Fixed-Start Nontermination in Homogeneous Nonnegative Equality Loops: Continuation Cores, Arithmetic Obstructions, and Decidable Frontiers develops a structural and arithmetic theory for deciding whether a prescribed initial state in a homogeneous nonnegative equality loop can generate an infinite integer trajectory. The problem arises naturally from balance-loop representations in numeration systems, but also belongs to the broader theory of integer linear-constraint loops and termination.
The paper introduces a span-relaxed continuation core, an effectively computable rational polyhedral cone that contains every infinite real or integer trajectory and stabilizes after at most rank(A) + 1 iterations. Within this core, failure of an exact integer successor is classified into three distinct mechanisms: a positive-cone obstruction, a lattice obstruction, or an affine-semigroup hole. A Hilbert-basis formulation also gives finite certificates for total viability of the continuation core.
The principal decidability result proves fixed-start nontermination decidable whenever the intrinsic physical continuation rank is at most one, regardless of the ambient matrix dimensions. In the difficult expanding case, a maximal-slope Apéry construction gives uniformly efficient predecessors; once the viability gcd stabilizes, numerical-semigroup conductors become uniformly bounded, forcing exact finite stabilization of the integer viability filtration. The paper also proves decidability whenever the coefficient continuation dimension is at most two.
Deterministic continuation cores are further connected to the Positivity Problem for integer linear recurrence sequences through an invariant orbit lattice. The resulting framework separates linear consistency, positive geometry, lattice congruences and semigroup holes, and identifies the first genuinely relational untreated case as a two-dimensional physical continuation core with one rational factorization degree of freedom.
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