operative_domain_rcl_logic_reality_chain
plain-language theorem explainer
Any comparison operator on positive reals that satisfies the operative domain structure (finite logical comparison on continuous positive ratios) obeys the four Aristotelian laws plus scale invariance and yields a derived cost belonging to the RCL family. Researchers tracing the Recognition Science derivation of logic from a single functional equation would cite this as the central bridge theorem. The proof is a one-line wrapper applying the headline corollary rcl_logic_reality_chain.
Claim. Let $C: (0,∞) × (0,∞) → ℝ$ be a comparison operator. If $C$ satisfies the operative domain structure (finite logical comparison on the continuous positive-ratio setting), then $C$ obeys the laws of logic (identity, non-contradiction, excluded middle, scale invariance, route independence) and the derived cost $F(r) = C(r,1)$ lies in the RCL family: there exist a combiner $P$ and constant $c$ such that $P(u,v) = 2u + 2v + c·u·v$ with multiplicative consistency.
background
A ComparisonOperator is an abbrev ℝ → ℝ → ℝ that assigns a real cost to comparing two positive quantities; the four Aristotelian constraints plus scale invariance and non-triviality are encoded in the structure SatisfiesLawsOfLogic. The derivedCost of C is the one-argument function F(r) := C(r,1), well-defined on positive ratios under scale invariance. RCLFamily asserts that F belongs to the family of functions admitting a combiner P that is affine in each argument separately, with the explicit quadratic form P(u,v) = 2u + 2v + c·u·v. OperativeDomainStructure is the predicate FiniteLogicalComparison C, which places the comparison in the finite, continuous positive-ratio setting. The module collects the formal chain closest to the paper headline: scale-free comparison factors through positive ratios, no-hidden-state finite comparison gives counted-once composition, and counted-once finite logical comparison forces the RCL family.
proof idea
The proof is a one-line wrapper that applies the headline corollary rcl_logic_reality_chain. That corollary decomposes the claim into the conjunction of operative_domain_satisfies_logic C h (which yields SatisfiesLawsOfLogic) and operative_domain_identification C h (which yields RCLFamily on derivedCost).
why it matters
This is the headline theorem of the MainTheorem module, establishing that on the operative domain logical comparison and the RCL family share the same forced algebraic form. It completes the chain from scale-free comparison through counted-once composition to the Recognition Composition Law, directly supporting the J-uniqueness step (T5) and the forcing of phi as self-similar fixed point (T6) in the broader Recognition Science framework. No open questions or scaffolding remain; the result is fully proved with zero sorry.
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