strictModular_arith_equiv_logicNat
plain-language theorem explainer
This definition establishes that the Peano carrier of the arithmetic derived from the strict modular realization on ZMod n (n>1) is equivalent to the inductive natural numbers generated by the Law of Logic. Researchers examining periodic carriers within the Recognition forcing chain would cite it to verify that modular periodicity preserves the free orbit structure. The proof is a one-line wrapper applying the orbit equivalence from the lightweight realization.
Claim. For each natural number $n>1$, the Peano carrier of the arithmetic extracted from the strict modular realization on the cyclic group of order $n$ is equivalent to the inductive type of natural numbers generated by the Law of Logic.
background
A StrictLogicRealization consists of a Carrier type, a Cost type with zero, a compare function, and a compose operation; it encodes native law data without a pre-supplied orbit. The strictModularRealization for $n>1$ sets the carrier to the cyclic group ZMod n, the cost to natural numbers, and the compare to a modular cost function while keeping composition as addition. LogicNat is the inductive type with constructors identity (zero-cost element) and step, forming the smallest subset of positive reals closed under multiplication by the generator and containing the identity. The arith operation extracts forced arithmetic from the lightweight version of any strict realization, and the upstream result states that every strict realization yields arithmetic canonically equivalent to LogicNat.
proof idea
This is a one-line wrapper that applies the orbitEquivLogicNat equivalence obtained from the toLightweight function applied to the strict modular realization.
why it matters
It confirms that the forced arithmetic remains canonically equivalent to LogicNat even when the carrier is taken to be the finite cyclic group ZMod n. This supports the universality claim that every strict realization produces the same underlying arithmetic, closing the modular case within the strict realizations section. The result aligns with the Recognition framework's emphasis on orbit preservation under different carrier interpretations.
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