Relative Minimal Module Basis for Structural–Regime Theory: Closure of Systemic Results, Constructive Irreducibility, and the Dynamical–Fisher–Statistical Interface

This paper studies a relative metatheoretical problem for structural–regime theory: under a common mathematical background and a fixed partition of semantic roles, does a theory spanning geometry, information, dynamics, and statistical observation possess an irreducible common modular basis? Here, “minimality” is not interpreted as the number of axioms, which can be altered arbitrarily by logical conjunction. Instead, we define a common background theory, semantic closure, role-preserving module representations, and a relative module-basis rank. Minimality is thereby recast as an irreducibility problem within a fixed class of semantic interfaces. We introduce four interface modules: the geometric interface B_G (Markov–Fisher naturality); the information–factor interface B_I (Markov information cascades, scale separation, nondegenerate mechanism roles, and factor-level distinguishability); the dynamical–boundary interface B_D (local structural stratification, nondegenerate critical sets, transverse crossing, and local normal forms); and the statistical-observation interface B_O (Fréchet differentiability of the statistical trace and nondegeneracy along the switching direction). The common background imposes only technical conditions concerning measurability, regularity, finite information quantities, and measure selection; when invariant measures are nonunique, the selected measure is explicitly specified by a measure selector. We obtain six groups of principal results. First, under the Markov cascade W → X → F → U, we establish the exact information-deficit decomposition H(W) − I(W;U) = H(W|X) + I(W;X|F) + I(W;F|U). Second, on a regular critical set, we prove that transverse crossing produces a sign change of a defining function, yielding local identification of a regime boundary. Third, if the parameter difference between two sides has a nonzero leading term aτ^q and the composite statistical trace T has a nonzero Fréchet derivative in the a direction, then the statistical visibility order is exactly preserved as q. Fourth, defining the Fisher intrinsic energy in the switching direction by E_sw = g_F(θ*)(a,a), we prove its coordinate invariance and monotone contraction under Markov coarse-graining. Fifth, we establish a joint-generation theorem for the four interfaces and construct directional witnesses for each interface, thereby proving strong irreducibility. Sixth, under a fixed role system, we define a relative module-basis rank and prove Rank_mod = 4. Furthermore, we package the prior results on Fisher–B6, information–factor structure, structural switching, periodic observability, and faithful statistical representation into an explicit core conclusion package of the theory, and prove relative semantic closure of this package under the four-interface basis. For results at the exactness level, we explicitly add calibration conditions rather than claiming an unconditional extension. The paper thus distinguishes two levels: “axiom reduction,” which studies conditional necessity for individual conclusions, and “joint metatheory,” which studies the irreducible semantic basis shared by multiple conclusions. This is a purely theoretical study and involves no empirical data, numerical simulation, or parameter fitting. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently determined by the author. Multiple AI tools assisted with code implementation, data presentation, and text drafting. The author bears full academic responsibility for all research content.