IndisputableMonolith.Foundation.LogicAsFunctionalEquation.MainTheorem
This module shows that no-hidden-state operative comparisons on positive ratios force the RCL family. Foundation researchers deriving the Recognition Composition Law from logical comparisons cite it when assembling the no-hidden-state, operative-domain, and positive-ratio components. The argument combines the three imported submodules into theorems that close the forcing chain.
claimNo-hidden-state operative comparison on positive ratios forces the RCL family: $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$.
background
The module lies in Foundation.LogicAsFunctionalEquation and packages the claim that no-hidden-state operative comparison forces the RCL family. Upstream, NoHiddenState defines no-hidden-state composition as the composite cost supplied by a counted-once resource expression in the two constituent costs, a formal version of no hidden route memory, no branch choice, no infinite series, and no reuse. OperativeDomain packages the chain finite logical comparison on positive ratios to encoded logical comparison to RCL family. PositiveRatioForcing states that scale-invariant comparison on positive magnitudes factors through the ratio x/y.
proof idea
The module imports NoHiddenState, OperativeDomain, and PositiveRatioForcing, then assembles them in sibling theorems such as no_hidden_state_logic_forces_rcl. Each theorem applies the counted-once composition property inside the operative-domain chain restricted to positive ratios, yielding the RCL family without additional hypotheses.
why it matters in Recognition Science
This module feeds BooleanRatioBridge, which supplies the finite Boolean bridge to positive ratios via likelihood ratios on positive event weights. It fills the step stated in the module doc-comment, connecting no-hidden-state operative comparison to the RCL family that appears in downstream Recognition Science derivations.
scope and limits
- Does not treat comparisons that reuse constituent costs or invoke hidden states.
- Does not extend the forcing beyond positive ratios.
- Does not derive numerical values for constants or spatial dimensions.
- Does not address non-operative or non-finite logical comparisons.