Frank Jet: A Disambiguation Problem

• Frank Jet is an ambiguous label that conflates the Frank-Read mechanism in nematic liquid crystals, FastJet in collider physics, and Frank t‐norms in probabilistic logic.
• In nematic systems, the Frank-Read mechanism describes a pinned disclination that bows under twist deformation and emits concentric loops to relieve stress.
• FastJet employs efficient sequential recombination algorithms for jet analysis while Frank t‐norms provide a coherent framework for probabilistic conjunctions in conditional events.

Searching arXiv for “Frank Jet” and closely related terms to determine whether it is an established term or a conflation. “Frank Jet” is not introduced as a technical term in the cited arXiv literature. Across the relevant sources, the proximate established terms are the Frank-Read mechanism in nematic liquid crystals, FastJet in collider jet finding, and Frank t-norms in the coherence-based semantics of compound conditional events. This suggests that “Frank Jet” is best treated as an ambiguous or conflated label rather than a recognized research concept. The three underlying topics belong to distinct domains—topological-defect dynamics, high-energy jet reconstruction, and probabilistic logic—and share nomenclature only at the lexical level, not through a common formalism (Long et al., 2022).

1. Terminological status and scope

The cited sources name three separate objects. In soft condensed matter, the relevant term is the Frank-Read mechanism, originally known from crystal plasticity and shown to have an analogue in nematic liquid crystals (Long et al., 2022). In collider phenomenology, the relevant term is FastJet, a C++ package and general-purpose jet-finding and jet-analysis library for modern collider analyses (Cacciari et al., 2011). In uncertainty formalisms for conditional events, the relevant term is Frank t-norms, together with the dual Frank t-conorms, used as probabilistic values of conjunctions and disjunctions under de Finetti coherence (Gilio et al., 2020).

Because these are distinct named constructions, “Frank Jet” has no stable denotation across the supplied literature. A plausible implication is that the expression arises from conflation: either “Frank” is carried over from Frank-Read or Frank t-norms, while “jet” is carried over from FastJet or from collider-physics usage of jets. No source among the cited works defines a hybrid object linking these meanings.

2. Frank-Read mechanism in nematic liquid crystals

In the nematic-liquid-crystal setting, the central object is a pinned disclination segment between two surface defects on one substrate in a nematic cell. Under applied twist deformation, the pinned segment bows, self-intersects, and emits a new disclination loop, while the original segment remains pinned. Repetition of the process generates concentric disclination loops, directly analogous to the emission of concentric dislocation loops by a crystalline Frank-Read source (Long et al., 2022).

The experiment uses a dynamic liquid-crystal cell with 5CB nematic liquid crystal between two glass substrates. The top substrate is a plano-convex lens with uniform planar anchoring; the bottom substrate has a photoaligned anchoring pattern containing two point defects of topological charge +1/2 and −1/2. These surface defects pin a vertical disclination arch. The reported surface defect spacing is about 100 μm, the cell gap is about 30 μm, and at zero twist the disclination spans the two surface defects. The control parameter is the relative in-plane rotation of the top and bottom substrate alignments, which imposes twist strain on the nematic director field (Long et al., 2022).

The observed sequence is explicit: the segment bows laterally, becomes a curved arc, self-intersects, and then snaps off a loop that expands away from the source. The smallest experimentally observed twist producing loop emission in a reasonable waiting time is about 144°, and at 160° loop emission occurs with essentially no waiting time. A further key result is periodicity: each additional 180° of applied twist can generate another loop, and each loop emission reduces the net twist deformation by $\pi$ radians (Long et al., 2022).

The theoretical treatment combines free-energy minimization and a Peach-Koehler-force picture. The free-energy model balances line tension energy of the disclination against twist elastic energy of the director field. One reported implicit relation for the bowed shape is

$$\delta\phi=\arctan(2\bar{y}_\text{top})+\frac{8\alpha \bar{y}_\text{top}}{1+4\bar{y}_\text{top}^2},$$

with

$$\bar{y}_\text{top}=y_\text{top}/w,\qquad \alpha=(Id)/(\pi K_{22}w).$$

The force-balance formulation yields

$r=\frac{Id}{\pi K(\delta\phi-\phi_B)}.$

A major scaling result is that the maximum stable effective stress is inversely proportional to the pinned length,

$$\sigma^\text{eff}_\text{crit}\approx \frac{2I}{\pi w},$$

or, using $I\approx K\pi^2$ for equal Frank constants,

$$\sigma^\text{eff}_\text{crit}\approx \frac{2K\pi}{w}.$$

This inverse-$w$ scaling is the direct analogue of the inverse pin-spacing law for crystalline Frank-Read sources (Long et al., 2022).

3. FastJet and the technical meaning of “jet”

In high-energy physics, “jet” refers not to a topological defect but to the output of a jet-finding procedure acting on particles or pseudojets. FastJet is a C++ package that provides jet finding and analysis tools, including efficient native implementations of widely used 2-to-1 sequential recombination jet algorithms for $pp$ and $e^+e^-$ collisions, as well as access to third-party algorithms through a plugin mechanism (Cacciari et al., 2011).

The library packages the algorithm name, parameters, and recombination scheme into a JetDefinition, and applies that definition through a ClusterSequence. For hadron collisions, FastJet natively implements the longitudinally invariant algorithms $$\delta\phi=\arctan(2\bar{y}_\text{top})+\frac{8\alpha \bar{y}_\text{top}}{1+4\bar{y}_\text{top}^2},$$0, Cambridge/Aachen, anti-$$\delta\phi=\arctan(2\bar{y}_\text{top})+\frac{8\alpha \bar{y}_\text{top}}{1+4\bar{y}_\text{top}^2},$$1, and generalized $$\delta\phi=\arctan(2\bar{y}_\text{top})+\frac{8\alpha \bar{y}_\text{top}}{1+4\bar{y}_\text{top}^2},$$2. The manual gives the standard distance measures. For longitudinally invariant $$\delta\phi=\arctan(2\bar{y}_\text{top})+\frac{8\alpha \bar{y}_\text{top}}{1+4\bar{y}_\text{top}^2},$$3,

$$\delta\phi=\arctan(2\bar{y}_\text{top})+\frac{8\alpha \bar{y}_\text{top}}{1+4\bar{y}_\text{top}^2},$$4

with

$$\delta\phi=\arctan(2\bar{y}_\text{top})+\frac{8\alpha \bar{y}_\text{top}}{1+4\bar{y}_\text{top}^2},$$5

Cambridge/Aachen uses

$$\delta\phi=\arctan(2\bar{y}_\text{top})+\frac{8\alpha \bar{y}_\text{top}}{1+4\bar{y}_\text{top}^2},$$6

anti-$$\delta\phi=\arctan(2\bar{y}_\text{top})+\frac{8\alpha \bar{y}_\text{top}}{1+4\bar{y}_\text{top}^2},$$7 uses

$$\delta\phi=\arctan(2\bar{y}_\text{top})+\frac{8\alpha \bar{y}_\text{top}}{1+4\bar{y}_\text{top}^2},$$8

and generalized $$\delta\phi=\arctan(2\bar{y}_\text{top})+\frac{8\alpha \bar{y}_\text{top}}{1+4\bar{y}_\text{top}^2},$$9 uses

$$\bar{y}_\text{top}=y_\text{top}/w,\qquad \alpha=(Id)/(\pi K_{22}w).$$0

The familiar special cases are $$\bar{y}_\text{top}=y_\text{top}/w,\qquad \alpha=(Id)/(\pi K_{22}w).$$1 for $$\bar{y}_\text{top}=y_\text{top}/w,\qquad \alpha=(Id)/(\pi K_{22}w).$$2, $$\bar{y}_\text{top}=y_\text{top}/w,\qquad \alpha=(Id)/(\pi K_{22}w).$$3 for Cambridge/Aachen, and $$\bar{y}_\text{top}=y_\text{top}/w,\qquad \alpha=(Id)/(\pi K_{22}w).$$4 for anti-$$\bar{y}_\text{top}=y_\text{top}/w,\qquad \alpha=(Id)/(\pi K_{22}w).$$5 (Cacciari et al., 2011).

FastJet is also explicitly an algorithmic-performance framework. The manual notes that naïve sequential recombination scales like $$\bar{y}_\text{top}=y_\text{top}/w,\qquad \alpha=(Id)/(\pi K_{22}w).$$6, whereas FastJet uses geometric acceleration, including nearest-neighbor and Delaunay-triangulation-based methods, to achieve expected $$\bar{y}_\text{top}=y_\text{top}/w,\qquad \alpha=(Id)/(\pi K_{22}w).$$7 performance for many algorithms. It provides several clustering strategies, including N2Plain, N2Tiled, N2MinHeapTiled, NlnN, and NlnNCam, with Best selecting an efficient strategy automatically (Cacciari et al., 2011).

In addition to clustering, FastJet includes a broad analysis stack: jet substructure, boosted-object tagging, pileup and underlying-event estimation, jet areas, and background subtraction or suppression. It supports active area, passive area, and Voronoi area definitions; median-based background estimators such as JetMedianBackgroundEstimator; and subtraction through the Subtractor transformer using

$$\bar{y}_\text{top}=y_\text{top}/w,\qquad \alpha=(Id)/(\pi K_{22}w).$$8

or, in 4-vector form,

$$\bar{y}_\text{top}=y_\text{top}/w,\qquad \alpha=(Id)/(\pi K_{22}w).$$9

Accordingly, within collider physics the technically correct lexical form is FastJet, not “Frank Jet” (Cacciari et al., 2011).

4. Frank t-norms and conditional-event conjunctions

A second possible source of confusion comes from Frank t-norms, which belong to probabilistic logic rather than collider or soft-matter physics. In the cited framework, a t-norm $r=\frac{Id}{\pi K(\delta\phi-\phi_B)}.$0 is a commutative, associative, monotone binary operation with $r=\frac{Id}{\pi K(\delta\phi-\phi_B)}.$1. The three extremal t-norms are

$r=\frac{Id}{\pi K(\delta\phi-\phi_B)}.$2

The Frank family $r=\frac{Id}{\pi K(\delta\phi-\phi_B)}.$3 interpolates among them: $r=\frac{Id}{\pi K(\delta\phi-\phi_B)}.$4 The paper states that $r=\frac{Id}{\pi K(\delta\phi-\phi_B)}.$5 is continuous in $r=\frac{Id}{\pi K(\delta\phi-\phi_B)}.$6, decreasing in $r=\frac{Id}{\pi K(\delta\phi-\phi_B)}.$7, and satisfies

$r=\frac{Id}{\pi K(\delta\phi-\phi_B)}.$8

The dual Frank t-conorm is

$r=\frac{Id}{\pi K(\delta\phi-\phi_B)}.$9

with the identity

$$\sigma^\text{eff}_\text{crit}\approx \frac{2I}{\pi w},$$0

(Gilio et al., 2020).

For conditional events $$\sigma^\text{eff}_\text{crit}\approx \frac{2I}{\pi w},$$1 and $$\sigma^\text{eff}_\text{crit}\approx \frac{2I}{\pi w},$$2, the conjunction is defined as a conditional random quantity,

$$\sigma^\text{eff}_\text{crit}\approx \frac{2I}{\pi w},$$3

where $$\sigma^\text{eff}_\text{crit}\approx \frac{2I}{\pi w},$$4, $$\sigma^\text{eff}_\text{crit}\approx \frac{2I}{\pi w},$$5, and $$\sigma^\text{eff}_\text{crit}\approx \frac{2I}{\pi w},$$6. Under logical independence, the paper proves

$$\sigma^\text{eff}_\text{crit}\approx \frac{2I}{\pi w},$$7

for every $$\sigma^\text{eff}_\text{crit}\approx \frac{2I}{\pi w},$$8. More generally, for $$\sigma^\text{eff}_\text{crit}\approx \frac{2I}{\pi w},$$9 logically independent conditional events, the coherent prevision of their conjunction satisfies the sharp Fréchet-Hoeffding bounds

$I\approx K\pi^2$0

and every coherent extension is representable as a Frank t-norm value $I\approx K\pi^2$1 for some $I\approx K\pi^2$2 (Gilio et al., 2020).

This domain-specific use of “Frank” therefore designates a parametric family of t-norms and t-conorms, not a jet algorithm and not a defect source.

5. Why the phrase is ambiguous

The ambiguity follows from the coexistence of three independent naming conventions:

Expression in the sources	Domain	Technical object
Frank-Read mechanism	Nematic liquid crystals / crystal plasticity analogy	Pinned defect segment emitting loops
FastJet	Collider physics	Jet-finding and jet-analysis library
Frank t-norms	Coherence-based probability theory	Parametric family for conjunction/disjunction previsions

Only FastJet contains the word “jet” as a technical noun, and only Frank-Read and Frank t-norms contain the word “Frank” as part of an established name. Thus “Frank Jet” has no direct one-to-one mapping onto a recognized formal object in the cited literature. A plausible implication is that the phrase may emerge from transcriptional error, memory compression, or cross-domain leakage between unrelated literatures (Cacciari et al., 2011).

The ambiguity is not merely stylistic. Each source attaches its terminology to a distinct ontology. In the nematic setting, the key entities are disclination lines, surface defects, twist strain, line tension, and effective stress (Long et al., 2022). In FastJet, the key abstractions are JetDefinition, ClusterSequence, PseudoJet, Selector, Transformer, and algorithm-dependent distance measures (Cacciari et al., 2011). In the conditional-event setting, the key objects are previsions, coherent assessments, Fréchet-Hoeffding bounds, and the family $I\approx K\pi^2$3 of Frank t-norms (Gilio et al., 2020). The same surface phrase therefore collapses incompatible technical vocabularies.

6. Disambiguation in scholarly usage

For precise scholarly usage, the term should be replaced by the intended established name.

If the intended topic is defect multiplication in nematics, the correct expression is Frank-Read mechanism in nematic liquid crystals, where a pinned disclination segment bows under twist and emits concentric disclination loops; the critical stress scales inversely with pin spacing, and the process depends on strain rate and temperature (Long et al., 2022).

If the intended topic is collider reconstruction, the correct expression is FastJet, the modular analysis framework that combines native sequential recombination algorithms, a plugin system for cone and legacy algorithms, substructure tools such as Filter and Pruner, taggers such as MassDropTagger and JHTopTagger, and background estimation/subtraction based on jet areas and median $I\approx K\pi^2$4 estimation (Cacciari et al., 2011).

If the intended topic is many-valued or coherence-based probabilistic logic, the correct expression is Frank t-norms or Frank t-conorms, which under logical independence provide the exact probabilistic semantics of conjunctions and disjunctions of conditional events and parametrize the coherent assessment sets between the Fréchet-Hoeffding bounds (Gilio et al., 2020).

In this sense, “Frank Jet” is best understood not as a concept but as a disambiguation problem.