We propose a new large $N$ limit which at the extreme ($N=\infty$) limit is dual in the bulk to a back-reacted traversable wormhole, by making use of an operator in the algebra at infinity, an algebra familiar in the literature from the study of quasi-local algebras. We also compute, from a purely algebraic perspective, the effects registered by a left universe observer due to a unitary fluctuation on the right universe of the traversable wormhole, and reproduce a result from an earlier computation by Maldacena, Stanford and Yang \cite{Maldacena:2017axo}.

We investigate string configurations in the deconfined phase of SU(3) gauge theory, which arise from the spontaneous breaking of the $Z_3$ center symmetry. These configurations form at the junctions of domain walls of the theory. The complex phase of the Polyakov loop changes by multiples of $2π$ on large spatial loops around the string, rendering them topologically stable. Using the Monte Carlo simulations of the partition function, we compute the free energy associated with these configurations. The simulations are performed on lattices with spatial dimensions $N_{x,y}=60, N_z=4$, and temporal extent $N_τ=2$. Our results show that the free energy of the $Z_3-$strings is dominated by the domain walls. Further near the transition point, thermal fluctuations cause the decay of domain walls as well as the $Z_3$ strings into confined-deconfined interfaces.

In this paper, we present a brief review and introduction to Quadratic Unconstrained D-ary Optimization (QUDO), Tensor Quadratic Unconstrained D-ary Optimization (T-QUDO) and Higher-Order Unconstrained Binary Optimization (HOBO) formulations for combinatorial optimization problems. We also show explicit encodings between these formulations and discuss their limitations. To help their understanding, we make some examples for the knapsack problem, traveling salesman problem and different combinatorial games. The games chosen to exemplify are: Hashiwokakero, N-Queens, Kakuro, Inshi no Heya, and Peg Solitaire. Although some of these games have already been formulated in a QUBO formulation, we are going to approach them with more general formulations, allowing their execution in new quantum or quantum-inspired optimization algorithms. This can be an easier way to introduce these more complicated formulations for harder problems.

This paper aims to devise the shape of the external electromagnetic field that drives the spin dynamics of radical pairs to a quantum coherent state through maximization of the triplet-born singlet yield in biochemical reactions. The model is a Schrödinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms. We introduce a one-parameter family of optimal control problems by coupling the Schrödinger system to a control field through filtering equations for the electromagnetic field. Fréchet differentiability and the Pontryagin Maximum Principle in Hilbert space are proved, and the bang-bang structure of the optimal control is established. A new iterative Pontryagin Maximum Principle (IPMP) method for the identification of the bang-bang optimal control is developed. Numerical simulations based on IPMP and the gradient projection method (GPM) in Hilbert spaces are pursued, and the convergence, stability, and the regularization effect are demonstrated. Comparative analysis of filtering with regular optimal electromagnetic field versus non-filtering with bang-bang optimal field ({\it Abdulla et al, Quantum Sci. Technol., {\bf9}, 4, 2024}) demonstrates that the change of the maxima of the singlet yield is less than 1\%. The results open a venue for a potential experimental work on magnetoreception as a manifestation of quantum biological phenomena.

Typically, trials investigate the impact of either an individual-level intervention on participant outcomes, or the impact of a cluster-level intervention on participant outcomes. Factorial designs consider two (or more) treatments for each of two (or more) different factors. In factorial trial designs, trial units (individuals or clusters) are each randomised to a level of each of the treatments; these designs allow assessment of the interactions between different interventions. Recently, there has been growing interest in the design of trials that jointly assess the impact of individual- and cluster-level interventions (i.e. multi-level interventions); requiring the development of methodology that accommodates randomisation at multiple levels. While recent work has developed sample size methodology for variants combining standard cluster randomisation and individual randomisation, that work does not apply to longitudinal cluster randomised trial designs such as the stepped wedge design or cluster randomised crossover design. Here we present dedicated sample size methodology for "split-plot factorial longitudinal cluster randomised trials" with continuous outcomes: allowing for joint assessment of individual-level and cluster-level interventions that allows for the impact of the cluster-level intervention to be assessed using any longitudinal cluster randomised trial design. We show how the power to detect given effects of the individual-level intervention, the cluster-level intervention, and the interaction between the two depends on standard results for individually-randomised trials and longitudinal cluster randomised trials. We apply these results to the SharES trial, which considered the effects of a patient- and clinician-level interventions for patients with breast cancer on patient knowledge about the risks and benefits of treatment.

We propose and demonstrate a fully passive discrete-state continuous-variable quantum key distribution (CV-QKD), which can eliminate all modulator side channels on the source side, using a local local oscillator (LLO). The CV-QKD system achieves a maximum transmission length of 100 km with a repetition rate of 1 GHz using specially designed phase rotation and discretization methods, and the corresponding secret key bit rate is 127 kbps, as estimated based on the amplitude of prepared states at the transmitter, as well as the first- and second-order moments of quadratures at the receiver by employing the convex optimization without imposing any assumptions on the quantum channel. The performance of the protocol is similar to that of modulated CV LLO protocols and better than those of passive discrete-variable and CV protocols. Our protocol is expected to play an important role in the quantum metropolitan area networks and quantum access networks with high realistic security.

We have analyzed LVK gravitational wave events that show some evidence of eccentricity from TEOBResumS modeling parameter estimations and have confronted them independently with full numerical generated waveforms from our bank of nearly two thousand simulations of binary black holes. We have used RIFT for Bayesian parameter estimation and found that GW200208_22 KDE estimates favor eccentricities $e_{20} = 0.198_{-0.180}^{+0.119}$ upon entering the LVK band at $\sim20$Hz within a $90\%$ confidence interval. Within this event analysis we employed 42 new targeted full numerical relativity simulations and we have thus found a top improved likelihood $\ln\mathcal{L}$ matching waveform, compared to model-based analysis, with an estimated eccentricity at 20Hz, $e_{20}=0.200$, thus reinforcing the eccentric hypothesis of the binary. We have also used our full bank of numerical waveforms on GW190620 finding that the KDE estimate favors eccentricities at 10 Hz in $e_{10}=0.190_{-0.186}^{+0.046}$. New specifically targeted simulations will be required to narrow these eccentricity ranges.

We consider the stationary Boltzmann equation with the angular cutoff cross section in a bounded convex domain under the incoming boundary condition. In this article, we discuss the fractional Sobolev regularity of the solution without assuming the positivity of the Gaussian curvature on the boundary. For a boundary data sufficiently smooth and close to the standard Maxwellian, the solution has $H^{1-}_x$ regularity for hard potentials and soft potentials ($-2 \leq γ\leq 1$), while $H^{((4 + γ)/2)-}_x$ regularity is obtained for very soft potentials ($-3 < γ< -2$). We first show the well-posedness of the linearized problem on a weighted $L^2$ space and develop the $L^2-L^\infty$ estimate without the stochastic cycle. We next investigate $H^s_x$ regularity of the solution to the linearized problem. The idea of the celebrated velocity averaging lemma plays a key role in our analysis. We finally derive a bilinear estimate to extend the result on the linearized problem to the weakly nonlinear problem.

Erdős first introduced the idea of covering systems in 1950. Since then, much of the work in this area has concentrated on identifying covering systems that meet specific conditions on their moduli. Among the central open problems in this field is the well-known odd covering problem. In this paper, we investigate a variant of that problem, where one odd integer is permitted to appear multiple times as a modulus in the covering system, while all remaining moduli are distinct odd integers greater than 1.

The Cremona dimension of a group $G$ is the minimal $n$ such that $G$ is isomorphic to a subgroup of the Cremona group of birational transformations of an $n$-dimensional rational variety. In this survey article, we give many examples that gives evidence to the conjecture that the Cremona dimension of a finite group over the field of complex numbers is less than or equal to the essential dimension of the group.

We theoretically investigate the impact of the anomalous magnetic moment (AMM) of Dirac fermions on the fermionic Casimir effect under magnetic fields. We formulate it as an extension of the well-known Lifshitz formula. From our formula, we find that the AMM increases the fermionic Casimir energy. In particular, when the AMM is large enough, the Casimir energy is significantly enhanced by the gapless behavior of the lowest Landau level. We also quantitatively estimate the Casimir energy from electron, muon, and constituent quark fields under magnetic fields and propose possible phenomena at finite temperature and fermion density.

Improving the coherence of superconducting qubits is essential for advancing quantum technologies. While superconductors are theoretically perfect conductors, they consistently exhibit residual energy dissipation when driven by microwave currents, limiting coherence times. Here, we report an empirical scaling relation between microwave dissipation and the superfluid density, a bulk property of superconductors related to charge carrier density and disorder. Our analysis spans a wide range of superconducting materials and device geometries, from highly disordered amorphous films to ultra-clean systems with record-high quality factors, including resonators, 3D cavities, and transmon qubits. This scaling reveals an intrinsic bulk dissipation channel, independent of surface dielectric losses, which we attribute to nonequilibrium quasiparticles trapped within disorder-induced spatial variations of the superconducting gap, with a density set by a universal material parameter. Our findings identify an empirical coherence limit associated with intrinsic material properties and provide a data-driven basis for materials selection in future superconducting quantum circuits.

Rational conformal field theories in 2d have partition functions built from holomorphic characters, whose classification can be addressed via the holomorphic modular bootstrap. This is facilitated by a special basis of ``quasi-characters'' that has been completely classified for rank-2. Suitably combining these to form admissible characters with non-negative integral coefficients $a_n$ depends crucially on the signs and growth of the quasi-character coefficients. We use Frobenius recursion relations for Modular Linear Differential Equations to estimate the growth with $c$ of these coefficients in the region $n\sim\frac{c}{12}$ that is inaccessible to Cardy asymptotics, and to prove rigorously that they have alternating signs that stabilise to a fixed sign at this order. This provides a practical path to obtain candidate RCFT partition functions at arbitrary Wronskian index.

The recently observed plasmonic skyrmions, as electromagnetic counterparts of topologically stable quasiparticles, hold significant promise as novel carriers for robust information transfer and manipulation of nontrivial light-matter interactions. However, their practical applications has been hindered by the lack of flexible tuning devices to encode these topological structures. Here, we present a programmable plasmonic skyrmion platform capable of encoding diverse skyrmion topologies, including Neel-type skyrmions and merons. Based on unprecedented ultra-fast coding feature, we synthesize harmonic skyrmions in the temporal dimension and, for the first time, applied skyrmions in communication and sensing applications. Specifically, we achieved highly robust and multi-channel wireless communications by using programmable topological skyrmions, providing a promising platform for communication in turbulent noise channels and extreme conditions. Furthermore, we implemented intelligent sensing across twenty animal models on the same platform, achieving high recognition accuracy. This methodology offers programmable and temporal insights into the skyrmions for their practical applications in next-generation wireless communication and intelligent sensing.

The Goeritz matrix is an alternative to the Kauffman bracket and, in addition, makes it possible to calculate the Jones polynomial faster with some minimal choice of a checkerboard surface of a link diagram. We introduce a modification of the Goeritz method that generalizes the Goeritz matrix for computing the HOMFLY-PT polynomials for any $N$ in the special case of bipartite links. Our method reduces to purely algebraic operations on matrices, and therefore, it can be easily implemented as a computer program. Bipartite links form a rather large family including a special class of Montesinos links constructed from the so-called rational tangles. We demonstrate how to obtain a bipartite diagram of such links and calculate the corresponding HOMFLY-PT polynomials using our developed generalized Goeritz method.

In a Galilean superfluid, the depletion of superfluid density with rising temperature can be attributed to thermally excited non-interacting phonons. For systems without Galilean symmetry, it has been shown [1] that ``phonon wind" is no longer responsible for the depletion of superfluid density. In this work, we develop the theory of superfluid density at low temperature ($T$) and provide detailed derivations of all results announced in [1]. Using Popov's hydrodynamic action, we show that the theory of low-temperature depletion in a $d$-dimensional quantum superfluid maps onto the problem of finite-size ($L$) corrections in a $(d+1)$-dimensional anisotropic (pseudo-)classical-field system with U(1)-symmetric complex-valued action. In addition to generalizing Landau's (canonical) formula, we develop the grand canonical theory, which in a broader context reveals a universal scaling, $T^{d+1}$ and $1/L^{d+1}$, for finite-$T$ and finite-$L$ effects of many thermodynamic quantities. We validate our theory with numeric simulations of interacting lattice bosons and the J-current model.

The proliferation of electric vehicles in recent years has significantly expanded the charging infrastructure while introducing new security risks to both vehicles and chargers. In this paper, we investigate the security of major charging protocols such as SAE J1772, CCS, IEC 61851, GB/T 20234, and NACS, uncovering new physical signal spoofing attacks in their authentication mechanisms. By inserting a compact malicious device into the charger connector, attackers can inject fraudulent signals to sabotage the charging process, leading to denial of service, vehicle-induced charger lockout, and damage to the chargers or the vehicle's charge management system. To demonstrate the feasibility of our attacks, we propose PORTulator, a proof-of-concept (PoC) attack hardware, including a charger gun plugin device for injecting physical signals and a wireless controller for remote manipulation. By evaluating PORTulator on multiple real-world chargers, we identify 7 charging standards used by 20 charger piles that are vulnerable to our attacks. The root cause is that chargers use simple physical signals for authentication and control, making them easily spoofed by attackers. To address this issue, we propose enhancing authentication circuits by integrating non-resistive memory components and utilizing dynamic high-frequency Pulse Width Modulation (PWM) signals to counter such physical signal spoofing attacks.

To meet the demands of future wireless networks, antenna arrays must scale from massive multiple-input multiple-output (MIMO) to gigantic MIMO, involving even larger numbers of antennas. To address the hardware and computational cost of gigantic MIMO, several strategies are available that shift processing from the digital to the analog domain. Among them, microwave linear analog computers (MiLACs) offer a compelling solution by enabling fully analog beamforming through reconfigurable microwave networks. Prior work has focused on fully-connected MiLACs, whose ports are all interconnected to each other via tunable impedance components. Although such MiLACs are capacity-achieving, their circuit complexity, given by the number of required impedance components, scales quadratically with the number of antennas, limiting their practicality. To solve this issue, in this paper, we propose a graph theoretical model of MiLAC facilitating the systematic design of lower-complexity MiLAC architectures. Leveraging this model, we propose stem-connected MiLACs as a family of MiLAC architectures maintaining capacity-achieving performance while drastically reducing the circuit complexity. Besides, we optimize stem-connected MiLACs with a closed-form capacity-achieving solution. Our theoretical analysis, confirmed by numerical simulations, shows that stem-connected MiLACs are capacity-achieving, but with circuit complexity that scales linearly with the number of antennas, enabling high-performance, scalable, gigantic MIMO.

We construct positional numeral systems that work natively over nonderived polyadic $\left( m,n\right) $-rings whose addition takes $m$ arguments and multiplication takes $n$. In such rings, the length of an admissible additive word and a multiplicative tower are not arbitrary (as in the binary case), but "quantized". Our main contributions are the following. Existence: every commutative $\left( m,n\right) $-ring admits a base-$p$ place-value expansion that respects the word length constraint in terms of numbers of operation compositions $\ell_{mult}=\ell_{add}(m-1)+1$. Lower bound: the minimum number of digits is greater than or equal to the arity of addition $m$. Representability gap: for $m,n\geq3$ only a proper subset of ring elements possess finite expansions, characterized by congruence-class arity shape invariants $I^{(m)}$ and $J^{(n)}$. Mixed-base "polyadic clocks": allowing a different base at each position enlarges the design space quadratically in the digit count. Catalogues: explicit tables for the integer rings $\mathbb{Z}_{4,3}$ and $\mathbb{Z}_{6,5}$ illustrate how ordinary integers lift to distinct polyadic variables. These results lay the groundwork for faster arity-aware arithmetic, exotic coding schemes, and hardware that exploits operations beyond the binary pair.

Quantum devices based on optically addressable spin qubits in diamond are promising platforms for quantum technologies such as quantum sensing and communication. Nano- and microstructuring of the diamond crystal is essential to enhance device performance, yet fabrication remains challenging and often involves trade-offs in surface quality, aspect ratio, device size, and uniformity. We tackle this hurdle with an approach producing millimeter-scale, thin (down to 70 nm) and highly parallel (< 0.35 nm/$\mathrm{μm}$}) membranes from single-crystal diamond. The membranes remain contamination-free and possess atomically smooth surfaces ($\mathrm{R_q}$ < 200 pm) as required by state-of-the-art quantum applications. We demonstrate the benefits and versatility of our method by fabricating large fields of free-standing and homogeneous photonic nano- and microstructures. Leveraging a refined photolithography-based strategy, our method offers enhanced scalability and produces robust structures suitable for direct use, while remaining compatible with heterogeneous integration through pick-and-place transfer techniques.

This paper establishes a Transition Path Theory (TPT) for Lévy-type processes, addressing a critical gap in the study of the transition mechanism between meta-stabile states in non-Gaussian stochastic systems. A key contribution is the rigorous derivation of the stochastic differential equation (SDE) representation for transition path processes, which share the same distributional properties as transition trajectories, along with a proof of its well-posedness. This result provides a solid theoretical foundation for sampling transition trajectories. The paper also investigates the statistical properties of transition trajectories, including their probability distribution, probability current, and rate of occurrence.

When a coherent electromagnetic wave passes through a beam splitter (BS), it is divided equally into two parts. However, the quantum noise associated with the resulting coherent states, despite being reduced in amplitude by half, remains fundamentally constrained by the quantum noise limit, independent of the intensity. By placing a mirror at the unused input port of the BS, a standing wave is formed in the vicinity of the mirror, which influences the vacuum fluctuations of the coherent state at the BS output. Using semi-classical and quantum mechanical approaches, we calculate the vacuum fluctuations induced by the mirror and demonstrate that the vacuum noise originating from the mirror side periodically reaches zero at the BS output. Leveraging this effect, we show that the vacuum fluctuations of the light split by the BS can be readily reduced below the quantum noise limit. Furthermore, through feedback mechanisms, the vacuum fluctuations of the electromagnetic field at the other output port can also be suppressed below the quantum noise limit. These findings provide a pivotal insight into the manipulation of electromagnetic noise, with broad implications for all experiments involving quantum noise control.

In \cite{Colding}, Colding proved monotonicity formulas for the Green function on manifolds with nonnegative Ricci curvature. Inspired by the sharp estimates relating the pinching of monotone quantities to the splitting function in \cite{cjn}, in this paper we investigate quantitative control obtained from pinching of Colding's monotone functionals. From the Green functions with poles at $(k+1)$-many independent points, $k$-splitting functions are constructed with regularity quantitatively controlled by the pinching. Moreover, the pinching at these independent points controls the distance to the nearest cone of the form $\mathbb{R}^k \times C(X)$.

Solutions to hyperbolic systems comprise waves propagating at finite speeds. When wave propagation is predominantly unidirectional, one-way wave equations can be used to evolve only the right-going solution by removing support for left-going waves. The One-Way Navier-Stokes (OWNS) approach, which was originally developed for systems of first-order hyperbolic equations, constructs one-way approximations to the linearized Navier-Stokes equations using a recursive filter to remove left-going waves. The computational cost scales with the number of recursion parameters, which must be carefully chosen to ensure accuracy and stability of the resulting one-way equation. Previous work has chosen parameters based on heuristic estimates of key eigenvalues, which requires trial-and-error tuning while also yielding slow error convergence. We propose a greedy algorithm for automatic parameter selection, which we show yields faster convergence and a net decrease in computational cost for linear and nonlinear disturbance evolution in boundary-layer flows. We review the OWNS projection (OWNS-P) and recursive (OWNS-R) methods, comparing their convergence properties, and show through our numerical analysis and experiments that OWNS-P yields superior convergence and stability properties. Although we demonstrate the method for Navier-Stokes equations, we perform our analyses on systems of linear first-order hyperbolic equations and emphasize that the greedy algorithm is applicable to such systems.

We generalize Bouc's construction of orthogonal idempotents in the double Burnside algebra to the setting of the double $\mathbb{C}^\times$-fibered Burnside algebra. This yields a structural decomposition of the evaluations of $\mathbb{C}^\times$-fibered biset functors on finite groups. We then construct a complete set of orthogonal idempotents in the category of $\mathbb{C}^\times$-fibered $p$-biset functors, leading to a categorical decomposition of this category into subcategories indexed by isomorphism classes of atoric $p$-groups. Furthermore, we introduce the notion of vertices for indecomposable functors and establish that the Ext-groups between simple functors with distinct vertices vanish. As an application, we describe a set containing composition factors of the monomial Burnside functor, thereby providing new insights into its structure. Additionally, we develop a technique for analyzing fibered biset functors via their underlying biset structures.

We introduce a novel approach to simulate the interaction between fluids and thin elastic solids without any penetration. Our approach is centered around an optimization system augmented with barriers, which aims to find a configuration that ensures the absence of penetration while enforcing incompressibility for the fluids and minimizing elastic potentials for the solids. Unlike previous methods that primarily focus on velocity coherence at the fluid-solid interfaces, we demonstrate the effectiveness and flexibility of explicitly resolving positional constraints, including both explicit representation of solid positions and the implicit representation of fluid level-set interface. To preserve the volume of the fluid, we propose a simple yet efficient approach that adjusts the associated level-set values. Additionally, we develop a distance metric capable of measuring the separation between an implicitly represented surface and a Lagrangian object of arbitrary codimension. By integrating the inertia, solid elastic potential, damping, barrier potential, and fluid incompressibility within a unified system, we are able to robustly simulate a wide range of processes involving fluid interactions with lower-dimensional objects such as shells and rods. These processes include topology changes, bouncing, splashing, sliding, rolling, floating, and more.

The electromagnetic field of standing-wave or ring cavities induces a spatially modulated, infinite-range interaction between atoms in an ultracold Fermi gas, with a single wavelength comparable to the Fermi length. This interaction has no analog in other systems of itinerant particles and has so far been studied only in the regime where it is attractive at zero distance. Here, we fully solve the problem of competing instabilities of the Fermi surface induced by single-wavelength interactions. We find that while the density-wave (superradiant) instability dominates on the attractive side, it is absent for repulsive interactions, where the competition is instead won by non-superradiant superfluid phases at low temperatures, with Fermion pairs forming at both vanishing and finite center-of-mass momentum. Moreover, even in the absence of such symmetry-breaking instabilities, we find the Fermi surface to be always nontrivially deformed from an isotropic shape. We estimate this full phenomenology to be within reach of dedicated state-of-the-art experimental setups.

We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data not belonging to $L^2(\mathbb{R}^n)$ in general. Our approach is based on the $L^p$-$L^q$ estimates of linear solutions and the fractional Leibniz rule in suitable homogeneous Besov spaces.

Dual-band plasmonic nanoantennas, exhibiting two widely separated user-defined resonances, are fundamental building blocks for the investigation and optimization of plasmon-enhanced optical phenomena, including photoluminescence, Raman scattering, and various nonlinear effects such as harmonic generation or sum-frequency generation, parametric down-conversion, etc. The nanoparticle-on-slit (NPoS) or nanoparticle-in-groove (NPiG) is a recently proposed dual-band antenna with independently tunable resonances at mid-infrared and visible wavelengths. It was used to enhance the corresponding sum- and difference-frequency generation processes from optimally located molecules by an estimated $10^{13}$-fold. However, the theoretical understanding of such structures and their eigenmodes remains poor, hindering further optimization and limiting broader applications. Here, we explore a diverse range of nanocavity-like quasi-normal modes (QNMs) supported by NPoS structures, examining the contributions of both their near-field (i.e., giant photonic density of states) and far-field (i.e., spatial radiation patterns) characteristics to frequency upconversion. We identify methods for independently tuning the visible and mid-infrared resonances while conserving a good mode overlap in the near field, which is essential for efficient nonlinear processes. Moreover, through mode analysis, we unveil an experimentally unexplored fundamental resonance with greater field enhancement and much-improved mode overlap with the mid-infrared field, which could, in principle, further boost the mid-infrared upconversion efficiency by 5-fold compared to existing results. This work helps to rationalize and optimize the enhancement of nonlinear effects across a wide spectral range using a flexible and experimentally attractive nanoplasmonic platform.

We construct the least superharmonic majorant of a continuous function $g$ on the $d$-dimensional unit ball ($d \geq 2$) via a canonical sequential scheme. While classical theory identifies this majorant with the value function of the optimal stopping problem for Brownian motion absorbed at the domain boundary, no comparable constructive approximation scheme has been available. We introduce branched harmonic majorants, obtained by arranging classical harmonic functions on smoothly bounded domains in a finite, depth-indexed branching structure, and prove two main results. First, the optimal stopping region is identified as the contact set between the gain function $g$ and the pointwise infimum of this family; the value function is recovered as the expected gain at the first exit time from the non-contact set. This yields a multidimensional generalisation of the Dynkin--Yushkevich concave-envelope theorem in which affine functions are replaced by branched harmonic majorants. Second, truncation in the branching depth produces a decreasing sequence of envelopes that converges pointwise to this infimum, yielding an explicit approximation scheme not present in classical formulations. Analytically, the branching structure relaxes the global majorisation constraint to a local constraint imposed on a decreasing sequence of non-contact sets, yielding a representation of the Perron envelope in terms of harmonic functions on smoothly bounded domains. Probabilistically, the construction corresponds to the sequential composition of stopping times and overcomes the localisation obstruction arising from the thinness of Brownian paths in dimensions $d \geq 2$.