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Abstract This paper develops the Calculation Methods of Economic Systems (CMES) as a rigorous macro-mathematical theory of economic value, grounded in four axiomatic pillars: topology, separability, measurement theory, and network effectiveness, with macro-structure expressed through nested groups of composites. The framework is developed as a purely logical system proceeding from axioms to theorems to corollaries, requiring no empirical calibration. It assumes strictly positive transaction costs, locally finite network topologies, and ratio-scale measurability of transaction value. The central object is the valuation function "Val" (⋅), whose existence is guaranteed by Debreu’s (1954) representation theorem and whose additive functional form is derived through topological joint separability (Debreu, 1960). We situate "Val" (⋅) within a self-contained poset and nested-group architecture of economic systems, where every transaction system—from a single firm to an entire global supply chain—is formally represented as a nested transaction set (a poset element) over four canonical role layers: Value-Adding (CE^v), Archived Information and Stored Material for Production (CE^a), Communication (CE^c), and Evaluation (CE^e) elements. All definitions and theorems required for this architecture are fully derived within this manuscript. The transformation scalar σ, which links any two such systems in the master equation, is shown to equal the total factor productivity ratio under explicit competitive conditions. We introduce and derive the Network Effectiveness metric Ξ_F, which quantifies how efficiently a sub-system integrates with a broader ecosystem, and demonstrate that Ξ_F acts as a structural multiplier on "Val" (⋅). Supply chains are characterized as ordered chains of inclusive nested groups, and market valuations are derived as aggregated "Val" (⋅) integrals over these chains, weighted by Ξ_F. The theory generates conditional falsifiable predictions at the macroeconomic level. The axiomatic core makes explicit the ratio-scale identification required for multiplying "Val" (CE) by transaction counts in the Master Equation (Axiom 8), establishing a commutative monoid structure under concatenation that collapses the admissible transformation group from positive affine to positive similarity mappings. Additive separability is derived through a non-differentiable topological proof route via the Thomsen–Debreu cancellation conditions, with explicit specification of the cancellation order required for the four-dimensional product space. A systematic assessment reveals that the framework’s most critical structural tension concerns the gap between interval-scale uniqueness guaranteed by the Debreu–KLST tradition and the ratio-scale identification required by the Master Equation’s multiplicative structure—a tension resolved herein through three complementary pathways. The paper further connects the CMES poset decomposition to Möbius inversion on locally finite posets, axiomatizes Ξ_F via centrality-theoretic frameworks, situates the supply chain decomposition as a CMES-specific analog of Hulten’s (1978) theorem, and parameterizes the Archived component’s accumulation dynamics to accommodate both Romer (1990) convex compounding and Jones (1995) diminishing returns. Keywords: transaction value; poset topology; measurement theory; network effectiveness; axiomatic foundations; supply chain valuation; market capitalization; macro-system dynamics; total factor productivity; Möbius inversion; monoid structure; ratio-scale identification; Hulten theorem
Aliabdali F. 2026. CALCULATION METHODS OF ECONOMIC SYSTEMS : Macro-System Economic Value. PREPRINTS.RU. https://doi.org/10.24108/preprints-3114615
