Papers

We prove that for a connected, geometrically unibranch variety $X$ over a perfect field of positive characteristic and a dense open subset $U \subset X$, the canonical homomorphism between the Tannaka duals of the categories of overconvergent isocrystals is a quotient map. We also show that the analogous statement for convergent isocrystals, and for all isocrystals, fails. The counterexample is based on Crew's construction of a unit-root $F$-isocrystal that is convergent but not overconvergent.

This paper relates algebraic K-theory of fields to polylogarithms via general linear groups. We focus on the case of number fields and prove that the motivic realisation map from the Goncharov Lie coalgebra to the motivic Lie coalgebra is an isomorphism. This implies the Goncharov universality conjecture and a structural result for special values of Dedekind zeta functions. We also construct explicit polylogarithmic cocycles representing nonzero multiples of the Borel classes.

We study Eratosthenes sieve as a discrete dynamic system. At each stage of the sieve there is a cycle of gaps ${\mathcal G}(p^\#)$ of length $\phi(p^\#)$ and span $p^\#$. There is a recursion ${\mathcal G}(p_k^\#)\longrightarrow {\mathcal G}(p_{k+1}^\#)$ that creates the next cycle from the current one. If we take initial conditions from the cycle ${\mathcal G}(p_0^\#)$, then for all constellations of span $|s| < 2p_1$, including gaps $g < 2p_1$, the driving terms of various lengths form Markov chains. These yield exact models for the populations $n_s(p_k^\#)$ for all further stages of the sieve. If $s$ is an admissible constellation of length $J$, then its population $n_{s,J}(p^\#)$ grows as $\Theta \left( \prod (q-J-1)\right)$. So we factor out the superexponential growth to obtain the exact model for the relative population $w_s(p_k^\#)$ of the constellation $s$ across all further stages of the sieve. $$ w_{s,J}(p_k^\#) \; = \; n_{s,J}(p_k^\#) \, / \, \prod_{J+1 < p \le p_k} (p-J-1) $$ The asymptotic value of the relative population is a constant ${w_{s,J}(\infty) \ge 1}$ that depends only on the odd prime factors that divide a span in $s$. Assuming that the instances of a constellation $s$ are approximately uniformly distributed in ${\mathcal G}(p_k^\#)$, we develop initial estimates of the number of instances $s$ that would occur in the interval of survival $\Delta H(p_k) = (p_k^2, p_{k+1}^2)$. We define a statistic $\eta_s(p_k)$, the quadratic density of the constellation $s$ over the interval $\Delta H(p_k)$. We show that the initial estimates $\widehat{\eta_g}(p)$ for prime gaps agree with samples up to $5.677\,E14$.

We consider a non-interacting Bose gas governed by a general one-particle Hamiltonian in the canonical ensemble. Our main results are large-deviation estimates for the number of particles in the Bose–Einstein condensate. We show that the left- and right-tail probabilities decay at different exponential rates as the number of particles tends to infinity.

We prove a Hall $\sigma$-subgroup analogue of the McKay conjecture for $\pi$-separable groups proposed by G. Navarro. This result simultaneously generalizes the classical McKay conjecture and its $\pi$-separable version. More precisely, if $G$ is $\pi$-separable, $p$ is a prime and $\sigma=\pi\cup\{p\}$, then it is known that there exists $H$ a Hall $\sigma$-subgroup of $G$. In this case, we prove that $|{\rm Irr}_{\sigma'}(G)|=|{\rm Irr}_{\sigma'}(\mathbf{N}_{G}(H))|$.

We give a short complex-analytic proof of a square-restricted form of local stable phase retrieval at the Gaussian in one-dimensional Fock space. The main estimate is a coercivity inequality for the map $F\mapsto F^2$: \[ \|F^2-F(0)^2|_{\mathcal{F}^2(\mathbb{C})} \lesssim \inf_{c\in\mathbb{R}}\||F|^2-c\|_{L^2(d \gamma)}. \] The proof uses a weighted derivative norm, two integrations by parts, and a weighted Cauchy inequality. The proof has been completely verified in Lean with the aid of Large Language Models.

We study approximate representations of locally compact quantum groups and prove stability results in the sense of Ulam in this context. Our main result is that compact and amenable discrete quantum groups are representation stable. We also show that an analogous stability result holds for unitary compressions of general amenable locally compact quantum groups without the assumption of compactness or discreteness.

For a fixed integer $n\ge 2,$ we consider the homogeneous polynomial $$ P(x_1, x_2, \ldots, x_{n+2})=\sum_{i=1}^{n} (x_2-x_1)^{i-1} x_1^{n-i} x_{i+2}. $$ We prove that, for any finite set $A$ of complex numbers, $$ \Bigl|\bigl\{P(x_1,x_2,\ldots,x_{n+2}): \, x_i\in A\bigr\}\Bigr|\gg |A|^{n}. $$ The implicit constant in $\gg$ may depend only on $n.$

Deoxyribonucleic acid (DNA) remains stable for millennia without degradation, can store information at densities orders of magnitude higher than current technologies, and is environmentally friendly due to its low energy requirement. While these advantages make it a promising candidate for storing archival data, DNA storage systems are constrained by high costs, particularly those associated with DNA synthesis. In this paper, we present a comprehensive cost model for calculating the cost of archiving data using various storage systems. Leveraging our model, we conduct a cost analysis of DNA storage versus leading magnetic tape and cloud-based archival storage services. Furthermore, this model can be customised with various parameters to reflect future cost declines and other cost-relevant trends. Our results indicate that, under current cost trajectories, DNA storage costs must fall by eight to nine orders of magnitude to become economically competitive with today's data archival technologies. Moreover, we explore multiple "what-if" scenarios over the coming decades, quantifying the rate of cost decline required to close the competitiveness gap relative to tape and cloud archives. Our findings underscore the critical need for accelerated innovation and investment in DNA synthesis technologies to reduce their cost and transform DNA storage into a practical archival solution.

Many variational principles for acoustics have appeared in the literature, often introduced in an ad hoc manner by picking and choosing terms in a Lagrangian density to yield a desired governing partial differential equation. In this chapter, we show that such guesswork is unnecessary, as the governing equations of acoustics can be derived systematically from the primitive Lagrangians of classical continuum mechanics. Furthermore, we note that using Eulerian coordinates is not the only way to derive variational principles in acoustics. To this end, we review the construction of an action in the Lagrangian frame for compressible nondissipative fluids. We explore the consequences of material relabeling symmetry and the associated pseudomomentum balance, together with the boundary and jump conditions it implies. Via perturbation expansions, we show how to recover the equations of linear acoustics from each variational principle, the material frame yielding in addition the second-order acoustic energy balance. We show how weakly nonlinear approximations with variational structure emerge in each frame, in both cases for a general barotropic fluid, with the nonlinearity carried by the fluid's parameter $B/A$ and the perfect-gas results recovered as a special case.

The Traveling Salesman Problem (TSP) asks for the shortest route to visit a set of cities exactly once. It combines a simple local rule - each city must be visited once - with a hard, global constraint- all cities must be traversed within a single cycle. We cast the problem within a probabilistic, information-theoretic framework. The coexistence of local and global constraints is precisely what makes the problem difficult to address in this framework: the local rule can be enforced through vertex-level constraints, whereas the global constraint cannot be captured by independent edge probabilities. We show that this obstacle can be overcome by defining a maximum-entropy probability distribution over graphs, in which edge costs and degree constraints generate an assignment-like ensemble, and a global term, describing the hard constraint, tilts this ensemble toward Hamiltonian cycles. To make the construction tractable, we derive a mean-field approximation in terms of edge occupancies and implement a differentiable cycle penalty that suppresses sub-tours. This leads to a self-consistent numerical procedure whose output is not only a candidate tour but also a probability matrix encoding competing edges and degenerate solutions. We test the method on synthetic ensembles and on TSPLIB instances. The algorithm converges to connected tours in polynomial time, matching the best-known solution in the majority of instances and remaining within a small relative gap otherwise. Beyond its competitive performance, the proposed framework offers a general approach for handling hard constraints while reducing hard combinatorial optimization problems to simpler ones.

This paper develops directional necessary optimality conditions for constrained optimization problems in Asplund spaces. Under directional metric subregularity, we first derive a directional optimality condition in terms of limiting subdifferentials. We then introduce sufficient conditions for directional metric subregularity and establish their relationships with directional pseudo-normality and quasi-normality. For systems with joint constraints, we further propose a joint criterion for directional metric subregularity and use it to obtain necessary optimality conditions. The results not only extend several finite-dimensional directional constructions to an Asplund-space setting, but also yield conclusions that are new even in finite dimensions.

Mixed-mode oscillations (MMOs), characterized by the alternation of small-amplitude oscillations (SAOs) and large-amplitude oscillations (LAOs), and bursting oscillations are common forms of complex oscillatory dynamics observed in systems with multiple timescales and have been widely studied across scientific disciplines. Mixed-mode bursting oscillations (MMBOs) combine features of MMOs and bursting, with LAOs organized into burst events. Most existing studies treat MMBOs as two-timescale phenomena, identifying distinct geometric mechanisms depending on how the timescales are grouped. In this work, we use a three-timescale biophysical cortical neuronal oscillator model to demonstrate that a three-timescale implementation of geometric singular perturbation theory (GSPT) provides stronger predictive insight into MMBO dynamics. This perspective unifies mechanisms previously identified from fast-slow analysis, while revealing the canard-delayed-Hopf (CDH) singularity as an organizing center for MMBOs near the singular limit. We perform a detailed bifurcation analysis of the full eight-dimensional model to determine how MMBOs are organized along families of isolas. We then combine GSPT with full-system bifurcation analysis to show how tuning the relative timescales induces transitions among MMOs, MMBOs, and bursting dynamics. Our results highlight the importance of studying MMBOs from a three-timescale perspective and demonstrate that combining GSPT with full-system bifurcation analysis can reveal organizing structures and mechanisms that are not apparent from either approach alone.

We show a birthday paradox for random non-backtracking walk on regular graphs of degree at least $3$: such a walk of length $k$ has high probability of self-intersecting when $k$ is significantly greater than $\sqrt n$, where $n$ is the number of vertices of the graph. This resolves a conjecture of Noga Alon and Yuval Peres for the fixed degree case.

Steering-based planners require solutions to state-to-state boundary value problems, which can be inaccessible for nonlinear platforms. Forward propagation evades the steering requirement, but the finite-sample behavior of the associated planners remains uncharacterized and their implementations underperform in practice. This paper develops a propagation-based kinodynamic planner with deterministic finite-sample near-optimality guarantees. We work within the large class of differentially flat nonlinear systems and show that a forward tree of locally dispersive control commands contains a near-optimal trajectory at a certified tree size. We provide a general mechanism to construct dispersive command sets for control-affine systems, which are necessary to implement the search algorithm prescribed by the theory. We show that covering the certified trajectory class irrespective of cost provably demands a tree exponentially sized in the problem horizon, and present a cost-conditioned dominance pruning procedure that retains near-optimality at a tree size polynomial in the horizon. We implement the resulting search algorithm, Dispersive Forward Tree search (DFT*), as breadth-first expansion of the forward tree, which maps naturally onto parallel hardware. We design efficient dispersive samplers for the unicycle, the trailer car, and the quadrotor and evaluate challenging planning tasks for these platforms. DFT* delivers consistently competitive and often substantially better solution quality than state-of-the-art kinodynamic planners at comparable solution times on embedded-tier processors, accelerating further as parallel compute is scaled. We also implement DFT* in a receding-horizon loop to demonstrate real-time planning in dynamic environments at embedded-tier compute budgets.

Let $M$ be a Weil-Petersson random hyperbolic surface of genus $g$, and let $\Gamma \subset \mathbb{Z}^{2g}$ be a lattice of prime index $q$. We study the distribution of primitive closed geodesics in homology classes mod $\Gamma$ in the large genus limit. Averaging over all lattices of index $q$, with $q \to \infty$, we compute all the centered moments of the corresponding weighted counting functions, and exhibit a transition between Poisson and Gaussian regimes (depending on whether $\frac{X}{q\log X}$, the expected number of primitive geodesics in a given homology class mod $\Gamma$, tends to $\lambda>0$ or $\infty$). We also study the unnormalized variance $G_M(X,\Gamma)$ of the counts among homology classes, and show that as $X \to \infty$, averaged over all lattices of prime index $q$, it is asymptotic to $X\log X$ in the large genus limit. These results are analogous to phenomena arising in the distribution of primes in arithmetic progressions.

We develop a general framework for instantaneous arithmetic computation using chemical reaction networks. Numerical values are encoded by ratios of species concentrations: an extended nonnegative value $a\in[0,+\infty]$ is represented by a computational pair $(A_0,A_1)$ through the ratio $a=a_1/a_0$. For this ratio encoding, our main theoretical result is a feedforward compositionality theorem: if a collection of modules satisfies two easily checked structural conditions, then any admissible finite feedforward composition computes the corresponding arithmetic expression instantaneously, meaning that every intermediate and output ratio is correct at every positive time. We make this framework concrete by constructing reaction network modules for four elementary operations—identification, inversion, multiplication, and addition—and verifying that each satisfies these structural conditions. As applications, we show that truncated power series and matrix products can be computed instantaneously using reaction networks whose size reflects the number of arithmetic operations in the underlying computation. We also extend the nonnegative ratio encoding to signed quantities by representing a real number as the difference of two nonnegative ratio-encoded values; this allows addition and multiplication over real-valued inputs. We discuss the practical implications of instantaneous computation, including its relationship to other resource constraints such as the number of chemical species required and the cost of reading out the final answer, which requires division and, in the signed case, rectified subtraction.

Flooding can damage healthcare facilities, interrupt local care capacity, and trigger costly patient evacuations. Addressing these risks requires long-term resilience investments under uncertainty. We develop a bi-objective two-stage stochastic optimization model that jointly determines permanent facility hardening and scenario-dependent evacuation decisions. The first objective minimizes expected evacuation, physical damage, and business-interruption costs. The second minimizes a service-disruption impact index that combines the duration and scale of service loss with a place-based social-vulnerability weight. We develop an exact Benders decomposition and a scalable Lagrangian-dual method with tailored primal recovery, embedding both within an adaptive procedure for constructing informative Pareto frontiers. A case study of 3,752 hospitals and nursing homes in Texas evaluates the framework under climate-informed tropical-cyclone flood scenarios. The results show that minimizing economic losses alone can systematically allocate less protection to facilities located in socially vulnerable areas, while moderate movement along the Pareto frontier can substantially reduce disruption impacts at limited additional expected cost. The exact decomposition solves all tested instances, including larger stress tests for which the extensive formulation becomes memory-limited, while the Lagrangian method provides substantially faster high-quality solutions.

Physics-Informed Neural Networks (PINNs) have recently gained considerable attention as a mesh-free framework for solving partial differential equations. Nevertheless, their performance deteriorates when applied to strongly coupled multiphysics systems, such as Biot's consolidation model, due to severely ill-conditioned optimization landscapes. In this work, we propose a robust PINN-based solver, termed FS-ENGD-PINN, which synergistically integrates physics-based decoupling with geometry-aware optimization. Specifically, the Fixed-Stress (FS) splitting scheme is employed to decompose the coupled poroelastic system into contractive mechanics and flow subproblems, thereby significantly improving training stability and convergence. To further accelerate optimization, we adopt Energy Natural Gradient Descent (ENGD), which approximates the Newton direction in function space effectively mitigates stiffness-induced slow convergence. Moreover, to address volumetric locking arising in the nearly incompressible regime, we incorporate a three-field mixed formulation with an additional total pressure variable into the PINN framework. Extensive numerical experiments demonstrate that the proposed FS-ENGD-PINN consistently outperforms standard PINN formulations in terms of accuracy and robustness, providing a unified and reliable learning-based solver for poroelasticity across a wide range of material parameters.

Motivated by the principle that positivity of the Beauville–Bogomolov–Fujiki (BBF) form controls the geometry of hyperkähler manifolds, we study BBF positivity from the perspective of bimeromorphic geometry, focusing on its relation to the Moishezonness of hyperfujiki manifolds (natural bimeromorphic analogues of hyperkähler manifolds). We prove that any Moishezon hyperfujiki manifold carries a big line bundle with positive BBF square. We also prove that the existence of a BBF-positive integral $(1,1)$-class implies Moishezonness for hyperfujiki $4$-folds and, more generally, for hyperfujiki $2n$-folds under a weak Kähler minimal-model condition. The proof mainly uses the theory of primitive symplectic varieties developed by B. Bakker–C. Lehn, a construction of small bimeromorphic models for certain $K$-trivial manifolds by I. Biswas–J. Cao–S. Dumitrescu–H. Guenancia, and the theory of pull-backs of reflexive differential forms established by S. Kebekus–C. Schnell. As an application, we give a Hodge-theoretic description of the Moishezon locus in certain smooth families by combining this criterion with the period theory developed by B. Anthes–A. Cattaneo–S. Rollenske–A. Tomassini.

Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect. It is known that if delta(X) is at least 4, then X is isomorphic to a product SxT where S is a surface, whereas if delta(X)=3, X admits a del Pezzo fibration with a prescribed structure. In this paper, we study the case where delta(X)=2. Our first result is that there exists a sequence of flips X–>X' such that X' is smooth and admits a conic bundle structure X'->Y, with rho(X)-rho(Y)=2. The conic bundle factors as X'->X”->Y, where X'->X” is a smooth blow-up, and f: X”->Y is again a conic bundle. Then we show that either f is smooth, or there exists a del Pezzo fibration g: X->T with rho(X)-rho(T)=3 and T smooth, admitting two possible generic fibers, and we describe the relative cone NE(g). Finally for one of the possible generic fibers we give a structure theorem for the del Pezzo fibration g; we show that T is Fano and that X can be reconstructed from T and a suitable divisor class on T. As an application, we construct a family of Fano 4-folds X exhibiting this behaviour.

Let \(D\in\{\C,\Hh,\Oct\}\) be a real composition division algebra of dimension \(d\in\{2,4,8\}\). We construct an explicit rank-\(d\) real vector bundle $ E_D\longrightarrow\RP^d$ by deleting one homogeneous coordinate on each standard affine chart, interpreting the remaining coordinates as an element of \(D\), and using the corresponding left-multiplication matrices as transition data. The bundle admits a matrix-determined trivialization away from the \(d+1\) coordinate points. Relative to this trivialization, each deleted point has local clutching map $S^{d-1}\longrightarrow\SO(d),\; u\longmapsto L_u$, which is respectively the complex, quaternionic, or octonionic Hopf clutching map. A section arising from the same matrices has exactly the coordinate points as nondegenerate zeros. Consequently, $w(E_D)=1+x^d$, and real \(K\)-theory together with Euler-class cancellation gives $E_D\cong\gamma^{\oplus d}$. Thus the construction supplies explicit framed-defect realizations of these familiar bundles. In particular, it gives a semialgebraic octonionic realization of \(\gamma^{\oplus8}\) on \(\RP^8\) with nine specified local Hopf defects. We also describe the associated Pfister quadratic bundle over arbitrary fields.

We study a thermodynamically consistent non-isothermal phase-field system coupling two order parameters and the inverse temperature through a fully non-diagonal Onsager mobility. The model describes the interaction of mass diffusion, heat conduction, and local phase relaxation while conserving mass and internal energy and producing entropy. Global generalised dissipative weak solutions are constructed using a fully discrete approximation. The discrete scheme conserves mass and internal energy and satisfies a discrete entropy inequality. An availability estimate, together with a dimension-adapted barrier, yields strict positivity at fixed mesh and uniform estimates. Compactness then allow passage to the continuous system. Concentration of the singular part of the internal energy is represented by a non-negative defect measure. The entropy-availability structure yields finite dissipation on the infinite time interval and the existence of stationary $\omega$-limit states. Finally, a relative-entropy argument establishes weak-strong uniqueness whenever the weak and strong solutions remain in a bounded thermodynamic state range.

We provide an elementary proof of the key aspect of Hopf's curvatura integra theorem. Namely, we show that the total curvature, i.e. the surface integral of the Gauss-Kronecker curvature B, is independent of the shape of the hypersurface. We accomplish this task by using the Calculus of Moving Surfaces to show that the rate of change of the total curvature vanishes under smooth changes in shape.

Muon has emerged as a strong optimizer for the matrix-valued parameters in large language model pretraining, approximately orthogonalizing its momentum with a few Newton-Schulz iterations. Existing theory either replaces this iteration with the exact polar factor it approximates, or treats its finite depth as an approximation error, and thus the iteration Muon actually runs can only hurt the guarantees. We show that finite Newton-Schulz can instead be beneficial for nonsmooth nonconvex optimization. To this end, we analyze Muon through the online-to-nonconvex conversion, which views the update rule as an online learner and converts its regret bound into a stationarity guarantee. The finite Newton-Schulz iteration smooths the discontinuous polar map into a Lipschitz map of the singular values, and Muon with finite Newton-Schulz can be regarded as an online learner with a smoothed spectral potential. This smoothing is exactly what the conversion needs: we prove that a Newton-Schulz depth growing only logarithmically in the target accuracy suffices for convergence to stationary points in nonsmooth nonconvex optimization, whereas Muon with the exact-polar update may fail to converge. The resulting sample complexity bounds match the best-known guarantees for nonsmooth nonconvex optimization and are optimal for smooth nonconvex optimization up to problem-dependent factors. The argument extends beyond Newton-Schulz to general spectral maps with the same smoothing property.

This paper deals with the computation of the sum of alternating series, whose general terms can be expressed by means of analytic functions. After rewriting the series as a weighted integral with the Abel weight, we employ the sinc (trapezoidal) rule and analyze the remainder term with respect to the number of quadrature points. We provide some numerical experiments to show the reliability of the derived error estimate and to test the algorithm for automatic summation with prescribed accuracy. All the Matlab codes used in the present paper can be found as open-source software at the authors' homepage.

We provide an overview of the problem of jointly testing two hypotheses and estimating a parameter of the selected model. Such problems arise in a variety of applications. First, we present a conceptual introduction to suboptimal and optimal procedures for joint detection and estimation. A numerical example illustrates the advantages of the optimal procedure over suboptimal ones. Next, we discuss how more advanced problem formulations affect the presented results. The second part covers joint detection and estimation in a sequential framework. First, we provide an introduction to sequential analysis through sequential hypothesis testing. Then, suboptimal and optimal sequential procedures for joint detection and estimation are discussed. A numerical example shows the advantages of optimal sequential procedures over suboptimal and optimal sequential procedures with a fixed number of samples. The third part discusses open problems and future research directions in joint detection and estimation.

We develop a nonperturbative, group-theoretic quantization of the effective guiding center theory of a charged particle drifting in a magnetic field in two spatial dimensions, and compare the resulting quantum theory with a corresponding coarse graining of the underlying microscopic theory. In the classical effective theory, the small gyro motion is not resolved, while the motion of the center of the gyro orbit remains observable. The reduced phase space is the physical space itself, so quantization leads to noncommuting spatial coordinates, and the effective theory loses access to the metric structure of physical space, retaining only its area structure. By formulating a prescription to match between the microscopic and effective quantum theories, we find that the predictions of the quantized effective theory are generally consistent with those of the microscopic theory. However, for closed isomagnetic contours the effective theory predicts a quantization of “radius”, a spatial discreteness absent from the microscopic theory. This illustrates that quantization of an effective theory may yield spurious nonperturbative predictions, even if that quantum theory shows no internal signs of breakdown.

The threshold theorem states that quantum computations can be made reliable below a physical error threshold, at the cost of additional physical qubits and circuit depth. Recent work has reduced these space and time overheads to polylogarithmic or nearly logarithmic scalings, but whether the cumulative spacetime overhead can be constant has remained unclear. Here, we show that even for the simplest task of preserving quantum information in a quantum memory, under an optimistic noise model and allowing general adaptive protocols, there is an unavoidable logarithmic contribution to the cumulative spacetime overhead. This additional cost can nevertheless be shared among many logical qubits, so sufficiently wide computations, including standard implementations of Shor's algorithm, may still achieve constant relative overhead. We further give a positive-rate CSS code construction that attains the memory bound, identify sufficient conditions under which the same scaling extends from quantum memory to fault-tolerant circuit implementations, and derive circuit-size bounds for subsystem spacetime codes. Our work establishes fundamental limits on the resources required for quantum fault tolerance.

The ${\cal M}$-theory dual of thermal QCD-like theories constructed in arXiv:1306.4339, 2004.07259 involves, as shown in arXiv:2211.13186, a closed $G_2$-structure seven-fold $\mathcal{M}_7$ supporting Contact 3-Structures (C3S) that induce a “transverse” $SU(3)$-structure. We prove that this induced structure belongs to the symplectic nearly half-flat (S(nearly)HF) class: $d\omega_\Phi=0$ exactly, while $d\Omega_+$ is suppressed as $\mathcal{O}(e^{-N^{1/3}/\mathcal{O}(1)})$ in the infrared "MQGP limit", leaving $W_\Phi^{SU(3)}=W_2^-$ as the sole non-vanishing Chiossi–Salamon torsion class and completing the $G$-structure classification of the MQGP dual initiated in arXiv:1507.02692,2004.07259, 2211.13186. As a physical application, Kaluza–Klein reduction of the eleven-dimensional Hodge dual of the differential of the $G_2$-structure positive three-form on $\mathcal{M}_7$, organised through the $G_2$/S(nearly)HF torsion data, yields a radial “magnetic field” $\mathcal{B}_{\rm KK}(r)$ whose squared norm factorises exactly into external and internal factors. $\mathcal{B}_{\rm KK}(r)$ satisfies a Bessel-type ODE with a regular singular point at the horizon; normalisability and infrared regularity select $J_2$, with harmonicity of the aforementioned internal factor recovered from the large-$N$ suppression. $\mathcal{B}_{\rm KK}$ is related to the magnetic field of the type~IIA flavour $D6$-branes via a non-linear power law, furnishing a top-down holographic mechanism for generating weak magnetic fields in finite-temperature string theory.