Papers

We establish a criterion of arithmetic $k$-polynomial spectral localization and $k$-polynomial dynamical localization in expectation for quasi-periodic long-range operators on $\ell^2(\mathbb{Z}^{d})$ with power-law hopping based on the quantitative $C^{k}$-reducibility of the dual Schrödinger cocycle. As the application, we prove both localization properties for power-law long-range perturbations of the Almost Mathieu Operators with sufficiently large couplings and Diophantine frequencies.

The increasing complexity in design and manufacturing has driven the need for advanced techniques for fast modeling problems with large-dimensional parameter spaces. Avoiding high-fidelity finite element models while achieving fast and accurate simulations in such contexts is challenging. Parametric model order reduction (pMOR) has drawn significant attention in recent years. Nevertheless, the curse of dimensionality in parameter spaces has severely limited its effectiveness. The active subspace (AS) approach has been successfully applied to pMOR for systems with many parameters. However, the balance between accuracy and compactness of the reduced model remains problematic for such systems with high-dimensional parameter spaces. It often results in models that are either not small enough or not accurate enough. In this paper, we propose an iterative active subspace (IAS) approach for parametric model order reduction, which, to some extent, addresses the trade-off between accuracy and reduced model size and achieves substantial computational gains compared to the original active subspace method.

This paper establishes an asymptotic formula for negative Sobolev norms as the fractional order tends to zero. In the Euclidean setting, under a mild boundedness condition, the product of the order and the norm raised to the power $p$ converges to a dimension-independent constant multiple of the corresponding $L^p$ norm. The proof relies on heat-kernel regularization, weak compactness, and an Abelian–Tauberian argument. The result is further extended to a general measure space equipped with a family of bounded and continuous operators that covering nonlinear and non-semigroup settings. We also present several applications in analysis and probability, including an absolute-continuity criterion for measures, a random-distribution regularity result, a construction of square-integrable local times for fractional Brownian motion, and limiting formulas for truncated maximal operators and martingales.

In this paper, an abstract framework is introduced for the derivation of reliable a posteriori error estimators for variational inequalities of the second kind. Under certain conditions (local) efficiency of the error estimators can also be derived. The framework is based on the estimation of a residuum which results from an equivalent mixed formulation and from the replacement of the Lagrange multiplier by its (possibly post-processed) discrete counterpart. The approach is applied to an idealized frictional contact problem and a Mosolov type model problem in an $hp$-finite element discretization setup. Several numerical experiments demonstrate the broad applicability of the theoretical findings.

Equivariant convolutional neural networks are usually built from a group acting globally on the space of signals. This hypothesis is inappropriate for many bounded or stratified domains: an ambient rigid motion may be admissible only on part of the domain, and the boundary introduces geometric types that are invisible to a transitive group action. We develop a theory of groupoid-equivariant neural networks in which the symmetry datum consists of a groupoid, a selected pseudogroup of local bisections, a measure, and input and output representation bundles. For integral channels on the object space, we prove a bisection-equivariant kernel theorem: equivariance is equivalent to a transport constraint on the two-point kernel, and its solutions are classified by one joint-stabilizer intertwiner on each orbit of pairs. As a case study we apply the theory to bounded planar domains. The resulting architecture is implemented through offline nullspace bases and sparse gather–transform–scatter operations. A Poisson–Dirichlet kernel study is used separately to assess boundary-aware inductive bias; the exact inverse is shown to preserve the global symmetries of the rectangle but not general proper local bisections. The numerical results show that the proposed architectures provide significant advantages when symmetries cannot be globally implemented by group actions and provide an accuracy improvement of at least one order of magnitude with respect to the models tested.

In this paper, we establish lower bounds for large values of the Hurwitz zeta function with rational parameter when the real part $\sigma \in [1/2,1]$. These results improve the earlier results of Ramachandra and Sankaranarayanan in 1989. On the critical line, our result recovers the corresponding lower bound of de la Bretèche and Tenenbaum (2019) for the Riemann zeta function, while for $1/2< \sigma \leq 1$, our lower bounds attain the same order as the corresponding lower bounds for large values of the Riemann zeta function. Our proofs are based on the resonance method.

It is known that the inverse of an invertible real square matrix satisfying the treeangle property, is a treediagonal matrix. A converse statement also holds. We show that the verbatim analogues are not true for the Moore-Penrose inverse, and obtain the precise structure of graphs corresponding to the Moore-Penrose inverse of matrices possessing the treeangle property.

Lee, Skipper, and Speakman (2018) asked for formulae or asymptotic expressions for the volumes of the strong and weak relaxations $\mathcal{P}_{m,n}\cap \mathcal{D}_{m,n}$ and $\mathcal{Q}_{m,n}\cap \mathcal{D}_{m,n}$ of the uncapacitated facility-location polytope with $m$ facilities and $n$ customers (their Problem~3, “Texas Hot”). We give exact formulae for all $m$ and $n$, and we give asymptotics that hold uniformly as $m+n\to\infty$, whatever the relative growth of $m$ and $n$. The volume ratio satisfies $Vol(\mathcal{Q}_{m,n}\cap \mathcal{D}_{m,n})/Vol(\mathcal{P}_{m,n}\cap \mathcal{D}_{m,n})\sim \prod_{j=2}^{n}\mleft(1+\tfrac{1}{(m-1)j}\mright)^{m}$. From this, we read off the behavior in every growth regime of $m$, and we identify a sharp transition at $m\asymp\ln n$. Below it, in a uniformly random point of the strong relaxation, each facility variable is close to $1$ with high probability, and the ratio grows superlinearly in $n$. Above it, in a uniformly random point of either relaxation, each facility variable is asymptotically uniform on $[0,1]$, and the ratio grows linearly in $n$.

This article introduces the LUneburg-lens-aided recoNfigurable Array (LUNA), an antenna architecture that unifies multiple-input multiple-output (MIMO) and network-controlled repeater (NCR) functionalities in the Luneburg lens-enabled hardware platform. A Luneburg lens, fabricated from graded-index dielectric materials, passively converts the radiation of a low-gain feed into a highly directional beam without active phase shifting, while a dense passive feed bank and a reconfigurable feed-selection network electronically switch the beam directions with minimal hardware complexity and power consumption. We commence by reviewing the basic principles and application history of Luneburg lenses in radar and wireless communications, which motivates their role in 6G-and-advanced networks. Then, we highlight how a Luneburg lens and a reconfigurable feed array construct both LUNA-MIMO and LUNA-NCR, where the lens and feed bank can be reused across functions and frequency bands. Case studies demonstrate that LUNA achieves the satisfactory spectral and energy efficiency with a few radio-frequency chains, and it also improves positioning performance for sensing tasks. Finally, some open problems and research directions are provided to inspire follow-up research on LUNA.

For $n\ge6$, $m=\binom n2$, let the complete-graph incidence family on $K_n$ have the vertices of $K_n$ as columns, its edges as rows, and the incidence graph as one-edge graph. The universal cell bound of Löfberg and Qi gives $z_2(m,n)\le Z(n):=\lfloor n(n-1)(n+2)/4\rfloor$. We implement the nested one-factorization construction of that family and determine exactly what it certifies. For $n=2q$ with $q$ an odd prime, $q\ge5$, the construction has no hole and the cross-factor transfer equations apply verbatim, giving $z_2=z_{SL}=z_{RL}=Z(n)$; $n=6$ is settled by a separate cyclic witness. For odd $n=2p+1$ the near-perfect one-factorization yields Hamilton paths closed into odd cycles, and the transfer argument breaks: that scheme certifies only $R(G_p)\le Z(n)$. The orders $n=7,8,9,12,13,16,17,18,20,21$ are settled by explicit configurations of a different shape, each attaining the cell bound and satisfying $(\mathrm{RW}3^+)$, obtained from a larger configuration by deleting vertex stars and repairing the restricted grid. At each of these orders $z_2=z_{SL}=z_{RL}=Z(n)$; all remaining orders are conjectural. The first settled order $n=7$ is the only one whose grid has a hole, grounded by a zero-companion rule. Machine-readable configurations and a certificate checker accompany the paper.

If $X$ is smooth projective complex variety, $D$ is a simple normal crossings divisor and $A$ an ample divisor with support $D$, then there is an associated filtration of the Lie algebra of on the affine variety $U=X\setminus D$ induced from the subspaces $T_X(\log D)(nA)\subset T_U.$ We show that the associated graded Lie algebra is finitely generated in a strong sense. We deduce as a consequence that vector fields on any smooth affine variety form a finitely generated Lie algebra.

This paper gives a criterion for an endopermutation Scott module to be slash indecomposable. In particular, under suitable assumptions, we only need to prove certain endopermutation Scott modules over local subgroups are slash indecomposable.

All the eigenvalues of an integral graphs are integers. Integral graphs are extremely rare. They form an asymptotically vanishing fraction $2^{-\Omega(n)}$ among all graphs on $n$ vertices. It makes the construction of a new family of integral graphs a challenging task. Also, most of the known infinite family of integral graphs rely on Cayley graphs over Abelian groups. In this article, we introduce a new family of integral graphs obtained from the groups. The construction of our graphs from groups is different from the construction of Cayley graphs. A spectral uniqueness theorem is established, which shows that each member of the infinite family is determined by its adjacency spectrum among all finite simple graphs. We also present recursive constructions that generates larger members of the family from smaller ones, providing a scalable class of integral graphs. Finally, we investigate algebraic realizations of these graphs as complements of Proper Prime Order Element Graphs of finite $2$-groups and obtain conditions characterizing such realizations. We also observe that the graphs obtained from different non-isomorphic groups have cospectral graphs.

We propose a bounded degree SOS+SONC hierarchy for constrained polynomial optimization, termed B-SOS+SONC. Starting from Lasserre's bounded-degree SOS framework, we enlarge the certificate cone from SOS to the recently introduced SOS+SONC cone, thereby combining the algebraic strength of semidefinite relaxations with the sparse structure captured by circuit polynomials. We show that, for each fixed certificate degree, the resulting hierarchy is complete, that is, its optimal values are monotone and converge to the global optimum. Moreover, we derive an explicit SDP-REP reformulation, so that each relaxation can be solved within a tractable convex optimization framework over semidefinite and relative entropy cones. Beyond the optimization hierarchy itself, we investigate structural properties of the SONC cone and introduce the notions of first-order and second-order SONC-convexity. This leads to a new sufficient condition for first-level exactness of the B-SOS+SONC hierarchy. Numerical experiments illustrate that the proposed hierarchy often yields tighter lower bounds than the B-SOS relaxation while remaining tractable.

We establish quantitative stability estimates for weighted Hardy inequalities in the local and fractional settings, showing that the Hardy deficit controls the distance of a normalized admissible function to the family of virtual extremizers. In the local case, for $p>1$, $\alpha\in[0,1)$, $0<p-\alpha p<N$, and $u\in C_c^1(\mathbb{R}^N)$ normalized by $\int_{\mathbb{R}^N}|u|^p|x|^{\alpha p-p}\,dx=1$, we prove \begin{equation*} \int_{\mathbb{R}^N}|\nabla u|^p|x|^{\alpha p}\,dx-\Big(\frac{N-p+\alpha p}{p}\Big)^p\int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{p-\alpha p}}\,dx \ge C\,\operatorname{dist}_{\alpha}(u,\mathcal{M}_\alpha)^{\max\{4,2p\}}, \end{equation*} where $\mathcal{M}_\alpha$ is generated by the virtual extremizer with scale-invariant distance. We extend this to the weighted fractional Hardy inequality: for $s\in(0,1)$, $\alpha=\alpha_1+\alpha_2\ge0$, $\alpha_1p,\alpha_2p\in(-N,sp)$, and $0<sp-\alpha p<N$, the fractional Hardy deficit satisfies \begin{equation*} \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}|x|^{\alpha_1p}|y|^{\alpha_2p}\,dx\,dy-\mathcal{C}\int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{sp-\alpha p}}\,dx \ge C\,\operatorname{dist}_{s,\alpha}(u,\mathcal{M}_{s,\alpha})^{\max\{4,2p\}}. \end{equation*} For $p\ge2$ the distance is measured on $u$, while for $1<p<2$ it is measured on a power-type transformation of $u$. Both recover the unweighted results when the weights vanish. We also prove weighted Sobolev and fractional Sobolev extension and embedding inequalities on bounded smooth domains avoiding the origin. The proof is rearrangement-free, combining these inequalities with scale-invariant Poincaré–Sobolev estimates on annuli, telescoping oscillation decompositions, and weighted Lorentz-space embeddings. The distinction between $p\ge2$ and $1<p<2$ stems from the convexity estimate used to obtain a remainder term from the deficit.

We study the behavior at infinity of the invariant Radon measure for the multidimensional affine stochastic recursion $V_n = A_n V_{n-1} + B_n,$ where $(A_n)_{n \geq 1}$ are positive random matrices, $(B_n)_{n \geq 1}$ are random vectors with nonnegative entries, and $(A_n, B_n)_{n \geq 1}$ are independent and identically distributed. In the critical regime where the top Lyapunov exponent of the random matrix products $A_n \cdots A_1$ is zero, Brofferio, Peigné and Pham [6] recently established the existence and uniqueness, up to multiplication by a constant, of an invariant Radon measure with infinite total mass. They proved that the tail behavior of this measure when applied to radial sets is governed by a slowly varying function. Our goal is to show that this slowly varying function is actually bounded. Moreover, we investigate directional tail behavior.

We prove that irreducible spherical type Artin groups are profinitely rigid within the class of all spherical type Artin groups. As part of the proof, building on ideas of Kolay for the $n$-strand braid group, we compute the smallest non-abelian quotient of every spherical type Artin group. In particular, given an irreducible spherical type Artin group, its smallest non-abelian quotient is always that of the corresponding Coxeter group, except for dihedral Artin groups of weight divisible by $4$, which all have $\mathfrak S_3$ as a non-abelian quotient. Parts of the proof also rely on the computation of the cohomological dimension of the profinite completion of certain Artin groups.

We present a new semi-implicit structure-preserving (SP) finite volume discretization for a unified first-order hyperbolic model of continuum mechanics. This framework provides a common mathematical description for fluids and solids and incorporates transport, viscous effects, heat conduction, and elastic deformations within a single system of hyperbolic partial differential equations. The coexistence of several physical mechanisms gives rise to multiple characteristic wave speeds, resulting in severe time-step restrictions for fully explicit discretizations. To overcome this difficulty, we develop a four-split scheme in which the governing equations are decomposed into convective, temperature, mechanical, and pressure subsystems. The convective subsystem is the only one that is advanced explicitly in time, while the remaining subsystems are treated implicitly in a sequential manner. As a consequence, the time step restriction of the proposed method depends only on the material velocity and is independent of the acoustic, shear, and thermal wave speeds that characterize the model. The scheme is designed to preserve key structural properties of the underlying equations on unstructured grids. In particular, a compatible vertex-staggered spatial discretization on triangles allows the curl-free involutions associated with the distortion field and thermal impulse to be respected whenever the relaxation source terms in the governing PDE are linear or absent. At the same time, the scheme is asymptotic preserving as it is consistent with the low Mach number limit and with the stiff relaxation limits of the GPR model that recover the classical Navier-Stokes-Fourier equations in fluid mechanics. A series of numerical experiments covering both fluids and solids demonstrates the accuracy, robustness, and multi-scale capabilities of the proposed approach.

We study Lipschitz mappings associated with group actions by similarities, extending the orbit-quotient approach for point-preserving isometric actions. Normalizing by the similarity ratios we obtain isometric representations whose fixed points are character-equivariant Lipschitz mappings. The corresponding quotients by closed spans of orbit differences provide canonical preduals and norm-preserving linearizations, without an amenability assumption. Applications include positively homogeneous, linear, and bilinear mappings. In particular, we prove that the space of positively homogeneous Lipschitz functions form a $1$-complemented subspace of the Lipschitz space (strenghthening the earlier existed result) and interpret existing projections onto linear and bilinear mappings as invariant-mean averages. For amenable groups, the same approach yields isometric realizations of the quotient spaces in the biduals of the original linearization spaces.

We revisit the modelling of the material decomposition problem in multispectral computed tomography and propose to leverage the known absorption spectra of different materials. With this we both derive new uniqueness conditions for the material decomposition and also offer a three-step reconstruction model. Our uniqueness criterion only requires certain characteristics of the bin sensitivity functions and the material attenuation spectra and hence, can be computed a-priori before measurements are taken.

For arithmetic quotients of Bruhat–Tits trees with finitely many cusps, we establish an explicit unitary correspondence between the spherical Eisenstein transform, with the Eisenstein series normalized by their constant terms, and the scattering transform of an associated Jacobi operator with finite core. Tracking the Haar measure, stabilizer weights, height coordinates, and cusp widths yields the Plancherel measure and shows that the absolutely continuous spectrum has multiplicity equal to the number of cusps. From a discrete Green identity we derive a matrix-valued Maass–Selberg formula for the Hermitian matrix $iS(\theta)^*\partial_\theta S(\theta)$, where $S(\theta)$ is the scattering matrix. Its trace is determined by $\det S(\theta)$, while the full matrix retains additional cusp-to-cusp information. After the corresponding change of normalization, the finite Schur complement obtained by eliminating the cusp rays agrees with the resonance matrix of Arends-Peterson-Weich. Using their resonance computations as input, we distinguish eigenvalues supported entirely in the finite core from poles of the scattering matrix. The Nagao and $\Gamma_0(T)$ quotients, together with a four-cusp quotient arising from an elliptic curve over $\mathbb F_3$, make the normalizations and matrix-valued conclusions explicit.

We study a dissipativity-based model predictive control (MPC) scheme for energy-optimal constrained output stabilization of a discrete-time nonlinear SISO port-Hamiltonian system described by difference and differential representation. For the optimal control problem to be solved in each MPC step, we establish a measure turnpike behavior. The turnpike set in our work is obtained from the underlying port-Hamiltonian structure without assuming existence of steady-states or periodic orbits. In particular, unlike existing results which typically establish the turnpike property w.r.t. a controlled forward invariant set such as optimal steady-states or optimal periodic orbits, we do not require the turnpike set to be forward control invariant. For MPC, we establish recursive feasibility and show practical stability to the turnpike set w.r.t. the MPC closed loop leveraging both dissipativity and the port-Hamiltonian structure. Finally, we demonstrate our results using two numerical examples.

We develop a nonlinear thermodynamic formalism for correspondences on compact metric spaces. A continuous energy is evaluated on empirical measures of successive pairs of states. The measure-theoretic pressure of a stationary transition pair is the sum of its kernel entropy and the energy of the joint distribution of two successive states. For forward expansive correspondences satisfying an ergodic-abundance condition, we prove a variational principle and the existence of nonlinear equilibrium pairs. We further prove that limits of the two-coordinate projections of asymptotically maximizing Gibbs ensembles are averages of the two-coordinate distributions of equilibrium pairs. When the energy depends on finitely many potentials on the space of admissible pairs, Legendre regularity assumptions allow us to identify nonlinear equilibrium pairs among linear equilibrium pairs through a self-consistency equation and a global maximization condition. Under these assumptions, real analyticity in one dimension and uniqueness of linear equilibrium pairs imply finiteness of the nonlinear equilibrium set. Finite-state examples exhibit continuous and first-order transitions, metastability, and phases that have identical one-coordinate marginals but distinct two-coordinate distributions. A criterion based on Taylor expansions determines the associated critical exponents

Complex-valued neural networks (CVNNs) are increasingly adopted for complex-valued data; however, they are often trained with first-order optimizers inherited from the real-valued case. The efficiency of these methods depends largely on the step size, and their step-size rules ignore the angular information available in the complex plane. We address step-size adaptation in the complex domain by introducing AURA (Angular Update Rate Adaptation), a per-parameter step-size adaptation that can be added on top of any first-order optimizer, and removed from it, without altering its update direction. AURA measures the agreement between consecutive updates of each complex parameter, in length, alignment, and sense of rotation, and enlarges the step when they are consistent and reduces it when they are not. It requires no additional gradient evaluations and only inexpensive vector operations per step. We combine AURA with Adam and Muon and compare the resulting methods with well-known first-order optimizers on four test cases of increasing complexity, ranging from the approximation of scalar complex functions to physics-informed training. Fully connected neural networks are used throughout this work. All hyperparameters other than the step size are held fixed across test cases; for one case, we also tune the hyperparameters of each optimizer under the same budget. Our empirical tests show that AURA improves the convergence of its base optimizer in most cases with a small per-step overhead, and we identify the conditions under which it fails to do so.

We propose a thresholding scheme for the half-harmonic map heat flow from the flat torus $\mathbb{T}^d$ into the unit sphere $\mathbb{S}^m$ in $\mathbb{R}^{m+1}$, whose iterates are given by convolution with the Poisson kernel, replacing the Gaussian kernel of the classical Merriman-Bence-Osher algorithm, and pointwise normalization. We prove that the piecewise constant interpolants subconverge to a weak solution of the half-harmonic maps heat flow that attains the initial map strongly in the energy space and satisfies the energy-dissipation inequality. For a spectrally truncated variant, convergence holds under a condition on the step size. A key ingredient is the derivation of the precise energy-dissipation identity directly from a refined within-step estimate, without resorting to De Giorgi interpolation typically used in the minimizing movement framework. This provides a direct way to recover the appropriate energy-dissipation property in the limit and, in turn, enables weak-strong uniqueness: every energy-dissipating weak solution coincides with the strong solution whenever the latter exists.

In this paper, it is shown that in spacetimes of maximal proper acceleration, there is a reduction of the mean lifetime of the muon. We consider several candidates for the maximal proper acceleration. For all of them, the induced corrections are small, but for the case when the maximal proper acceleration is identified with the Schwinger's acceleration, the effect is detectable in future muon collider facilities.

We study SPDEs associated with $2$D active scalars driven by incompressible transport noise of Kraichnan type, with regularity exponent $\alpha\in (0,1)$; our examples include the Euler, SQG and IPM systems. We investigate whether a number of “turbulent” phenomenologies, which are well understood in the linear Kraichnan model, persist in this nonlinear setting, uniformly in vanishing viscosity approximations. First, for suitable values of $\alpha$ and initial data in $L^p_x$, we establish anomalous regularization estimates, measured in appropriate endpoint Besov-type spaces of regularity $\beta=\beta(\alpha,p)>0$. Remarkably, these results allow for some scaling supercritical regimes of the parameters $\alpha,p$; on the other hand, for (sub)critical parameters, we recover the same regularity exponent $\beta=1-\alpha$ as in the linear case. Second, in the (sub)critical case, we further prove anomalous integrability, namely solutions becoming instantaneously $L^\infty_x$-valued at positive times, uniformly in the viscosity; moreover, in this case we establish strong existence and pathwise uniqueness of solutions to the inviscid SPDE, which are recovered as the unique vanishing viscosity limit. Finally, in the $2$D Euler case, for $\alpha\in (0,1/2)$, we establish anomalous dissipation of enstrophy and sharpness of anomalous regularization.

In this paper, we systematically exhibit the roles of logistic source on global boundedness and convergence of classical solutions, and global existence and eventual smoothness as well as convergence of generalized solutions for a singular chemotaxis urban crime model. Roughly speaking, this work shows that suitably (can be precise) strong logistic damping ensures boundedness and convergence of classical solutions, and weak logistic damping ensures global existence of generalized solution and its eventual smoothness as well as convergence in two and three dimensions. The former in particular implies that strong logistic damping can prevent the formation of crime hotspots (spatial patterns). Previous known results in this respect are thus significantly improved and extended.

We establish a structure theorem for measures satisfying left-invariant PDE constraints in Carnot groups. Our generalization, that takes inspiration from the paper arxiv:1601.06543 by the first named author and F. Rindler, requires the introduction of the hypoelliptic wave cone, a sub-Riemannian version of the wave cone from the theory of Compensated Compactness, and it employs classical results in Harmonic Analysis on homogeneous spaces. Then, we utilize this result to extend the Rank-one Theorem for functions of bounded horizontal variation ($BV_H$) to all Carnot groups. This is achieved by considering a suitable curl-type left-invariant operator, that provides necessary differential constraints for horizontal gradients, and by studying the hypoellipticity of said constraints.

We introduce a Green–potential heat flow for maps from a closed Riemannian surface. Inspired by Parker–Wolfson renormalization~\cite{ParkerWolfson1993,Parker1996}, the conformal factor is reconstructed at each time from an energy-based source determined by the current map. The resulting parabolic–elliptic system incorporates renormalization into the evolution while leaving the Dirichlet energy functional and its harmonic critical points unchanged. We establish local well-posedness for $H^3$ initial data and a continuation criterion in terms of energy concentration. For sufficiently large coupling, we exclude finite-time bubbling and obtain global solutions. At infinite time, the energy identity provides only weighted $L^2$ control of the tension field. Localizing to regular regions of concentration charts yields harmonic profiles along subsequences of prescribed times. We represent these profiles in the flow's packet charts, organize them into a finite bubble tree, and decompose the limiting energy into harmonic component energies and neck energies. A critically normalized radial example illustrates how the modified flow can prevent finite-time blow-up and produce harmonic bubbles with no-neck property.