Papers

In a previous manuscript we addressed the possibility of generating a reflection in a region of spacetime inside a null surface under electromagnetic gauge transformations. In this manuscript we will deal with the possibility of full inversions in a region of spacetime inside a null surface under electromagnetic gauge transformations. Since new tetrads whose construction depends on electromagnetic gauge were introduced it has been proved that the physical kinematic nature of spacetime can be altered as proved previously. In this case we will see how to create a field configuration that will reverse the flow of time. We will prove that we can turn a future directed timelike vector into a past directed timelike vector by physical means. Experiments devised in order to achieve time reversal or full inversion have been never discussed before.

Symmetry-plane correlations (SPCs) are key observables sensitive to the medium's transport properties and are driven by participant-plane correlations (PPCs) in the nuclear overlap region. This study explores the possibility of nuclear-geometry-driven SPCs in Oxygen–Oxygen (OO) and Neon–Neon (Ne–Ne) collisions at $\sqrt{s_{\rm NN}}=5.36$ TeV using nuclear geometry simulations based on Nuclear Lattice Effective Field Theory (NLEFT) and Projected Generator Coordinate Method (PGCM) configurations. We investigate $\langle \cos[4(\psi_2 - \psi_4)]\rangle_{\rm GE}$ and $\langle \cos[6(\psi_3 - \psi_6)]\rangle_{\rm GE}$ in OO and Ne–Ne collisions at $\sqrt{s_{\rm NN}}=5.36$ TeV using the A Multi-Phase Transport (AMPT) model. We find that Ne–Ne collisions exhibit larger $\langle \cos[4(\psi_2 - \psi_4)]\rangle_{\rm GE}$ values than OO collisions, whereas $\langle \cos[6(\psi_3 - \psi_6)]\rangle_{\rm GE}$ is larger in OO than in Ne–Ne collisions. This behavior indicates a strongly deformed shape of the $^{20}$Ne nucleus and a tetrahedral structure of the $^{16}$O nucleus. We also explore SPCs for events with tip-tip and body-body collision configurations, which further support these findings.

The global topology of spacetime, though invisible to local curvature measurements, leaves signatures on the correlation functions of quantum fields. We study these signatures using an Unruh-DeWitt particle detector operating in four-dimensional Minkowski spacetime with two spatial directions periodically identified, yielding a spatial topology $\mathbb{R}\times T^2$. We compute detector transition rates for three trajectories: uniform inertial motion, uniform proper acceleration directed along one of the compact axes, and uniform proper acceleration along the non-compact axis. Our results show how a local quantum measurement can reveal features of the large-scale spatial topology.

Ultra high energy cosmic rays have been see coming from the direction of the local cosmic void. We use this fact to argue that at least some of these these cosmic rays are relatively light magnetic monopoles and that their relative fraction above 1020 eV can be found from full sky observations.

Boundary conformal field theory (BCFT) provides a universal framework for critical phenomena in the presence of boundaries. We determine BCFT data for the normal and ordinary boundary universality classes of the $1+1$-dimensional boundaries of the $2+1$-dimensional $O(2)$ and $O(3)$ Wilson-Fisher fixed points, realized microscopically by a bilayer Heisenberg model on the fuzzy sphere. Using the fuzzy-sphere state-operator correspondence, we obtain boundary operator spectra, identify low-lying boundary primary operators, extract operator-product-expansion (OPE) data, and estimate the boundary central charges for both boundary conditions. For the normal boundary condition, the universal amplitudes $a_\sigma$ and $b_t$ extracted from one- and two-point functions agree quantitatively with Monte Carlo benchmarks where available. For both $N=2$ and $N=3$, we find a positive extraordinary-log exponent $\alpha$, providing independent microscopic evidence for extraordinary-log boundary criticality. Our results extend fuzzy-sphere BCFT spectroscopy beyond the Ising universality class to continuous $O(N)$ symmetry.

We present a modified gravitational framework in which the standard Einstein field equations are sourced by a classical matter sector coupled to a Higgs-type scalar field $\chi$ modeling a dynamic vacuum substrate. By introducing a phenomenological covariant coupling we implement a physical displacement principle where massive baryonic matter drives the vacuum field away from its vacuum expectation value. We show that this coupling leads to a field-dependent modulation of a particle's inertial rest mass alongside a spatial fifth force, yielding localized violations of the Einstein Equivalence Principle while preserving universal free fall for fundamental point masses. In the weak-field, non-relativistic limit, this interaction manifests as a Yukawa-type correction to the Newtonian potential. We test the viability of this framework against local gravitational constraints, including planetary perihelion precession and Eötvös parameter limits. Finally, we model the steady-state, non-relativistic spherical accretion of dust, demonstrating that the competing effects of vacuum-induced mass modulation and fifth-force acceleration yield distinct density and velocity profiles.

The chiral algebra bootstrap (CAB) is a novel bootstrap program for form factors in quantum-integrable self-dual gauge theories, some of which in turn are helicity amplitudes in the corresponding gauge theories. The singularities that recursively generate a given (loop-level) form factor are holomorphic collinear splitting functions, equivalently celestial chiral algebra OPEs, of the self-dual theory. In this note, we apply the chiral algebra bootstrap to the simple example of self-dual 4d $\mathscr{N}=4$ super Yang-Mills (SDSYM). We use a combination of twistor space input, Koszul duality, supersymmetry, and associativity to obtain the all-loop holomorphic collinear splitting functions for SDSYM. We also use associativity to provide a simple proof of the conjecture that there are no double-poles in the loop-level OPEs for this theory. We conclude by computing several form factors, including both a reproduction of several known results and novel form factors up to two loops involving insertions of powers of the anti-self-dual field strength. These form factors compute a supersymmetric version of Higgs amplitudes in the self-dual sector. Detailed sample computations are provided to familiarize the reader with the CAB method.

We present a dual gravity interpretation of the complex reparametrization mode $\psi(u)$ that governs the soft dynamics of double-scaled SYK in the presence of a time-dependent Maldacena-Qi coupling. We find that the dual gravity system takes the form of 2+1-dimensional Einstein-de Sitter gravity with an energy distribution localized on a dS$_2$ slice within dS$_3$. The effective SYK equations of motion take the form of the Israel junction conditions across the dS$_2$ slice. We study the 1D effective action of the SYK soft mode and show that it coincides with the effective action derived from 3D Einstein-de Sitter gravity with conformal boundary conditions on $\mathscr{I}^\pm$. The boundary conditions split $\mathscr{I}^\pm$ into two hyperbolic $k=-1$ slices, and the holographic screen is placed at the intersection. We adapt the Gibbons-Hawking calculation of the Schwarzschild-de Sitter entropy to the case with $k=-1$ boundary conditions and find that it reproduces the semiclassical DSSYK entropy. The boundary-to-boundary Green functions in 3D de Sitter are equal to the square of DSSYK two-point functions. We give an alternative holographic interpretation of our results in terms of 3D AdS gravity with two time directions.

We present an invariant relational path-integral quantization framework for general-relativistic gauge field theories based on the Dressing Field Method. The construction implements an automatic anomaly-cancellation mechanism that encompasses Bardeen-Wess-Zumino counterterms. The resulting framework unifies invariant schemes across contexts ranging from electroweak theory to cosmology, and is amenable to lattice implementations, key to high-precision tests in both domains.

We conjecture, and show in a plethora of examples, that the sphere partition function of 3d $\mathcal{N}=4$ Chern-Simons-matter theories equals a sum of twisted traces on tensor products of Verma modules over the quantization of the moduli spaces of vacua. This extends a conjecture of Gaiotto-Okazaki to Chern-Simons-matter theories. We also show that the partition function of every Abelian gauge theory with higher charges has such twisted trace decomposition, and uncover new Abelian dualities between theories with and without Chern-Simons couplings.

We present SubTropica, a Mathematica package that performs symbolic integration of multi-polylogarithmic integrals using recent advances in tropical geometry. It focuses on the class of linearly-reducible Euler integrals, such as Feynman integrals, and expands them using a tropical subtraction scheme. The engine behind it is HyperIntica, a native Mathematica package for hyperlogarithm integration that can be used independently. This paper documents both packages and illustrates their usage on examples from across different physics applications. Additionally, we introduce an AI-driven library of Feynman integrals, which catalogs diagrams discussed in the literature and serves as a database for computed results. Its online version is available at: https://subtropi.ca and features a graphical user interface for diagram input and retrieval of records.

Building on recent work introducing the idea of Quasilocal Probability in curved spacetime, we develop its observational implications for black hole ringdown in detail. We show that horizon-induced probability flux leads to an effective non-Hermitian dynamics producing three distinctive signatures, which are correlated multi-mode deviations, weak amplitude dependence and a mismatch between waveform damping and energy accounting. These effects arise from a single boundary-flux mechanism and therefore exhibit a constrained, low-dimensional structure not expected in generic modified gravity scenarios. We demonstrate that while individual deviations may be mimicked, their combined pattern provides a robust discriminator of quasilocal probability. We further argue that upcoming gravitational wave observations can probe these signatures at meaningful precision. We also establish that black hole ringdown is a novel arena to test whether quantum mechanical Hermiticity is really a fundamental property or an emergent symmetry in quantum gravity.

We apply harmonic analysis to study the $T\bar{T}$-deformed torus partition function. We first express the CFT partition functions in terms of Maass waveforms, including the Eisenstein series and cusp forms. These basis functions turn out to deform in a very simple way under the $T\bar{T}$-deformation. The spectral decomposition provides a numerically stable and efficient method to compute the partition function at finite values of the deformation parameter $\lambda$, allowing us to clearly resolve the analytic structure of the partition function as a function of $\lambda$. The resulting deformed partition function exhibits a Hagedorn singularity. Building on harmonic analysis approach, we propose a natural analytic continuation beyond the Hagedorn singularity, which enables us to compute the full partition function for any value of $\lambda$.

Sufficiently strong first-order phase transitions (FOPTs) in the early Universe can simultaneously produce an observable stochastic gravitational wave background (SGWB) and a large-scale primordial magnetic field (PMF). The recent $3.8\sigma$ evidence for a non-zero intergalactic magnetic field from anisotropic pair-halo searches using Fermi-LAT data further motivates a cosmological origin. We investigate an FOPT-origin of both cosmic signatures, namely, PMF and SGWB, and their correlation, within a minimal axion-like particle (ALP) framework in which a global $U(1)$ symmetry is spontaneously broken through radiative corrections, with the ALP sector coupled to the Standard Model (SM) via a Higgs-portal. We compute the present-day PMF amplitude and coherence length for both maximally helical and non-helical configurations, accounting for inverse cascade effects. For maximally helical configurations, we find peak field strengths up to $B_0 \sim 10^{-9}$ G at coherence length $\lambda_0 \sim 10^{-3}-10^{-1}$ Mpc, consistent with lower bounds on the IGMF inferred from blazar observations by MAGIC, H.E.S.S. and Fermi-LAT. We show that the ALP parameter region consistent with $\gamma$-ray blazar data (assuming maximally helical fields) simultaneously produces stochastic GW signals detectable at future space-based interferometers, such as LISA, etc., over the ALP decay constant range $10^3~\text{GeV} \lesssim f_a \lesssim 10^5~\text{GeV}$. We demonstrate that these correlated constraints can be directly mapped onto effective ALP couplings to SM particles, e.g., photons, gluons, and fermions. This establishes a multi-messenger complementarity between cosmological observables and laboratory/astrophysical ALP searches, with combined constraints preferring relatively heavy ALPs, $m_a \gtrsim 0.1~\text{GeV}$, accessible to next-generation intensity and energy-frontier experiments.

The eigenstates of two test-masses (where each test-mass is placed inside of a harmonic trap) separated by a distance, can get entangled where gravity acts as the mediator of entanglement and it has been argued in \href{https://doi.org/10.48550/arXiv.2511.07348}{arXiv:2511.07348 [quant-ph]} that this entanglement of masses cannot be generated without the underlying quantum nature of gravity. In this work, we consider two non-relativistic Bose-Einstein condensates (formed inside of harmonic trap potentials with identical trapping frequencies) separated by a distance. We take a linearized quantum gravity model and investigate the generation of entanglement while gravitons serve as the mediator of entanglement. The entanglement is generated between the phonon modes of the two condensates, and we observe that for very low separation distance, the entanglement generated is significantly higher than that observed for the quantum gravity induced entanglement of masses or QGEM protocol; however, the fall of entanglement is faster than the two-particle case for two separated Bose-Einstein condensates. We observe that when the number of particles in the condensate is increased, the degree of entanglement for a smaller separation distance becomes substantially higher compared to the case discussed in \href{https://doi.org/10.1103/PhysRevD.105.106028}{Phys. Rev. D 105 (2022) 106028}, which allows for a more robust experimental proposal using this quantum gravity induced entanglement of phonons or QGEP protocol.

The universal $2$-parameter vertex algebra $\mathcal{W}_{\infty}$ of type $\mathcal{W}(2,3,\dots)$ is a classifying object for vertex algebras of type $\mathcal{W}(2,3,\dots,N)$ for some $N$; under mild hypotheses, all such vertex algebras arise as quotients of $\mathcal{W}_{\infty}$. In 2017, Gaiotto and Rapčák introduced a family of such vertex algebras called $Y$-algebras, and conjectured that they fall into groups of three that are mutually isomorphic. This is a common generalization of both Feigin-Frenkel duality and the coset realization of principal $\mathcal{W}$-algebras in type $A$, and was proven in 2021 for the simple $Y$-algebras (i.e., one label is zero) by the first and third authors. In this paper, we extend this entire story to the $\mathcal{N}=2$ superconformal setting. First, we prove the 2013 conjecture of Gaberdiel and Candu that there exists a universal $2$-parameter vertex algebra $\mathcal{W}^{\mathcal{N}=2}_{\infty}$ which is an extension of the $\mathcal{N}=2$ superconformal algebra, and has four additional generators in weights $i, i + \frac{1}{2}, i + \frac{1}{2}, i+1$, for each integer $i > 1$. This admits many $1$-parameter quotients which we call $\mathcal{N}=2$ supersymmetric $Y$-algebras, and we prove the dualities among these algebras which were conjectured in 2018 by Prochazka and Rapčák. A special case is the coset realization of the principal $\mathcal{W}$-algebra $\mathcal{W}^k(\mathfrak{sl}_{n+1|n})$ which was conjectured in 1992 by Ito. As a corollary, we obtain the strong rationality of $\mathcal{W}_k(\mathfrak{sl}_{n+1|n})$ for $k = -1 + \frac{1}{n+a+1}$ for all positive integers $n,a$, and we describe its module category. This generalizes Adamović's 1999 result on $\mathcal{N}=2$ minimal models, which is the case $n=1$.

We study the instanton counting in four dimensional $\mathcal{N}=2$ supersymmetric gauge theories on the blow-up of $\mathbb{C}^2$: we start by formulating the instanton moduli space as a quiver variety, which we regularise by introducing two stability parameters, thus endowing it with a structure of infinitely many chambers separated by walls. Within a given chamber, we formulate the instanton partition function as a contour integral, which can be evaluated using the Jeffrey-Kirwan residue prescription. We characterise the physically relevant contributions in terms of bipartite oriented graphs and show that they can more efficiently be classified in terms of combinatorial objects called super-partitions. Within a given chamber, only certain types of super-partitions contribute and we show that the corresponding selection criteria are equivalent to stability conditions that have previously been proposed in the literature. We use this formalism to compare how the instanton counting changes when moving across walls between neighbouring chambers and provide explicit expressions for the corresponding partition functions. In a limiting chamber and using our approach, we show how to reproduce the Nakajima-Yoshioka blow-up formula.

BPS states in holographic CFTs are usually classified into supergravitons, namely BPS fluctuations around empty AdS, and black-hole microstates, which appear above an energy threshold. In AdS$_3$/CFT$_2$, however, this picture is incomplete because of additional degrees of freedom, called singletons, associated with boundary diffeomorphisms. We present a general procedure for extending the BPS spectrum of supergravitons by dressing them with singletons, thereby defining a generalized, gravity-sector Hilbert space that admits decomposition into affine multiplets of the full superconformal algebra. This extends the procedure previously proposed in arXiv:2505.14888 which was applicable only at low levels, by removing that limitation. We apply the new procedure to the D1-D5 CFT ${\rm Sym}^N(T^4)$ and explicitly construct affine multiplets in the gravity sector for the $N=2$ theory up to level $h=2$. We find that, at the free orbifold point, the gravity-sector spectrum agrees with the CFT up to $h=\frac12$. Upon turning on a deformation, however, states at $h=1$ lift and the agreement improves to $h=\frac32$. Interestingly, the lifting occurs between states in the gravity sector, involving mixtures of supergravitons and singletons, and stringy states. We conjecture that, upon deformation, the gravity-sector Hilbert space becomes the monotone Hilbert space while its complement becomes the fortuitous Hilbert space.

We investigate timelike geodesics in asymptotically flat regular black holes supported by a phantom scalar field characterized by a scalar charge $A$. This parameter removes the central singularity and continuously deforms the Schwarzschild geometry while preserving asymptotic flatness. We derive the equations of motion for massive test particles and classify bounded and unbounded trajectories in terms of the conserved energy and angular momentum. We determine circular and critical orbits, including the innermost stable circular orbit (ISCO), and analyze the transition between capture and scattering. We show that the scalar charge modifies the location of the unstable and stable circular orbits, the ISCO, and the threshold angular momentum for scattering, exhibiting a nontrivial dependence on the radial coordinate. Their physical scales are naturally described in terms of the invariant areal radius $R(r)=\sqrt{r^2+A^2}$. In the weak-field regime, we compute the perihelion precession and obtain corrections proportional to $A^2$, allowing us to constrain the scalar charge from Solar System observations. We also analyze the motion with vanishing angular momentum and show that, while the qualitative structure of the trajectories remains connected to the Schwarzschild limit $A\to 0$, the quantitative deviations encode the geometric effects of the scalar hair.

The interaction between a Nambu-Goto string and a Kerr black hole gives one of the methods of energy extraction from a rotating black hole. Although the properties of such processes have been well studied for rigidly rotating strings, little is known for non-rigidly rotating strings. In this paper, we study time evolution of a Nambu-Goto string on the equatorial plane of a Kerr spacetime, which sticks on the horizon and extends to spatial infinity. The time evolution is studied by the series expansion with respect to $t$ and the numerical simulations, which give reliable results for $t\lesssim 4M$ and $t\lesssim 38M$, respectively, where $M$ is the black hole mass. Since the angular velocity of the string on the horizon must coincide with the horizon angular velocity to keep the timelike property, the string is dragged into rotation and coils around the black hole. The negative energy is observed to fall into the black hole, but the positive energy follows after that, meaning that the energy extraction occurs for a short period of time. In the outside region, a wave is generated and propagates to the distant region carrying the extracted energy. After the propagation of the wave, the system approaches the time-independent configuration found by Boos and Frolov, and the total extracted energy is estimated as $E_{\rm ext}\lesssim \mu M$, where $\mu$ is the tension of the string.

We study finite-node conifold degenerations of Calabi–Yau threefolds from the point of view of interacting light sectors. Although each ordinary double point contributes a rank-one local vanishing sector, the corrected global object need not assemble as a freely independent sum of nodewise pieces. Using the corrected perverse and mixed-Hodge-module degeneration package, the global gluing law for corrected extension classes, and the rigid-flexible atom decomposition on the \(F\)-bundle side, we define an interacting multi-node light-sector package and prove a block-reduced structure theorem. In the block-separated cycle family, the finite-node package separates into two logically distinct layers: relation collapse, controlled by a common relation lattice on the corrected-extension, smoothing, and resolution sides, and residual interaction among the surviving global sectors, controlled by a reduced block interaction matrix on the transport and atom sides. The result isolates the geometric and Hodge-theoretic precursor of coupled conifold light states and provides the mathematical input for a later multi-node reformulation of Strominger's conifold mechanism.

In closed quantum systems, Krylov complexity admits a geometric description; operator growth is equivalent to Hamiltonian flow in an emergent phase space whose structure is fixed by the Lanczos coefficients. We show that this picture survives, albeit in a fundamentally altered form, once the system is coupled to an environment.Using a Schwinger-Keldysh formulation of the full counting statistics of the Krylov position, we derive an effective action for operator growth under Lindblad dynamics. Even for the minimal case of dephasing, the phase-space dynamics ceases to be Hamiltonian; environmental coupling generates diffusion in the variable conjugate to Krylov depth, converting deterministic trajectories in to stochastic ones. The hyperbolic mechanism underlying exponential complexity growth is therefore broadened and, beyond a parametrically controlled scale, destroyed.This identifies dissipation as a relevant perturbation of the chaotic Krylov fixed point and reveals operator growth in open systems as a problem of stochastic dynamics in an emergent phase space.

We present a non-Abelian model for magnetic monopoles in inhomogeneous media, based on a generalization of the standard 't~Hooft-Polyakov model. The medium is described by spatially dependent couplings in the gauge and scalar sectors, constrained by $P(|\Phi|,r)M(|\Phi|,r)=1$ so that the Bogomol'nyi-Prasad-Sommerfield (BPS) bound is preserved. For static spherically symmetric configurations, we study the first-order monopole equations for the class of generalized permeabilities $M(H,r)=f(r)/H^\alpha$. For the power-law profile $f(r)=r^\beta$, we determine the domain in the $(\alpha,\beta)$ plane where regular BPS solutions exist. On the line $\alpha=1$, the system becomes exactly integrable, with closed-form monopole solutions in an inhomogeneous background. Away from this analytical sector, the solutions are constructed numerically. The model supports a rich spectrum of configurations, including effectively point-like monopoles, compact-core monopoles, hollow monopoles, shell-like structures, and multi-shell monopoles characterized by multiple concentric peaks in the energy density.

In this work, we present a detailed comparison of the SuSAv2 (SuperScaling Approach version 2) and RDWIA (Relativistic Distorted-Wave Impulse Approximation) models with measurements of charged-current neutrino-induced single-pion production from different experiments (T2K, MINERvA and MiniBooNE), studying the differences between the two theoretical descriptions. The neutrino energy range in these experiments spans from hundreds of MeV to roughly 20 GeV, and the nuclear targets are mainly composed of $^{12}$C. The SuSAv2 model uses the single-nucleon inelastic structure functions from the ANL-Osaka DCC model, which allows for a separation of pion production channels, distinguishing between the $\pi^+$, $\pi^-$ and $\pi^0$ final states. In the RDWIA approach, the Hybrid model developed by the Ghent group is used for the description of the boson-pion-nucleon vertex.

We study the drag force acting on a heavy quark in a holographic plasma with rotational anisotropy and finite density. The bulk dual is the CCLP black hole of five-dimensional minimal gauged supergravity, characterised by two independent rotation parameters and electric charge. In the neutral Kerr–AdS limit, we use the principal Killing string to obtain an exact drag force for arbitrary rotation parameters. The resulting force is purely tangential but generically anisotropic, reducing to the viscous form only in the equal-spin sector. We then analyse stationary strings in the charged CCLP background perturbatively in the slow-rotation regime. A regularity analysis of the Lorentzian worldsheet fixes the angular integration constants that would otherwise remain ambiguous, yielding a finite renormalised transverse drag force with a smooth Kerr–AdS limit. We also show that, in the equal-spin sector, worldsheet regularity selects a unique co-rotating equilibrium quark and compute its renormalised free-energy shift.

We study black-hole quasinormal modes by applying the complex scaling method (CSM) to the perturbation equations of Schwarzschild and Reissner–Nordström black holes. The method converts the outgoing-wave boundary condition into a non-Hermitian eigenvalue problem, allowing quasinormal-mode frequencies to be computed within a common spectral framework. We first benchmark the method for the Schwarzschild Regge–Wheeler equation and then extend it to the Reissner–Nordström family, including the extremal limit. Our results show that CSM provides a unified and flexible approach to the computation of black-hole quasinormal frequencies.

Large families of confining holographic QFTs, described by Einstein-Dilaton gravity, are considered on constant-curvature manifolds in the presence of a $\theta$-angle. The space of ground states of such theories is explored as a function of the UV parameters, namely the dimensionless curvature and the $\theta$ angle. The free energy is computed, and the phase structure is determined. For constant negative curvature manifolds, we find solutions dual to single QFTs as well as solutions describing interfaces. The single QFTs exhibit an infinite family of saddle points, with the leading one dominating the gravitational path integral and no phase transitions present. For constant positive curvature manifolds, like de Sitter, the ($\theta$-angle, curvature) phase diagram exhibits both first and second order phase transitions, as a function of the class of theories considered. We also show that when $\theta=0$, a holographic Vafa-Witten-like theorem can be proven.

In this paper, we develop in detail the off-shell recursion for planar loop integrands in Yang-Mills theory. Starting from the classical equations of motion solved with the perturbiner method, we derive an exact transfer-matrix representation of the pure-gluon sector. We then include the ghost contributions to the loop kernels based on \cite{Tao:2025fch}. Finally, as an example, we work out the two-loop recursion in detail and conclude a general recursion strategy for two-loop planar integrands whose external legs are gluons.

We extend our gauge-covariant stochastic neural-field framework by promoting architecture-level parameters to slow stochastic variables evolving in function space. Our effective theory is formulated in terms of classical commuting fields and provides symmetry-constrained diagnostics of marginality and finite-width effects through the maximal Lyapunov exponent, the amplification factor, and dressed spectral kernels. On top of this dynamics, we introduce a Markovian evolutionary scheme compatible with the local $U(1)$ structure of the effective model. By using a minimal implementation, the genotype is reduced to the weight-variance parameter $\sigma_w^2$, and the fitness functional combines spectral agreement, marginal stability, and a symmetry-constrained critical anchor. Comparing three evolutionary models, we find that only the fully symmetry-constrained Ginibre $U(1)$ version robustly approaches a narrow near-marginal regime and reproduces the predicted low-frequency finite-width spectral behavior. These results support the use of symmetry-guided effective stability diagnostics as practical principles for stochastic architecture search in controlled settings.

Light scalars generically mediate a fifth force incompatible with local tests of gravity unless their couplings are parametrically suppressed or screening mechanisms are introduced. We demonstrate that such suppression can arise from symmetry. We propose a $Z_2$-symmetric mirror extension of the Standard Model within a bi-conformal gravity construction, where spontaneous breaking of scale invariance produces a light scalaron as a pseudo-Nambu-Goldstone boson. This scalaron couples to the difference of trace anomalies between the Standard Model and mirror sectors. We find a parameter-independent correlation between the fifth-force strength $\alpha$ and the scalaron mass $m_\sigma$, with the proportionality set by QCD observables and the electroweak scale. The Standard Model predicts $\alpha \lesssim 10^{-3}$ above 10~cm scales for $m_\sigma \lesssim 10^{-6}$ eV, which is directly in the target window of next-generation experiments. In contrast to environmental screening mechanisms, this suppression mechanism follows directly from symmetry rather than nonlinear scalar dynamics.