Axiom Dynamics 1: Order Structure of Axiom Bases—Theoretical Lattices, Minimal Bases, and Derivation Cycles

An axiom set is not a list of isolated assertions but a generating set of a theoret
ical lattice. This paper establishes a theory of the order structure of axiom bases
on finite Horn deduction systems. A theory is defined as a fact set closed under
deduction; the closure operator satisfies the three axioms of extensionality, mono
tonicity, and idempotence; all theories form a complete lattice under inclusion, with
intersection as meet and closure-union as join. A modularity audit yields the first
structural negation: on the seven-closed-set lattice of the main system, the modular
law fails in two cases and distributivity fails in eight cases; on the ten-closed-set
lattice of the acyclic control, the modular law fails in ten cases and distributivity
fails in forty cases. Thus a cyclic dependency graph is not a necessary condition
for non-modularity; non-modularity need not be explained by cycles alone. Base
theory yields the second structural negation: the universe has exactly two minimal
bases, of cardinalities one and two, respectively, so “the size of some minimal base”
is not a well-defined invariant; base cardinality is defined as the minimum of the
cardinalities of all minimal bases, and pointwise readings are registered as empty-set
zero, atomic-level theory one, binary theory two, and universe one. The universe
base cardinality is smaller than that of its binary subtheory, showing that base car
dinality is not a monotone function of theoretical inclusion. The derivation-cycle
theorem yields the third structural conclusion: pole collapse, base ambiguity, failure
of anticommutativity, and cyclic dependency graphs do not form a fully equivalent
relation but a spectral band—pole collapse implies base ambiguity, base ambigu
ity and failure of anticommutativity are equivalent to each other in finite systems,
and failure of anticommutativity implies a cyclic dependency graph. Each positive
assertion in the band is supported by proof-theoretic arguments; the failure of the
two outer converses is confirmed by explicit witnesses: there exists a system that
is cyclic while none of the four pathologies occurs, and there exists a system with
base ambiguity and failure of anticommutativity while pole collapse does not occur.
The nonempty-premise convention is an indispensable boundary for the link “pole
collapse implies base ambiguity” in the band: in systems containing empty-premise
rules, 4,456 cases of single-point collapse appear, all originating from fact rules and
unrelated to cycles. Within the four boundaries—four atoms, at most four rules,
conclusion not in premises, premises nonempty—the co-occurrence matrix of all
24,158 rule sets is compatible with the spectral-band structure: on the acyclic side,
5,365 systems have zero occurrence of the four pathologies; on the cyclic side, of
18,793 systems, 4,398 have no pathology, 9,096 have ambiguity and anticommu
tativity without collapse, and 5,299 have all four; the numbers of systems of the
two types “collapse without ambiguity” and “ambiguity with anticommutativity
holding” are both zero. The decisive reading is: cyclicity is a necessary but not
sufficient condition for failure of anticommutativity; cyclicity grants only the eli
gibility for pathologies to occur, not the guarantee that they actually occur; pole
collapse lies at the top of the band and is the strongest item in the cost account
of cyclicity. The interface with theoretical cartography is limited to the structural
correspondence at the level of closure axioms: deductive closure and anticommuta
tive closure share the three axioms, but deductive closure generally does not satisfy
the anticommutativity condition; the executability of the bridge takes finiteness,
decidability, and negative controls as admission conditions, and the readings of the
specific matching procedure must be registered independently.