zero
plain-language theorem explainer
The definition sets the zero element of the logic-forced naturals to the identity constructor, which encodes a self-comparison at zero J-cost. Researchers building Peano-style arithmetic from the Recognition functional equation reference this base when introducing successor and addition operations. It is a direct abbreviation that aliases the identity case of the inductive type.
Claim. Let $N_L$ be the inductive type whose constructors are the identity element and the step operation. Define zero in $N_L$ as the identity constructor, which represents the zero-cost comparison of a quantity with itself.
background
LogicNat is the inductive type that encodes the natural numbers forced by the Law of Logic. Its identity constructor stands for the zero-cost element (the multiplicative identity in the orbit), while the step constructor adds one iteration of the generator. The structure mirrors the smallest subset of positive reals closed under multiplication by the generator and containing 1. The module ArithmeticFromLogic imports LogicAsFunctionalEquation and constructs arithmetic primitives on top of this type. Upstream, ObserverForcing.identity supplies the canonical identity event at the J-cost minimum where the state equals 1.
proof idea
One-line definition that directly aliases the identity constructor of LogicNat.
why it matters
This definition supplies the zero base case required to derive successor, addition, and induction inside ArithmeticFromLogic. It therefore feeds the subsequent construction of integers and rationals from logic that appear as sibling declarations. The placement aligns with the early steps of the forcing chain that extract arithmetic before reaching the phi-ladder and the eight-tick octave.
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