forcingFn_zero
plain-language theorem explainer
Between any two strict Law-of-Logic realizations, the universal forcing map on the forced naturals sends zero to zero. Anyone assembling the forced-semiring certificate or proving that the map preserves addition, multiplication, or one cites this fact. The proof is a one-line wrapper of the strict universal-forcing equivalence's zero-preservation lemma.
Claim. For any two strict Law-of-Logic realizations $R$ and $S$, the universal forcing map $F_{R,S}\colon\mathbb{N}_{\mathrm{Logic}}\to\mathbb{N}_{\mathrm{Logic}}$ satisfies $F_{R,S}(0)=0$.
background
A strict Law-of-Logic realization packages a carrier, a cost type with a distinguished zero-cost, a comparison map, and a composition operation, with no externally supplied orbit. From any such realization one extracts Peano data on the forced naturals.
LogicNat is the inductive type forced by the Law of Logic: identity is the zero-cost element (multiplicative identity in the orbit), and step is one further iteration of the generator. Its two-constructor shape mirrors the orbit ${1,\gamma,\gamma^2,\ldots}$ as the smallest positive-real subset closed under multiplication by $\gamma$ and containing $1$.
The universal forcing map between two strict realizations is the unique structure-preserving equivalence of their forced arithmetic. In this module it is re-presented as an endomap of LogicNat (the forced carriers are definitionally that type). Upstream, the canonical forcing development already records that this equivalence sends $R$'s forced zero to $S$'s forced zero.
proof idea
One-line wrapper. It applies the upstream lemma universal_forcing_map_zero on the pair of strict realizations $(R,S)$, which states that the strict universal-forcing equivalence sends $R$'s Peano zero to $S$'s Peano zero. Unfolding the definition of the forcing endomap on LogicNat yields the claim.
why it matters
Zero-preservation is the base case for the forced-semiring package. Downstream, addition and multiplication preservation are proved by induction that takes this fact (together with successor preservation) as input; one-preservation is the same pattern at the unit. The forced-semiring certificate bundles the map with bijectivity and the four structure laws, listing zero-preservation explicitly among its fields.
In the broader Recognition Science foundation this sits inside universal forcing: any two strict realizations of the Law of Logic are connected by a unique arithmetic equivalence, so the forced carrier is unique up to unique isomorphism. That uniqueness underwrites the later forcing chain (T0–T8) in which the cost functional, $\varphi$, the eight-tick octave, and $D=3$ are derived rather than postulated.
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