A general classification of the replication dynamics with a unique fixed point in the interior of simplex S_N

We present an algorithm for computing all evolutionarily stable strategies in nondegenerate normal-form games with three or more players.

Persistence models often conflate propagation, survival, and cross-scale loss. The Replicator-Optimization Mechanism (ROM) is a replicator-mutator template separating baseline weight, bounded survival, and a transfer kernel at a declared scale. Its equation conserves mass but guarantees neither invariance, convergence, a potential, nor a preferred scale. For finite static density-independent continuous time, an irreducible weighted kernel yields a unique positive Perron-Frobenius composition; discrete-time power convergence needs primitivity. The componentwise ranking proved here is guaranteed under exact uniform-residual transfer. Strong lumpability gives universal first-order transfer closure, and blockwise effective fitness gives an exact quotient. An institutional instantiation uses normalized stakes, signed preference-decision alignment, information loss, and descriptive effective voice. It specifies conditional survival, not legitimacy or normative authority. A companion mixed-motive MARL battery reports exploratory evidence against the implemented proxy ratio in its environment: a positive signed target-coordinate-correlation effect survives held-out evaluation under shared-state contention, while a reduced feasible-centred frozen-policy crossing reverses the predicted correlation-noise interaction. The treatment varies ideal-point correlation inside a fixed reward family, not objective- or reward-function alignment. Lean checks mapped algebraic identities and scalar monotonicities, not the stationary theorem, empirical mapping, or normative bridge. ROM is an assumptions ledger and model-construction discipline, not a cross-substrate law.

We show how large-deviation statistics allows one to obtain reliable estimates of the multiplicity of stable fixed-points in a model of neural ordinary differential equations previously employed in computational tasks. The result is obtained by developing a suitable perturbative method in the amplitude of the disorder. It turns out that for not-too-large coupling strengths there are no qualitative differences between the symmetric case, when the dynamics is a purely gradient evolution, and the asymmetric case, when limit cycles and chaos can, in principle, arise. The selection of this specific model is dictated by pedagogical reasons, but we are confident that the approach can be extended to other many-degree-of-freedom dynamical models characterized by different classes of random coupling matrices.