Symmetry

Sequential dynamical systems (SDSs) are discrete dynamical systems defined by a graph, a collection of local update functions, and an update order. A central problem in their theory is to determine the maximum number of periodic orbits of a given period that can appear in the phase space of a system on a fixed graph. Extending Defant’s correspondence between period-2 orbits and binary codes to the ternary setting, we recently showed that the maximum number of period-3 orbits of an SDS on the complete graph equals the maximum size of a ternary code with a minimum Hamming distance of at least four. Building on that structural correspondence, the present paper provides a stage-by-stage computational analysis of the associated graph construction, exact search, SDS realization, consistency checking, and candidate-verification procedures. The admissibility graph realizing the extremal quantity as a clique number can be constructed in time, and a supplied clique of size k can be processed through the construction, consistency-checking, and orbit-verification stages in time under the stated assumptions. This is polynomial in the explicit candidate parameters. Substituting $k\le 3^{m-1}$ yields the m-only upper bound ; this reparameterization does not imply non-polynomial verification in the explicit input size. Determining the extremal value from scratch, however, requires solving a Maximum Clique problem. By the Moon–Moser theorem, an N-vertex graph can have as many as $3^{N/3}$ maximal cliques, with $N=3^{m-1}$ in the present setting; thus an enumeration-based exact strategy may face a doubly exponential family of maximal cliques in m. This general bound is not a prediction of the search tree observed for the tested instances. Since the SDS graph and the corresponding coding graph have vertex sets of equal size, equality of vertex counts alone gives no immediate asymptotic advantage at the level of the presented worst-case search-space bound; it does not imply identical practical difficulty. The SDS formulation is therefore best understood as a structured framework for constructing and checking candidate solutions rather than as a faster exact algorithm. We do not claim a new complexity-class characterization, lower bound for the admissibility-graph family, or general-purpose Maximum Clique algorithm.

Affiliation Correction [...]

Hidden-state probes are widely used to study safety-related behavior in language and multimodal models, yet their results can change with the data sources, behavior labels, extracted layer, and probe family. We introduce TriSAIL, an evaluation protocol that records these choices and tests how they affect the resulting claim. TriSAIL uses three response-derived states—benign, non-refusal jailbreak, and borderline/refusal—and assigns train, validation, and test data by source. We extract hidden states from the final input position in the first generation forward pass, before a generated token is returned. The operating layer is selected using validation data only. We evaluate the protocol on five text-only LLMs and six MLLMs with logistic, kNN, and SVM probes. The LLM results vary substantially across held-out attack families, probe families, and source assignments. The MLLM probes achieve high label separability, while source-ID decoding, prediction–source association, and residualization tests show strong source alignment in the same representations. Because the main MLLM matrix couples sources and labels, these results describe source-sensitive separability. Source-transfer conclusions use source-held-out columns; mixed accuracy is a descriptive overall summary. TriSAIL reports these results with probe sensitivity and source diagnostics so that each score is interpreted under the conditions that produced it.