Entropy Geometry and Condensation in Wealth Allocation

Abstract

We develop a statistical framework for wealth allocation in which equilibrium-like statistics follow from unbiased counting of admissible configurations rather than postulated exchange rules. Each agent is described by a value--wealth map $V_i(w)$, whose local resolution fixes the microscopic weight through a Jacobian relation. In a closed system, the microcanonical marginal and a reservoir expansion yield an emergent canonical distribution for the regular sector. Its partition sum gives a general condensation criterion: if this sector has finite wealth capacity, excess wealth concentrates on a small subset of agents. We extend the construction to open systems with variable wealth and agent number and to weak quasistatic driving. The global constraint determines an evolution equation for the common parameter $λ(t)$, while simultaneous changes in total wealth and value--wealth geometry produce a unified first-order response. The susceptibility $χ_W=\sum_i\mathrm{Var}_i(w)$ equals the Fisher information of the joint canonical family, and Legendre duality gives $\mathrm{d}s^2=χ_W\mathrm{d}λ^2=χ_W^{-1}\mathrm{d}W^2=-\mathcal{S}''(W)\mathrm{d}W^2$. For power-law critical tails, finite capacity requires $p>2$; within this regime, $χ_W$ diverges for $2 3$, while the critical boundary lies at finite Fisher--Rao distance. We also derive a qualified Cram'er--Rao duality and an open-system mixed-response relation. Contact-geometric, Airy-scaling, and stochastic-dynamical interpretations are identified only as conjectures or future work. The time-dependent results are quasistatic and do not determine microscopic relaxation times, while the information geometry describes the canonical family.