IndisputableMonolith.Foundation.LogicAsFunctionalEquation.MainTheorem
This module shows that no-hidden-state operative comparisons on positive ratios force the RCL family. Foundation researchers cite it when tracing how logical comparisons produce the Recognition Composition Law without hidden route memory or reuse. The module assembles results from PositiveRatioForcing, NoHiddenState, and OperativeDomain into a single forcing statement.
claimNo-hidden-state operative comparison on positive ratios forces the Recognition Composition Law family $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$.
background
The module sits in the LogicAsFunctionalEquation section and packages the final link of the operative-domain chain. NoHiddenState supplies the counted-once resource rule: the composite cost equals a single-use expression in the two constituent costs, ruling out hidden memory, branching, or reuse. OperativeDomain encodes the progression from finite logical comparison on positive ratios through encoded comparison to the RCL family. PositiveRatioForcing establishes that scale-invariant comparison on positive magnitudes factors through the ratio $x/y$.
proof idea
This module imports PositiveRatioForcing, NoHiddenState, and OperativeDomain and combines their statements to obtain the forcing of the RCL family by no-hidden-state operative comparison. It contains no internal proofs.
why it matters in Recognition Science
The module feeds BooleanRatioBridge, which supplies the discrete-to-continuous link from finite Boolean events with positive weights to positive likelihood ratios. It completes the foundation step stated in the module doc-comment: no-hidden-state operative comparison forces the RCL family.
scope and limits
- Does not treat comparisons that allow hidden route memory or reuse.
- Does not extend the forcing to non-positive magnitudes.
- Does not derive numerical constants or the phi-ladder from this step alone.