IndisputableMonolith.Foundation.LogicAsFunctionalEquation.OperativeDomain
OperativeDomain defines the operative-domain structure as finite logical comparison applied to continuous positive ratios. Researchers tracing the logic-as-functional-equation chain cite it to bridge finite comparison results to the full RCL derivation. The module supplies the required domain identification through definitions and supporting lemmas.
claimAn operative domain is a structure in which finite logical comparison holds on the positive reals, satisfying the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$.
background
The upstream FiniteLogicalComparison module packages the sharpened result that finite logical comparison on positive ratios forces the RCL family, naming the finite-pairwise-polynomial condition as its algebraic content. This module extends the setting to continuous positive ratios by introducing the operative-domain structure. The local theoretical setting is the positive-ratio domain where scale-free comparison operates without hidden states, as required for the main theorem package.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
This module feeds the MainTheorem package, which collects the formal chain: scale-free comparison factors through positive ratios; no-hidden-state finite comparison gives counted-once composition; counted-once finite logical comparison forces the RCL family. It supplies the domain identification step in the logic-as-functional-equation development.
scope and limits
- Does not derive the RCL from scratch.
- Does not handle discrete or negative ratio cases.
- Does not compute physical constants such as alpha or G.
- Does not address the eight-tick octave or spatial dimension forcing.