IndisputableMonolith.Foundation.LogicAsFunctionalEquation.MainTheorem
No-hidden-state operative comparison on positive ratios forces the RCL family. Researchers tracing the origin of the J-function from logical comparisons cite this module as the central forcing step. The module assembles the result by importing PositiveRatioForcing for ratio coordinates, NoHiddenState for counted-once composition, and OperativeDomain for the identification to RCL without internal lemmas.
claimNo-hidden-state operative comparison on positive ratios $x, y > 0$ forces the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$, where $J$ satisfies the scale-free fixed-point properties.
background
The module sits inside the LogicAsFunctionalEquation development that derives the Recognition Composition Law from finite logical comparisons. No-hidden-state composition means the composite cost is supplied by a counted-once resource expression in the two constituent costs; this is a formal version of no hidden route memory, no branch choice, no infinite series, and no reuse of a constituent comparison. OperativeDomain packages the chain finite logical comparison on positive ratios to encoded logical comparison to RCL family. PositiveRatioForcing shows that scale-invariant comparison on positive magnitudes factors through the ratio $x/y$.
proof idea
This is a packaging module with no internal proofs. Its argument structure is the direct composition of the three imported modules: PositiveRatioForcing supplies the universal property of ratio coordinates, NoHiddenState enforces the counted-once resource constraint, and OperativeDomain completes the identification of the resulting functional equation with the RCL family.
why it matters in Recognition Science
The module supplies the main forcing result that feeds BooleanRatioBridge, which extends the same logic to finite Boolean events with positive weights. It fills the central step in the LogicAsFunctionalEquation chain and directly supports the Recognition Composition Law as the bridge from discrete logic to continuous physics. It touches the T5 J-uniqueness and RCL landmarks in the forcing chain.
scope and limits
- Does not extend to non-positive ratios or signed quantities.
- Does not derive numerical values for constants such as alpha or G.
- Does not address the eight-tick octave or spatial dimensions.
- Does not provide explicit constructions for the J function beyond the RCL family.