zero_add
plain-language theorem explainer
zero_add establishes that left-addition by the identity leaves any element of the naturals forced by the Law of Logic unchanged. Researchers building the additive monoid on the orbit generated by the functional equation cite it when assembling Peano structure from the Recognition primitives. The proof is a direct induction on the two-constructor type, applying the successor-addition rule in the step case and reflexivity at the base.
Claim. For every $n$ in the natural numbers forced by the Law of Logic, $0 + n = n$, where $0$ is the identity constructor of that type.
background
LogicNat is the inductive type whose constructors are identity (the zero-cost multiplicative identity in the orbit) and step (one further iteration of the generator), mirroring the smallest subset of positive reals closed under multiplication by the golden ratio and containing 1. Addition is defined recursively on this type, with the companion lemma add_succ stating that left-addition distributes over successor. The module ArithmeticFromLogic derives these arithmetic primitives directly from the functional equation without external axioms, anchoring zero at the identity event of minimum J-cost.
proof idea
The proof proceeds by induction on the LogicNat structure. The identity base case reduces immediately to reflexivity. In the step case the add_succ lemma rewrites the left-hand side to succ (zero + n), after which the induction hypothesis yields succ n.
why it matters
This theorem supplies the left-unit law for addition on the forced naturals and is invoked in downstream constructions such as PhiInt, the canonical PhiRing object, and the logarithmic derivative of the inverse fine-structure constant. It completes one link in the chain from the Law of Logic to the arithmetic layer that supports the eight-tick octave and D = 3 spatial dimensions. No scaffolding remains; the result is fully proved.
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