Topness: Collider & Topological Perspectives

• Topness is a concept with distinct definitions: one as a collider kinematic observable and another as a topology-aware geometric goodness-of-fit measure.
• In collider phenomenology, topness improves background rejection by quantifying event consistency with top-quark decay hypotheses, enhancing signal sensitivity in stop and double-Higgs searches.
• In topological data analysis, topness evaluates the emptiness and structuredness of projected point clouds using Vietoris–Rips complexes, detecting nonlinear geometric features.

Searching arXiv for the specific works and closely related uses of “Topness” across fields. Topness is not a single universal concept but a family of technically distinct notions that appear in several research areas. In collider phenomenology, it denotes a reconstruction-based discriminator that measures how well an event fits a top-quark decay hypothesis, especially dileptonic $t\bar t$ backgrounds in searches for stops or double-Higgs production (Graesser et al., 2012, Kim et al., 2018). In topology-aware data analysis, it denotes a geometric goodness-of-fit score built from Vietoris–Rips complexes and interpreted as a measure of emptiness or nontrivial structure in projected point clouds (Hernández et al., 2018). In broader usage, the term can function informally as shorthand for the degree to which a system exhibits nontrivial topology, but such usage does not imply a common formal definition across fields (Lantagne-Hurtubise et al., 2019).

1. Terminological scope and disambiguation

The principal technical usages of topness in the supplied literature fall into two categories: a collider-kinematics observable and a topology-aware data-analytic index. These two meanings are operationally unrelated. In the first, topness is a constrained fit to a top-quark decay topology; in the second, it is an area-based score derived from a simplicial complex built on data points.

Domain	Object called topness	Operational meaning
Collider phenomenology	Event-level kinematic discriminator	Compatibility with a $t\bar t$ hypothesis
Topological data analysis	Geometric goodness-of-fit index	Emptiness or structuredness of a projected point cloud
Informal topology language	Shorthand only	Degree of nontrivial topological character

Several nearby terms are distinct and should not be conflated with topness. In toric geometry, a top is a lattice polytope cut out from a 4-dimensional reflexive polytope by a hyperplane and used to construct building blocks for twisted connected sums of compact $G_2$ manifolds; this is a toric-combinatorial object, not a measure of “topness” (Braun, 2016). Likewise, in function-space topology, tightness, supertightness, Id-fan tightness, and $T$-tightness are cardinal or selection-type properties of $(C(X),\tau^s_{\mathfrak B})$, again unrelated to collider or data-analytic topness (Chandra et al., 2022).

2. Topness in semileptonic stop searches

In supersymmetry searches, topness was introduced as a kinematic discriminator for rejecting semileptonic backgrounds from top-quark pair production when one lepton is missed, especially in searches for the asymmetric stop decay $\tilde t \to t\,\chi^0$ and $\tilde t \to b\,\chi^\pm$ (Graesser et al., 2012). The central problem is that dileptonic $t\bar t$ events with one unreconstructed lepton can mimic a signal containing one isolated lepton, $b$-jets, and large missing transverse energy.

The construction assumes that the event arose from dileptonic $t\bar t$ and asks how well the invisible momenta can be chosen to satisfy the expected mass shells. The paper defines

$t\bar t$0

and then

$t\bar t$1

The minimization incorporates transverse momentum conservation, the neutrino masslessness condition $t\bar t$2, and the on-shell condition $t\bar t$3. The weighting parameters are

$t\bar t$4

Operationally, small values correspond to events that can be reconstructed as dileptonic $t\bar t$5, whereas larger values correspond to events that fit that hypothesis poorly. The implementation uses one identified isolated lepton, two jets taken as $t\bar t$6-jet candidates, and missing transverse momentum. When two $t\bar t$7-tagged jets are present they are used directly; if only one $t\bar t$8-tag is found, the two hardest untagged jets with $t\bar t$9 and $G_2$0 are used in the pairing procedure. The minimization is performed with 10 iterations of the Nelder–Mead algorithm per event.

The observable was introduced because standard variables such as $G_2$1, $G_2$2, or $G_2$3 were developed mainly for more symmetric topologies and do not fully exploit the partially reconstructed dileptonic-top structure of the dominant background. In the benchmark asymmetric-stop search, the cut flow shows that after several standard cuts the significance is around $G_2$4, while after the topness cut $G_2$5 it rises to about $G_2$6 for the reference point $G_2$7 GeV, $G_2$8 GeV at $G_2$9 and 8 TeV. The study also reports that for the asymmetric $T$0 signal, topness performs best among the variables compared, and for the symmetric $T$1 signal it is competitive with $T$2 (Graesser et al., 2012).

3. Topness in double-Higgs analyses

In double-Higgs searches, topness was reformulated for the $T$3 final state as a measure of consistency with dileptonic $T$4 production, which is the dominant background to $T$5 (Kim et al., 2018). The essential interpretation is explicit: topness provides a degree of consistency for a given event to dilepton $T$6 production. Small topness means that the event looks like $T$7; large topness means that the event is less consistent with $T$8 and therefore more signal-like.

The observable is defined by minimizing a $T$9-like function over the unknown neutrino momenta, subject to the missing-transverse-momentum constraint $(C(X),\tau^s_{\mathfrak B})$0. Because there are two possible $(C(X),\tau^s_{\mathfrak B})$1-jet–lepton pairings,

$(C(X),\tau^s_{\mathfrak B})$2

The objective function penalizes deviations from two top-mass constraints $(C(X),\tau^s_{\mathfrak B})$3 and two $(C(X),\tau^s_{\mathfrak B})$4-mass constraints $(C(X),\tau^s_{\mathfrak B})$5, with $(C(X),\tau^s_{\mathfrak B})$6 GeV and $(C(X),\tau^s_{\mathfrak B})$7 GeV. The minimization is performed numerically with MINUIT.

The utility of the variable follows directly from event topology. In $(C(X),\tau^s_{\mathfrak B})$8, the final state naturally satisfies two $(C(X),\tau^s_{\mathfrak B})$9-mass constraints, two top-mass constraints, and the missing-momentum condition. In the signal $\tilde t \to t\,\chi^0$0, the $\tilde t \to t\,\chi^0$1-jets come from a Higgs boson rather than from top decays, so the same set of constraints is generically incompatible with the event. The analysis therefore combines topness with a signal-side discriminator called Higgsness and with subsystem variables $\tilde t \to t\,\chi^0$2, $\tilde t \to t\,\chi^0$3, and $\tilde t \to t\,\chi^0$4, using the strategy

$\tilde t \to t\,\chi^0$5

The paper reports a substantial gain in sensitivity. After baseline cuts, the significance is $\tilde t \to t\,\chi^0$6. After applying Higgsness, topness, $\tilde t \to t\,\chi^0$7, $\tilde t \to t\,\chi^0$8, and $\tilde t \to t\,\chi^0$9, the significance increases to $\tilde t \to b\,\chi^\pm$0, quoted as about $\tilde t \to b\,\chi^\pm$1 at $\tilde t \to b\,\chi^\pm$2. Over the same sequence, the $\tilde t \to b\,\chi^\pm$3 background drops from $\tilde t \to b\,\chi^\pm$4 fb after baseline cuts to $\tilde t \to b\,\chi^\pm$5 fb, and total background falls from $\tilde t \to b\,\chi^\pm$6 fb to $\tilde t \to b\,\chi^\pm$7 fb. The study states that this optimized kinematic strategy performs better than CMS neural-network or boosted-decision-tree-based approaches and improves the final significance by at least about a factor of two relative to earlier studies (Kim et al., 2018).

4. Neutrino-solution topness and feature engineering

A later double-Higgs analysis retains the same basic interpretation of topness but modifies the implementation and exploits the minimization outputs more aggressively for feature construction (Alves et al., 15 Sep 2025). Here topness again distinguishes events compatible with $\tilde t \to b\,\chi^\pm$8 from $\tilde t \to b\,\chi^\pm$9 and Drell–Yan–like backgrounds. The observable is defined as

$t\bar t$0

with $t\bar t$1 built from two top-mass and two $t\bar t$2-mass penalties. The resolution-like parameters are $t\bar t$3 GeV and $t\bar t$4 GeV.

The principal methodological change is that the analysis does not impose the measured missing-transverse-momentum constraint in the minimization. Instead, the two neutrino three-momenta are left unconstrained by $t\bar t$5, with the energies fixed by the massless condition $t\bar t$6. The minimization is performed numerically with the simplex algorithm, specifically SciPy/Nelder–Mead. The paper argues analytically that for $t\bar t$7 events the constraints cannot be simultaneously satisfied, so the minimum can occur near a non-smooth point at nearly zero neutrino momentum. This produces low-mass peaks in quantities reconstructed from the topness solution.

The neutrino solutions are then reused to define additional observables. The paper explicitly introduces a neutrino-pair invariant mass from the topness solution, $t\bar t$8, the ratio

$t\bar t$9

a ratio $b$0 built from reconstructed scales $b$1 and $b$2, and a modified mass variable $b$3 involving $b$4 in the Higgs-pair rest frame. The paper emphasizes that these are largely ratios of kinematic quantities and are therefore less sensitive to overall normalization uncertainties such as luminosity.

In the cut-based analysis, the baseline selection requires

$b$5

The optimized cuts include

$b$6

along with additional cuts on $b$7, $b$8, $b$9, and $t\bar t$0. The reported expected significance is $t\bar t$1 at $t\bar t$2 fb$t\bar t$3, about $t\bar t$4 better than the benchmark taken from the literature. In the multivariate analysis, $t\bar t$5 and $t\bar t$6 remain important according to SHAP-based feature-importance studies, and a profile-likelihood treatment of the boosted-decision-tree score reaches $t\bar t$7 if the background systematic uncertainty is about $t\bar t$8. The study stresses, however, that this shape-based gain degrades quickly as the background uncertainty grows (Alves et al., 15 Sep 2025).

5. Topness as a topology-aware goodness-of-fit measure

In topological data analysis, topness denotes a geometric goodness-of-fit index intended to play a role analogous to $t\bar t$9, but for topological and geometric structure rather than variance explained (Hernández et al., 2018). The construction is variablewise: for each input variable $t\bar t$00, one studies the projected point cloud $t\bar t$01, reconstructs a geometric object with a Vietoris–Rips complex, and measures how much of the surrounding rectangular domain remains empty. If the cloud fills the rectangle almost completely, the index is near $t\bar t$02; if the points organize into a structured pattern leaving large voids, the index approaches $t\bar t$03.

The method first rescales each bivariate projection $t\bar t$04 to the unit square $t\bar t$05. It then computes all pairwise Euclidean distances, chooses $t\bar t$06 as an empirical quantile of the distance matrix, constructs the neighborhood graph

$t\bar t$07

with

$t\bar t$08

and expands it to the Vietoris–Rips complex

$t\bar t$09

The paper explicitly adopts the clique-complex viewpoint, following the two-phase procedure of first building the neighborhood graph and then computing the simplicial expansion.

For the projection $t\bar t$10, the bounding rectangle is

$t\bar t$11

and the index is

$t\bar t$12

The interpretation is that $t\bar t$13 measures emptiness: if the complex fills the box, $t\bar t$14 is near $t\bar t$15; if the complex occupies only a small part of the box, the index is near $t\bar t$16. The measure is designed to detect nonlinear, non-monotonic, circular, hole-containing, disconnected, or manifold-like structures that classical linear or polynomial goodness-of-fit can miss.

The implementation is provided in the TopSA package in R, using the TDA package for barcode and complex estimation and sf for spatial objects and area estimation. The examples include a linear model $t\bar t$17, a “circle with one hole,” multiple separated circles, and the Ishigami function. In the “circle with one hole” example, classical $t\bar t$18 is near zero while $t\bar t$19 is around $t\bar t$20 for $t\bar t$21, indicating hole detection. For the Ishigami model, the reported geometric scores are around the 50–60% range across variables. The paper also records important limitations: the choice of $t\bar t$22 is critical and not fully solved, the neighborhood graph construction is $t\bar t$23, clique enumeration can be expensive, and the method is described mainly for bivariate projections rather than multivariate interactions (Hernández et al., 2018).

6. Informal and extended uses of the term

Beyond its formal uses as a kinematic or geometric index, topness can function as an informal shorthand for topological character. In the classification of non-magnetic crystalline compounds, topology is diagnosed from symmetry representations, band representations, elementary band representations, and symmetry indicators. The large-scale survey of the International Crystal Structure Database covers more than 26,000 materials and reports that about 27–30% are topological, roughly 12% are insulating topological materials, about 15–18% are semimetals, and the outcome amounts to around 8000 new candidate topological compounds (Lantagne-Hurtubise et al., 2019). In this setting, talk of the “topness” of a crystalline solid refers to whether its occupied bands are compatible with an atomic insulator or instead exhibit topological band structure inferred from symmetry fingerprints.

A different but related terminological neighborhood appears in gravitational theory. “Topological stars,” or top stars, are smooth horizonless static solutions of five-dimensional Einstein–Maxwell theory that reduce to spherically symmetric solutions of Einstein–Maxwell–Dilaton theory in four dimensions. For scalar perturbations, the analysis finds three classes of quasi-normal modes—prompt ringdown modes, long-lived metastable modes, and “blind” modes—and all computed frequencies have negative imaginary parts, suggesting linear stability. The same work reports zero static Love numbers, nontrivial dynamical Love numbers, and vanishing dissipative effects at linear order (Bianchi et al., 2023). These results concern topological stars rather than a scalar quantity called topness.

This suggests a useful distinction. In collider phenomenology and topology-aware data analysis, topness is a formal observable with an explicit computational definition. In materials and gravitational discussions, by contrast, analogous language is usually descriptive and refers more loosely to nontrivial topological structure than to a named invariant or discriminator. The shared intuition is structural rather than mathematical: in each case, “topness” points to the extent to which hidden organization is present and diagnostically useful, but the underlying objects, algorithms, and interpretations remain domain-specific.