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In data clustering, it is often desirable to find not just a single partition into clusters but a sequence of partitions that describes the data at different scales (or levels of coarseness). A natural problem then is to analyse and compare, as a whole, the (not necessarily hierarchical) sequences of partitions that underpin such multiscale descriptions. Using tools from topological data analysis, we introduce here the Multiscale Clustering Filtration (MCF), a filtration of abstract simplicial complexes that encodes arbitrary cluster assignments in a sequence of partitions across scales of increasing coarseness. We prove the stability of the MCF with respect to small changes in the sequence of partitions, and establish that its persistence diagram (PD) provides an interpretable representation of the multiscale clustering: at dimension 0, it captures violations of the refinement order, and at higher dimensions, it tracks the emergence and resolution of higher-order inconsistencies between cluster assignments. We also introduce an equivalent construction of the MCF via a nerve complex filtration, and show that this nerve-based MCF has computational advantages in certain settings, aiding scalability. Further, we show that in the hierarchical case, the MCF reduces to a Vietoris-Rips filtration of an ultrametric space, and we establish that pairwise measures, such as conditional entropy or zigzag persistence, are fully determined by a graph representation of the Sankey diagram and thus fail to describe the higher-order inconsistencies across scales that MCF captures. Using synthetic and real-world data from stochastic block models, single-cell transcriptomics and temporal social interactions, we illustrate how the MCF PD serves to characterise and compare multiscale clusterings beyond pairwise or hierarchical baseline methods.

A computational path records the steps used to establish an equality. Topologies on computational equality witnesses can depend on the chosen generators and execution schedule, even when their algebraic and homotopical meanings agree. We study continuous presentations of finite computational paths whose declared rewrites are realized by endpoint-fixed homotopies. We compare a topology retaining the full signed trace with an observable topology recording endpoints, length, and exact path realization. Continuous finite-word interpretations that are inverse up to scoped rewriting preserve trace-sensitive quotient arrows and their final composable-pair spaces. Observable quotients behave differently. Over a point with a nonempty primitive alphabet, we classify their topology: it records length parity when all named relations preserve parity, and is otherwise indiscrete. Adjoining a conservative abbreviation can therefore turn a discrete two-element quotient into an indiscrete space, even though ordinary multiplication remains continuous. A circle presentation also gives nonhomeomorphic quotients under binary and equal-slot timing. Geometric completeness alone does not identify the trace-sensitive and observable topologies. Multiplication is continuous on the quotient of composable representatives. We characterize when this topology agrees with the ordinary pullback topology and give an open-projection criterion. Universal presentations recover classical fundamental-groupoid topology. Normal forms, primitive interchange, and a Cayley covering prove completeness for circle, torus, product, and rose presentations. A literal-word correspondence relates the semantics to the reflexivity, symmetry, and transitivity fragment of equality derivations. Selected results are formalized in Lean, with the realization models and proof coverage stated explicitly.