Axiom Dynamics 7: Large-Scale Machine Checking of a Lightweight Terminological Ontology for the Deadlock Criterion on Real Rule Bases

Lightweight terminological ontologies are usually implemented as fragments of
Horn rule systems, whose axiom sets are required in practice to be consistent and
incrementally maintainable. This paper examines the execution criterion of axiom
transitions within the candidate space of leaf-premise single-antecedent rules: given
a baseline rule system and a candidate batch, the question is whether there ex
ists an ordering of the batch such that, when the candidates are applied one by
one, every step produces a nonempty newly added object-level derived pair. Under
the conditions that the baseline derivation relation is acyclic, candidate premises
are leaves, and candidate rules have single antecedents, the blocking relation con
stitutes a strict partial order among distinct candidates; under the condition that
the batch has no inter-candidate chain, the folding lemma guarantees that every
multi-hop derivation contains a single-hop blocker, whence every batch without an
inter-candidate chain admits a legal sequential execution, and a legal order is ex
plicitly constructed in polynomial time as a linear extension of the blocking partial
order; when a chain exists this construction may fail, and its existence is registered
as an open problem. The semantics of proper subfamilies with equivalent closure is
therefore located as ledger-redundancy declaration internal to a batch, rather than
as a criterion for sequential infeasibility. At the quantitative level, when there is no
inter-candidate chain the batch gain equals the union of the candidate tail sets, and
coverage enumeration agrees with the general formula under this condition; when
a chain exists the coverage formula may omit bridging pairs, and this boundary
is fixed by a strict synthetic counterexample. Machine checking on four real OBO
corpora registers the zero-redundancy reading of the single-axiom redundancy sur
vey, the negative result of candidate-pool scale and equivalence-class compression
ratio 1.0, and, on a 12-candidate demonstration batch, the reading of the inclusion
minimal redundancy-declaration subfamily obtained by running all 4096 subsets.
The theoretical conclusions hold within the boundaries of the Horn fragment, leaf
premises, single antecedents, and an acyclic baseline; the sequential-feasibility con
clusion additionally requires that the batch have no inter-candidate chain; universal
directional propositions are borne by proofs, while machine checking bears finite
counterexample search and implementation cross-verification.