le_def
plain-language theorem explainer
The order on LogicNat is characterized exactly by the existence of an additive witness k that carries n to m. Researchers building Peano-style arithmetic inside the Recognition Science derivation from the functional equation reference this equivalence to establish monotonicity and induction. The proof is a direct reflexivity step on the biconditional, confirming that the declared predicate matches its stated property by definition.
Claim. For elements $n, m$ of the inductive type of natural numbers generated by the identity and step constructors, $n$ is less than or equal to $m$ if and only if there exists $k$ such that $n + k = m$.
background
LogicNat is the inductive type of natural numbers forced by the Law of Logic, with the identity constructor supplying the zero-cost multiplicative identity and the step constructor supplying successive iterations of the generator. Its two-constructor structure mirrors the orbit {1, γ, γ², γ³, ...} as the smallest subset of ℝ₊ closed under multiplication by γ and containing 1. The module ArithmeticFromLogic constructs arithmetic primitives directly from this type after importing the functional-equation foundation.
proof idea
The proof is a one-line wrapper that applies Iff.rfl. Reflexivity succeeds because the predicate ≤ on LogicNat is introduced precisely as the existential statement involving addition, so the biconditional holds definitionally.
why it matters
This equivalence anchors the ordering relation inside the arithmetic layer built from the Law of Logic, supporting sibling constructions such as addition and the induction principle. It supplies the ordering needed for later steps in the Recognition Science program that derive the forcing chain and the phi-ladder from the same functional equation. No downstream uses are recorded yet, leaving open its precise integration with the eight-tick octave and spatial dimension results.
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