Beyond Finite Kleene Convergence: Integral Cores and a Collatz Barrier for Cone-Monotone Integer Relations

Beyond Finite Kleene Convergence: Integral Cores and a Collatz Barrier for Cone-Monotone Integer Relations

This paper studies exact weakest nontermination for homogeneous nondeterministic integer relations of the form Bλ = Aλ′ under a cone-monotonicity condition.

A companion paper characterized exactly when the descending predecessor sequence stabilizes after finitely many steps. Paper XI goes beyond that boundary. It identifies a strictly larger, effectively recognizable class for which the exact weakest nontermination set remains computable even when finite Kleene convergence fails.

The main construction is a canonical core-face decomposition. When the full-cone deterministic quotient is nonintegral, the relation is compressed to its exact forward-integral coefficient core, retaining the lattice congruences required by infinite computations while strictly reducing physical dimension. When the ambient quotient is integral but nonintegral proper closed faces remain, shadow-free bad faces are isolated recursively and finite convergence is restored on their complement.

For every core-face-decomposable system, the recursion terminates after at most the physical dimension and flattens to a finite exact normal form. The weakest nontermination precondition is therefore effectively definable in existential Presburger arithmetic and is effectively semilinear. The earlier finite-Kleene class appears exactly as the depth-zero case.

The paper also develops explicit quantitative bounds for the size of the decomposition tree, the number and size of Hilbert-basis generators introduced during core reduction, and the syntactic size of the resulting Presburger formulas. An exhaustive exact census of 4,608 small two-dimensional systems shows that the new core-reduction and shadow-free mechanisms solve 646 of the 1,320 systems rejected by finite Kleene convergence, reducing the unresolved fraction by 48.94%.

The boundary of the method is arithmetically significant. In physical dimension two, a shadowed bad face can already realize the integer map n ↦ floor(4n/3) on the first adjacent lattice layer. Exact nontermination for a specific explicit relation is thereby equivalent to an unresolved generalized-Collatz reachability problem. Physical dimension two is minimal for this obstruction.

Main results:

• canonical finite core-face decomposition;
• exact dimension-reducing integral-core compression preserving lattice congruences;
• relative finite convergence outside shadow-free bad faces;
• finite exact normal form for the weakest nontermination set;
• effective existential-Presburger and semilinear representation;
• explicit decomposition-tree, Hilbert-basis and formula-size bounds;
• complete classification in physical dimension one;
• exhaustive 4,608-system computational census;
• a dimensionally minimal generalized-Collatz barrier in physical dimension two.

Verification material: The accompanying repository contains the exact-arithmetic verifier used to reproduce the 4,608-system census reported in Section 15. The computational results are validation data only; no theorem depends on the census.