting needed for the Recognition functional equation. The module is imported by MainTheorem to assemble the chain from scale-free comparison through counted-once composition to the RCL family. Content consists of definitions and identification lemmas with no proofs.
claimAn operative domain is a structure realizing finite logical comparison on positive real ratios, thereby satisfying the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$.
background
This module belongs to the LogicAsFunctionalEquation section of the Foundation layer. It extends the upstream FiniteLogicalComparison result, whose doc-comment states that finite logical comparison on positive ratios forces the RCL family, by naming the continuous positive-ratio case as an operative-domain structure. The finite-pairwise-polynomial condition is retained as the algebraic content of logical comparison without hidden states.
proof idea
This is a definition module containing the OperativeDomainStructure definition together with the lemmas operative_domain_satisfies_logic and operative_domain_identification; no proofs are present.
why it matters in Recognition Science
The module supplies the operative-domain structure to the MainTheorem package. That package assembles the formal chain closest to the paper's headline: scale-free comparison factors through positive ratios, no-hidden-state finite comparison yields counted-once composition, and counted-once finite logical comparison forces the RCL family. It thereby supports the T5 J-uniqueness and T6 phi fixed-point steps.
scope and limits
- Does not remove the finite-pairwise-polynomial condition.
- Does not address non-positive ratios or infinite sets.
- Does not derive the eight-tick octave or D=3.
- Does not compute constants such as alpha inverse.