Discrete Operator Theory of Distribution Dynamics (DOTDD) v1.0

The DOTDD extends the Operator Theory of Distribution Dynamics (OTDD) to discrete and constrained distribution spaces. Three canonical kernel geometries — birth-death chains, doubly stochastic matrices, and triangular matrices — produce three canonical attractor regimes covering Poisson, Binomial, Geometric, and Zipf distributions. Two structural theorems are proven: (1) the α-invariance theorem, showing that equilibrium distributions are independent of the dissipation parameter α; and (2) the Leonardo–DOTDD equivalence, establishing that Leonardo's branching rule (β=2) corresponds to α=0 in the Allocation Law and to a doubly stochastic kernel in DOTDD. Empirical validation against Fortune 500 corporate salary data and tree branching measurements confirms the framework. A new diagnostic measure, the Overhead Deviation Index (ODI), quantifies structural deviation from the physical optimum established by natural branching systems.