IndisputableMonolith.Foundation.LogicAsFunctionalEquation.NoHiddenState
The NoHiddenState module states a composition law for the cost function C under the no-hidden-state condition. It asserts existence of a counted-once resource expression that recovers the composite cost from the two constituent costs. Researchers tracing the formal derivation of the Recognition Composition Law cite this step. The module imports the normal-form counted-once syntax and states the relevant lemmas without embedding proof bodies.
claimThere exists a counted-once resource expression $E$ such that $E(C(x),C(y))$ equals the composite cost $C(xy)$.
background
This module belongs to the LogicAsFunctionalEquation namespace and imports LinearLogicBridge. The upstream module supplies the resource-sensitive syntax for counted-once comparisons; its doc-comment states that each constituent comparison appears at most once, restricting scalar monomials to 1, u, v, and u*v. The no-hidden-state condition requires that the composite cost is obtained solely by evaluating such an expression on the input costs, with no additional hidden variables.
proof idea
This is a module that aggregates the no-hidden-state composition results. It declares the composition law and the sibling lemmas no_hidden_state_implies_counted_once and no_hidden_state_comparison_forces_rcl, relying directly on the counted-once syntax imported from LinearLogicBridge.
why it matters in Recognition Science
The module supplies the no-hidden-state composition step imported by the MainTheorem package. The downstream doc-comment identifies the formal chain closest to the paper headline: 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 therefore advances the link from logical comparisons to the Recognition Composition Law.
scope and limits
- Does not derive an explicit closed form for the resource expression E.
- Does not treat comparisons that allow hidden states.
- Does not address the phi-ladder, mass formulas, or dimensional forcing.
- Does not contain the full RCL derivation or the main theorem package.