Papers

This paper is concerned with the construction of the bosonic universal character (UC) hierarchy, whose tau functions are investigated within the framework of charged free bosons. The Lie algebra corresponding to the bosonic UC hierarchy is $\mathfrak{\widehat{gl}}_{2\infty}$. It is shown that the tau functions of bosonic UC hierarchy can be represented based on a series of ordered exponential operators. Furthermore, we derive correlation functions of the bosonic UC hierarchy, which can be expressed as the product of UCs. It is worth noting that the correlation functions of the bosonic UC hierarchy are the inverse of the correlation functions of the generalized phase model.

This paper investigates the uniqueness, connectedness and openness properties of spaces of Kähler forms on closed $4$-manifolds, extending discussions in \cite{Li08,Sal13}. Motivated by the Streets–Tian conjecture concerning the existence of Kähler metrics on Hermitian-symplectic complex manifolds, we also study holomorphically tamed symplectic forms, which are symplectic forms tamed by some integrable complex structure. We formulate a parallel question and relate it to the corresponding questions for Kähler-type symplectic forms.

Noise-induced order (NIO) is a paradigmatic example of nontrivial noise-induced phenomena, characterized by pronounced spectral peaks and pseudoperiodic dynamics. We show that this periodicity is governed by the Ruelle-Pollicott resonances of the annealed transfer operator, independently of dynamical stability. Exact results for an analytically solvable model and numerical results for a modified Lasota–Mackey map show excellent agreement between resonance-based predictions and empirical power spectra across a broad range of noise amplitudes. Independent transitions in stability, diagnosed by the Lyapunov exponent, and statistical periodicity, diagnosed by the Ruelle-Pollicott resonances, give rise to three distinct types of NIO.

Summation error depends on partial-sum order, which standard worst-case bounds omit. To capture this dependence, we derive an exact mean-square error (MSE) recurrence for a binary reduction tree T under conditionally unbiased rounding. With unit roundoff u, the constant-nu model sets the local variance at pre-rounding value x to nu u^2 x^2. Its leading tree-dependent cost for the input vector p is p^T K_T p, where the common-ancestor kernel K_T counts the internal ancestors shared by each pair of leaves. For i.i.d. inputs of mean mu and variance tau^2, this expected cost is tau^2 Lambda_1(T) + mu^2 Lambda_2(T), where Lambda_1 is total leaf depth and Lambda_2 sums squared internal-subtree sizes; Lambda_1 governs centered inputs, while Lambda_2 captures nonzero means. We use these statistics to characterize optimal tree topologies and schedules. Balanced and sequential trees attain the centered extrema. For k inputs, optimal two-stage sequential blocking yields root-mean-square (RMS) error scaling as k^{3/4}. For fixed-stage hierarchies, geometric schedules are optimal for centered inputs, whereas the optimal noncentered stage exponents halve successively. For independent centered inputs with unequal variances, Huffman coding minimizes variance-weighted depth over free leaf assignments. We extend the kernel to matrix multiplication through operand Gram matrices. We then test the approximation under round-to-nearest using exact residuals. Across binary64, binary32, and software-emulated binary16 and bfloat16, the model recovers the ordering among tree topologies; K_T tracks AR(1) partial-sum costs. For GEMM, independently calibrated predictions differ from measurements by at most 3% on the tested grid. A reduction tree extracted from an array library predicts the measured RMS scaling. However, stagnation and bias in positive low-precision sums limit the model's applicability.

We prove that monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial. These results extend to finite products. Using M.~Watanabe's Schubert duality, we deduce that stretched Schubert structure constants are eventually polynomial. This proves a conjecture stated by I.~Pak and Z.~Slonim. We also prove that the positive key expansion of every affine-code family of Schubert polynomials eventually has fixed coefficients and affine indices. The proofs combine Demazure operators and vector partition functions with P.~Magyar's orthodontic formula and the generalized Demazure character theory of V.~Lakshmibai, P.~Littelmann, and P.~Magyar.

We study low-precision computation of C=AB with both factors quantized. We derive an exact finite-dimensional identity for the expected squared product error under independent, zero-mean entrywise errors with known variance fields; it holds exactly for non-overloading subtractive dither and for independent stochastic rounding, and we empirically assess deterministic round-to-nearest (RTN). Using the product-preserving equivalence AB=(AT)(T^{-1}B), we formulate contraction-gauge preconditioning: jointly choosing a factor representation and its sharing pattern before quantization. Preconditioning can reduce product error but may require extra transformed, quantized copies of the opposite operand: a shared transform needs one copy, a block-specific transform up to one per block. Within the bounded family of positive diagonal gauges (folds), a geometric program computes a globally optimal shared fold and a linear program decides whether the identity fold is already optimal. For other families we derive computable selection statistics – tail index for scaling, profile spread for partitioning, coherence and weighted-Gram energy for rotations, slice-energy covariance for hierarchy depth – with upper bounds for ranking heuristic candidates. Across twelve linear products from a trained three-block image classifier, median within-product rank correlations between dither-model predictions and deterministic-RTN errors are 0.937 at 8 bits and 0.918 at 4 bits. The GP fold cuts held-out product error over the identity fold by 18.0% (8-bit) and 20.5% (4-bit) in geometric mean, beats a SmoothQuant-style grid baseline at both precisions and on ten of twelve products, and lowers composed logit MSE by 15.4% and 26.4%. We thus provide exact stochastic product-error accounting, certified selection within the diagonal family, and a common objective for evaluating reusable transform candidates under RTN.

Low-complexity channel state information acquisition is crucial for extremely large-scale multiple-input multiple-output (XL-MIMO) systems. However, practical deployments of non-square uniform planar arrays (UPAs) in hybrid-field environments face prohibitive computational complexity and degraded estimation accuracy due to limited elevation angle-of-arrival (AoA) resolution and deteriorated channel sparsity. To tackle these challenges, we propose a low-complexity channel estimation framework. First, an antenna-domain extrapolation scheme synthesizes a virtually enlarged vertical aperture via the spatial correlation among adjacent elements, breaking the elevation resolution limit. The framework then disentangles the parameter coupling by transforming the two-dimensional joint search into two sequential one-dimensional searches. Specifically, elevation AoAs are extracted via an extrapolation-enhanced discrete Fourier transform-Newtonized orthogonal matching pursuit (NOMP) algorithm along the virtually enlarged vertical uniform linear array (ULA), while azimuth AoAs, ranges, and gains are acquired utilizing a discrete fractional Fourier transform-NOMP algorithm along a horizontal ULA. A subspace fitting-driven path matching algorithm pairs these decoupled parameters. To overcome the accuracy bottleneck of the antenna-domain scheme, a correlation-domain extrapolation scheme is further developed by exploiting the structural properties of the spatial correlation matrix to decouple the near-field quadratic and azimuth phase components, yielding a noise-suppressed virtual array. Numerical results validate the effectiveness of the proposed framework.

Extremely large-scale multiple-input multiple-output (XL-MIMO) is crucial for next-generation communication systems. In practice, the deployment of non-square uniform planar arrays (UPAs) fundamentally alters wavefront characteristics and induces anisotropic beamfocusing capability along different axes due to the aperture disparity originating from the non-square array geometry. To fully uncover the performance impact of such non-square geometry and thus unleash the potential of the non-square UPAs, we investigate the anisotropic near-field characteristics, fundamental limits, and channel estimation for non-square UPA-enabled XL-MIMO systems. First, we derive the effective beamfocusing distances for the long and short axes of the array. Interestingly, the radiation space of a non-square UPA can be partitioned into three regions, i.e., the fully near-field, the anisotropic near-field, and the far-field regions, and the anisotropic region asymptotically dominates the overall near-field space as the array aspect ratio increases. Then, the asymptotic effective degree of freedom for non-square UPA-enabled XL-MIMO systems is provided, which reveals that distance-domain multiplexing is governed by the long-axis aperture in the large array aspect ratio regime. Furthermore, the closed-form Cramer-Rao bound for distance estimation and the three-dimensional (3D) position error bound (PEB) are derived to reveal the geometry-induced performance trade-offs among distance, azimuth, and elevation estimation, based on which the optimal array aspect ratio that minimizes the 3D PEB is determined. Finally, by exploiting the anisotropic wavefront properties, we design a 3D anisotropic near-field codebook to facilitate low-complexity channel estimation for non-square UPAs. Numerical results validate that the proposed codebook achieves comparable accuracy to the 3D polar-domain codebook at reduced complexity.

Given an elliptic curve with Weierstrass equation $y^2=f(x)$, and a positive definite binary quadratic form $Q(u, v)$. We show that there are infinitely many $d$ in the set represented by the quadratic forms in the genus of $Q$ such that the twisted elliptic curve $dy^2=f(x)$ has analytic rank one.

The rearrangement number $\mathfrak{rr}$ is the least cardinality of a collection of permutations of $\omega$ such that every conditionally convergent real series is disrupted by some permutation in the collection. Blass, Brendle, Brian, Hamkins, Hardy, and Larson proved that $\max\{\operatorname{cov}(\mathcal N),\mathfrak b\}\leq\mathfrak{rr}\leq\operatorname{non}(\mathcal M)$ and asked whether $\mathfrak{rr}=\operatorname{non}(\mathcal M)$. We prove in ZFC that $\mathfrak{rr}=\operatorname{non}(\mathcal M)$.

This paper employs extreme value theory to establish the asymptotic distribution of the $k$-th maximum order statistics of signal to noise ratio (SNR) for a $\kappa-\mu$ fading channel with independent and non-identically distributed (i.n.i.d.) parameters. Since $\kappa-\mu$ encompasses well-known distributions such as Rice, Rayleigh, and Nakagami-m, the order statistics for these are also derived as special cases. We demonstrate the practical significance of our results by showcasing their applicability in several applications, including antenna selection in MIMO systems, backscatter systems, reconfigurable intelligent surfaces, and UAV-assisted relay selection systems. Furthermore, by utilizing the $k$-th maximum order statistics, we derive expressions for outage probability and average throughput of $k$-th maximum order statistics. Comprehensive Monte Carlo simulations are carried out to verify the accuracy of the proposed results.

We develop refinements of inner products, the Gram-Schmidt process, root system, Hilbert space and Maschke's Theorem in the context of zero divisor mathematics. We then introduce the concept of respecting nilpotent step representations of Lie algebras and give the new refinement of Ado's Theorem.

We study a one-parameter family of smooth projective genus-three hyperelliptic curves over algebraically closed fields of odd characteristic, with the parameter different from zero and one. We translate superspeciality into the simultaneous vanishing of two truncated hypergeometric polynomials arising from the Cartier operator. For their monic greatest common divisor, we prove squarefreeness and show that every irreducible factor has degree at most two. We then determine its behavior according to the characteristic modulo eight: in one residue class every nonconstant irreducible factor is quadratic; in two residue classes the greatest common divisor is trivial; and in the remaining residue class a distinguished linear factor always occurs.

In 1986, Deligne and Mostow constructed a ball quotient $\mathbb{B}^2 / \Gamma$ biholomorphic to the complex projective plane $\mathbb{P}^2$ whose branch locus is a line arrangement. In this paper, we show that if $\mathbb{P}^2$ is realized as a ball quotient whose branch divisor $D$ is an arrangement of smooth pairwise normal-crossing curves, then the orbifold $(\mathbb{P}^2,D)$ is isomorphic to either the Deligne-Mostow example or a certain degree 9 cover of it. This classification of "ball quotient structures" on $\mathbb{P}^2$ generalizes the $\mathbb{P}^1$ case due to Poincaré.

We study the lifting problem for mod $2$ characteristic maps over simplicial $3$-spheres. Using a bad-block partition of the universal complex $X(\mathbb{Z}_2^4)$, we prove an avoidance criterion for liftability. We show that every simplicial $3$-sphere with at most $20$ vertices has the toric lifting property. We also obtain image-size and join-type results, and prove sharpness of the image-size bound in the universal-complex sense.

We investigate subsonic time-periodic solutions to the radially symmetric non-isentropic Euler system with nonzero angular velocity in an annulus. Under the assumption that the boundary conditions are dissipative on the inner circle and non-dissipative on the outer circle, we establish the existence and stability of time-periodic solutions which are close to steady radially symmetric subsonic spiral flows. It is worth noting that there is no smallness restriction on the background subsonic solutions. The main difficulties arise from the derivative loss caused by the coupling with the radial entropy derivative and the additional zeroth-order terms induced by the nonzero angular velocity in the diagonalized system. One of the key ingredients of the analysis is to rewrite the radial derivative of the entropy as a derivative along the first or fourth characteristic direction. Another one is to control the accumulation of perturbations along characteristic curves by suitably restricting the width of the annulus.

We study inequalities for orbifold second Chern classes of compact normal analytic varieties in Fujiki's class. We prove Miyaoka's inequality for singular varieties in Fujiki's class with nef canonical divisor, as well as the semipositivity of the orbifold second Chern class for varieties with nef anti-canonical divisor. To prove these results, we establish generic nefness theorems for tangent and cotangent sheaves and an orbifold Bogomolov–Gieseker inequality for mixed polarizations.

Let $L\subseteq\mathbb Z^4$ be a rank-three lattice possessing an admissible basis, and let $S=K[x_1,\ldots,x_4]$ be the polynomial ring over a field $K$. We show that the associated lattice ideal $I_L \subset S$ is generic and that $S/I_L$ is a Cohen–Macaulay ring. We also identify the neighbors of the origin in $L$ and the indispensable binomials of $I_L$. Moreover, we prove that the Betti numbers of $S/I_L$ are $\beta_0(S/I_L)=1$, $\beta_1(S/I_L)=7$, $\beta_2(S/I_L)=12$, and $\beta_3(S/I_L)=6$. In addition, we show that $I_L$ is a set-theoretic complete intersection over an arbitrary field.

For a finite set $A\subset\mathbb{Z}$, write $hA$ for its $h$-fold sumset, and let \[ R(h,k)=\{|hA|:A\subset\mathbb{Z},\ |A|=k\}. \] We determine the part of $R(h,4)$ lying between $4h+2$ and $6h-4$: for $h=4$ the only value is $5h-1$, while for $h\geq 5$ the only values are $5h-1$ and $5h+1$. This proves Rajagopal's conjectured gap $5h\notin R(h,4)$ for every $h\geq 4$. For $h\geq 6$, it also yields the new missing interval $[5h+2,6h-4]$, which lies outside Rajagopal's general excluded set. Lev's lower bound for the successive growth of multiple sumsets reduces the problem to normalized sets of affine diameter five, of which there are only six. Reflection and four elementary exact sumset computations finish the classification.

The Vietoris–Rips complex $\mathrm{VR}_\epsilon(X)$, the Dowker complex $\mathrm{D}_R(X,Y)$, and its flagified Dowker–Rips variant $\mathrm{DR}_R(X,Y)=\mathrm{F}(\mathrm{D}_R(X,Y))$ are simplicial complexes constructed from metric data or witness relations. They are useful in topological data analysis because they encode topology through combinatorial data derived from pairwise information, but at a fixed scale they retain little of the underlying geometry. Unlike alpha complexes or mesh-based discretizations, Rips-type complexes carry no canonical primal–dual cell structure, which is the ingredient used by the discrete exterior calculus Hodge star to encode metric information. We address this gap by equipping a Rips-type complex $K$ with diagonal geometry-induced Hodge stars represented by positive simplex weights $W_k=\operatorname{diag}\{w_k(\sigma):\sigma\in K_k\}$, which define weighted inner products on $k$-cochains. The resulting weighted discrete Hodge Laplacian $\Delta_k^W$ has kernel dimension equal to the $k$th Betti number of the underlying complex, while its nonzero spectrum is governed by the chosen geometric weights. The central issue is therefore not the existence of a weighted Laplacian, since any positive diagonal weights define one, but the design of weights that encode meaningful metric or witness geometry. We focus on two computable choices: simplex-volume weights, based on Euclidean simplex volumes, and soft witness weights, based on a Dowker-style support function $s_t(\sigma;Y)$ that quantifies higher-order witness support lost under flagification. We prove positivity, weighted self-adjointness, and Betti-number preservation for arbitrary positive diagonal weights, establish an asymptotic decay-rate characterization for soft witness support, and describe spectral descriptors derived from $\Delta_k^W$ for comparing geometry-aware Hodge spectra on Rips complexes.

We study averaging-learning dynamics without an exogenous ground truth: the reference signal is generated endogenously by the population. The dynamics combine a time-varying averaging matrix, a learning-source matrix, and a learning matrix, typically diagonal. Dobrushin-type contraction controls the decay of oscillations and yields asymptotic agreement. Under a summability condition, the backward products converge exponentially to rank-one limits. With summable perturbations, the process converges to a random consensus state. For i.i.d. perturbations, we prove a central limit theorem for the centered process: the limiting Gaussian law is supported on the agreement direction. Thus, despite the multi-agent dynamics, the long-time fluctuations are asymptotically one-dimensional. We also record a pairwise Dobrushin formulation that clarifies the dynamic agreement-class geometry underlying the one-class regime.

In combinatorial game theory, the choice of play convention is a fundamental aspect of the theory. The two most widely studied conventions are normal play, in which the player who makes the last move wins, and misère play, in which the player who makes the last move loses. Another well-known variant is play with a single shared pass. In such games, at most one pass may be used in total during the game: once the pass has been used, neither player may pass thereafter. Moreover, once a terminal position has been reached, passing is no longer allowed. Recall that, for misère play, one sometimes defines SG values (or Sprague-Grundy values) in the same recursive manner as in normal play but assigns the terminal position the value 1. Also, SG values can be considered for normal-play games with a pass. In this work, we propose transformations that allow both misère play and play with a pass to be treated as normal-play games. As a consequence, we show that there is a certain duality between a generalization of misère play and a generalization of play with a pass.

Regularity and depth of binomial ideals generated by adjacent $2$-minors together with those arising from finite lattices are studied.

Qualitative independence hypergraphs provide a useful combinatorial framework for analyzing the existence and structure of covering arrays. In this work, we study the uniform and \emph{almost-uniform $3$-qualitative independence hypergraphs} $3\text{-}UQI(n,2)$ and $3\text{-}AUQI(n,2)$, and establish a structural correspondence between these families and merged Johnson graphs, with emphasis on their cores. Focusing on the smallest unresolved instance, $3\text{-}QI(8,2)$, we classify all of its strongly independent sets and determine its strong independence number. Using this, along with its strong chromatic number and the size of the largest $3$-clique, we show that $3\text{-}QI(8,2)$ is a core. For $n>8$, we further identify sufficient conditions under which $3\text{-}UQI(n,2)$ and $3\text{-}AUQI(n,2)$ are cores.

In this document, we discuss how cyclic power operations for periodic complex bordism can be understood through algebraic geometry, following from the theory of power operations for Morava $E$-theory developed by Ando, Hopkins, and Strickland. Using this perspective, we describe how one computes power operations on the $E$-homology of $BU$ and $BU\times \mathbb{Z}$, where $E$ is a 2-periodic even $\mathbb{E}_\infty$ ring. Then we provide some explicit example computations at $p=2$; in particular, we compute the action of the additive operations $\Delta$ on the indecomposables $\widehat{Q}(E_0(BU))$ for $E=KU, E_2$.

We determine the strong independence number of the qualitative independence hypergraph, $3\text{-}QI(11, 2)$, using a technique that involves considering its vertices as subsets of $\{1,2, \ldots, 11\}$ and assessing them as intersecting set systems. This gives the maximum size of colour classes in any strong colouring and thus, a lower bound on the strong chromatic number of $3\text{-}QI(11,2)$. We leverage this bound along with an upper bound of the strong chromatic number of $3\text{-}QI(10,2)$, to consequently, establish that the covering array number of $3\text{-}QI(11, 2)$, $CAN(3\text{-}QI(11,2),2) = 11$ and give a sufficient condition for a hypergraph $H$ to have $CAN(H, 2)=11$.

Given observations $\mathbf x=(x_1,\dots,x_n)$, Gaffke (2005) defined \[ K_n(\mathbf x)=\mathbb{P}_{\mathbf D}\!\left\{\sum_{i=1}^n x_iD_i\le 1\right\}, \qquad (D_0,D_1,\ldots,D_n)\sim\mathrm{Dirichlet}(1,\ldots,1), \] and conjectured that it is a $p$-value whenever the inputs are independent e-values. Recently, Vlassis and Thomas (2026) proved this conjecture. Inverting the tests for observations in $[0,1]$ gives the confidence interval studied by Learned-Miller and Thomas (2020), which reduces to Clopper–Pearson for Bernoulli data. We give a finite- and large-sample account of Gaffke's test and interval. First, for every $\mathbf x\in[0,\infty)^n$ and every elementary symmetric polynomial $e_k$, \( K_n(\mathbf x)e_k(\mathbf x)\le {n\choose k}, \) so the Gaffke $p$-value never larger than the SymPol $p$-value of Ming et al. (2026). However, Gaffke's p-value is inadmissible. For $n=2$, we construct a valid rule that is strictly smaller on mixed configurations and is the unique admissible rule that dominates $K_2$. A neutral-face extension proves inadmissibility of $K_n$ for every $n\ge2$. If one independent uniform random variable is allowed, there is an even simpler full-dimensional improvement: on the upper orthant, where $K_n(\mathbf x)=1/\prod_i x_i$, replace it by $U/\prod_i x_i$. The equal-tail Gaffke confidence interval $I_n$ is nevertheless first-order asymptotically efficient: for iid observations on $[0,1]$ with unknown variance $\sigma^2>0$, \[ \sqrt n\,\operatorname{Width}(I_n)\longrightarrow 2\sigma z_{1-\alpha/2}\qquad\text{almost surely}. \] Our simulations also find that, among a variety of bounded-mean intervals considered, the Gaffke interval is the shortest, including comparisons with a recent empirical Berry–Esseen procedure having the same first-order Gaussian target.

We construct a master $2$-Toda tau sequence for the stationary relative theory of $(\PP^1,0,\infty)$ with any fixed finite lists of negative contacts. The negative-contact blocks are encoded by finite products of exponentials of infinite-wedge Lie algebra elements, with the block at $\infty$ taken as the adjoint of the entire corresponding left block. After adjoining independent positive-profile Toda times, the resulting charged-vacuum matrix elements form a genuine $2$-Toda tau sequence. Following the standard $Q,u$ specialization, its positive signed-degree coefficients recover the stationary relative Gromov–Witten invariants with the prescribed positive and negative contacts, which occur in and are constrained by the induced coefficientwise Hirota equations. The master tau sequence is larger than the enumerative sector: it generally contains nonzero, non-vacuum signed-degree-zero coefficients, as well as auxiliary nonzero lattice components, for which no enumerative interpretation is asserted. Consequently the Hirota equations form a coupled master system and do not, without additional input, close on the positive-degree invariants alone. The construction uses the large-root definition of negative contact, the Johnson–Kumaran–Wu operator formula for the tube, and the Kyoto/Plücker construction of the $2$-Toda hierarchy.

Kawauchi conjectured that the Conway polynomial of an amphicheiral knot factors as $\nabla_K(z)=f(z)f(-z)$ for some integer polynomial $f(z)$. In joint work with Hartley, he showed this was true for strongly amphicheiral knots, and Hartley used the JSJ decomposition of the knot exterior to generalize to all negative amphicheiral knots. Ermotti–Hongler–Weber were the first to publish a counterexample to the general case. Independently, in 2006 the author had conjectured a statement which is equivalent to the statement that $\nabla_K \equiv f(z)f(-z) \pmod 4$ for amphicheiral knots, based on certain patterns he noticed in finite type invariants. In this paper we prove this mod 4 version of Kawauchi's original conjecture as a consequence of a stronger integral statement. The mathematical content of this paper was produced with the help of Claude Fable 5.

We establish improved lower bounds on the minimax expected regret of stochastic bandit convex optimization for $1$-Lipschitz functions on the $d$-dimensional Euclidean ball. For time horizons $n\ge d^{10/3}$, we prove a lower bound of $\Omega(d^{4/3}\sqrt{n})$, the first nontrivial bound that exceeds the $d\sqrt{n}$ dependence of linear bandits, showing that stochastic bandit convex optimization is fundamentally harder than linear bandits. For $d^2\le n\le d^{10/3}$, we obtain a lower bound of $\Omega(\sqrt{d}n^{3/4})$, matching the regret of the algorithm of Flaxman et al. (2005), establishing its optimality in this regime. The hard class of convex functions we construct takes the following form in dimension $2d$: for an action $a=(a^1,a^2)\in \mathbb{B}^{2d}$, each function is the scaled soft maximum of a "tube", $r^{-1}\|W^\star a^1-\frac{r}{8\varepsilon}a^2 \|$ (hyperparameterized by $\varepsilon,r$), and a squared distance function, $\frac12\|a^1-u^\star\|^2-\frac12\|u^\star\|^2$. Here $u^\star\in\mathbb{R}^d$ is the unknown target determining the minimizer, while $W^\star\in\mathbb{R}^{d\times d}$ hides the region in which the quadratic curvature is observable. Indeed, observations reveal substantial information about $u^\star$ only when the learner acts near the hidden tube $a^2\approx \frac{8\varepsilon}{r}W^\star a^1$; away from it, the tube branch masks the quadratic branch. Thus the learner must pay to uncover the geometry encoded by $W^\star$ before it can effectively exploit the curvature that identifies $u^\star$. Formalizing this tradeoff yields a sample complexity lower bound of $\Omega(\frac{d^{5/2}}{\varepsilon^2}\wedge\frac{d^2}{\varepsilon^4})$ for finding an $\varepsilon$-optimal action, and ultimately the $\Omega(d^{4/3}\sqrt{n}\wedge\sqrt{d}n^{3/4})$ regret lower bound. The proof was developed by GPT-5.5 Pro and GPT-5.6 Sol Pro under the authors' guidance.