Complete Historical Distinguishability in Two-Agent Delayed Asynchronous Consensus

Abstract

We study a scalar two-agent asynchronous averaging process with bounded discrete communication age. At each primitive action, one agent replaces its current value by a fixed convex combination of its own current value and a delayed value of the other agent, while the other agent holds. A maximum age \(D\) gives an augmented history of dimension \(M=2(D+1)\). Every admissible infinite future that reaches consensus induces a linear functional on that history. Finite-word observability and minimal realization are established switched-system concepts. Here the output family consists of asymptotic endpoints; its static row-space interpretation is elementary and is not claimed as new. The contribution is a protocol-specific rank theorem. For every \(D\ge 0\) and every self-weight \(\theta\in(0,1)\), we construct \(2(D+1)\) admissible period-two futures whose consensus endpoint functionals are linearly independent. The future-functional span is therefore the complete dual of the augmented history space. Consequently, no nontrivial linear historical compression can reproduce the consensus endpoint for every admissible future. The same scalar lower bound holds for continuously differentiable encoders and decoders on a nonempty open set of independently initialized histories. For complete probability rows with \(m\ge2\) objective coordinates, the corresponding intrinsic initialized-history dimension is \(M(m-1)\). These counts do not automatically apply to histories constrained by a preceding lawful execution. Finally, equality of consensus endpoints under every admissible future is equivalent to equality of the complete augmented histories; an affine intervention can therefore achieve schedule-robust restoration only by exactly repairing the full history. Exact rational computations corroborate the witness construction for delays through 12 and six rational self-weights. The result concerns the declared two-agent recurrence. It does not characterize larger graphs, packet availability, nonlinear discontinuous encodings, hardware, or publication priority.