Arithmetic Completeness for Two-Dimensional Cone-Monotone Integer Relations

This paper develops an exact structural theory of weakest nontermination for two-dimensional cone-monotone homogeneous integer equality relations of the form Bλ = Aλ′, with nonnegative integer coefficient states and integral full canonical quotient. The first main result removes the nonnegativity constraint and studies the resulting signed predecessor sequence as the forward domain sequence of an additive linear relation on Z^n. Finite attainment of this stable domain is characterized exactly: the signed sequence stabilizes after finitely many steps if and only if the canonical rational deterministic quotient is integral over Z. In the integral case, the stable signed lattice is effectively computable. Intersecting this stable lattice with the nonnegative orthant produces an affine semigroup. Standard normalization and conductor theory, combined with cone monotonicity, then yields an exact deep-interior saturation theorem: beyond an effectively computable translated cone, weakest nontermination is precisely membership in the stable signed lattice. Thus the entire deep interior is effectively Presburger-definable. In physical dimension two, the complement of the conductor region decomposes into two bounded facet strips and a finite corner. Good strips are shown to have effectively Presburger weakest-nontermination sets. Every remaining bad strip reduces exactly, after a finite prefix, to a finite-branch guarded residue-affine system with a common expanding multiplier of the form d n′ = m n + c, where m > d ≥ 2 and gcd(m,d) = 1. The resulting arithmetic reduction is complete. Weakest nontermination for every relation in the integral-full-quotient two-dimensional class is a finite union of Presburger pieces and inverse images of survivor sets for these common-multiplier residue-affine systems. Conversely, every system in the residual arithmetic class can be realized on a forward-invariant tail of a two-dimensional cone-monotone equality relation with integral full quotient. Consequently, universal semilinearity and effective Presburger synthesis of weakest-nontermination sets for the geometric class are equivalent to the corresponding problems for the residual arithmetic survivor class. The result isolates the precise source of non-Presburger complexity in the two-dimensional integral case: the deep interior and all good boundary regions are tame, while every surviving arithmetic obstruction is forced onto bad shadowed faces and reduces to one-dimensional common-multiplier dynamics.