strict_arith_universal_initial
plain-language theorem explainer
This definition supplies a canonical equivalence between the Peano carrier of arithmetic derived from any strict logic realization and the LogicNat inductive type. Researchers formalizing invariance of forced arithmetic in Recognition Science would cite it to anchor the Strict Universal Forcing result. The construction is a direct one-line wrapper that routes through the lightweight realization and its orbit equivalence.
Claim. For any strict logic realization $R$, the carrier of the Peano structure in the arithmetic derived from $R$ is canonically equivalent to LogicNat, the inductive type whose constructors identity and step generate the orbit of the multiplicative generator starting from the zero-cost element.
background
The Strict/Invariance module establishes that native Law-of-Logic data determines a derived free orbit and that all such orbits are canonically equivalent. StrictLogicRealization encodes a strict realization of the Law of Logic; its arith field extracts the derived arithmetic whose Peano carrier is the object of interest. LogicNat, defined upstream in ArithmeticFromLogic, is the inductive type with constructors identity (the multiplicative identity, zero-cost element) and step (one iteration of the generator), exactly mirroring the smallest subset of positive reals closed under multiplication by the generator and containing 1.
proof idea
The definition is a one-line wrapper that applies the orbit equivalence after converting the input strict realization to its lightweight form via toLightweight.
why it matters
The definition anchors the Strict Universal Forcing theorem by exhibiting the canonical equivalence of all derived arithmetics to LogicNat. It supplies the base invariance step that later results in the module rely upon to equate free orbits across realizations. In the Recognition Science framework it realizes the claim that arithmetic is forced uniformly from logical data, independent of auxiliary physical constants or dimensional choices.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.