Persistence/Transition-Variety Framework

• The Persistence/Transition-Variety Framework is a schema that jointly examines persistent structures, transition maps, and admissible state varieties across fields like TDA and dynamical systems.
• It formalizes persistence using methods such as zigzag persistence, interleavings, and continuation, turning invariants into coordinates for analyzing state transitions.
• The framework’s applications span from phase stratification in cosmology and spin systems to routine behavior and economic transitions, highlighting layered, structured change.

Searching arXiv for papers related to persistence/transition/variety frameworks in TDA, dynamics, and state-transition analysis. arXiv search query: "persistence transition varieties interleavings persistence diagrams zigzag persistence continuation point clouds routine transitions identity persistence" The “Persistence/Transition-Variety Framework” (Editor’s term) denotes a comparative schema in which persistent structure, the mechanisms that transport or revise that structure, and the space of admissible states or regimes are studied together rather than separately. In topological data analysis this schema appears in zigzag persistence, interleavings, continuation, and localization-aware diagram representations; in dynamical systems it appears as stratification by bifurcation loci and phase-regime boundaries; and in applied sequence analysis it appears as stable state distributions together with transition laws across time (0812.0197, Gameiro et al., 2015, Acharya et al., 2020, Cotsakis, 7 Apr 2026).

1. Conceptual architecture

At its most general, the framework separates three questions. First, what persists: homology classes, diagram points, routine types, institutional effects, or identity constraints. Second, what counts as a transition: a non-monotone map in a zigzag, a parameter crossing in a phase portrait, a continuation step in persistence space, a shift between daily routine types, or a revision of an operative role. Third, what is the relevant variety: an interval decomposition, an affine variety of interleavings, a phase stratification, or a family of clustered behavioral states.

A foundational formalization is given by zigzag persistence, which replaces a monotone filtration

$$X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X_n$$

with a mixed-direction sequence

$$X_1 \leftrightarrow X_2 \leftrightarrow \cdots \leftrightarrow X_n,$$

and proves that the resulting zigzag module decomposes into interval modules $I(b,d)$ (0812.0197). Here persistence is no longer tied to one-directional growth; it is defined across additions, deletions, unions, intersections, and other non-monotone transitions.

A second canonical formalization appears in the study of interleavings. For interval modules $M$ and $N$, the collection of all $\epsilon$-interleavings is organized as an affine variety $V_{M,N}^{\epsilon}$, and the smallest $\epsilon$ for which this variety is nonempty is the interleaving distance (Acharya et al., 2020). In that setting, persistence is encoded by the existence of compatible morphisms, transition is encoded by variation in $\epsilon$, and variety is literal algebraic-geometric structure.

Continuation of point clouds gives the same triad in inverse form. A point cloud $u$ is mapped to a persistence vector $$X_1 \leftrightarrow X_2 \leftrightarrow \cdots \leftrightarrow X_n,$$0, a target $$X_1 \leftrightarrow X_2 \leftrightarrow \cdots \leftrightarrow X_n,$$1 is chosen, and the transition $$X_1 \leftrightarrow X_2 \leftrightarrow \cdots \leftrightarrow X_n,$$2 is lifted to configuration space by pseudo-inverse Newton continuation,

$$X_1 \leftrightarrow X_2 \leftrightarrow \cdots \leftrightarrow X_n,$$3

thereby turning a persistence-space path into a geometric path of realizations (Gameiro et al., 2015). This suggests a general principle: persistence data can be treated not only as invariants, but as coordinates on a transition problem.

2. Representation of persistent structure

A central requirement of the framework is a representation in which persistent structure is comparable across instances while still retaining enough localization to support transition analysis. Several papers realize this requirement in distinct but compatible ways.

The Persistence Atlas introduces a domain-based representation of critical-point layouts. For a critical point set $$X_1 \leftrightarrow X_2 \leftrightarrow \cdots \leftrightarrow X_n,$$4, the persistence map is

$$X_1 \leftrightarrow X_2 \leftrightarrow \cdots \leftrightarrow X_n,$$5

with $$X_1 \leftrightarrow X_2 \leftrightarrow \cdots \leftrightarrow X_n,$$6 and $$X_1 \leftrightarrow X_2 \leftrightarrow \cdots \leftrightarrow X_n,$$7, so geometrically robust critical points contribute more strongly and over broader support (Favelier et al., 2018). Distances between ensemble members are then defined by $$X_1 \leftrightarrow X_2 \leftrightarrow \cdots \leftrightarrow X_n,$$8 distances between these maps, and spectral embedding plus clustering identifies dominant layout patterns. Here persistence is tied to topological saliency, while variety is realized as a multimodal decomposition of ensemble structure.

The persistence transformation adds positional information to one-dimensional Morse-style persistence. For a maximum $$X_1 \leftrightarrow X_2 \leftrightarrow \cdots \leftrightarrow X_n,$$9 matched to $I(b,d)$0, the transformation records

$I(b,d)$1

and the reduced transformation records

$I(b,d)$2

Its main stability result is

$I(b,d)$3

so localization is added without sacrificing Wasserstein stability (Klaila et al., 2023). This is important for a persistence/transition framework because silent rearrangement of features in the domain is itself a kind of transition.

A computationally efficient Hilbert-space embedding is provided by the square-block construction for persistence diagrams. Each diagram point $I(b,d)$4 is replaced by a square $I(b,d)$5 of side length $I(b,d)$6, and the diagram is mapped to

$I(b,d)$7

With a weighting function $I(b,d)$8, the associated persistence surface is

$I(b,d)$9

and pixel integration over a partition $M$0 gives coordinates

$M$1

The resulting vectorization is stable under Wasserstein perturbations and yields bounds such as

$M$2

under boundedness and Lipschitz hypotheses on $M$3, $M$4, and $M$5 (Chan et al., 2021). The paper is explicit that this representation is not equivalent to bottleneck distance; it is instead a computationally favorable summary.

3. Transition mechanisms and comparison morphisms

Within the framework, transitions are not merely observed changes; they are structured morphisms, continuations, or compatibility relations.

For $M$6-indexed persistence modules, the generalized comparison framework for interleavings replaces constant shifts by monotone translation maps. A $M$7-interleaving is given by

$M$8

such that

$M$9

This yields induced matchings between persistence diagrams and interval-wise bounds sharper than a single bottleneck constant (Harker et al., 2018). In the bijective case, matched endpoints satisfy

$N$0

The resulting theory is explicitly transition-oriented: the control data are not only distances, but the maps that tell how one parameterization deforms into another.

The affine-variety construction for interval interleavings pushes this further. For interval modules, all possible progressions of $N$1 are classified. If $N$2, the progression is origin, axis, origin; if $N$3, it is axis, origin; if $N$4, it is either hyperbola, plane, axis, origin or hyperbola, plane, origin (Acharya et al., 2020). These progressions encode which constraints die first: cross-Hom births, self-map vanishing, or final loss of all cross-maps. Variety is therefore not an ornament; it contains recoverable information about how interleavings come into existence.

Continuation methods provide the constructive analog. The persistence map $N$5 is decomposed as $N$6, where $N$7 sends point-cloud coordinates to simplex birth radii and $N$8 extracts persistence coordinates. Under genericity assumptions, $N$9 is $\epsilon$0, and local realization sets $\epsilon$1 are smooth manifolds of dimension $\epsilon$2 when $\epsilon$3 is surjective (Gameiro et al., 2015). This gives a precise local model of transition varieties as fibers of the persistence map.

4. Stratifications, phase regimes, and transition sets

In several literatures, the framework becomes a stratification of parameter or state space into persistence domains separated by transition loci.

In scalar-field cosmology with exponential potential $\epsilon$4, $\epsilon$5, the local phase portrait is organized by five loci in the $\epsilon$6-plane: $\epsilon$7 Near these loci the paper computes translated jets, centre(-like) reductions, and normal forms, then constructs an explicit stratification for the exponential class and a pull-back stratification for the quadratic case (Cotsakis, 7 Apr 2026). Persistence domains are the connected components of the complement of the organising set; transition varieties are the codimension-$\epsilon$8 subsets where hyperbolicity or normal hyperbolicity is lost.

In the long-range XY model, persistence means persistence of the BKT transition under algebraically decaying couplings $\epsilon$9. The paper argues for a three-phase organization consisting of a magnetized low-temperature phase, an intermediate BKT or quasi-long-range ordered phase, and a high-temperature disordered phase, and it states that there is no direct magnetized-to-disordered transition (Walther et al., 10 Nov 2025). In this setting, variety is phase content, and transition structure is determined by the coupling between spin waves and vortices rather than by a bare long-range confinement argument.

A nonequilibrium analog appears in the disordered contact process, where the absorbing/active threshold is

$V_{M,N}^{\epsilon}$0

while a persistence-specific threshold satisfies

$V_{M,N}^{\epsilon}$1

Below $V_{M,N}^{\epsilon}$2, the paper identifies a Griffiths phase with

$V_{M,N}^{\epsilon}$3

at $V_{M,N}^{\epsilon}$4, it reports activated scaling; and for $V_{M,N}^{\epsilon}$5, local persistence obeys

$V_{M,N}^{\epsilon}$6

with continuously varying, and near $V_{M,N}^{\epsilon}$7, effectively complex exponents (Bhoyar et al., 2020). This is a particularly clear example of the framework: a single control parameter organizes multiple transition varieties depending on which observable is tracked.

5. Behavioral, institutional, and identity-oriented applications

The same triad of persistence, transition, and variety appears in domains far from topology.

In the study of daily routines, each day is represented by 13 features spanning sleep, mobility, and device use, pooled days are clustered by a Gaussian mixture model, and the resulting routine types define a day-level state space. The paper reports that daily life typically resolves into approximately eight routine types, that each person maintains a characteristic distribution over those types, and that both occupancy distributions and transition matrices are more similar within individuals than between individuals across observation windows spanning weeks to months (Luong et al., 26 Apr 2026). Here persistence is the temporal stability of the routine signature, transition is the adjacent-day routine matrix, and variety is the distribution over routine types.

The Narrative Continuity Test generalizes the same structure to AI identity persistence. It defines five necessary axes: Situated Memory, Goal Persistence, Autonomous Self-Correction, Stylistic & Semantic Stability, and Persona/Role Continuity, and argues that current stateless LLM architectures systematically fail to support them (Natangelo, 28 Oct 2025). The central requirement is explicitly diachronic: commitments formed at $V_{M,N}^{\epsilon}$8 must constrain behavior at $V_{M,N}^{\epsilon}$9 beyond what is reconstructible from the local prompt. This is a non-topological but exact analogue of persistence under transition.

Macroeconomic work on transition economies uses an institutional version of the framework. Inflation persistence is modeled through lagged inflation and interaction terms with institutional variables, so that persistence becomes conditional on institutional configuration rather than a common constant. In the reported dynamic-panel results, both wage rigidity and exchange-rate regime rigidity tend to dampen inflation persistence, with the exchange-rate effect described as particularly strong and robust (Tanevski et al., 16 May 2026). Variety here is institutional cohort structure; transition is propagation of inflation across periods; persistence is the carry-over coefficient conditioned by that structure.

A philosophical counterpart is provided by the General Formal Ontology, which distinguishes presentials, continuants, and processes as pairwise disjoint kinds of entities (Herre, 2013). In the paper’s formulation, continuants persist, processes are temporally extended and fundamental, and each material continuant is linked to an associated process whose process boundaries coincide with the presentials exhibited by the continuant. This suggests a layered reading of the framework in which persistent entities, transition processes, and manifestational variation are different ontological strata rather than competing descriptions.

6. Scope, limits, and methodological cautions

The framework is broad, but the meaning of “persistence” is not uniform across the cited literatures. In the persistence transformation it is elder-rule endurance of local maxima (Klaila et al., 2023); in interleaving theory it is existence of compatible morphisms (Harker et al., 2018, Acharya et al., 2020); in routine analysis it is stability of occupancy and transition signatures (Luong et al., 26 Apr 2026); in AI continuity it is maintenance of commitments across interaction gaps (Natangelo, 28 Oct 2025); and in inflation dynamics it is the lagged transmission of price growth (Tanevski et al., 16 May 2026). A unified framework therefore remains analogical unless a specific semantics is fixed.

Several constructions are explicitly local or conditional. Point-cloud continuation is local, depends on genericity assumptions, and can fail at rank-deficient points or realizability boundaries (Gameiro et al., 2015). The scalar-field cosmology framework is a local phase-portrait and normal-form analysis near organising loci rather than a complete global classification (Cotsakis, 7 Apr 2026). The long-range BKT result is analytic and perturbative in $\epsilon$0, and the paper itself states that a complete analysis of the full RG equations is left open (Walther et al., 10 Nov 2025).

Other frameworks are descriptive rather than fully dynamical. The Persistence Atlas provides a latent geometry of ensemble members and cluster-wise confidence regions, but it does not define transition probabilities or temporal trajectories; its transition content is therefore implicit (Favelier et al., 2018). The daily-routine model is a “bag-of-days” clustering stage followed by empirical transition matrices, not a jointly estimated hidden Markov model (Luong et al., 26 Apr 2026). The Narrative Continuity Test is a conceptual framework for evaluation rather than an operational benchmark with fixed scoring rules (Natangelo, 28 Oct 2025).

These limits do not weaken the comparative value of the framework. They indicate, rather, that persistence/transition/variety is best understood as a family of structurally related research programs: one concerned with stable objects or constraints, one with the morphisms or perturbations that transport them, and one with the stratified spaces in which those transports change type.