Papers

We study noncooperative multi-vehicle games among heterogeneous autonomous vehicles, where each vehicle adopts a decentralized closed-loop policy based on its own state, and optimizes an objective that depends on other vehicles through potentially asymmetric interaction weights. We develop an $\alpha$-potential game framework that reduces the computation of an approximate Nash equilibrium (NE) to the minimization of a single auxiliary $\alpha$-potential function. We explicitly construct this $\alpha$-potential, establish the existence of its minimizers, and characterize the equilibrium approximation error $\alpha$ in terms of interaction asymmetry. We further introduce vehicle-specific scaling to reduce the effective interaction asymmetry, thereby tightening the equilibrium approximation and, in important cases, recovering an exact NE despite asymmetric interactions. We also derive social-efficiency guarantees for the potential-selected policies, revealing how the interaction structure shapes worst-case efficiency. Numerical experiments demonstrate the flexibility of the framework in capturing heterogeneous vehicle interactions, collision and obstacle avoidance, lane changing and overtaking under different traffic configurations, and priority-based intersection crossing.

In graphical two-choice allocation, each arriving ball is assigned to one endpoint of a random edge. We study rules whose decision is a monotone function of the two endpoint loads, allowing edge-dependent thresholds and fresh randomization. Such a rule has an exact unit-discrepancy coupling: adding one ball to the initial state produces one tagged discrepancy at every later time. We represent the tag by conditional-expectation projections on the marked edge space and obtain diffusive displacement bounds. A transport-volume inequality then converts slow propagation of influence into lower bounds for the load gap. On the cycle with $n$ vertices, from every initial distribution and at every physical time $t\ge 1/n$, the expected gap is at least a constant times $\min\{\sqrt n,t^{1/4}\}$, and the gap exceeds this scale with probability at least $1/8$. After exactly $k\ge1$ allocations, the corresponding scale is $\min\{\sqrt n,(k/n)^{1/4}\}$. No stationarity, symmetry, recurrence, or moment assumption is used. A smoothed threshold rule in the same class has expected gap $O(\sqrt n\log n)$ up to any fixed polynomial time horizon, so the saturated cycle bound is sharp within the class up to a logarithmic factor. The general inequality also yields a lower bound of order $\sqrt{L/K}$ on the $L\times K$ rectangular torus $C_L\square C_K$; combined with a strategy-independent logarithmic bound, this gives order $\sqrt{L/K}+\log(LK)$. These results separate endpoint-local rules from global-information strategies that achieve polylogarithmic gaps on cycles. All numbered results are verified in Lean.

We study finite-blocklength lossy transmission of a Gaussian memoryless source over memoryless, possibly nonstationary block erasure channels under an excess mean-squared distortion criterion. We derive computable nonasymptotic achievability and converse bounds and establish matching third-order, channel-uniform normal approximations. Our achievability bound exactly evaluates the ensemble-average excess-distortion probability of a specified random coding scheme and improves corresponding specializations of known general one-shot bounds. Our converse conditions on the erasure pattern and source energy and combines spherical-cap and volume bounds, with the spherical-cap term providing the geometric prefactor needed for the matching third-order term. In our channel-uniform normal approximation, we show that for fixed distortion ratio, target excess-distortion probability, and block size, the sufficient and necessary information-balance conditions have the same dispersion and $\frac{1}{2}\log k$ terms, where $k$ is the source blocklength, and differ only by bounded remainders that are uniform over the channel blocklength and erasure-probability profile. Numerical evaluations compare the nonasymptotic JSCC bounds and their common third-order normal approximation with an optimized symmetrized SSCC achievability benchmark.

We study the problem of sampling from a continuous density $\pi\propto \exp(-V)$ on $\mathbb R^d$, where $V\in C^2(\mathbb R^d)$ has a $\beta$-Lipschitz gradient and $\pi$ satisfies a logarithmic Sobolev inequality with constant $\alpha^{-1}$, and write $\kappa = \beta/\alpha$. We introduce the Gaussian cloud sampler, which achieves total variation accuracy $\varepsilon$ using $\widetilde O(\kappa d^{1/5}\,\text{polylog}(1/\varepsilon))$ gradient queries in expectation. The algorithm uses first-order rejection sampling (FORS) to correct the law of smoothed Picard HMC trajectories. To do so, we represent the iterates of the ideal Picard iteration via Gaussian clouds, whose centers are never evaluated, and we develop a suite of likelihood correction gadgets which only use samples from this indirect cloud representation.

A classical 1977 result by Roter states that a Riemannian manifold of dimension at least four with parallel Weyl curvature must be locally symmetric or conformally flat. We establish Roter's theorem in the symplectic category: namely, a Fedosov manifold with parallel symplectic Weyl curvature must be locally symmetric or Weyl-flat.

We introduce the notion of $K_{\mathfrak{q}}$-algebra, a conjectural axiomatization of the fusion algebra of rotation-equivariant BPS line defects in a 4d ${\cal N}=2$ Supersymmetric Quantum Field Theory. We also introduce the notion of RG flow of $K_{\mathfrak{q}}$-algebras, a conjectural axiomatization of the action of Seiberg-Witten type RG flows on line defects. These definitions enrich the theory of cluster algebras, $K$-theoretic Coulomb branch algebras, skein algebras and quantum groups.

We study a Kyle–Back model in which the informed trader has dynamic mean–variance preferences, leading to a time-inconsistent trading problem that is formulated as an intrapersonal game. The model combines two equilibrium requirements: a trading–pricing equilibrium between the informed trader and the market maker, and a time-consistent equilibrium among the trader's successive selves. Following the framework of Cho (2003), we consider both risk-neutral and risk-averse informed traders. In the risk-neutral case, we derive the equilibrium trading strategy and price impact explicitly. In the risk-averse case, we establish existence and characterize the equilibrium through a coupled system of nonlinear ordinary differential equations. Our main technique is to reduce the equilibrium conditions to a forward–backward ODE system and resolve the resulting boundary conditions by a shooting argument. Economically, the mean–variance preference reshapes the intertemporal trade-off between exploiting current private information and preserving future informational advantage, generating a declining term structure of price impact and shifting both information revelation and informed-trading profits toward earlier stages of the trading horizon.

We investigate Jordan structure and dynamical limits in six families of integrable spin-$\frac{1}{2}$ chains obtained from the boost automorphism method. Local operator algebras connect nilpotent evolution on finite periodic chains to central defects in spectral additivity, polynomial unipotent $R$-matrices and constant braid limits. For conservative endpoint erosion, we determine the characteristic and minimal polynomials and show that the Jordan partition can change while both polynomials remain unchanged. We also identify the deletion limits that distinguish the parity obstruction to removing signs by diagonal similarity in Class 5 from the parity nilpotent dependence in Class 6. Matching expansions and fragmentation recursions yield exact amplitudes and survival probabilities for processes with killing. In addition, we establish criteria excluding conservative Markov realizations and determine which conservative completions of specified reaction densities admit regular $R$-matrices of difference form.

Let $\Sigma_{1} = \mathbb{N}^*$ and $\Sigma_{k+1} = \overline{\mathbb{N}^* + \Sigma_{k}}$ be the cumulative depth filtration of $\beta\mathbb{N}$, the analogue for $(\mathbb{N},+)$ of a chain of closed ideals that Protasov and Protasova studied for discrete groups, where strict descent follows from a theorem of Lutsenko and Protasov. For every $k$ we give an explicit set whose closure meets $\Sigma_{k}$ but not $\Sigma_{k+1}$. Fix the doubly exponential sequence $e_{n} = 2^{2^{n}}$, partition it into $k$ subsequences $E_{0}, \dots, E_{k-1}$ by the residue of the index modulo $k$, and set $A_{k} = E_{0} + \cdots + E_{k-1}$. We prove that any sum $q_{0} + \cdots + q_{k-1}$ of free ultrafilters with $E_{t} \in q_{t}$ lies in $\Sigma_{k} \setminus \Sigma_{k+1}$. The engine is a master lemma, proved by induction on $j$: if a sum $F_{1} + \cdots + F_{j}$ of subsequences of $\{e_{n}\}$ with pairwise disjoint index sets belongs to a free ultrafilter $s$, then $s \notin \Sigma_{j+1}$. The proof rests on a single rigidity of the doubly exponential sequence: a fixed difference forces the largest index in any shift-intersection, once it is large, to cancel within its own subsequence, which makes every shift-intersection descend by at least one level. The same witnesses lie in the gaps of the pure filtration.

Imputation-based causal estimation is typically viewed as relying exclusively on an outcome model, in contrast to augmented inverse-probability weighting, whose consistency is protected by fitting two nuisance models. This paper argues that this view can be misleading by highlighting a hidden dual weighting structure in least-squares sieve regression imputation. Although only outcome regressions are explicitly fitted, the resulting imputation estimator admits an exact weighting representation whose induced weights balance every function in the sieve space and the corresponding population weighting functions are the $L^2$ projections of the inverse propensity scores onto the same sieve space. This projection structure yields an implicit form of double robustness and, under standard sieve approximation and growth conditions, asymptotic linearity with the efficient influence function. Thus, weighting, covariate balance, double robustness, and semiparametric efficiency can all emerge from imputation alone through the geometry of least-squares projection.

We develop a three-stage adaptive procedure for principal component analysis (PCA) when high-dimensional observations are collected sequentially and additional sampling incurs a cost. The procedure balances PCA compression loss against sampling cost while selecting the retained dimension through a prescribed explained-variance criterion. Starting from a pilot sample, an intermediate stage updates the PCA quantities before determining the final sample size, thereby avoiding reliance on unknown population eigenvalues. Under suitable regularity conditions, we establish both first- and second-order efficiency relative to the population oracle. Comparison with the corresponding two-stage rule shows that the additional recalibration yields sharper second-order control and reduces the influence of the pilot stage on the final sampling decision. The theory allows the ambient dimension to exceed the sample size under appropriate covariance and spectral conditions. Simulation studies demonstrate the strong finite-sample performance of the procedure across increasing dimensions and several dense covariance structures. As a real-data application, we conduct a retrospective study of gene-expression data from 32 cancer-type cohorts in The Cancer Genome Atlas, illustrating both cost-effective early stopping and settings in which additional observations are recommended.

In a bulk ferromagnet, magnetic energy is six orders of magnitude weaker than elastic energy. Slenderness removes this disparity. In a thin ferromagnetic rod, the two energies scale differently with aspect ratio, and an ordinary nickel wire deforms nonlinearly even under modest remote fields. We develop a three-dimensional variational theory of a ferromagnetic Kirchhoff rod under gravity and magnetic loading and study it through bifurcation analysis, computation, and experiment. A uniform field can stabilize a vertical rod against buckling under its own weight, the classical Greenhill instability. We give sharp criteria for clamped and pinned supports in terms of an Airy-type principal eigenvalue, and give a fine description of the character of the bifurcation landscape, which has qualitatively different features than its non-magnetic counterpart. When subject to the field generated by a permanent magnet, we identify a rich bifurcation landscape as the magnet is rotated quasistatically along a circle of radius slightly larger than the rest length of the rod. We prove that instability occurs only through a planar mode and that each snap is a nondegenerate saddle-node. We also locate the two folds that produce hysteresis. Experiments on $50\,\mu$m nickel wires confirm the Greenhill threshold, restabilization at ~$30$ mT for both clamped and pinned supports, as well as the salient features of the hysteresis loop such as switching angles, width, and geometry of folds accurately. Our analysis and experiments offer principled design rules for field-stabilized filaments and remotely switched bistable actuators. Our analysis also offers a well-conditioned numerical method is valid near the switching events, where the direct numerical minimization is strongly ill-conditioned.

Consider the shifted truncations of the exponential series and their Galois groups: $$F_{n,a}(x) := \sum_{j=0}^{n}\frac{x^j}{(a+j)!}\qquad G_{n,a}:= \text{Gal}(F_{n,a}/\mathbb{Q}).$$ In this paper we prove that for every fixed integer $a>0$, the set of $n$ for which $G_{n,a}=S_n$ has natural density one, thus answering a question of Katz and Rivin.

Many systems read the latent space of a variational autoencoder (VAE) as discrete codes, and rely on each latent being read back as the code it was written as. Shannon (1948) took such a shared codebook as given; inside a VAE nothing enforces it. We prove that no item of a standard training report can reveal whether the written and read codes agree on encoded data. We construct two VAEs that reconstruct every input perfectly, infer exactly, and match on every per-example ELBO and every code histogram, yet agreement is one in the first and zero in the second. We then show that the ELBO's own terms bound how far a measured agreement can move: the variational gap bounds the shift to exact inference, and the rate bounds the shift to generation from the prior. For Gaussian posteriors, a mean-field encoder pays an exact price in the gap that depends on its choice of axes. On probabilistic PCA the gap certifies a trained encoder in all 75 settings we test, at agreement between 0.76 and 0.97. On 240 trained VAEs, a floor set by the rate and the dataset size rules out rate-side certification in advance for 66 models, and none of them certifies. The results yield a simple protocol, Certify-or-Measure, for any pipeline that reads a VAE as codes.

We give the geometric classification of complex 3-dimensional Novikov-Poisson and Gelfand-Dorfman algebras. In particular, we prove that the variety of complex 3-dimensional Novikov-Poisson algebras is 12-dimensional; it has 4 irreducible components and has no rigid algebras; the variety of complex 3-dimensional Gelfand-Dorfman algebras is 11-dimensional; it has 10 irreducible components and has 3 rigid algebras. As an additional result, we have the geometric classification of complex 3-dimensional Novikov algebras: it is 9-dimensional, and it has $4$ irreducible components with one rigid algebra.

Let $p$ be a complex polynomial of degree $n\ge2$, and let $\sigma_2(p)$ be the root mean square distance of its zeros from their centroid, including multiplicities. We prove that every zero of $p$ lies within distance $\sigma_2(p)$ of a critical point. This proves Borcea's quadratic conjecture and, in particular, gives a new proof of Sendov's conjecture. Two polynomial integration identities prove the result analytically for $n\ge2048$. For the remaining degrees, we derive explicit necessary inequalities, including a variable integration endpoint and an exact centered coefficient identity. Directed integer arithmetic certifies a finite covering of the resulting parameter domains. Degrees $4$–$26$ use $150\,845$ certified three-dimensional regions; degrees $27$–$2047$ use a further $27\,429$ regions, with degree intervals allowed. The cases of degree 2 and 3 were already covered by Khavinson–Pereira–Putinar–Saff–Shimorin.

We study foliations which force every transverse flow to have a closed orbit. We fully characterize this property among $C^2$ finite depth foliations on atoroidal 3-manifolds. We also show that for many foliations, this property can be broken by a $C^0$-small modification of the foliation. We produce closed orbits using the theory of endperiodic maps, and we construct $C^1$ flows without closed orbits by inserting dynamical plugs transverse to foliations along curves of attracting holonomy.

Keith numbers are integers that reappear in sliding-window recurrences generated from their own digits. We introduce Keith mirrors, a subclass defined by saturation at the center of the recurrence's incidence profile. For an even index m, let C_d(m) denote the sum of the two central incidence counts. We prove that the central deficit m-C_d(m) is exactly the distance from m to the interval [2d,2d+2], which forces every Keith mirror of order d>=3 in base B>=3 to return at 2d+2. A digit identity and analytic capacity bounds then reduce the classification to finitely many cases. Exact enumeration, independently checked against the original recurrence and incidence definition, yields precisely 35 mirrors of order at least three, while order two consists of two explicit infinite families. We also classify exactly eight cross-base triple alignments satisfying R=C=x_1+x_d and record the conditional reconstruction property induced by incidence symmetry.

Surface braids are a generalization of braids to surfaces in the 4-disk. They are described by planar graphs called charts, and sequences of braid words called braid systems, that represent monodromies of branch points in projections. Yang-Baxter operators (YBOs) have been used extensively in knot theory for constructions of Jones and other polynomials, defined by braid group representations. We propose to use YBOs for constructing surface braid invariants through braid systems. The intersections of eigenspaces are used for this construction. We further use 3-cocycles of the Yang-Baxter cohomology theories via charts, in an analogue of quandle cocycle invariants but more generally utilizing tensor products of modules. An approach towards defining knotted surface invariants through closures of surface braids is discussed.

Let $m_T[0,1)$ denote the number of Laplacian eigenvalues of a tree $T$ that are strictly less than $1$. Guo, Xue and Liu \cite{GXL} proved that every tree $T$ of diameter $d$ satisfies $m_T[0,1)\ge \lceil(d+1)/3\rceil$, and that this bound is sharp when $d\equiv2\pmod3$. It is classical that every graph of diameter $d$ has domination number at least $\lceil(d+1)/3\rceil$. In this paper, for any diameter $d$, $m_T[0,1)=\lceil(d+1)/3\rceil$ if and only if $\gamma(T)=\lceil(d+1)/3\rceil$; we characterize all trees satisfying this necessary and sufficient condition, and also prove that almost all trees satisfy $m_T[0,1)\ge\lceil(d+1)/3\rceil+1$.

For multivariate polynomial bases of total degree at most n, a column of the Vandermonde matrix is identified by a multi-index of exponents. In this work, we give a direct, recursive map from the scalar column index to this multi-index, based on a prescribed ordering in which exponents of the last variable are increased at the block level and the same rule is applied recursively to the remaining variables. The map is first derived explicitly in the bivariate case, where it yields closed-form column indices for the univariate Vandermonde matrices, and is then extended recursively to arbitrary dimension by means of a binomial counting function, with a proof of correctness by induction. The procedure provides random access to any total-degree column without forming the full tensor-product matrix, and is particularly advantageous for streaming construction and matrix-free evaluation. It is basis independent and applies directly to Chebyshev–Vandermonde matrices. For the full dense matrix, nested loops over the exponent multi-indices achieve the same asymptotic cost; the gain of the present approach lies in exact indexing and the avoidance of superfluous intermediate arrays.

A connected graph is called $\ell_1$-embeddable if its shortest-path metric admits an isometric embedding into the $\ell_1$-space; equivalently, in the finite case, it admits a scale embedding into a hypercube. This paper gives a complete parameter-level classification of $\ell_1$-embeddability for regular quadrilateral maps on the torus and the Klein bottle. We treat the two toroidal families $Q_{k,m,r}$ and $Q_{k,m,e}$, obtained from suitable identifications of finite square grids, and, using the known isomorphisms among Klein-bottle quadrilateral maps, the three essential Klein-bottle families $Q_{k,m,a}$, $Q_{k,m,b}$, and $Q_{k,m,h}$. The $\ell_1$-embeddable toroidal maps are exactly $Q_{k,m,0}, Q_{k,2,1}, Q_{k,1,e}, Q_{5,2,e}, Q_{2i,2,e} (i\ge 2)$, and the $\ell_1$-embeddable Klein-bottle maps are exactly $ Q_{2,2,a}, Q_{4,2,a}, Q_{2,m,b}, Q_{1,2,h}.$ All other members of these five families are not $\ell_1$-embeddable.

Multi-agent LLM systems pair a sender with broad context and an executor with a limited local view. We study when a short message improves the executor's next decision, when raw context is preferable, and when a stronger sender helps. Our framework, receiver-relative bounded coordination, expresses message utility as receiver gain minus protocol tax. Compression beats raw context when tax savings exceed losses from omitted information and decoder mismatch. Even Bayes-sufficient compression can fail when a bounded executor cannot use its surface form. A three-stage decomposition separates externalization, absorption, and action closure, explaining how errors remain after the correct content reaches the receiver. Under a single-crossing condition, sender upgrades help above a receiver-burden threshold. Across six benchmarks, the same Qwen protocol raises ContextBench joint accuracy from $0.633$ to $0.775$ but lowers ToolSandbox from $0.889$ to $0.653$. Fixed-message replay reveals closure failures despite correct artifact recovery. These results guide an inference-time selector that improves the accuracy-cost frontier on the evaluated communication regimes.

Hambleton and Kreck described the group of based homotopy self-equivalences of a closed oriented $4$-manifold by a commutative braid of exact sequences built from bordism groups and the Postnikov $2$-type. We construct the analogous braid for closed oriented $BO\langle n+1\rangle$-manifolds $M$ of dimension $2n$ or $2n-1$ whose fundamental group is finite or infinite cyclic and which satisfy one condition on the action of $\pi_1$ on the homotopy of the fibre of $M\to P_n(M)$. We compute $\operatorname{Aut}_\bullet(S^1\times S^{m-1})\cong(\mathbb{Z}/2)^2$ for $m\ge4$, and $\operatorname{Aut}_\bullet(S^3\times S^3)\cong(\mathbb{Z}/12)^2\rtimes SL_2(\mathbb{Z})$ with the standard action modulo $12$. Further, $\operatorname{Aut}_\bullet(S^3\times S^4)\cong(\mathbb{Z}/2\oplus\mathbb{Z}/24)\rtimes(\mathbb{Z}/2)^2$ has order $192$, that $\operatorname{Aut}_\bullet(S^4\times S^4)\cong D_4\times\mathbb{Z}/2$ has order $16$, and that for connected sums of $r\ge2$ copies of $S^3\times S^3$ the kernel contains a subgroup $(\mathbb{Z}/12)^{2r}$ on which the permutations of the summands act non-trivially. For $S^p\times S^p$ the kernel is a sum of two copies of $\pi_{2p}(S^p)$ modulo Whitehead products with $\iota_p$, and the image is determined by an obstruction on the Whitehead square. In particular $\operatorname{Aut}_\bullet(S^5\times S^5)$ is isomorphic to the theta subgroup of $SL_2(\mathbb{Z})$ and $\operatorname{Aut}_\bullet(S^7\times S^7)$ is an extension of $SL_2(\mathbb{Z})$ by $(\mathbb{Z}/120)^2$.

We show that cohomological and homological goodness are equivalent. We prove that if $G$ is a coherent even Artin group or an even Artin group whose underlying graph does not contain 4-clique or a LERF Artin group then $G \otimes G$ and $G \wedge G$ are residually finite. Furthermore for $G$ a RAAG both $G \otimes G$ and $G \wedge G$ are residually $p$-finite for any prime integer $p$. For $G$ a limit group or a Coxeter group we show that $G \wedge G$ is residually finite. For $G$ a soluble group of type $FP_{\infty}$ we prove that the groups $\X(G)$, $\nu(G)$, $G \wedge G$ are virtually residually $p$-finite for almost all primes $p$. For $G$ a generalised Baumslag-Solitar group of rank $n \geq 1$ that is residually finite, we show that $G \wedge G$ is residually finite. For $G = SL_n(\Z)$ and $G = GL_n( \mathbb{Z})$, $n \geq 3$, we show that $G \wedge G$ is residually finite.

We study the query complexity of optimization with exact scalar-value information. For globally $L$-smooth convex functions on $\mathbb R^d$ with a minimizer in a Euclidean ball of radius $R$, we prove the lower bound $\Omega(d\min\{d,\sqrt{LR^2/\varepsilon}\})$ for adaptive randomized algorithms in the stated accuracy range. Queries are unrestricted, and the bound contains no logarithmic factor. It holds for quadratics and matches the minimum of two elementary upper bounds on that class. Prescribed strong convexity introduces the additional cap $\sqrt{L/\mu}$. The proof compares the mean of a matrix with the mean of its inverse on arbitrary affine slices and bounds posterior moments after adaptive measurements. We extend these information bounds to saddle-point problems with independent block dimensions. For compact bilinear games, the coupling alone yields $\Omega(r\min\{r,\sqrt{CR_xR_y/\varepsilon}\})$ queries, where $r=\min\lbrace d_x,d_y\rbrace$. A strongly convex–strongly concave construction yields a condition-dependent lower bound even with known isotropic diagonal Hessians. Separate reductions establish scalar query lower bounds under mixed value and partial-gradient access, including for outer objective accuracy. A Euclidean-image radius yields smooth and nonsmooth extensions to independently specified norms and known penalties.

We show that, given a partially defined metric $d:R\rightarrow\mathbb{R}$ where $R\subset X^2$ is a closed equivalence relation on a compact metrizable space $X$, there is a metric $\hat{d}:X^2\rightarrow \mathbb{R}$ such that $d = \hat{d}|_{R}$

We show that there exists a finitely generated linear group whose non-abelian tensor square is not linear. On the other hand, the non-abelian tensor square of the braid group $B_n$, $n>3$, is linear.

We derive a short-maturity expansion for up-and-out put barrier option prices under continuous stochastic volatility when the strike and the barrier approach the spot at the diffusive scale. Assuming joint weak convergence of the normalized terminal return, the relative volatility fluctuation, and the running maximum, together with uniform integrability, we show that the leading term is the time-inhomogeneous Black-Scholes barrier price fitted to the forward variance curve. The first model-dependent correction is of order $\theta^{H+1/2}$, where $\theta^H$, $H \in (0,1/2]$, is the order of the relative volatility fluctuation, and is represented explicitly through killed Brownian transition densities. For regular volatility models, where $H= 1/2$, the coefficient is determined by the short-maturity at-the-money implied-volatility skew. For rough volatility models with $H < 1/2$, the coefficient reduces to a one-dimensional integral. Numerical experiments show that the correction materially improves the Black-Scholes approximation.

We establish the sharp asymptotic and symmetry theory for the critical quasilinear Hartree equation \[ -\Delta_pu=\big(|x|^{-\sigma}*u^{p^*_{\sigma}}\big)u^{p^*_{\sigma}-1} \quad\text{in }\mathbb R^N, \qquad p^*_{\sigma}=\frac{p(2N-\sigma)}{2(N-p)}, \] in the complementary range $2p<\sigma<N$, beyond the quasilinear theory previously available for $\sigma\le2p$ [Math. Ann., 391(2):2653–2708, 2025]. This range is structurally different: $p^*_{\sigma}<p$, so the effective potential obtained by factoring out $u^{p-1}$ becomes singular as $u\to0$. Nevertheless, whenever $p^*_{\sigma}>1$, every nonnegative finite-energy weak solution is automatically locally bounded. Our proof provides a quasilinear regularity mechanism adapted to this complementary range, where the potential-theoretic route available for $\sigma\le2p$ breaks down. For positive finite-energy solutions with $p^*_{\sigma}>1$ we moreover uncover a sharp transition governed by the sign of $\sigma(p-2)+2p$. If this quantity is positive, the classical $p$-harmonic profile persists. At the threshold, \[ u(x)\simeq |x|^{-\gamma_h}(\log|x|)^{1/(p-p^*_{\sigma})}, \qquad \gamma_h=\frac{N-p}{p-1}, \] with the matching two-sided gradient rate. Below the threshold, \[ u(x)=A_*|x|^{-\gamma_*}+o(|x|^{-\gamma_*}), \qquad \gamma_*=\frac{2(N-p)(\sigma-p)}{p(\sigma-2p)}<\gamma_h, \] and the rescaled gradients converge uniformly to the corresponding radial profile. In all three regimes the solutions are radially symmetric about some point and strictly decreasing in the radial variable.