Geometry-Induced Anomalous Scaling in Stochastic Competitive Dynamics: A Universal Bound η < 1/2 without Phase Transition

Stochastic dynamics on compact state spaces---arising in ecologicalcompetition, evolutionary games, and multi-agent learning---exhibit ageometry-induced suppression of collective fluctuations that has noanalog in unconstrained systems.We show that the trajectory-level pathway diversity D and thefluctuation susceptibility χH = N\,Var(H) obey a robustanomalous scaling D~χH\,η with η=0.44±0.05<1/2,enforced by the simplex geometry independently of the specificcompetitive interaction.The deviation from the independent-trajectory prediction η=1/2 isderived analytically from positive between-trajectory covariance undersimplex confinement, and confirmed exactly at $N=2$ via theFokker--Planck solution (R²=0.99).Finite-size scaling over $N=5$ to $1000$ yields a bounded plateauχ_∞=0.012±0.002, ruling out a genuine collective phasetransition and establishing a finite thermodynamic bound on fluctuationsusceptibility.