ofInt_neg
plain-language theorem explainer
Embedding the additive inverse of a verifier integer yields the same PRC integer as pointwise negation of the embedded value. Anyone simplifying signed-orbit arithmetic or the PRC integer layer would cite this simp fact. The proof pushes equality through injectivity of the integer display map and closes by simplification.
Claim. For every integer $n\in\mathbb{Z}$, the PRC integer obtained by embedding $-n$ equals the pointwise negation (swap of positive and negative orbit parts) of the PRC integer obtained by embedding $n$.
background
In Primitive Recognition Calculus, integers are realized as equivalence classes of signed orbits: pairs of distinction-natural components recording positive and negative parts. The display map sends such a class to an ordinary verifier integer. Pointwise negation on a signed orbit swaps the two components; the corresponding operation on PRC integers is written as that swap on representatives.
The embedding of a verifier integer routes its positive and negative parts through the $\delta$-orbit construction on distinction naturals, then packages them as a signed orbit. Upstream, injectivity of the display (K4.8) says distinct PRC integers have distinct verifier displays, so equal displays force equal PRC integers. That injectivity is the bridge used whenever an identity is easier to check after projecting to $\mathbb{Z}$.
proof idea
Apply injectivity of the PRC-to-verifier integer display. It remains only to check that both sides display to the same ordinary integer; a single simplification step discharges that equality from the definitions of the embedding and of pointwise negation.
why it matters
This is a basic commutation law between verifier negation and the PRC integer embedding. It keeps the signed-orbit model aligned with ordinary integer arithmetic, so later lemmas can move freely between the two presentations without ad hoc case splits on sign. No downstream users are recorded yet; the lemma sits as simp infrastructure in the IntegerRational layer of the foundation, supporting any argument that treats PRC integers as a faithful lift of $\mathbb{Z}$. It does not itself touch the forcing chain (T0–T8) or the Recognition Composition Law, but it stabilizes the integer substrate those layers eventually use.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.