Papers

We present a numerical study of the ultrametric properties of the set of RNA secondary structures with the maximum number of base pairs (energetically degenerate minima) within the maximum matching model (Nussinov algorithm). Using 18 natural small nuclear RNAs as examples, we show that the degree of nontrivial ultrametricity varies widely. We consider the optimization problem for the degree of nontrivial ultrametricity of RNA secondary structures with the maximum number of base pairs under a fixed nucleotide composition. It is found that permuting the nucleotide sequence can strongly change the degree of ultrametricity, indicating the key role of nucleotide order in shaping the hierarchical properties of RNA secondary structures.

Statistical procedures rarely retain all features of the observed data. A sufficient statistic removes information irrelevant to a parameter; a maximum likelihood estimate compresses an empirical objective into an optimizing point; and a hidden state in a sequential model compresses past observations into a learned representation. This article develops these practices under the unified notion of target-oriented statistical compression: a useful summary preserves what matters for an inferential, predictive, or decision-relevant target, rather than every detail of the realized data path. The central object is the conditional target process \(M_n=\E(Z\given\G_n)\), where \(Z\) is the target and \(\G_n=\sigma(T_n)\) is the information retained by the compression map \(T_n\). When \((\G_n)\) is a decreasing filtration, \((M_n)\) is a reverse martingale with limit \(M_\infty=\E(Z\given\G_\infty)\). Exact sufficiency corresponds to lossless compression, while approximate summaries such as penalized estimators, principal components, and neural-network hidden states produce reverse quasi-martingale defects measuring coherence loss across compression levels. The diagnostic \(r_n=|M_n-M_{n-1}|\) is treated as an observable stability proxy, not as an unbiased estimator of the theoretical defect. Boundary degeneracy in sequential binary problems is developed as a central application. Practical boundary claims require joint assessment of boundary closeness, uncertainty control, and trajectory stability. The companion paper \citet{chang2025rm} develops the corresponding stopping procedures, finite-sample bounds, and numerical evidence; the present paper provides the broader theoretical infrastructure and extends the framework to Gaussian, Poisson, and quasi-martingale monitoring problems.

Conforming mapped Lagrange finite elements of any fixed degree are analysed for the Poisson problem on exact curved triangulations in two dimensions. The element map is factorised into an anisotropic affine core map and a smooth curved correction. Assuming compatible edge parametrisations, semi-regular affine cores and uniform bounds on the curved corrections, we obtain anisotropic $L^2$- and $H^1$-interpolation estimates with constants independent of the aspect ratios of the cores. The estimates retain lower-order terms induced by the non-affine maps, and these terms cannot be omitted in general. For sufficiently regular solutions, the Galerkin errors on the exact domain converge with the optimal orders in the energy norm and, on convex domains, in the $L^2$ norm. A fixed smooth diffeomorphism maps a semi-regular family of polygonal triangulations to exact curved triangulations satisfying these assumptions, even when the core aspect ratios are unbounded. An explicit family with an exact circular boundary arc is given. Numerical experiments with linear and quadratic elements confirm the predicted rates. At a fixed horizontal mesh width, the errors remain bounded as the aspect ratio increases.

In this paper, we study maximal Cohen-Macaulay sheaves on closures of minimal nilpotent orbits in simple Lie algebras. For singularities of type $A_n$, we first classify vector bundles on their symplectic resolutions whose pushforwards are maximal Cohen-Macaulay. We then construct equivariant maximal Cohen-Macaulay sheaves via irreducible representations of the stabilizer group. We compare these two approaches in the case of maximal Cohen-Macaulay Weil divisors, and extend the equivariant construction to the classical types $B_n$, $C_n$, and $D_n$. Finally, we formulate the construction for an arbitrary simple Lie algebra and carry it out explicitly in the exceptional cases.

We use the connective formal group law to define a one-parameter ($\beta$-)deformation of the motivic Segre classes of Schubert cells in the $d$-step flag variety. This $\beta$-deformation specializes to the motivic Segre classes of Schubert cells when $\beta=1$ and to the Segre-Schwartz-MacPherson classes of Schubert cells when $\beta=0$ (though the $\beta$-deformations are not simply the $\beta$-homogenizations of the motivic Segre classes). We define rational function representatives for the $\beta$-deformed classes in the $d=1$ case in terms of a solvable lattice model, and we prove a combinatorial formula for the structure constants in the $\beta$-deformed basis in the $d=1$ case using Knutson-Tao puzzles. Unexpectedly, the structure constants are built out of entries of $R$-matrices for the multi-parameter quantum group of type $\widehat{a}_2$ (which suggests an action of multi-parameter quantum groups on equivariant connective $K$-rings of Nakajima quiver varieties, with their convolution algebra structures). We show that our $\beta$-deformations arise from canonical elements in a quotient of the torus-equivariant algebraic cobordism ring of the cotangent bundle of the flag variety by proving that the canonical elements satisfy a GKM type condition (in future work, we will show that these canonical elements also lie in the torus-equivariant connective $K$-ring, in the sense of Karpenko–Merkurjev, of this cotangent bundle).

We prove that two fixed univariate functions, namely, an arbitrary continuous non-affine function and a particular affine function, are sufficient to approximate continuous functions of one variable under the operations of addition and composition. The same fixed functions can also be used to approximate multivariate continuous functions, provided that the coordinate functions are also available. We also show that the number of generators can be reduced from two to one. We construct a specific continuous function that generates a dense class in the univariate setting and, together with the coordinate functions, in the multivariate setting.

With the widespread application of location-based services, fingerprint-based localization has demonstrated advantages in environments with complex signal propagation. Deep learning has significantly improved the efficiency of both offline training and online matching in localization processes. However, existing fingerprints only contain terminal position information without capturing motion states, and neural network designs have not fully incorporated structural features such as fingerprint sparsity. In this paper, we propose a triple-beam fingerprint (TBF) incorporating Doppler information and design a Transformer-based localization and orientation awareness network (LOA-Net) to simultaneously estimate user position and motion direction in massive multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. We first show the correlation between TBF and multipath information, and investigate the collinearity of different TBFs, demonstrating that TBF is an effective small-size sparse fingerprint. Then, we propose LOA-Net containing a mask-augmented detection Transformer for regression (MaskDETR-Reg) module and a fusion-enhanced Transformer for direction classification (Fusion-TDC) module to process angle-delay domain information and Doppler domain information, respectively. Finally, in the simulation of indoor scenarios defined in 3GPP 38.901, the proposed method achieves significantly better localization accuracy than weighted $K$-nearest neighbors (WKNN), 2D and 3D convolutional neural networks (CNNs), and achieves satisfactory motion direction estimation accuracy.

Randomized zeroth-order methods are classically analyzed in expectation, but a black-box Markov conversion can give misleading high-probability guarantees, in particular by forcing the finite-difference smoothing radius to shrink with the confidence parameter. This paper gives a direct finite-horizon high-probability analysis of a two-query Gaussian finite-difference method for deterministic objectives with Lipschitz gradients. The method uses the classical two-point estimator together with the normalized stepsize \(\eta_t=1/(4L\norm{\bu_t}^2)\). We prove that it finds an \(\varepsilon\)-suboptimal point with probability at least \(1-\delta\) using \(\cO((dL/\mu)\log(1/\varepsilon)+\log(1/\delta))\) function queries under strong convexity, subject to an explicit finite-difference smoothing-radius condition. We also establish high-probability guarantees for smooth convex objectives under a level-set distance-to-solution radius condition and a pathwise smoothing-radius condition. For lower-bounded smooth non-convex objectives, the trajectory average is certified in stationarity with \(\cO(L\Delta_0(d+\log(1/\delta))/\varepsilon)\) function queries. The proofs combine lower-tail bounds for adaptive sums of Gaussian directional projections with upper-tail bounds for accumulated finite-difference smoothing errors.

Generative models have emerged as a powerful paradigm for solving physics systems and modeling complex spatiotemporal dynamics. However, achieving high physical accuracy without incurring high computational cost remains a fundamental challenge, as existing approaches face a critical speed-fidelity trade-off. In this work, we introduce Recursive Flow Matching (RecFM), a generative framework for forecasting complex spatiotemporal dynamics. RecFM enforces self-consistency to align trajectories across discretization scales, reducing discretization errors and improving performance across metrics for physics-based tasks. To our knowledge, this is the first method to achieve high-fidelity one- and few-step (2-4 step) dynamic generation for scientific systems with performance comparable to state-of-the-art multi-step solvers. Across challenging scientific benchmarks, RecFM achieves up to a 20$\times$ speedup over leading diffusion-based emulators while improving predictive accuracy. Furthermore, RecFM reduces mean squared error by over 15% compared to vanilla flow matching, offering a scalable and efficient solution for real-time scientific emulation.

Gaussian processes (GPs) provide a principled Bayesian framework for uncertainty estimation, but their computational complexity severely limits scalability to large datasets. We propose SIKA-GP, which accelerates GP inference using sparse inducing kernel approximations based on a dyadic ordered template basis, incurring only ${O}(\log M)$ complexity dependence on the number of inducing points. Our approach constructs compact and expressive kernel representations from sparsely activated bases, enabling efficient tensorized GPU computation and seamless integration with modern large-scale models. SIKA-GP can be naturally embedded into Bayesian neural networks (BNNs) with sparse activations, yielding significant speedups in both training and inference without sacrificing predictive performance. The method naturally extends to deep feature learning, addressing the scalability challenges introduced by deep architectures and high-dimensional feature representations. Empirical results on vision and transformer-based language benchmarks demonstrate that our approach consistently delivers fast and accurate GP models, providing a principled path toward scalable kernel learning.

The nonlinear stability of Minkowski spacetime has been one of the central achievements in the mathematical theory of general relativity and, more broadly, in the analysis of nonlinear geometric wave equations. Since the seminal work of Christodoulou-Klainerman, the problem has shaped fundamental advances in our understanding of decay, dispersion, and the intricate interplay between geometry and analysis in the Einstein vacuum equations. This survey presents an overview of the main ideas and techniques underlying the stability theory of Minkowski spacetime. We emphasize the role of decay assumptions, geometric foliations, energy identities, and gauge choices in the global analysis. Particular attention is devoted to exterior stability results, minimal decay regimes, and the borderline case, where the failure of spacetime integrability for nonlinear interactions reveals subtle threshold phenomena. Our goal is to provide a coherent perspective on the evolution of the field, to clarify the structural mechanisms behind the known results, and to outline some of the central open problems that remain in the borderline regime.

In 1994, Frieze and Teng proposed an integer linear programming formulation of the NP-Complete Exact Partition problem, whose LP-relaxation they claimed was non-degenerate. Contrary to their claim, we show how an instance of Exact Partition can produce a degenerate polytope, and study conditions for which this can happen. We then give details of one of the smallest such degenerate Frieze-Teng polytopes, along with a closely related non-degenerate Frieze-Teng polytope that encodes an equivalent problem. We note that for the purposes of the complexity results in the literature that use their formulation, these degenerate polytopes can be avoided via a simple preprocessing step.

Let $(M^3, g, \mathbf{k})$ be a complete asymptotically flat initial data set satisfying the dominant energy condition, and let $m$ denote its ADM mass. The generalized Penrose conjecture asserts that the area of an outermost generalized apparent horizon $N\subset M$ satisfies $|N| \leq 16 \pi m^2$. In this paper, we establish this inequality for each connected component of $N$ in the case where $\mathbf{k}$ is proportional to the metric $g$. Our approach is based on a new geometric evolution, which we call the $\sigma $-inverse mean curvature flow, together with a novel monotonicity formula that may be of independent interest.

The continuity of the injectivity radius of a compact manifold under $C^2$ perturbation of the Riemannian metric was originally proved by P. Ehrlich (Composito Math., 1974), and later the proof was simplified by T. Sakai (Math. J. Okayama Univ., 1983). Using this continuity, jointly with J. Itoh and S. Prasad (J. Math. Anal. Appl., 2025), we proved the Hausdorff stability of the cut locus of a point, when both the point and the metric are perturbed. In the present article, we extend both these results to submanifolds. We first show that the injectivity radius of a submanifold depends continuously on the metric. Then, we obtain the Hausdorff stability of the cut locus of the submanifold, under $C^2$ perturbation of the metric. In fact, we allow the submanifold to be perturbed in the Whitney $C^2$ sense as well.

Annual oil and gas exploration planning involves selecting a limited portfolio of drilling and appraisal-related projects before geological outcomes are known. This decision is affected by uncertainties in geological success, reserve size, and economic value, while also subject to budget, well-count, success-rate, and reserve-reliability requirements. A strategy based only on expected value is therefore insufficient, as early drilling results may change the value of subsequent follow-up opportunities. This study develops a posterior-informed two-stage stochastic multi-objective optimization framework for exploration well selection under uncertainty. The first stage selects a here-and-now portfolio of frontier traps, appraisal projects, and mature appraisal units. After first-stage outcomes are observed, the second stage determines scenario-dependent recourse projects, including follow-up appraisal, reserve upgrading, conversion-to-proved reserves, rolling extension, and data re-evaluation projects. Geological learning is modeled using a logit-scale posterior updating mechanism that links first-stage success or failure to the success probabilities of related recourse projects. The model maximizes expected net present value and minimizes conditional value-at-risk, while imposing chance constraints on drilling success rate and individual and joint reserve targets. To solve the model, sample average approximation is combined with NSGA-II for first-stage portfolio search and a scenario-wise constrained 0-1 optimization procedure for second-stage evaluation. A numerical case study shows that the proposed framework provides an interpretable risk-return frontier and supports adaptive exploration planning under geological learning, downside-risk control, and reserve-reliability requirements.

We revisit congruence zeta functions of smooth projective varieties over finite fields in the framework of Scholze's Berkovich motives. Via this formalism and categorical traces, we construct a new zeta function, and show that it agree with classical one

Let $r\in\mathbb{C}$, let $K$ be a finite extension of $\mathbb{Q}$, let $I_K$ be the monoid of integral ideals in the ring of integers $\mathcal{O}_K$ of $K$, and let $\chi$ be a Dirichlet character. Then define the twisted ideal divisor function $\sigma_{r, K, \chi} : I_K \rightarrow \mathbb{C}$ by $$\sigma_{r,K,\chi}(I) = \sum_{J \mid I} N(J)^{-r}\chi(N(J)),$$ where $N$ denotes the ideal norm. For real $r>1,$ we study the number of connected components $C_{r, K, \chi}$ of the closure $\overline{\sigma_{r,K,\chi}(I_K)}$, writing $C_{r,K}$ when $\chi$ is the principal character modulo 1. We prove that $C_{r,K,\chi}$ is finite when $\chi$ is real-valued. When $K = \mathbb{Q}$, we show that for fixed $r > 1,$ every sufficiently large positive integer is realized as $C_{r,\mathbb{Q},\chi},$ and if $r$ is sufficiently large, then every positive integer is realized as $\chi$ varies. For finite Galois extensions $K$ over $\mathbb{Q}$, we exhibit new exponential lower bounds for $C_{r,K},$ and we prove that for every fixed integer $s \geq 2$, the values $C_{r,K}$ are unbounded as $K$ ranges over degree-$s$ extensions of $\mathbb{Q}$.

In 1980, Paul Erdős posed the following problem: For every positive integer $n,$ determine a nonhamiltonian graph of order $n$ having the maximum number of Hamilton paths. We solve the more general problem of determining the nonhamiltonian graphs of order $n$ having the maximum number of paths of length $k$ for given integers $n$ and $k$ with $1\le k\le n-1.$ The case $k=n-1$ gives a solution to Erdős's problem and the case $k=1$ corresponds to a theorem due to Ore and Bondy.

Let $(\mathcal{E}, \mathbb{E}, \mathfrak{s})$ be an extriangulated category. Motivated by the theory of hereditary algebras, we introduce the notion of a hereditary-type subcategory $\mathcal{W}\subseteq \mathcal{E}$. We prove that the quotient $\mathcal{E}/\mathcal{W}$ is a quasi-abelian category, that is, an additive category with kernels and cokernels in which kernels are stable under pushouts and cokernels are stable under pullbacks. Moreover, we show that $\mathcal{E}/\mathcal{W}$ is abelian if and only if $\mathcal{W}$ is a cluster tilting subcategory in a suitable relative extriangulated structure. Several examples are provided to illustrate the main results, showing that our approach both recovers known abelian hearts and yields new abelian or quasi-abelian quotients beyond classical settings.

Let $f: T\to \{ 0,1 \}$ be a Boolean function on the Boolean half-slice, $T$, \ie elements of $\{0,1\}^n$ with Hamming weight $n/2$. We show that if $f(x)+f(y)=f(x+y)$ holds with probability $\frac{1+\delta}{2}$ over a uniform pair $(x,y)$ such that $x,y,x+y\in T$, then $f$ agrees with some linear function on at least $\frac{1+\delta}{2}-o(1)$ fraction of the points in $T$. More generally, we show that if $f$ passes the natural $k$-query BLR test with probability $\frac{1+\delta}{2}$ for any $k\geq3$, then it must agree with some affine function at $\frac{1+\delta^{\frac{1}{k-2}}}{2}-o(1)$ fraction of the points in $T$. The only other known linearity test for the slice in the low soundness regime (i.e., when $\delta$ can be arbitrarily small) was given by Kalai, Lifshitz, Minzer, and Ziegler [FOCS'24]. Our result improves upon this result in two significant ways: firstly, it works for $k=3$ queries, instead of requiring $k\geq4$; secondly, our result is sharper, e.g., when $k=4$, we are able to conclude an agreement of $\frac{1+\sqrt{\delta}}{2}-o(1)$ instead of $\frac{1+c\sqrt{\delta}}{2}$ for $c\approx.0035$. In particular, our result matches (up to the $o(1)$ term) the conclusion one obtains over the full hypercube via the classical BLR analysis. Our main technical contribution is a new dense model theorem using bounds on Krawtchouk polynomials. Using these Krawtchouk polynomial bounds, we also obtain a simple $k$-query test ($k\geq 5$) that avoids any use of the dense model machinery. This simplified test naturally extends to the slice over the $q$-ary hypercube, giving the first such result over larger alphabets.

We generalize some results of Campana-Păun regarding foliations, slope stability, and positivity of log canonical bundles on smooth projective varieties to the case of smooth proper DM stacks admitting projective coarse moduli spaces. This paper is the third in a series aiming to generalize results of Popa-Schnell and Wei-Wu on Viehweg hyperbolicity to the setting of DM stacks, and in particular, to certain KSBA moduli spaces.

This note studies the 1D stochastic heat equation driven by a one-dimensional Brownian motion. We prove that the associated Markov process satisfies the strong Feller property under mild non-degeneracy conditions. The approach combines Malliavin calculus with the moment method from PDE control theory.

Defect modes localize vibrations within phononic-crystal bandgaps. Prescribing their frequencies alone does not ensure separation from competing modes. We use frequency-dependent selection weights to combine attraction of target-adjacent modes with repulsion of competing in-gap modes. Only the central defect cell is optimized within a previously designed, fixed host bandgap. Cell-level Craig–Bampton reduction retains physical interface coordinates, allowing the host contribution to be assembled once while the defect basis and matrices are updated. Across three numerical cases, the reduced-order method accelerates defect optimization by approximately 13–20 times. A comparison with attraction-only optimization shows that the combined objective moves competing modes away from the target and increases the minimum target-to-non-target frequency separation. Full-order reanalysis shows that the optimized modes remain close to the prescribed frequencies and localized around the defect. A finite-array calculation shows enhanced response near the target relative to the defect-free host under the prescribed loading.

In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new class of problems for these and related additive bases. The problems are designed, in part, to be susceptible to solution by AI.

Many problems in nonlinear and statistical physics are formulated through represented flows, including physical-space vector fields, phase-space drift fields, and truncated renormalization-group beta functions. We introduce a complementary representation-dependent diagnostic for testing whether finite-separation flow geometry is stable across observational resolution. For two separated points, states, or theories, the method compares the direction of the corresponding vector-field increment after the field has been smoothed at two resolutions. Averaging this normalized comparison over sampled separations gives a coherence matrix tied to the chosen variables, coarse graining, metric, and sampling protocol; it is a consistency test, not a coordinate-invariant quantity. We demonstrate the diagnostic in three settings. Synthetic divergence-free fields with identical Fourier amplitudes, spectra, and scalar two-point correlations nevertheless produce distinct coherence matrices, showing that second-order statistics do not determine cross-resolution increment geometry. Lorenz phase-space tests show that a smooth coordinate wrinkling changes represented drift geometry without changing the underlying dynamics, and that a weak model perturbation lowers finite-separation coherence even when local stretching proxies remain closely matched. Finally, for functional renormalization-group flows of the three-dimensional \(O(1)\) scalar theory, projected \(M=4,5,6\) LPA beta fields remain internally coherent, while cross-truncation coherence decreases as higher-order coupling directions are activated. The diagnostic provides a practical field-level check of how representations, models, and truncations preserve finite-separation flow geometry, complementing rather than replacing standard local, spectral, or fixed-point diagnostics.

We give a recognition and reconstruction theorem for finite strictly convex tetrahedra in ${\rm dS}^3$ and ${\rm AdS}^3$ with spacelike, timelike, or null faces. The input consists of four nontrivial based ${\rm SO}^+(1,2)$ holonomies satisfying the closure relation. We construct their Gram data and give a global condition ensuring that the reconstructed tetrahedron lies within the required geometric domain. This condition is strict copositivity of the signed inverse Gram form on the outward branch. Together with nondegeneracy of the Gram data, it guarantees a unique finite tetrahedron up to ambient isometry, with exactly the prescribed Levi–Civita face holonomies. We express this criterion as finitely many polynomial inequalities in trace coordinates, leading to an identification of the finite tetrahedra and a subset of the relative ${\rm SL}(2,\mathbb R)$ character varieties of the four-holed sphere. We also discuss the degenerate Gram strata, vector closure in the curvature-zero limit, and the projective dual tetrahedra, which include ideal and hyperideal tetrahedral sectors. The results provide classical geometric input for quantizing Lorentzian curved tetrahedra and for quantum gravity models with a nonzero cosmological constant.

We produce refined index obstructions, generalizing recently constructed index obstructions due to de Jong and Perry, for topologically trivial Brauer classes on smooth and projective complex varieties. We show that our refined obstructions are more stringent than previous obstructions and, as a consequence, we produce more counterexamples to the integral Hodge conjecture. Throughout this work, we focus on algorithmic aspects of these obstructions and we illustrate many of these aspects through the concrete examples of complex abelian varieties.

We study the fluctuations of the number of real roots of random polynomials with independent, nonzero-mean coefficients. Such non-centered ensembles arise naturally in signal-plus-noise models and in random perturbations of deterministic polynomials. While Ibragimov and Maslova (1971) established the leading asymptotics of the expected number of real roots for non-centered polynomials with i.i.d. coefficients, the corresponding variance asymptotics and central limit theorem have remained open for more than fifty years. This stands in sharp contrast to the centered case, where the fluctuation theory is now well understood across a wide range of ensembles. We resolve this gap by developing novel comparison principles that reduce the fluctuation theory of a non-centered ensemble to that of its centered counterpart. These principles yield sharp variance asymptotics and central limit theorems for broad classes of ensembles, including Kac and hyperbolic polynomials, their derivatives, and related extensions. In particular, for both Kac and hyperbolic polynomials, the leading variance constant equals exactly one-half of that in the centered case, reflecting asymmetric suppression of fluctuations across the two regions where roots concentrate. Our results provide the first comprehensive fluctuation theory for the number of real roots of non-centered random polynomials.

In this thesis we use the Beauville-Bogomolov decomposition to compute the LLV algebra of smooth projective complex varieties admitting a holomorphic symplectic form, generalizing known results from hyperkähler and abelian varieties. Using this explicit computation, we prove for many such varieties that Orlovs conjecture holds, which states that for two derived equivalent smooth projective varieties there exists an isomorphism of rational cohomology preserving the grading and Hodge structure. Moreover, we prove that this conjecture holds for all four-dimensional smooth projective varieties admitting a holomorphic symplectic form.

The topological gap $\Delta$ – the excess $H_1$ total persistence of a critical point cloud over a density-matched null – scales as $\Delta \sim L^{d+\eta}$. We derive this analytically: the spectral integral $I(\alpha) = \sum_{k\neq 0} S_{\mathrm{conn}}(k)\,|k|^{\alpha}$ scales as $L^{2-\alpha-\eta}$ when IR-dominated, giving $I(-2\eta) \sim L^{d+\eta}$. The decomposition $I(-2\eta) = I_0 \cdot I_{\mathrm{shape}}$ separates volume ($I_0 \propto N(1-m^2) \sim L^d$) from anomalous dimension ($I_{\mathrm{shape}} \sim L^{\eta}$); the volume factor accounts for the magnetization-driven per-configuration variance of $\Delta$. We prove the mechanism requires $d < 2 + \eta$ (IR dominance), confining it to $d = 2$ for physical systems; in $d = 3$ the spectral integral is UV-dominated, explaining why density normalization is needed. An $\alpha$-sweep for Potts $q = 4$ at $L = 32$–$256$ finds $\alpha_{\mathrm{opt}}$ in $[-0.75, -0.5]$, consistent with $-2\eta_{\mathrm{Ising}}$ and inconsistent with $-2\eta_{q=4} = -1$; we flag this as tentative pending $L \geq 1024$ confirmation. The $\langle m^2 \cdot I(-2\eta)\rangle$ hyperscaling product is dominated by the correlation $r(m^2, I) \approx -0.98$ via the shared $I_0$ amplitude, so we report it as a covariance-correction analysis. Under a heuristic argument extending Divol–Polonik to inhomogeneous Poisson intensities, the bare PH kernel is flat; the effective kernel acquires $k$-dependence only at criticality. The per-configuration agreement between $\Delta$ and $I(-0.5)$ is primarily a magnetization correlation: $R^2 = 0.91$ at $L = 256$ collapses to $R^2 \approx 0$ once $|M|$ is partialed out. Per-configuration evidence corroborates the $I_0$ Parseval identity but not the $|k|^{-2\eta}$ shape factor; the latter is established by ensemble $L$-scaling.