ComparisonOperatorOn
plain-language theorem explainer
ComparisonOperatorOn names the binary operator type C : K → K → K on an arbitrary type K, supplying the primitive needed to state the Law of Logic on a generic ordered field. Foundation researchers cite it when packaging the Aristotelian conditions and scale invariance that any ambient field must satisfy. The declaration is a direct one-line type alias with no proof content.
Claim. For any type $K$, a comparison operator is a function $C : K → K → K$.
background
DomainBootstrap moves from the Law of Logic (stated in LogicAsFunctionalEquation) to a bootstrap theorem for ordered fields. The comparison operator C is the primitive that lets one write the four structural conditions: identity (C x x = 0 for positive x), non-contradiction (C x y = C y x), scale invariance under positive multiplication, and distinguishability. Upstream, K is the dimensionless bridge ratio defined as φ^{1/2}. Cost functions in MultiplicativeRecognizerL4 and ObserverForcing derive J-costs from comparators on positive ratios. The module resolves the chicken-and-egg between the ambient real line and the recovered LogicReal by invoking Archimedean and Dedekind completeness to force canonical isomorphism to ℝ.
proof idea
This is a one-line abbreviation that directly equates ComparisonOperatorOn K with the function type K → K → K. No lemmas or tactics are invoked; the definition simply names the operator for downstream use in IdentityOn, NonContradictionOn, ScaleInvariantOn, and the LogicSupported structure.
why it matters
ComparisonOperatorOn supplies the operator type inside the LogicSupported structure, which collects the conditions under which the Law of Logic can be stated on an ambient field. It is the common root for derivedCostOn, DistinguishabilityOn, and the remaining bootstrap definitions that feed the uniqueness theorem forcing any such field to be isomorphic to ℝ. In the Recognition framework this step makes explicit the residual analytic input (Archimedean completeness) required to close the loop with the phi-ladder and eight-tick constructions.
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