IndisputableMonolith.Foundation.UniversalForcing.NaturalNumberObject
NaturalNumberObject module defines the Lawvere natural-number object as the universal recursion property for Peano structures inside universal forcing. Researchers tracing forced arithmetic objects cite it when showing that any two LogicRealizations yield canonically equivalent initial algebras. The module supplies only definitions and supporting lemmas; no proofs appear.
claimA triple $(N, z, s)$ is a Lawvere natural-number object when, for every pointed endomap $(X, x, f)$, there exists a unique morphism $h : N o X$ satisfying $h(z) = x$ and $h \circ s = f \circ h$.
background
UniversalForcing states that any two Law-of-Logic realizations produce canonically equivalent forced arithmetic objects because those objects are initial Peano algebras. The present module imports that result together with the Strict.DiscreteBoolean realization, whose carrier orbit is periodic yet whose forced arithmetic is the free iteration object.
The module encodes the Lawvere NNO via the structure whose recursor field supplies the unique primitive recursion map. This characterization makes no reference to the native natural-number type and serves as the universal property any Peano structure must satisfy.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module supplies the NNO definition that TimeAsOrbit uses to claim the temporal sequence Tick is the natural-number object forced by recognition. It also feeds UniversalForcingSelfReference, which shows that the Universal Forcing Meta-Theorem itself fits the Law-of-Logic structural shape and thereby closes the framework reflexively.
scope and limits
- Does not construct an explicit carrier for any NNO.
- Does not derive the NNO from the T0-T8 forcing chain.
- Does not prove isomorphism to the native natural numbers.
- Does not address constants or mass formulas in RS units.