ComparisonOperator
plain-language theorem explainer
The type of maps assigning a real cost to any pair of positive quantities serves as the entry point for deriving the laws of logic from the Recognition functional equation. Researchers tracing the arithmetic-from-logic chain cite this definition as the root object. It is introduced by a direct type abbreviation with no additional computation required.
Claim. A comparison operator is any function $C : ℝ → ℝ → ℝ$ such that, for positive inputs, $C(x, y)$ returns the cost of comparing the quantities $x$ and $y$.
background
The module treats the emergence of logic as a consequence of a functional equation satisfied by cost functions. The basic object is a map that takes two positive real numbers and returns a real number interpreted as their comparison cost. This map is required to obey four structural constraints that encode the Aristotelian laws of logic.
proof idea
This declaration consists of a single-line type abbreviation that identifies the comparison operator with the function space from the reals to the reals to the reals. No lemmas are applied; the abbreviation simply names the type for use in subsequent statements.
why it matters
The definition supplies the root type for the entire logic-as-functional-equation development. It is used to construct the generator of the laws of logic and to establish that the real numbers support logic. It also enters the analysis of continuous combiners that recover bilinearity under additional smoothness hypotheses. In the broader Recognition Science setting it realizes the cost map whose properties force the phi-ladder and the spatial dimension count.
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