Goldbach-Lemoine Descent Graphs

We introduce the Goldbach–Lemoine Descent Graph, a discrete dynamical object formed through iterated additive decomposition. Using witness pairs selected at each step from a constrained search window according to a fixed policy, we construct a directed acyclic graph rooted in the source integer. Unconstrained Descent Graphs exist wherever the Goldbach and Lemoine conjectures hold; we empirically verify the existence of our constrained variants for all integers up to 109. Over a set of randomly sampled integers between 5x104 and 5x108, and across five policy frameworks (four deterministic and one random null), we instantiate the graphs and probe several key properties. For all integers N ≥ 98 in our tested range, the graphs—across policies—terminate at a common set of ten non-consecutive primes. Aggregating the graph-wide branching weights at each sink yields a composition vector that constitutes an arithmetic signature of N. After ILR embedding, these signatures form policy-dependent manifolds. Despite the abundance of additive representations, deterministic selection produces sharply differentiated low-dimensional geometry, modular concentration phenomena, and long-range serial dependence. Under randomized witness selection these structures disappear. The results demonstrate that in the context of additive partitions, arithmetic selection rules alone can induce coherent geometric and dynamical organization in integer indexed data.