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 theoretical lattice. This paper establishes a theory of the order structure of axiom baseson finite Horn deduction systems. A theory is defined as a fact set closed underdeduction; the closure operator satisfies the three axioms of extensionality, monotonicity, and idempotence; all theories form a complete lattice under inclusion, withintersection as meet and closure-union as join. A modularity audit yields the firststructural negation: on the seven-closed-set lattice of the main system, the modularlaw fails in two cases and distributivity fails in eight cases; on the ten-closed-setlattice of the acyclic control, the modular law fails in ten cases and distributivityfails in forty cases. Thus a cyclic dependency graph is not a necessary conditionfor non-modularity; non-modularity need not be explained by cycles alone. Basetheory yields the second structural negation: the universe has exactly two minimalbases, 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 thecardinalities of all minimal bases, and pointwise readings are registered as empty-setzero, atomic-level theory one, binary theory two, and universe one. The universebase cardinality is smaller than that of its binary subtheory, showing that base cardinality is not a monotone function of theoretical inclusion. The derivation-cycletheorem yields the third structural conclusion: pole collapse, base ambiguity, failureof anticommutativity, and cyclic dependency graphs do not form a fully equivalentrelation but a spectral band—pole collapse implies base ambiguity, base ambiguity and failure of anticommutativity are equivalent to each other in finite systems,and failure of anticommutativity implies a cyclic dependency graph. Each positiveassertion in the band is supported by proof-theoretic arguments; the failure of thetwo outer converses is confirmed by explicit witnesses: there exists a system thatis cyclic while none of the four pathologies occurs, and there exists a system withbase ambiguity and failure of anticommutativity while pole collapse does not occur.The nonempty-premise convention is an indispensable boundary for the link “polecollapse implies base ambiguity” in the band: in systems containing empty-premiserules, 4,456 cases of single-point collapse appear, all originating from fact rules andunrelated to cycles. Within the four boundaries—four atoms, at most four rules,conclusion not in premises, premises nonempty—the co-occurrence matrix of all24,158 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, of18,793 systems, 4,398 have no pathology, 9,096 have ambiguity and anticommutativity without collapse, and 5,299 have all four; the numbers of systems of thetwo types “collapse without ambiguity” and “ambiguity with anticommutativityholding” are both zero. The decisive reading is: cyclicity is a necessary but notsufficient condition for failure of anticommutativity; cyclicity grants only the eligibility for pathologies to occur, not the guarantee that they actually occur; polecollapse lies at the top of the band and is the strongest item in the cost accountof cyclicity. The interface with theoretical cartography is limited to the structuralcorrespondence at the level of closure axioms: deductive closure and anticommutative closure share the three axioms, but deductive closure generally does not satisfythe anticommutativity condition; the executability of the bridge takes finiteness,decidability, and negative controls as admission conditions, and the readings of thespecific matching procedure must be registered independently.