Influence Capacity Beyond GDP: An Axiomatic Three-Axis Decomposition

<div> <span>Cross-national comparisons that rest on Gross Domestic Product silently assume that one dollar of value added corresponds to one dollar of mobilizable, system-shaping capability. The assumption fails sharply when the comparison spans different industrial structures, different state capacities, and different network positions in the global production and rule-making system. This paper develops an axiomatic framework that replaces GDP, in its role as a proxy for national power, with a three-axis influence functional I(S) = PI^a * MU^b * NU^c, where PI is the productive-capacity axis (defined via useful work / exergy throughput), MU is the mobilizable-surplus axis (defined as a control-theoretic reachability rate over factor allocations), and NU is the network-position axis (defined as a percolation-based perturbation rate on the technology sets of other systems). The Cobb-Douglas form is derived along three converging routes: (i) a representation theorem showing any influence functional satisfying a commensurability-prior axiom plus five representation axioms (continuity, monotonicity, zero-axis annihilation, base-point-independent multiplicative scaling, log-additivity) takes this form; (ii) a Tullock-contest microfoundation in which states compete in three independent contests with distinct battlegrounds; (iii) a budget-feedback dynamic microfoundation deriving the linear coupled ODE that supports the Cobb-Douglas hegemonic-dynamics interpretation from a capability-evolution principle. The dynamics admit a meta-theorem on collapse sequence under which the slowest-decay axis crosses its threshold first; this unifies a strict PI -&gt; NU -&gt; MU ordering for self-productive industrial-era hegemons with an information-era inversion (NU -&gt; PI -&gt; MU) and an administered-external-base regime. The framework is calibrated against eleven historical and contemporary dynamic cases (Habsburg Spain, Dutch Republic, Pax Britannica, late Soviet Union, Qing China, Ottoman Empire, Mughal India, Tang China, Roman Empire, Mongol Empire, Russia 2014-2024) plus the Saudi-US 1970 static cross-section as a twelfth supplementary check, with sub-national portability tested against three corporate dyads (Apple/Samsung, Microsoft/Google, TSMC/Intel) and three documented historical firm collapses (BlackBerry, Nokia, Kodak). Under an ex-ante regime-assignment protocol that locks regime tags from primitives observable at the latest pre-crossing date, the eleven cases yield 9 strict matches + 1 joint-shock partial (Tang) + 1 explicit violation (Mughal under lock-date central-treasury MU); the structural-ordering theorem rejects the random-ordering null at p approx 4e-6, with a 4-case selection-bias-corrected extension yielding p approx 1.88e-4 on 15 cases. Four pre-registered falsifiable predictions for the US-China transition are reported with explicit failure thresholds: (P1) 2028 dollar reserve share &gt;= 52%; (P2) 2030 advanced-semiconductor alliance share &gt;= 80%; (P3) 2032 RMB reserve share &lt;= 8%; (P4) 2045 China/US overall ratio in [0.69, 0.95] (upper bound 0.95 grid-conditional on the displayed sensitivity sweep). Two further regime-discriminating predictions are added: (P5) Saudi Arabia AEB-discriminating NU-first ordering by 2030 conditional on hegemonic-claimant trajectory; (P6) Germany SP-discriminating PI-first ordering by 2035 in the deindustrialization trajectory. Joint failure of any two of (P1)-(P3) within five years, or (P4) outside the displayed grid envelope, constitutes framework rejection rather than calibration revision.</span> </div>