HRT-Boost: Gravity, Collider, & ML Applications

• HRT-Boost is a polysemous term that unifies three disparate constructs across gravity, collider physics, and machine learning.
• In semiclassical gravity, it denotes a boost operator derived from extremal HRT surfaces that alters extrinsic curvature and entanglement wedges.
• In jet physics and machine learning, it enables dynamic jet tagging for hyper-boosted heavy objects and stage-wise gradient boosting with hinge regression trees.

HRT-Boost is a polysemous technical term used in at least three distinct research programs. In semiclassical gravity, it denotes the Hamiltonian flow generated by the renormalized Hubeny–Rangamani–Takayanagi area, interpreted as a relative boost across an HRT surface (Kaplan et al., 2022). In collider phenomenology, it denotes the “Hyper-Boosted Radius Transverse-Boost” jet tagger for hadronically decaying heavy objects at multi-TeV transverse momentum (Larkoski et al., 2015). In tabular machine learning, it denotes the ensemble extension of Hinge Regression Trees, combining node-level damped Newton optimization with stage-wise functional gradient descent (Li et al., 22 May 2026). Because the same label is attached to unrelated constructions, precise domain-specific usage is essential.

1. Terminological scope

The shared string “HRT-Boost” conceals different expansions of “HRT,” different meanings of “boost,” and different mathematical objects. In gravity, HRT refers to Hubeny–Rangamani–Takayanagi surfaces and “boost” is a Lorentzian matching transformation across an extremal surface. In jet physics, HRT expands to “Hyper-Boosted Radius Transverse-Boost” and “boost” refers to the large transverse boost of a heavy particle. In machine learning, HRT expands to Hinge Regression Tree and “boost” refers to gradient boosting over tree learners (Kaplan et al., 2022, Larkoski et al., 2015, Li et al., 22 May 2026).

Usage of HRT-Boost	Domain	Core object
HRT-area boost	Semiclassical gravity	Hamiltonian flow generated by $A_{\rm HRT}/(4G)$
Hyper-Boosted Radius Transverse-Boost	Jet substructure	Dynamic-radius, track-based tagger
Hinge Regression Tree Boost	Tabular regression	Boosted ensemble of oblique piecewise-linear trees

A frequent source of confusion is that the three usages are not minor variants of one another. They belong to different literatures, use different acronyms, and address different technical problems. The common surface form of the term is therefore chiefly a nomenclature collision rather than a sign of conceptual continuity.

2. HRT-Boost in semiclassical gravity

In Einstein–Hilbert gravity minimally coupled to matter, one picks a boundary achronal region $R$ with HRT surface $\gamma_R[g]$, the bulk codimension-2 extremal surface homologically anchored on $\partial R$. The renormalized area functional is

$$A_{\rm HRT}[R] := {\rm Area}[\gamma_R] = \int_{\gamma_R} d^{D-2}w\,\sqrt{q(w)},$$

and the rescaled generator

$H_R \equiv \frac{A_{\rm HRT}[R]}{4G}$

plays the role of a “boost Hamiltonian” on the covariant phase space (Kaplan et al., 2022).

The central structural fact is that extremality of $\gamma_R[g]$ implies

$$\delta_\gamma A[\gamma,g]\vert_{\gamma=\gamma_R[g]}=0.$$

As a result, when computing Poisson or Peierls brackets involving $A_{\rm HRT}$, one may hold the surface fixed and vary only the bulk metric. In canonical variables $(h_{ij},\Pi^{ij})$ on a Cauchy slice $R$0, $R$1 depends only on $R$2, not on $R$3, so

$R$4

The nontrivial action is on the extrinsic curvature: $R$5 with $R$6 the unit $R$7-normal to $R$8. Equivalently, the finite flow by parameter $R$9 satisfies

$\gamma_R[g]$0

This identifies the action of $\gamma_R[g]$1 as a localized change in the normal-normal component of the extrinsic curvature. In the language used in the paper, the shift reproduces the “kink transformation” of Bousso–Chandrasekaran–Rath–Shahbazi. In a local Rindler frame near $\gamma_R[g]$2, the transformation glues the two entanglement wedges with a relative Lorentz boost of rapidity

$\gamma_R[g]$3

For any bulk field $\gamma_R[g]$4, one may write

$\gamma_R[g]$5

which effects no change deep inside either wedge but implements the matching across $\gamma_R[g]$6. The physical interpretation is therefore a relative boost between the two entanglement wedges separated by $\gamma_R[g]$7, matching the idea that the area of an extremal surface is dual to the modular Hamiltonian.

3. Algebra, renormalization, and constrained HRT surfaces

The gravitational HRT-Boost has a nontrivial algebraic realization in pure $\gamma_R[g]$8 with planar boundary. There all solutions are obtained by boundary conformal transformations of the Poincaré vacuum, so one can compute Poisson brackets of HRT areas by expressing each area as a functional of the boundary stress tensor and using the Virasoro algebra. For two boundary intervals $\gamma_R[g]$9, the resulting bracket is a piecewise constant function with the characteristic property that disjoint or nested intervals commute, whereas overlapping intervals do not (Kaplan et al., 2022).

The same work also clarifies the renormalization issue for asymptotically AdS spacetimes. When $\partial R$0 reaches the AdS boundary, the bare area diverges. One introduces a regulator $\partial R$1, subtracts local covariant counterterms $\partial R$2, and then sends $\partial R$3. The renormalized $\partial R$4 is finite as a function on phase space, and because the counterterms are $\partial R$5-numbers, they do not affect the Hamiltonian flow. However, the finite flow produces a boundary stress tensor containing terms such as $\partial R$6 and $\partial R$7, signaling infinite energy in the dual CFT. Finite one-sided boosts are therefore UV-singular, while infinitesimal boosts remain well-defined.

A further development appears in the theory of constrained HRT surfaces. For boundary subregions $\partial R$8 and $\partial R$9 on a common boundary Cauchy surface, the constrained HRT surface $$A_{\rm HRT}[R] := {\rm Area}[\gamma_R] = \int_{\gamma_R} d^{D-2}w\,\sqrt{q(w)},$$0 is obtained by a maximin construction restricted to bulk Cauchy slices containing $$A_{\rm HRT}[R] := {\rm Area}[\gamma_R] = \int_{\gamma_R} d^{D-2}w\,\sqrt{q(w)},$$1. It decomposes as

$$A_{\rm HRT}[R] := {\rm Area}[\gamma_R] = \int_{\gamma_R} d^{D-2}w\,\sqrt{q(w)},$$2

with one piece in the entanglement wedge of $$A_{\rm HRT}[R] := {\rm Area}[\gamma_R] = \int_{\gamma_R} d^{D-2}w\,\sqrt{q(w)},$$3 and the other in that of $$A_{\rm HRT}[R] := {\rm Area}[\gamma_R] = \int_{\gamma_R} d^{D-2}w\,\sqrt{q(w)},$$4. In this setting, the area $$A_{\rm HRT}[R] := {\rm Area}[\gamma_R] = \int_{\gamma_R} d^{D-2}w\,\sqrt{q(w)},$$5 generates a boundary-condition-preserving kink transformation, and the boost parameter $$A_{\rm HRT}[R] := {\rm Area}[\gamma_R] = \int_{\gamma_R} d^{D-2}w\,\sqrt{q(w)},$$6, denoted $$A_{\rm HRT}[R] := {\rm Area}[\gamma_R] = \int_{\gamma_R} d^{D-2}w\,\sqrt{q(w)},$$7, is canonically conjugate to the area via

$$A_{\rm HRT}[R] := {\rm Area}[\gamma_R] = \int_{\gamma_R} d^{D-2}w\,\sqrt{q(w)},$$8

(Dong et al., 2023).

For fixed-area states of subregion $$A_{\rm HRT}[R] := {\rm Area}[\gamma_R] = \int_{\gamma_R} d^{D-2}w\,\sqrt{q(w)},$$9, the path-integral analysis of the $H_R \equiv \frac{A_{\rm HRT}[R]}{4G}$0 Rényi entropy yields

$H_R \equiv \frac{A_{\rm HRT}[R]}{4G}$1

When $H_R \equiv \frac{A_{\rm HRT}[R]}{4G}$2 and $H_R \equiv \frac{A_{\rm HRT}[R]}{4G}$3 intersect at a constant boost angle, this becomes

$H_R \equiv \frac{A_{\rm HRT}[R]}{4G}$4

The same analysis shows that the von Neumann entropy differs, because the limits $H_R \equiv \frac{A_{\rm HRT}[R]}{4G}$5 and $H_R \equiv \frac{A_{\rm HRT}[R]}{4G}$6 do not commute. A common misconception is therefore that the constrained HRT area directly computes the entanglement entropy in the fixed-area state; in the cited construction it computes the Rényi entropy at $H_R \equiv \frac{A_{\rm HRT}[R]}{4G}$7, while the von Neumann entropy is an average of $H_R \equiv \frac{A_{\rm HRT}[R]}{4G}$8 over the boost distribution.

4. HRT-Boost in collider jet substructure

In collider phenomenology, HRT-Boost is a tagger for hyper-boosted heavy objects, introduced for top quark identification in the regime where transverse boosts reach several TeV and the angular separation of decay products becomes comparable to individual calorimeter cells (Larkoski et al., 2015). The method rests on three core ideas: dynamically scaling the jet radius inversely with transverse momentum, using calorimetric or particle-flow information only for total jet energy and momentum, and using charged-track measurements for all substructure observables.

The dynamic-radius prescription begins by clustering the event with anti-$H_R \equiv \frac{A_{\rm HRT}[R]}{4G}$9, $\gamma_R[g]$0, using Winner-Take-All recombination to obtain a fat jet $\gamma_R[g]$1. One measures $\gamma_R[g]$2 calorimetrically or via particle flow and defines

$\gamma_R[g]$3

The constituents of $\gamma_R[g]$4 are then reclustered with anti-$\gamma_R[g]$5 and radius $\gamma_R[g]$6, and the hardest subjet is taken as the candidate top jet. The choice $\gamma_R[g]$7 captures $\gamma_R[g]$8 of the top-decay products.

Because calorimeter granularity becomes inadequate at $\gamma_R[g]$9, HRT-Boost uses calorimetry only to determine the total four-momentum $$\delta_\gamma A[\gamma,g]\vert_{\gamma=\gamma_R[g]}=0.$$0 or $$\delta_\gamma A[\gamma,g]\vert_{\gamma=\gamma_R[g]}=0.$$1 of the scaled-radius jet and ignores calorimetric angular substructure. A track jet is formed from charged tracks inside $$\delta_\gamma A[\gamma,g]\vert_{\gamma=\gamma_R[g]}=0.$$2, and its mass is

$$\delta_\gamma A[\gamma,g]\vert_{\gamma=\gamma_R[g]}=0.$$3

To correct for missing neutral energy, the mass is rescaled as

$$\delta_\gamma A[\gamma,g]\vert_{\gamma=\gamma_R[g]}=0.$$4

Substructure discrimination is performed with track-based observables, particularly $$\delta_\gamma A[\gamma,g]\vert_{\gamma=\gamma_R[g]}=0.$$5-subjettiness and the energy-correlation ratio $$\delta_\gamma A[\gamma,g]\vert_{\gamma=\gamma_R[g]}=0.$$6. The paper uses $$\delta_\gamma A[\gamma,g]\vert_{\gamma=\gamma_R[g]}=0.$$7 with axes found by exclusive $$\delta_\gamma A[\gamma,g]\vert_{\gamma=\gamma_R[g]}=0.$$8+WTA, and $$\delta_\gamma A[\gamma,g]\vert_{\gamma=\gamma_R[g]}=0.$$9 with $A_{\rm HRT}$0.

The validation is performed in Delphes with CMS-like and FCC-like detector models. For each jet $A_{\rm HRT}$1 bin, the procedure applies the dynamic radius $A_{\rm HRT}$2, requires

$A_{\rm HRT}$3

and then cuts on $A_{\rm HRT}$4 and/or $A_{\rm HRT}$5. In fast simulation, calorimetry-only discrimination essentially disappears at $A_{\rm HRT}$6, whereas track-based HRT-Boost maintains an almost $A_{\rm HRT}$7-independent tagging efficiency $A_{\rm HRT}$8 at mistag rates $A_{\rm HRT}$9 up to $(h_{ij},\Pi^{ij})$0. At $(h_{ij},\Pi^{ij})$1 in the FCC detector model, the paper reports that at $(h_{ij},\Pi^{ij})$2, gluon-jet rejection is $(h_{ij},\Pi^{ij})$3 via $(h_{ij},\Pi^{ij})$4 and $(h_{ij},\Pi^{ij})$5 of light quarks via $(h_{ij},\Pi^{ij})$6.

The method’s contamination mitigation follows directly from the radius shrinkage. Since ISR, UE, and pile-up enter as uncorrelated soft background and scale with jet area, taking $(h_{ij},\Pi^{ij})$7 suppresses their contribution to the mass. Final-state radiation inside the dead cone remains inside $(h_{ij},\Pi^{ij})$8, wide-angle FSR is reduced by the same shrinkage, and the WTA axis reduces recoil sensitivity. The paper also notes that HRT-Boost can be combined with standard grooming, such as Soft Drop, if pile-up is extreme.

5. HRT-Boost in tabular machine learning

In tabular regression, HRT-Boost is the ensemble extension of the Hinge Regression Tree framework (Li et al., 22 May 2026). At an internal node with data $(h_{ij},\Pi^{ij})$9 and augmented features $R$00, the base learner fits two linear predictors $R$01 by minimizing

$R$02

where

$R$03

or the corresponding $R$04 variant. With the partition

$R$05

the objective is smooth quadratic on each subset. The gradient and Hessian are block-diagonal, and because $R$06 is locally linear on each partition, the Gauss–Newton approximation is exact.

This yields a damped Newton update

$R$07

where the pure Newton direction decouples to ordinary least-squares corrections on the two branches. The backtracking variant starts from $R$08, shrinks by $R$09, and guarantees monotonic decrease of the node objective. The theoretical results establish that once the partition stabilizes, the iterates converge to the exact OLS solution for that partition, and that HRT is a universal approximator with explicit $R$10 approximation rate on compact domains under the stated assumptions.

HRT-Boost places this base learner inside a standard squared-loss boosting framework. For empirical risk

$R$11

the algorithm initializes $R$12, computes pseudo-residuals $R$13, fits an HRT $R$14 to the residuals, and updates

$R$15

with learning rate $R$16. The paper defines the realized fit coefficient

$R$17

and proves the stage-wise empirical risk reduction bound

$R$18

Hence the empirical risk never increases and decreases strictly whenever $R$19.

Empirically, the paper reports that single-tree HRT matches or outperforms recent oblique-tree baselines on 12 real-world regression datasets while remaining substantially shallower and using far fewer leaves. As an ensemble, HRT-Boost is compared with RF, AdaBoost, Scikit-GBM, XGBoost, LightGBM, TabNet, and TabM, delivering the lowest or highly competitive RMSE on most datasets. A distinctive claim is model compactness: total leaf count is often much smaller, with the example Abalone result reported as $R$20 leaves for HRT-Boost versus $R$21 for LightGBM. Appendix H further reports inference FLOPs per sample in the low-thousands, contrasted with $R$22 FLOPs for deep tabular models.

6. Comparative interpretation and nomenclature

The three uses of HRT-Boost are technically unrelated, but their internal meanings of “boost” are sharply different in ways that track each field’s formalism. In gravity, the boost is a literal relative Lorentz boost of rapidity $R$23 between entanglement wedges across an HRT surface (Kaplan et al., 2022). In jet physics, the boost is the large transverse boost of a heavy resonance, which motivates the inverse-$R$24 radius scaling and track-dominant substructure measurements (Larkoski et al., 2015). In machine learning, the boost is stage-wise functional gradient descent over Hinge Regression Tree weak learners (Li et al., 22 May 2026).

This difference matters for interpretation. In the gravitational literature, HRT-Boost is an operatorial statement about the covariant phase space and, in later work, about Rényi entropies in fixed-area states (Dong et al., 2023). In collider physics, it is a detector-aware tagging strategy designed for ISR/FSR/UE/pile-up robustness in the hyper-boosted regime. In tabular learning, it is a compact-model construction with theoretical guarantees on node-level descent, universal approximation, and stage-wise risk reduction. A plausible implication is that the term should always be expanded on first use, since the acronym alone is insufficient to identify the intended object.