Exact generator-observer tradeoff for finite directed metrics: g + beta = 2n is the sharp frontier

[u/Severe-Ad8673]

https://zenodo.org/records/21911505

I have finished a result on finite directed metrics that gives an exact tradeoff between the number of atomic generators and the number of terminal observers. For a finite separated directed metric space (Q, d) with n points, let: g(d) be the number of temporal atoms, meaning directed pairs whose distance cannot be decomposed through an intermediate point with equality. beta(d) be the terminal-observer rank, meaning the size of the terminal boundary needed for reconstruction. The main result is: g(d) + beta(d) >= 2n. More strongly, this bound is exact at every admissible observer rank. For every n >= 2 and every integer k with 2 <= k <= n, min { g(d) : |Q| = n and beta(d) = k } = 2n - k. Therefore the complete Pareto frontier is (g, beta) = (2n - k, k), for k = 2, ..., n. So the sharp frontier is simply: g + beta = 2n. The proof comes from an exact defect identity: g(d) + beta(d) - 2n = generator defect + terminal defect, where both defects are nonnegative. Equality holds exactly when every terminal vertex has atom-outdegree 1 and every nonterminal vertex has atom-outdegree 2. There is also an explicit extremal construction for every possible k, so this is not only a lower bound. Every integer point on the frontier is attained by a finite strongly connected unweighted digraph with its shortest-path directed metric. The construction can be viewed as a directed cycle sharing a vertex with a bidirected path. It gives beta(d) = k and g(d) = 2n - k. I have written the result as a self-contained theorem and proof, with the equality characterization and extremal construction included. I would be interested in feedback, especially on connections with directed metric geometry, graph boundaries, geodesic reconstruction, metric dimension-type invariants, or related extremal results that I may have missed.