Papers

We present a new family of exact solutions representing accelerating, rotating, point particle and black hole solutions in three-dimensional Einstein gravity. We investigate their geometric properties and the global embedding in three dimensional anti-de Sitter space. Our results extend the catalogue of exact solutions in three-dimensional Einstein gravity and provide a useful setting for the study of black hole thermodynamics, asymptotic symmetries, and holography.

We show that for static & spherically symmetric black hole geometries, the angular components of the Brown-York tensor on a finite radius boundary can be interpreted as the product of the Tolman-Ehrenfest temperature and a quasi-local entropy. This entropy vanishes at spatial infinity and monotonically increases as radius decreases, taking its maximum value on the horizon where it is equal to the Bekenstein-Hawking entropy. Thus, the behaviour of this entropy provides explicit verification of the covariant entropy conjecture.

We introduce a new set of boundary conditions for three-dimensional Einstein gravity with negative cosmological constant. The canonical generators plainly realize the semidirect sum of Virasoro and $u(1)$ Kac-Moody algebras in its standard form, without the need of additional redefinitions of the global charges. We show that requiring Euclidean BTZ and thermal AdS$_3$ to be diffeomorphic precisely induces the warped modular $S$-transformation of the chemical potentials. The corresponding warped Cardy formula then exactly reproduces the Bekenstein-Hawking entropy, where the ground state is identified as Euclidean AdS$_3$. An explicit comparison with the boundary conditions of Brown-Henneaux and Compere-Song-Strominger is also addressed.

In this paper we consider some general aspects of Euclidean conformal field theories on curved spaces of dimension $D\ge 3$. We first look at the OPE of scalar primary fields on conformally flat spaces. Extending the results of \cite{Konechny:2026bqg}, we give a general construction of the descendants' contributions to the OPE in the scalar channel. We then discuss CFTs on non-conformally flat spaces in the ambient space formalism. We investigate the short-distance behaviour of three-point functions on such spaces using the general ansatz proposed by Parisini, Skenderis, and Withers \cite{Parisini:2022wkb, Parisini:2023nbd}. We find that some additional corrections need to be added to the ansatz to ensure the existence of a local covariant OPE.

We provide a first-principles definition of Cotton tensors on conformal Carroll geometries by gauging the conformal Carroll algebra. On the mathematical side, this analysis yields a conformally covariant object designed to quantify deviations from conformal flatness in Carrollian geometries. On the physical side, it offers a powerful geometric tool to describe gravitational radiative degrees of freedom at the conformal boundary of four-dimensional asymptotically flat spacetimes. Unlike the Bondi news, whose fully-covariant definition requires amendment by supplementary data that we identify with the Geroch tensor, the Carroll–Cotton tensors offer the natural geometric objects to quantify deviations from stationarity due to the emission of gravitational waves.

General relativity is a century-old subject. Yet modern explorations through generalized symmetries, scattering amplitudes, and effective field theory have raised open problems, motivating a reassessment of conventional views on gravitation, spacetime, and spin. First, do generalized symmetries exist in dynamical gravity as a low-energy EFT? Second, is there a field-theoretic explanation for the tree-level double copy relationship between general relativity and Yang-Mills theory? Third, what are the exact equations of motion or Lagrangians of four-dimensional spinning black holes in external fields, in point-particle effective theory? In this dissertation, three different perspectives on gravity are developed to shed light on each of these puzzles. In Part I, we view gravity as a gauge theory of Lorentz group to establish a one-form symmetry of dynamical gravity. In Part II, we investigate how far gravity can be viewed as a gauge theory of diffeomorphisms, in connection with color-kinematics duality and double copy. In Part III, we view four-dimensional gravity as a nonlinear interaction between self-dual and anti-self-dual parts. We apply this view to the derivation of four-dimensional spinning black hole solutions as well as their dynamics in point-particle effective theory. First, we derive a new self-dual black hole metric from classical double copy and show that the Kerr metric represents a pair of self-dual and anti-self-dual Taub-NUT solutions, elevating the Newman-Janis algorithm to a rigorous derivation. Second, we propose a dynamical probe counterpart of Newman-Janis algorithm. We give a Lagrangian derivation of spinning black hole Compton amplitudes that exhibit correct factorizations and are free of spurious poles for all helicity configurations while developing a chiral formalism that manifests and maximally utilizes the simplicity of the self-dual sector.

Positroid varieties $\Pi_f$ provide a decomposition of the Grassmannian $Gr(k,n)$; they can be enumerated using bounded affine permutations ($f$) which have bijections with a number of interesting combinatorial objects. Furthermore, (the nonnegative part of) positroid varieties parameterize the space that is integrated over when calculating amplitudes in planar N=4 supersymmetric Yang-Mills theory. The main question we answer in this thesis is whether a positroid variety $\Pi_f$ has any geometric singularities. We show that it is sufficient to check singularity at the $T$-fixed points ($\lambda$) and we can obtain the multiplicity at these points $\lambda$ by calculating the equivariant cohomology of $\Pi_f$ restricted to the point. We give 2 ways of doing this: (1) A diagrammatic way using affine pipe dreams and (2) A computational method. For (2), we have written code that does the computation and outputs the multiplicity (files at josephflueg.github.io). We have included tables listing the multiplicities of all the points on positroid varieties up to $n=6$. We also describe an ordering on pairs $(\Pi_f,\lambda)$ given by deletion/contraction that interacts nicely with smoothness, and describe the relationship between affine pipe dreams and finite pipe dreams. In Part II of this thesis connects with physics of planar N=4 SYM. We briefly introduce quantum field theory, leading singularities, and positroids in N=4 SYM. We then explain Britto-Cachazo-Feng-Witten (BCFW) recursion and the BCFW bridge decomposition of an on-shell diagram. We describe how to build an on-shell diagram on a pipe dream using BCFW bridges. Finally, we work out some combinatorics related to inverse soft factors for on-shell diagrams and explore whether singularities in positroid varieties have relevance to amplitudes. This is a revised version of my thesis originally written in August 2024.

The article constructs the Lagrangian analog of the amplituhedron, studies its geometric properties and BCFW decompositions.

The cusp anomalous dimension is the universal logarithmic contribution to the free energy of a line defect with a cusp, and provides a useful observable characterizing defect dynamics. We generalize this notion to cusped line defects in the presence of a planar conformal boundary or interface. The geometry is characterized by three angles, and the distance of the cusp from the boundary or interface which separates short-distance and long-distance regimes. The resulting free energy contains two large logarithmic terms: a short-distance contribution governed by the usual vacuum cusp anomalous dimension, and a new long-distance contribution sensitive to the boundary or interface. Using reflection positivity for transverse deformations of the defect rays, we derive several constraints on the coefficients of these logarithmic terms, which can be packaged into negative semi-definiteness of their angular Hessians. We also derive constraints on the continuation of these coefficients to Lorentzian kinematics. We compute these coefficients and verify the constraints explicitly in free scalar and Maxwell theories and in planar $\mathcal N=4$ super-Yang-Mills theory at weak coupling for various choices of boundary conditions. At strong coupling, we compute the logarithmic coefficients in the holographic D3-D5 defect CFT and find a Gross-Ooguri-like phase transition between competing string worldsheets driven by the geometry of the cusp relative to the interface. We also observe a close connection of the cusp free energy in this setup with holographic corner entanglement entropy.

We study the quantum transport of Dirac electrons in graphene subjected to a polyadic Cantor-structured electrostatic potential. Using the superperiodic potential formalism, we obtain a closed-form expression for the transmission probability. As the Cantor stage increases, the transmission spectrum evolves from a sparse set of superlattice resonances to a near-transparent regime, with the polyadic order setting the rate of this evolution. The angular response depends on the doping configuration, showing distinct behavior in the $n$–$n$–$n$, $n$–$p$–$n$, and Dirac-point cases. In the near-transparent regime, the transmission follows double-logarithmic scaling laws with respect to four independent control parameters: the Cantor stage, the potential height, the initiator length, and the angle of incidence. By combining these individual scaling relations, we establish a unified scaling law governing quantum transport. These results show that the hierarchical self-similarity of the potential governs the transport properties of such systems.

General relativity coupled to multiple scalar fields is a diffeomorphism-invariant constrained system. Consequently, a naive counting of the perturbative degrees of freedom unavoidably overestimates the true number of physical modes propagating in the theory, as gauge redundancies and constraint equations remove non-dynamical ones. While this problem has been solved for linear fluctuations, this work presents the first explicit calculation of all large-scale gauge-invariant phase-space variables in multifield inflation and at second order in perturbation theory, in a Hamiltonian language. Building upon the well-known Sasaki-Mukhanov variables, we show how to construct a finite-dimensional basis of quadratic corrections which are invariant under gauge transformations. Although our procedure is generic to any number of fields and at any scale, we restrict to super-Hubble scales for their explicit solution, which we deliver. Henceforth, we prove that it is possible to recover the usual flat-gauge and comoving-gauge fluctuations as large-scale gauge-invariant combinations, making for a robust consistency check of the gauge-fixed procedure to connect theoretical predictions above the horizon to observations. We derive the quadratic and cubic Hamiltonian of multifield inflation in a gauge independent manner, then we gauge fix our theory by going into the flat gauge, and we show perfect agreement with the literature on this topic, usually based on a Lagrangian approach. After these concrete steps, we propose a more formal proof of the existence of gauge-invariant variables at quadratic order, and we provide a sketch of the procedure that should allow to go to higher orders in perturbation theory.

We test the Sharpened Distance Conjecture (SharpDC) for CFTs, as well as the Refined Distance Conjecture (RDC), in AdS$_5$ gravitational spacetimes, using the CFT dual. The former provides a lower bound on the exponential mass decay rate of the tower of states becoming light at infinite distance in the conformal manifold, while the latter bounds the bulk field range that can be traversed before the exponential behaviour of the tower kicks in. We prove the SharpDC for any large $N$ Lagrangian supersymmetric gauge theory with a finite number $M$ of gauge factors and vanishing one-loop $\beta$-function. We also investigate whether it universally holds in the limits of large central charge, finding that it does not: We exhibit concrete examples in Class S theories where the conjecture is violated in the $M\rightarrow\infty$ limit with $N$ fixed, where the size of the bulk internal geometry is much larger than the AdS curvature scale. We also test the RDC in the moduli space of Class S theories without punctures in the large $N$ limit, showing that it amounts to a precise upper bound on the Weil-Petersson diameter of the moduli space of Riemann surfaces of genus $g$. This bound is consistent with all the mathematical literature up to date, but so far remains unproven.

We prove explicitly the n-point K-identities we proposed previously in the calculation of one tensor hard string scattering amplitudes (HSSA). These K-identities were the key to obtain the stringy scaling behavior in the saddle point calculation of the n-point HSSA and, on the other hand, to consistently match with the calculation of decoupling of zero norm states. Moreover, we introduce a G function to generate an infinite set of generalized K-identities (GKI). The lowest order set of these GKI is the scattering equations (SE) used in the calculation of field theory amplitudes in the CHY formalism. The next to leading order set of these GKI is the K-identities used previously in the calculation of one tensor HSSA. We expect that the higher order sets of these GKI can find applications in calculating other HSSA.

In cosmology, the Hubble tension has become one of the most serious challenges in the last decade. Assuming that all the observational data are right, one of the ways out is to modify the standard $\Lambda$CDM cosmology. In the literature, the theoretical modifications are mainly made in the early and the late/local universes. But on both sides, some no-go arguments have been proposed recently. In particular, the general no-go arguments for the late-time modifications are mainly based on the PAge parameterization. In the present work, we note that there are some flaws in the (original) PAge parameterization, and then propose various generalized PAge (GPAge) parameterizations, which can be extended to $z>z_{\rm CMB}$ and hence the cosmic microwave background (CMB) can be taken into account, while they are also more accurate at low redshifts. With these GPAge parameterizations, we revisit the Hubble tension by using not only the late-time observations but also the observational data of CMB. We find that the Hubble tension could be alleviated (or even resolved) in GPAges significantly different from the standard $\Lambda$CDM cosmology, while they are overwhelmingly preferred over $\Lambda$CDM by the observational data in terms of all the information criteria. Surprisingly, we find that the new physics in the mid-time might be the key to resolve the Hubble tension.

We discuss the tension one encounters in trying to solve, simultaneously, the $U(1)$ and strong-$CP$ problems in QCD with light quarks and a small $\bar{\theta}$ angle within QCD itself. We do so by considering the low-energy expansion of a $1$PI effective action $\Gamma$, a functional of a properly chosen set of gauge-invariant (composite) operators. After enforcing Ward-identity constraints on $\Gamma$, we prove, under minor assumptions, that $CP$ violation at small non-vanishing $\bar{\theta}$ can be avoided only if the anomaly-induced contribution to the pseudoscalar mass matrix is completely negligible when compared to its non-anomalous counterpart - that is, only if the $U(1)$ problem is not solved. The topological susceptibility in QCD and a precise generalization of the one in pure Yang-Mills theory emerge as the most important players in establishing the above tension and in determining the corrections to various known results at leading order in $\bar{\theta}$.

A proof is given that the six fixed points of the massless 1-loop renormalization group running of the CKM matrix in the Standard Model found previously at 1-loop are fixed points to all orders in perturbation theory. A previous proof required an extra assumption on the running which is eliminated in the proof presented here.

Applications of the concept of a hydrodynamic attractor have been mostly constrained to the context of hydrodynamization in heavy ion collisions. Deepening the understanding of the general phenomenon requires generalizing it to other contexts in order to see which facettes are truly universal and not a peculiarity of equilibrating conformal Bjorken flow. As part of this endeavor, this work aims to study the behaviour in a system that does not hydrodynamize, but nevertheless loses memory through expansion. We find that the system goes through three stages. At early times, expansion drives the ratio $P_L/\epsilon$ to converge to $-1$ as $\tau^{-2}$. This memory loss is partially restored at intermediate times, when interaction dominated degrees of freedom through expansion dominated evolution grow quadraticallly in time. At late times, interactions shuffle of information in state space without significant memory loss or restoration.

The classification and regression of particle collision events constitute a persistent computational challenge in experimental high energy physics, where large volumes of simulated data must be processed with both speed and precision. This work carries out a systematic comparison of four classical machine learning architectures, support vector machines (SVM), artificial neural networks (ANN), convolutional neural networks (CNN), and long short-term memory (LSTM) networks against their quantum counterparts: quantum SVM (QSVM), quantum neural networks (QNN), quantum CNN (QCNN), and quantum LSTM (QLSTM). All models are trained on simulated proton-proton collision events with electron-positron and muon-antimuon final states from the CERN Open Data portal, using transverse-momentum components as input features and transverse-momentum magnitude as the regression target. Classical architectures, and in particular the CNN and LSTM, achieve marginally better quantitative performance under current hardware and dataset constraints. Quantum models, however, reach competitive accuracy with substantially fewer trainable parameters: the QCNN reproduces the performance of the deep classical CNN using only four qubits and a circuit of depth three, pointing to a genuine parameter-efficiency advantage on near-term quantum devices. A baseline analysis confirms that the regression problem is non-trivial for shallow polynomial fits, supporting the relevance of the architectural comparison. These results characterize the trade-offs between classical and quantum approaches under realistic, resource-constrained conditions and provide a benchmark for future studies on actual quantum hardware.

Astrophysical black holes exist within non-vacuum environments, and their motion through these environments generically result in a force on these black holes through processes such as dynamical friction and Bondi accretion. We explore these forces numerically for black holes moving through a complex scalar field of mass $m$ with constant acceleration $a$, while also revisiting the constant velocity case considered in the existing literature. The former is modeled by the C-metric, while we use Painlevé-Gullstrand metric with an additional divergence and vorticity free velocity field to mimic the constant velocity motion. Our simulations reveal several novel and interesting aspects of the drag force in both cases. For the constant velocity case, the force saturates at late times, with a value that depends on velocity in a manner distinct from known dependence. For the constant acceleration case, the force increases to a maximum value $F_\star$ and then reduces drastically, with $F_\star \approx 0.3~ m a$. Moreover, the density wake in this case shows revivals separated by decreasing time intervals.

In three-dimensional compact U(1) lattice gauge theory color confinement can be understood analytically through the dynamics of magnetic monopoles. However, its continuum limit is pathological, since the ratio of the glueball masses to the square root of the string tension vanishes as the continuum limit is approached. We investigate a simple extension of the Wilson action in which the total number of lattice monopoles is coupled to an additional parameter $\mu$. By tuning $\beta$ and $\mu$ simultaneously, we identify a line of constant physics along which the ratio of the lightest glueball mass to the square root of the string tension remains constant. We further show that the same scaling is satisfied, within numerical uncertainties, by the other low-lying glueball masses considered in this work. Along this trajectory, the string tension in lattice units decreases with increasing $\beta$, thus suggesting that the modified lattice action may provide a regularization of three-dimensional compact U(1) gauge theory with a physically well-behaved continuum limit.

We study holographic ensembles in type IIB string theory on the Schur twisted AdS$_5 \times S^5$ background, and its corresponding boundary partition function: the Schur index of $\mathcal{N}=4$ SYM. In the supergravity limit the two natural ensembles are either at fixed flux on $S^5$, corresponding to a boundary theory at fixed rank, or its conjugate choice of fixed potential on the boundary of AdS$_5$, corresponding on the boundary to a grand canonical ensemble of theories at fixed chemical potential. Due to the self-dual nature of the five-form in type IIB supergravity the choice of ensembles is not seen in the standard bulk action, but rather by the coupling of an additional topological term. We compute the bulk partition function to one loop, including the Kaluza-Klein determinant and the measure of the ensemble-changing sum, and show that depending on the sign of this topological term it matches the perturbative boundary partition function in either the canonical or the grand canonical ensemble.

We develop a universal framework for computing generating functions in quiver gauge theories in the large $N$ limit. We apply this method to the superconformal index of 4d $\mathcal{N}=2$ gauge theories, as well as to the 0-instanton index of 5d $\mathcal{N}=1$ theories, both for finite quivers of arbitrary shape and in the long quiver limit, including both special unitary and orthosymplectic gauge groups, demonstrating its versatility. In the Hall–Littlewood limit, we show that the series expansion of the resulting index captures the expected Higgs branch chiral ring generators. Our exact expressions in the long quiver limit provide new predictions for the supergravity index of the holographically dual warped AdS$_5$ and AdS$_6$ backgrounds.

We study Seiberg duality for quiver gauge theories with fundamental, bifundamental, and rank-two tensor matter. The existence of a Seiberg dual description places strong constraints on the spectrum of undressed mesons, which is naturally encoded in a factorization equation for the matrix Hilbert series. We derive this factorization from a closed-form large-$N$ superconformal index for general quiver gauge theories with these matter contents. We apply the method to two-node quivers and recover known $SU$–$SU$ and $SO$–$USp$ product-group dualities. We also discuss finite-$N$ consistency conditions from anomaly and operator-spectrum matching.

The gauge-theoretic formulation of gravity, enriched with the observation that the tangent group of a curved manifold need not have the same dimension as the manifold itself, provides a natural framework for unifying gravity with internal interactions. In the present work we study the unification of Einstein gravity with the $SO(10)$ grand unified theory within an $SO(1,17)$ gauge theory. As a central result, we derive Einstein gravity with a cosmological constant from the spontaneous symmetry breaking (SSB) of an $SO(1,5)$ gauge theory to the Lorentz group via two independent routes: one using two scalar fields in the fundamental representation $\mathbf{6}$ of $SO(6)$, and one using a single scalar in the adjoint $\mathbf{15}$. Both yield a Gauss-Bonnet + Einstein-Hilbert + cosmological constant action. Embedding the $SO(1,5)$ gravitational sector in $SO(1,17)$ and breaking the full gauge group yields Einstein gravity coupled to $SO(10)$. The Weyl-Majorana condition reduces the fermion family degeneracy to two, in contrast to the four families obtained in the $SO(2,16)$ conformal gravity unification.

Quantum Yang-Mills theory in Euclidean AdS$_4$ with Neumann boundary conditions is studied as a function of the complex couplings $\frac{1}{g_\pm^2} \equiv \frac{1}{g^2}\mp \frac{i \theta }{8 \pi^2}$. We define a "self-dual limit" by $g_- \to 0$ with $g_+ ~{\rm fixed}$. We argue that the limiting theory is well-defined perturbatively and equivalent to the BF formulation of self-dual Yang-Mills theory with a particular boundary condition relating $B$ and $F$ at the boundary $\partial$AdS$_4$. $g_+$ is the loop-counting parameter of the self-dual theory. This is in stark contrast to flat space, where $\theta$ has no effect on perturbative dynamics. In the self-dual limit, it is shown that all tree-level zero-plus and single-plus boundary correlators vanish. A closed form expression is given for any number of gluons for the double-plus tree correlators. Finally, we show that in the flat space limit of AdS$_4$, the total energy poles in the tree boundary correlators correctly reduce to the single-minus gluon amplitudes.

We investigate thermodynamic criticality in a spatially flat Friedmann-Lemaître-Robertson-Walker universe with a non-extensive Loop Quantum Gravity inspired entropy on its apparent horizon. Using the full Kodama-Hayward temperature and the unified first law, we derive the modified Friedmann dynamics and construct the corresponding horizon equation of state. We see that a finite physical critical point only appears in the non-extensive branch $q>1$, while in the Bekenstein-Hawking limit $q\to1$, the critical point is continuously pushed to $v_c\to\infty$ and $T_c\to0$. Along the same spinodal curve, both the constant-pressure heat capacity and the isothermal compressibility become unbounded, and the extremum of this curve is located at the critical point. Below the critical temperature, the Gibbs free energy develops several thermodynamic branches, and the phase coexistence for different horizon states is verified by the equality of temperature, pressure, and Gibbs free energy. The critical exponents are $(\alpha_{cr},\beta_{cr},\gamma_{cr},\delta_{cr}) =(0,1/2,1,3)$, indicating that the system belongs to the standard mean-field universality class despite the non-algebraic form of the equation of state. The normalized Ruppeiner curvature diverges precisely on the spinodal curve and shows critical scaling $R_N\sim-|v-v_c|^{-4}$ on the critical isotherm, and $R_N\sim-|t|^{-2}$ on the critical isochore. Finally, the critical expansion scale is given by $H_c^2=2(\sqrt{5}-2)|\beta|$, which implies that the thermodynamic critical point occurs when the entropy deformation is of order unity. These results establish a self-consistent critical structure for the effective thermodynamic state space of the cosmological apparent horizon and retain the mean-field critical universality.

We consider two-loop corrections to two-point functions of protected scalar operators in ABJM theory. We infer a compact finite-rank formula valid for operators of arbitrary dimension and multi-trace structure. The result is governed by the exact tree-level metric and a simple kernel on the Young lattice, while the planar limit reduces to a simple partition-theoretic rule. The required integrals organize into uniformly transcendental combinations, providing evidence for uniform transcendentality of these protected two-point functions.

Nambu–Jona-Lasinio (NJL) models of color-superconducting quark matter suffer from cutoff artifacts once temperature or quark chemical potential become comparable to the model cutoff. These artifacts can be removed employing a renormalization-group (RG) consistent regularization scheme. In this article we perform a systematic study of neutral beta-equilibrated quark matter within the three-flavor NJL model with RG consistent regularization. Varying the coupling constant in the scalar diquark channel, we map out the phase diagram in the plane of chemical potential and temperature, and identify the gapless domains. We particularly focus on the melting pattern of the color-flavor locked (CFL) phase. At larger couplings the CFL phase melts through a so-called $d$SC phase, as expected from leading-order Ginzburg–Landau analyses. Lowering the coupling, the phase structure becomes markedly richer: While at large densities the CFL phase still melts through a $dSC$ phase, we find a $uSC$ phase at lower chemical potential. These phases, $uSC$ and $dSC$ meet at a doubly critical point whose existence had been anticipated long ago but was never demonstrated explicitly in a model.

We establish that vacuum entanglement entropy, regulated by weak spacetime curvature, imposes a fundamental lower bound on radial resolution for quantum excitations. From the area law, weak gravity introduces a natural UV regulator via the perturbative Green's function. Geometric projection yields a minimal radial scale $\Delta r_{\min} \propto r_s^2/R$, where $r_s$ is the Schwarzschild radius and $R$ the boundary radius. Consistency with the Compton wavelength defines a global mass scale $m_{\text{geo}} \equiv \hbar R/(c r_s^2)$. Particle masses satisfy $m = \alpha m_{\text{geo}}$, where $\alpha$ encodes vacuum structure. For Earth, $m_{\text{geo}} \approx 2.84 \times 10^{-32}$ kg. We determine $\alpha$ via spectral analysis of curvature-induced mode shifts, implemented in Python with $N=3000$ radial points and $n_{\text{modes}}=1000$. The calculation yields $\alpha(2) \approx 33$, matching the electron benchmark without adjustable parameters, ruling out exponential ansätze in favor of slow power-law growth $\alpha(l) \propto l^{1.27}$. Crucially, this resolution limit endows the quantum vacuum with the properties of a granular refractive medium. This elegantly explains a fundamental particle-wave asymmetry: massive excitations with $\lambda_C \ll \Delta r_{\min}$ propagate in the geometric optics limit, following macroscopic gradients as classical geodesics. Conversely, photons with $\lambda \ll \Delta r_{\min}$ interact directly with the medium's granularity, leading to phase decoherence and anomalous dispersion. This framework reframes mass as a probe of vacuum-imposed resolution limits, yielding distinct, falsifiable predictions for high-energy photon propagation.

We derive the late-time ramp of the finite-temperature spectral form factor in the double-scaled SYK model directly from cactus diagrams, which is introduced as a multi-trace generalization of chord diagram in Ref \cite{Berkooz:2020fvm}. Although single-trace observables admit an exact chord-diagram description, multi-trace sums are obstructed by the chord-intersection weights $q_{IJ}$, which cannot be averaged independently to $q$. Our key idea is that the non-analytic contribution responsible for the ramp is controlled by the large-order tail of the Cactus-diagram expansion and is insensitive to finite changes in its low-order analytic terms. Decomposing the contribution with $n$-cross-trace-pairings Cactus diagram into two buds $B_n$ and a kernel $\mathcal K_n$, we determine their large-$n$ asymptotics with fixed $0\leq q<1$ and resum the resulting tail. For $\beta_L=\beta+it$ and $\beta_R=\beta-it$, we obtain \begin{equation*} Z_s^{\mathrm{sing}}(\beta+it,\beta-it) =s_p c_N \frac{|t|}{2\pi} \int_{E_{min}}^{E_{max}} e^{-2\beta E} dE+O(1), \qquad s_p=\begin{cases}2,&4\mid p,\\1,&4\nmid p.\end{cases} \end{equation*} This result reproduces the linear ramp predicted by random matrix theory, including its temperature dependence and symmetry factor, and agrees with the semiclassical predictions. It provides a direct microscopic origin of the ramp within the general $q$-deformed quantum algebra. While the plateau lies beyond the scope of the present analysis and calculation.