Papers

Let $f$ be a Steinhaus random multiplicative function and let $g$ be a polynomial of degree $d$. Write $S_N=N^{-1/2}\sum_{n\le N}f(n)\,e(g(n))$. We prove a quantitative dichotomy for the moments of $S_N$: for each integer $s\ge 2$, either $\mathbb{E}|S_N|^{2s}$ is close to the Gaussian moment $s!$, or the coefficients of $g$ can be approximated by rationals with a small denominator. In particular, if the coefficients of $g$ satisfy a Diophantine condition, then $S_N$ converges in law to a complex normal distribution with mean $0$ and variance $1$. Lean 4 code for the proofs is provided, for convenience of verification.

Bourneuf, Charbit and Thomassé [BCT25] showed that the domination number of a tournament can be bounded as a function of its fractional dichromatic number. The function proved was exponential and the tools were based on VC-dimension. In this paper, we present two new proofs of this theorem. The first proof is based on a reduction to the problem of bounding the domination number of a $(1/2-\epsilon)$-majority tournament, for which [BCT25] and Charikar, Ramakrishnan and Wang [CRW26] gave tight bounds. This proof yields the same exponential bound on the domination number as in [BCT25]. The second proof gives a quasilinear bound for the domination in terms of the fractional dichromatic number. It was obtained via AI and was inspired by the recent book proof of the existence of a Condorcet Winning Set of size five due to Ramakrishnan [Ram26].

We study the nonlinear stability of the quadratic plane Poiseuille flow $U(y)=(y^2,0)$ for the two-dimensional incompressible Navier–Stokes equations on $\mathbb T\times\mathbb R$. We prove that for initial perturbations of size $O(\nu^{2/3})$ in a $\nu$-dependent weighted Sobolev space, the flow is globally nonlinearly stable. In particular, the nonzero streamwise modes experience enhanced dissipation and decay exponentially on the characteristic time scale $O(\nu^{-1/2})$, while the zero mode remains uniformly controlled. Thus, in this weighted Sobolev topology, \(O(\nu^{2/3})\) is a sufficient stability scale without logarithmic loss.

For a finite group $G$, define the field $\mathbb{Q}(G) := \mathbb{Q}(\{\chi(g) : \chi \in \operatorname{Irr}(G) , g \in G\})$. In this article we compute generating sets for the fields $\mathbb{Q}(G)$ when $G$ is a finite general or special linear group, a finite general or special unitary group, or one of the simple groups $\operatorname{PSL}_n(q)$ or $\operatorname{PSU}_n(q)$. We then apply these results to classify all number fields $F$ of degree $2$ or $3$ over $\mathbb{Q}$ such that $F = \mathbb{Q}(S)$ for some non-abelian simple group $S$.

We study forward orbits in $\mathbb Z^2$ generated by the expanding affine maps $(x,y)\mapsto(mx+i,ny+j)$, where $m\ge n\ge2$ are integers and $(i,j)$ ranges over a nonempty digit set $\Lambda\subseteq\{0,\ldots,m-1\}\times\{0,\ldots,n-1\}$. We obtain explicit formulae for the mass, Beurling, discrete packing, and Assouad dimensions, the discrete Hausdorff dimension and its lower variant, and the lower entropy index. When $m=n$, all these dimensions equal $\log_m\#\Lambda$. When $m>n$, they can differ, and several depend on the starting point through the sizes of the endpoint columns of $\Lambda$. The two Hausdorff dimensions coincide and admit a pressure variational formula. Our proofs use finite symbolic models and approximate squares to relate digit counts to coverings in centered windows, translated windows, and annuli. We also characterize invariant lattice sets and determine their dimensions. Finally, we compare the orbit dimensions with semigroup growth and the dimensions of the compact dual attractor. In particular, the mass and Beurling dimensions need not equal the semigroup growth exponent, in contrast to the corresponding one-dimensional theory.

We employ the set-theoretic methods yielding the consistency of the existence of nontrivial pairwise disjoint covers of the unit interval (or equivalently $\mathbb{R}^n$ for $n\in\mathbb{N}\setminus\{0\}$) by less than $2^\omega$ closed sets in the context of covers of infinite dimensional Banach spaces $\ell_1(\kappa)$ for infinite $\kappa$ under geometric conditions arising in tiling theory and in approximation theory. Specifically, for $\kappa=\omega_1, \omega_2$, using side-by-side Sacks forcing we prove the consistency of an arbitrarily large continuum above $\kappa$ with the existence of a highly disconnected proximinal set in $\ell_1(\kappa)$, built from norm-compact pieces separated by a common positive distance, and the existence of a normal disjoint tiling of $\ell_1(\kappa)$ by star-shaped bodies of the form $X+B$, where $X$ is norm compact and $B$ is the unit ball. We also prove that if $\kappa$ is an uncountable cardinal of countable cofinality, then $\ell_1(\kappa)$ does not admit any normal disjoint tiling by bodies of the form $X+B$, where $X$ is closed and norm-separable and $B$ is the unit ball. These results complement a result of Klee of 1981 obtained for $\kappa$ satisfying $\kappa^\omega=\kappa$, recent results of De Bernardi, Russo, Sezgek and Somaglia, and a $\mathsf{ZFC}$ a machine-discovered result (included in the appendix) that there are no nontrivial discrete Chebyshev sets in $\ell_1(\kappa)$ when $\kappa<2^\omega$.

In this paper we study a class of 1-homogeneous solutions of the one-phase problem coming from the Cartan isoparametric cubic polynomials, which exist only in dimensions $d=5,8,14,26$. Our main result shows that they are unstable in dimensions $d=5,8,14$, and minimizing in dimension $d=26$. This provides the first example of a singular minimizing cone in the Alt-Caffarelli problem that is neither axially symmetric nor of Lawson-type.

We extend the notion of robust viscosity solutions to semilinear rough partial differential equations (RPDEs) of the form \[ \partial_t u + \mathcal{L}_t u + f(t,x,u,\sigma^{\top}\nabla_x u) + h(t,x,u)\,\diamond dW = 0, \quad u(T,x)=\xi(x), \] driven by a discontinuous path $W$ of finite $q$-variation for some $q<2$. This includes, in particular, typical paths from a broad class of Lévy processes, for instance from $\alpha$-stable Lévy processes with $\alpha<2$. Our notion of solution is formulated through a solution map that agrees with the classical viscosity solution for smooth drivers and is continuous in the terminal condition and the driver, with both the driver and the solution paths viewed in a suitable Skorokhod-type decorated-path metric \cite{chevyrev_superdiffusive_2024}. These requirements naturally imply that robust viscosity solutions must employ Marcus-type jumps, and we establish well-posedness, a flow property, and a stochastic representation in terms of the rough BSDEs with discontinuous Young drivers \cite{becherer_rough_2026}. For Lévy processes of finite $q$-variation with $q<2$, our notion of RPDE solutions provides a pathwise interpretation for corresponding SPDEs, which are seen to be Markov processes in an infinite-dimensional function space.

We prove uniqueness of the blow-up at every singular point of the one-phase free boundary problem for which one blow-up has an isolated singularity. The result applies to Lipschitz stationary solutions and does not require any integrability assumption on the cone. In this sense, our result completes the picture for uniqueness of tangent cones at isolated singularities in the one-phase problem. Inspired by Simon's work in the minimal surface setting, we prove an infinite dimensional Łojasiewicz inequality for the spherical Weiss' energy. This yields an epiperimetric inequality for non-minimizing solutions, which leads to the uniqueness result.

We study a new quasilinear system of balance laws (i.e. conservative first-order PDEs with source terms) for the numerical simulation by a front-capturing approach of viscoelastic athermal multi-material flows with sharp interfaces (like a mixture of immiscible fluids and solids). It naturally extends with an energy balance that is meaningful to stability and thermodynamics. In particular, the unequivocal definition of solutions to Cauchy problems can be guaranteed in some cases, thanks to a symmetric-hyperbolic formulation.

We study a one-parameter family of double octic Calabi-Yau threefolds with two degenerations of type $K$. For each of these degenerations we construct a semistable model and compute the limiting mixed Hodge structure. We show that the $\ell$-adic Galois representation on the limiting cohomology is isomorphic (up to semisimplification) to the direct sum of the Galois representation attached to a weight-three modular form associated with a singular K3 surface and its Tate twist. This gives an explicit realization of the expected relation between type $K$ degenerations of Calabi-Yau threefolds and K3 surfaces.

We study a non-stationary queueing system with multiple parallel servers and a finite buffer capacity, over a finite planning horizon. Motivated by an operation that is central to an EV (electric vehicle) battery-swapping station – how to best manage the charging of batteries swapped off from EVs, we develop a continuous-time MDP (Markov decision process) approach to the problem. The main technical challenge is to allow both the arrival rate and the service-cost rate to be functions of time with jumps at discrete time points, key features of the EV swapping station's operation. We derive an exact and explicit threshold structure that characterizes the optimal control policy. Moreover, we give conditions under which the thresholds are monotone over time, and with respect to key system parameters such as service and buffer capacities. Extensive numerical experiments are carried out to supplement as well as validate the theoretical results, and also to illustrate the robustness of the threshold structure in practical implementations.

Successive weaving segments (SWSs) on urban expressways are bottlenecks prone to recurrent congestion and collisions, requiring fine-grained active traffic management (ATM). Existing approaches struggle to balance the adaptive performance of data-driven optimization with the resilience and transferability of model-based control. We propose a hybrid framework to coordinate lane-level variable speed limits (VSLs) and ramp metering across SWSs. First, we reconstruct L-METANET, a lane-level macroscopic traffic flow model that captures free and forced lane changes. Second, we combine XGBoost-SHAP with a random-parameters binary logit (RPBL) model to derive analytical equations for merging and diverging collision risks and formulate system cost and reward functions. Third, we develop MPC-STMAPPO, a hierarchical controller integrating model predictive control (MPC) and multi-agent reinforcement learning (MARL). Its upper MPC layer uses L-METANET for long-horizon rolling optimization and generates baseline commands; its lower spatiotemporal MAPPO (ST-MAPPO) layer, enhanced with Mamba cells and graph attention, produces residual actions for short-horizon adjustment. Real-world experiments on the 18-km Eastern Expressway in Changchun, China, show that L-METANET accurately reproduces lane-changing-induced flow redistribution and capacity drops, with state evolution aligned with ground truth. XGBoost-SHAP-RPBL achieves AUCs above 0.80 in most tasks, outperforming conventional logit models. MPC-STMAPPO converges faster and performs better across multiple metrics than MPC- and MARL-based baselines. Under randomly fluctuating demand, it also significantly outperforms pure MARL in generalization, demonstrating strong potential for industrial deployment.

We study the exact Euclidean ball-proximal point method for proper, closed, convex functions, where each iteration minimizes the objective over a ball centered at the current point. Retaining the objective gap in the decrease of squared distance yields sharper bounds on objective values, stationarity, and the symmetric Bregman distance to a minimizer. For constant radius $t>0$ and initial distance $D_0>0$ to the solution set, the objective gap after $K$ iterations is at most its initial value multiplied by $\exp(-2K^2t^2/D_0^2)$. We characterize convergence for arbitrary positive radius sequences. If their sum diverges, the method reaches a minimizer in finitely many iterations whenever one exists; otherwise, the objective values converge to the infimum and the iterates escape every bounded set. If the radii are summable, the iterates converge to a possibly nonoptimal point. Self-contraction gives finite trajectory length for bounded iterates and a constant-radius termination bound $O_d(1+D_0/t)$, whose implicit constant depends only on the dimension. A polyhedral family shows that the dimension-independent quadratic bound remains asymptotically sharp. We also identify a minimum successful geometric decay factor and analyze adaptive radius rules based on subgradients or objective gaps, epigraph reformulation, and relaxed updates. Together, these results strengthen the foundations and convergence guarantees of the method without assuming smoothness or strong convexity.

We prove the existence of a non-trivial continuous stationary reversible dynamics on the directed landscape as a subsequential scaling limit of Brownian last passage percolation under the Ornstein–Uhlenbeck dynamics. Strong passage-time stability estimates from the companion paper Bhatia '26, together with static endpoint regularity, give tightness at the critical dynamical scale $n^{-1/3}$. To establish non-triviality of the subsequential limiting dynamics, we show that, in dynamical BLPP, the fractional overlap between geodesics at times zero and $a n^{-1/3}$ tends to one as first $n\to\infty$ and then $a\downarrow0$. The proof refines the excursion argument of Ganguly-Hammond '24, using strong passage-time stability and the fact that a fixed dynamical time is increasingly subcritical at finer spatial scales.

Let $d \ge 2$, let $\alpha_1,\dots, \alpha_n > 0$, and put $A = \sum_k \alpha_k$. We determine, up to constants depending only on $d$, the smallest possible $L^1$ norm in the unit ball $\mathbb{B}^d\subset \mathbb{R}^d$ of the Coulomb field generated by charges of strengths $\alpha_k$ placed on the unit sphere $\mathbb{S}^{d-1}$. More precisely, we prove \[ \inf_{x_1,\dots,x_n \in \mathbb{S}^{d-1}} \int_{\mathbb{B}^d} \left| \sum_{k=1}^n \alpha_k \frac{x_k-x}{|x_k - x|^d} \right| dx \asymp_d A^{-1/(d-1)} \sum_{k=1}^n \alpha_k^{d/(d-1)}. \] The lower bound follows from an $L^1$ Lipschitz trace estimate and an explicit weighted tent function on the sphere. For the upper bound we use a partition of the sphere into cells of masses $\alpha_k/A$ and diameters $O_d \left((\alpha_k/A)^{1/(d-1)} \right)$. For the unit weights the sharp order is $n^{(d-2)/(d-1)}$, and in dimension three it is $\sqrt{n}$.

We show that under mild assumptions a functor is a prefibration if and only if it has terminal objects in its fibers and admits precartesian liftings of monomorphisms. We use this to show that the following conditions on a pointed category $\mathbb{C}$ are equivalent: (a) The category of morphisms of $\mathbb{C}$ admits generic split extensions; (b) Each functor category of $\mathbb{C}$ with finite domain category admits generic split extensions; (c) $\mathbb{C}$ admits normalizers and generic split extensions; (d) The kernel functor from the category split extensions in $\mathbb{C}$ to $\mathbb{C}$ is a prefibration. In addition we show that the category of morphisms of a pointed protomodular $\mathbb{C}$ admits generic split extensions if and only if for each morphism $f:X\to Z$ the functor sending each object $B$ to the isomorphism class of split extensions in $\mathbb{C}^{\mathbf{2}}$ of $(B,B,1_B)$ with kernel $(X,Z,f)$, is representable.

We prove that the sheaf $\infty$-topos associated to a geometric site in the sense of Lurie can be written as a partially lax limit of smaller sheaf $\infty$-topoi. This is thus a formalization of Lurie's vision for fractured $\infty$-topoi to axiomatize the relation between gros and petit topoi. As a consequence, we realize the $\infty$-topos of $\kappa$-small condensed anima as a partially lax limit of sheaf $\infty$-topoi over extremally disconnected spaces, marked at the open embeddings. Moreover, we deduce a gros version of the étale exodromy theorem.

We consider Brownian last passage percolation (BLPP) in which the Brownian increment process on each unit horizontal interval is independently resampled at rate one. By combining strong passage-time stability estimates with a multiscale analysis of static near-optimal paths, we show that, for every $\varepsilon>0$, the union of all geodesics between two KPZ-scale rectangles of transverse width of order $n^{2/3}$ and longitudinal length of order $n$, separated by a distance of order $n$, visits at most $n^{1+\varepsilon}$ unit horizontal cells in the bulk during the critical time interval $[0,n^{-1/3}]$, both in expectation and with stretched-exponentially high probability. We also obtain the quantitative bound $n\exp\{C(\log\log n)^2\}$ on the expected hitset size, with a corresponding failure probability at most $Ce^{-c(\log n)^2}$. Using this, we establish that the set of times admitting a non-trivial bigeodesic has almost surely zero Hausdorff measure for the subpolynomially decaying gauge $H(r)=\exp\{-L(r)^2\log L(r)\}$, where $L(r)=\log\log(1/r)$, as $r\downarrow0$. In particular, this set almost surely has Hausdorff dimension zero. For each fixed deterministic non-axial direction, we further show that almost surely no time admits a bigeodesic in that direction.

We study the hard-core model on recursively defined sequences $(G_n)_{n\geq0}$ of graphs with a fixed number $k\geq 1$ of labeled vertices in each graph. The next graph in the sequence is constructed by taking a fixed number $m\geq 2$ of copies of the previous graph, connecting these copies by identifying some labeled vertices according to a fixed rule, and afterward choosing $k$ labeled vertices in the resulting graph, again in accordance with a fixed rule. Examples of such sequences include the Sierpiński gasket graphs, hierarchical lattices, and many more. We prove that, when the vertex degrees of the graphs $G_n$ are uniformly bounded and the distances between the labeled vertices in $G_n$ diverge, the complex zeros of the univariate independence polynomials $Z_{G_n}(\lambda)$ avoid a neighborhood of the non-negative real axis. By the Lee–Yang theory this implies that no phase transitions occur for the hard-core model on these recursive sequences of graphs, independently of the starting graph $G_0$. The proof relies on the study of the dynamical properties of a one-parameter family of rational maps $F_\lambda$ on the $(2^k-1)$-dimensional complex projective space induced by the graph recursion operator. The dynamical framework developed in this paper can be naturally extended to other classical models in statistical mechanics (such as the Ising or Potts models) and to more general notions of graph recursions.

This paper studies stochastic algorithms for minimizing paraconvex functions, a function class that generalizes weakly convex functions and includes, for instance, Hölder smooth functions and compositions of convex functions with Hölder smooth maps. We first establish the convergence of the stochastic subgradient method and the stochastic proximal subgradient method for unconstrained and composite paraconvex optimization, respectively. The analysis is then extended to a broad family of algorithms that access the objective through stochastic models satisfying certain approximation-quality and continuity assumptions. The main principle that underlies the convergence analysis is that the algorithms under consideration can be interpreted as perturbed descent methods on the generalized Moreau envelope of the original paraconvex problem.

We study the multifractal analysis of self-affine measures supported on planar carpets of type Gatzouras–Lalley. We work primarily with Olsen's Hausdorff-dimensional variant of the $L^q$ spectrum, which we denote by $\vartheta(q)$. We prove a variational formula for $\vartheta$, from which it follows that multifractal formalism holds at all values $q$ for which $\vartheta(q)$ is differentiable. However, in general, $\vartheta$ need not be differentiable and the multifractal formalism may fail. In fact, the multifractal spectrum may be non-differentiable and have jump discontinuities in the interior of its support, and there may be multiple phase transitions at both negative and positive values of $q$.

We study the loss of observability for a class of strongly coupled parabolic systems as the coupling parameter tends to zero. The system is observed through a fixed matrix $B$ on a measurable subset of space-time with positive measure. Under the Kalman rank condition, we determine the exact blow-up rates of the optimal observability cost. Different components of the state, as well as different combinations of them, may have different blow-up rates, and these rates are determined by the algebraic structure of the pair $(B,K)$. The full-state observability cost is determined by the largest of these rates. We also show that the worst-case minimum $L^\infty$-norm of null controls has the same asymptotic behavior.

The set-coloring Ramsey number $R(k;r,s)$ is the least $N$ such that every assignment of an $s$-element subset of $[r]$ to each edge of $K_N$ yields a copy of $K_k$ whose edges share a common color. For every fixed prime power $q$, we construct infinitely many positive integer triples $(r,j,s)$ with $j\sim(q-1)^{-2/3}r^{1/3}$ and $s=(1-1/q)(r-j)$ such that $R(q+1;r,s)=\Theta_q(r^{4/3})$. For $q=3$, this answers in the affirmative a question of Conlon, Fox, Pham and Zhao, showing that polynomially superlinear growth for $R(4;r,2(r-j)/3)$ already occurs at the scale \(j=\Theta(r^{1/3})\). Moreover, along the same sequence, the maximum size of a $q$-ary code of length $r$ and minimum Hamming distance at least $s$ is $(1+o(1))(q-1)^{4/3}r^{4/3}$.

We study heterotic $G_2$ structures near a torsion-free standard embedding, to first order in $\alpha'$ and to all orders in formal deformations. The tangent connection is the Hull connection determined by the metric and flux. We construct a locally invertible family of differential operators relating the supersymmetry and instanton $BPS$ equations to the Euler-Lagrange equations of the heterotic superpotential. Using this comparison and the physical gauge symmetries, we construct a homological vector field locally formally conjugate to the Hamiltonian vector field of the superpotential. On a closed manifold in the fixed gerbe sector, the torsion equation implies vanishing of the leading flux. We prove ellipticity of the linearized deformation complex with mixed differential-order weights and construct a finite-dimensional cyclic $L_\infty$ model with a formal effective potential.

In this paper, we establish an explicit isomorphism between Drinfeld and R-matrix presentations of the quantum affine superalgebra associated with the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. This result can be viewed as a supersymmetric analogue of the isomorphism between two presentations of the quantum affine algebra $U_{q}(\widehat{\mathfrak{gl}(n)})$ proposed by Frenkel-Ding. Furthermore, employing the quantum Berezinian of $U_{q}(\widehat{\mathfrak{gl}(m|n))}$, we construct the R-matrix presentation of the quantum affine superalgebra $U_{q}(\widehat{\mathfrak{sl}(m|n))}$ and prove the isomorphism between its R-matrix and Drinfeld presentations. Our proof based on the commutative relation between Gaussian generators, $\mathbb{A}$-form and quantum Berezinian.

We solve the perpetual liquidation problem for a geometric multi-skew Brownian motion carrying local-time pushes upward at a support level and downward at a resistance level, a model of technical analysis that is Markov in the price alone. Three geometries arise, separated by a closed-form criterion: the continuation band lies below resistance, straddles it, or pins its upper edge there. Selling into resistance is a conclusion rather than an assumption. Identification is asymmetric: exercise behaviour determines the support parameters exactly, while the strength of resistance is recoverable only on an interval with a closed-form endpoint.

Low-power error-correcting cooling (LPECC) codes and constant-power error-correcting cooling (CPECC) codes provide error correction while controlling power consumption and thermal effects in on-chip buses. In this paper, we study binary CPECC and LPECC codes with \(e=w-3\). For CPECC codes, we extend the applicability of the upper bound previously obtained by Zhao and Zhang from the quadratic-order condition \(w\ge 2t(t+1)+2\) to \(w\ge w_0(t)\), where \(w_0(t)\sim \sqrt{2}\,t^{3/2}\). Using Steiner systems, we show that the CPECC bound is attainable and asymptotically tight for fixed \(t,w\). For LPECC codes, we establish the new upper bound \(\left\lfloor\frac{\binom{n+2}{3}}{\binom{w+t}{3}}\right\rfloor\) for \(w\ge \mu(t)\), where \(\mu(t)\sim \sqrt{2}\,t^{3/2}\). This bound is strictly smaller than the previous bound of Zhao and Zhang whenever both apply, and is asymptotically tight for fixed \(t,w\) in the stated range.

Long contexts are central to modern transformer systems, but most expressivity results choose a different network for each fixed sequence length. We study whether one masked transformer can approximate causal token-to-token maps uniformly over sequences of arbitrary length sampling a fixed normalized horizon. To relate sampling resolutions, we model tokens by $\alpha$-Hölder sequences or, more generally, a common modulus of continuity. Our notion of continuity across resolutions characterizes the causal families admitting uniform approximation on these compact input classes by a single transformer with length-independent parameters. The result extends to the infinite-length mean-field limit, where tokens form continuous curves and masked attention becomes a causal time integral. For bounded regression with target maps satisfying a $\beta$-smooth stability condition defined using regular test functions, quantitative approximation yields a generalization bound: exact empirical risk minimization over suitably sized bounded-weight transformers gives root mean-square prediction error $O((\log\log N/\log N)^{\beta/(d+2)})$ from $N$ iid labeled sequences. The bound holds at fixed confidence on the same sampling distribution, with $d$ the token dimension and no maximum-length factor. Finally, experiments on physical time series support the Hölder-regular token model at observed scales, with dataset-dependent fitted exponents, whereas text input embeddings provide a contrasting case. Native and dense sampling, shuffled controls, and refinement checks delimit this empirical regularity regime.

The kissing-number problem is a classical problem in discrete geometry whose exact solution is known in only a few dimensions. Recent artificial-intelligence approaches have begun to discover improved configurations through large-scale numerical and combinatorial search, but converting such searches into general mathematical constructions and rigorous proofs remains challenging. Here we use Qiushi Engine, an autonomous multi-agent research system, to investigate kissing numbers and obtain new lower bounds in nineteen dimensions: $25$, $27$, $32$–$39$, $43$, $45$, and $49$–$55$. The resulting constructions arise from distinct structural mechanisms, including coordinated motions of contact layers, labelled direction reuse, joint support exchanges, signed-code replacements, cross-shell lattice constructions, low-overlap lattice isometries, and spherical-design moment certificates. They yield sharp capacities for parameterized signed-code models, deterministic image-union guarantees, and exact section and projection counts controlled by embedded root systems and anchor-graph statistics. These methods yield, among others, $K(25)\ge197580$, $K(27)\ge201567$, $K(38)\ge591900$, $K(43)\ge2553792$, $K(45)\ge7380090$, and $K(55)\ge53301140$. The autonomous system carried out the construction searches, mathematical analysis and computational verification, while each final result was reduced to explicit mathematical arguments and independently checkable finite certificates. Our results illustrate how autonomous research systems can move beyond optimization within fixed formulations to discover new mathematical representations and constructions at scale.