Papers

We prove that the $L$-packets for $p$-adic $\mathrm{G}_2$ constructed by Gan and Savin satisfy the endoscopic character identities. In the course of the proof, we also prove twisted endoscopic character identities for functorial lifts from $\mathrm{G}_2$ to $\mathrm{PGSO}_8$ with triality. As a byproduct, we also prove a global multiplicity formula for the discrete automorphic spectrum of $\mathrm{G}_2$ coming from the cuspidal automorphic spectrum of $\mathrm{PGL}_3$.

Solvency games are a gambling problem on infinite-state Markov decision processes in which the state $n \in \mathbb{N}$ represents an investor's fortune. In every round, the investor chooses an action from a finite action set, and every action yields a distribution over integer-valued gains in an interval $\{-\ell,\ldots,m\}$. The risk-averse investor wants to minimise the probability of eventual ruin (reaching a fortune $\le 0$). It was shown in [Berger et al.] that memoryless deterministic optimal strategies exist, but they are not eventually constant in general. Even in the special case of gains in $\{-2,\ldots,1\}$, the optimal strategy may need to make use of two different actions at arbitrarily high fortunes. We show that optimal strategies in solvency games need not be ultimately periodic in general (thus disproving a 2012 conjecture of Kučera). Already in the case of gains in $\{-3,\ldots,1\}$, it is possible for the optimal strategy to be unique but aperiodic. For gains in $\{-2,\ldots,1\}$, there always exists an ultimately periodic optimal strategy whose tail is constant or alternates between two actions. Finally, we show that the optimal strategy is computable if it is unique. Moreover, (some) optimal strategy can always be computed in the case of gains in $\{-\ell,\ldots,1\}$ for any $\ell \in \mathbb{N}$. Computability in the general case however remains open.

Buckling of thin cylindrical shells under axial compression, a classical example of a subcritical instability, is highly sensitive to small imperfections, with minute geometric variations causing large changes in buckling threshold. To uncover the origin of this sensitivity, we use numerical continuation and bifurcation analysis while systematically varying the depth and size of a single localized Gaussian defect. We show that the instabilities of the imperfect shell originate from localized equilibria already present in the perfect shell. By breaking translation symmetry, the defect pins these equilibria and changes how they connect to the imperfect base state. Small changes in defect geometry can thereby switch the bifurcation that triggers buckling, producing non-monotonic and discontinuous changes in buckling threshold and abrupt changes in buckling mode. Imperfection sensitivity is therefore not simply sensitivity to imperfection magnitude, but sensitivity of the underlying bifurcation structure to imperfection geometry.

Let $p_n$ be the $n$th prime, and let $r,s\ge2$ be fixed multiplicatively independent integers. We count the integers up to $x$ whose prime factors all have the form $p_{r^a s^b}$ with integers $a,b\ge0$. Our asymptotic formula for this count has relative error $o(1)$ and is explicit down to the multiplicative constant. For $(r,s)=(2,3)$ these integers are the prime codes of the ordinals below $\omega^{\omega^2}$, so the formula settles that case of the counting problem of Vernaeve, Vindas and Weiermann.

We present an efficient wideband construction of directional $\mathcal{H}^2$-matrices for the electrical field integral equation that, in contrast to existing constructions, supports not only box trees but also geometry-adaptive cluster trees. To accommodate geometry-adaptive cluster trees, we determine the number of directions from the electrical size of each cluster (instead of the level of a box tree), construct the directions using Spherical-Fibonacci points, and establish a hierarchy between the direction sets of a cluster and its children through an angular nearest-neighbor mapping. We construct the nested directional representation using the incomplete adaptive cross approximation, for which we introduce a robustified tree-mimicry pivoting strategy that prevents premature convergence for block-structured matrices arising from certain geometries and meshes. Numerical results demonstrate that the proposed approach achieves the desired accuracy, requires no more storage than the octree-based construction and substantially less when the geometry or discretization is poorly matched to octree clustering, and exhibits the expected $\mathcal{O}(N\log N)$ scaling for high-frequency problems.

We prove that the twisted endoscopic character identities between triality $\mathrm{PGSO}_8$ and the endoscopic group $\mathrm{G}_2$ imply the Hiraga-Ichino-Ikeda conjecture on formal degrees for $\mathrm{G}_2$. In the course of proving the main result, we also establish some fundamental results on representation theory of $\mathrm{PGSO}_8$ with triality. A key feature of our approach is an application of the adjoint group of type $E_6$.

Let $V$ be a complex symplectic vector space and let $\mathbb{P}(V)$ carry the induced contact structure and symplectic polarity. We study the loci of hypersurfaces containing codimension-two cones supported on hyperplanes. The starting point is the following characterization: a projective subvariety $X\subset\mathbb{P}(V)$ which is a hypersurface of degree at least two in a hyperplane $H$ is involutive if and only if it is a cone whose vertex contains the polar point $\sigma(H)$. We give a short proof, construct the corresponding incidence spaces, compute their dimensions and express the incidence degrees as Segre-class integrals. In $\mathbb{P}^3$ the incidence map is birational for every $d\ge m\ge 2$ except $(m,d)=(2,2)$, and the same holds in $\mathbb{P}^5$; the quadratic case is exceptional in every dimension, the incidence having generic degree $2n$ in $\mathbb{P}^{2n-1}$. We also give a sufficient criterion for birationality in higher dimension, closed formulas in $\mathbb{P}^3$ and coefficient formulas in $\mathbb{P}^5$. For quadric cones in $\mathbb{P}^5$ the degree is the product of shifted binomial factors and an irreducible polynomial of degree 36, and we explain where the shifted factors come from.

Given red and blue points drawn from Poisson point processes of equal intensities, we match them by locally minimizing a cost function. In our paper, we identify all possible measurable cost functions on $\mathbb{R}^d$ such that the resulting matching is scale invariant. Our cost functions can be asymmetrical, which we refer to as unbalanced cost functions. The investigation is motivated by arXiv:2012.07129 where the authors consider scale invariant cost functions with added regularity assumptions. Furthermore, we discuss how different the unbalanced matchings are compared to these more regular cost functions for points on $\mathbb{R}$.

What can and cannot be predicted by a model when only limited information is available? We answer this question by constructing a ladder of information limits that organizes prediction tasks according to their information requirements: from memoryless and memory-augmented trajectory forecasts to event prediction, stationary statistics, and generative laws. The analysis, which is independent of the model structure or architecture, identifies the minimum attainable error along with the information limitations that give rise to it. The approach distinguishes information hidden in unresolved variables from information recovered through time-delayed observations and connects these contributions to Mori–Zwanzig memory. The ladder also reveals why Lyapunov growth alone cannot characterize reduced-order prediction error, why stationary statistics may remain predictable even when individual trajectories become unpredictable, and why stochasticity can represent (but cannot recover) missing information. Numerical studies of the Kuramoto–Sivashinsky equation and the Lorenz system illustrate these results across the ladder. This unified perspective provides a principled language for understanding the fundamental limitations of ROMs for a given prediction task and for clarifying whether improved predictions require richer observations, greater precision in the input, or additional memory.

We define a quadratically enriched pushforward for proper morphisms between smooth, proper Deligne–Mumford stacks over an arbitrary field using Grothendieck–Serre duality. We apply this pushforward to define quadratically enriched Euler classes and Euler numbers of relatively oriented vector bundles on smooth, proper Deligne–Mumford stacks. Finally, we define quadratically enriched fundamental classes for smooth, proper schemes over an arbitrary field, and we use these to propose a definition of quadratically enriched quantum $K$-invariants when the moduli space of stable maps is smooth, tame, and relatively oriented.

In evidence-based policymaking, typically one experimental sample is observed, then a learned policy recommendation is implemented at scale. Policies learned from the experimental data can perform well in expected welfare, yet random sampling in the experiment can produce recommendations with poor welfare outcomes. In this paper, we ask: how should policy learning algorithms balance expected welfare against sampling risk? Our main contribution is to show that algorithmic stability plays a central role in characterizing and navigating the tradeoff. Intuitively, if a policy learning algorithm's recommendation remains stable when one experimental unit is replaced, then that algorithm has limited sampling risk. We propose a method for policy learning called policy-vote bagging, which learns treatment decisions on many subsamples then averages their votes into treatment probabilities. Relative to using one subsample, averaging across subsamples preserves expected welfare and improves expected utility for a risk-averse researcher. We derive sharp bounds linking estimation accuracy, subsample size, and welfare variation, including an exact guarantee under CARA utility.

The exact capacity of the binary deletion channel remains unknown despite decades of work on achievable rates and converse bounds. We establish a computer-assisted approximation whose error is below $0.0095$ bits per transmitted bit, uniformly over all deletion probabilities. The mean certified error bound, with uniform weighting of deletion probability, is below $0.006522$. The estimate is the midpoint of explicit lower and upper bounds. For the converse, a stationary-source reduction is combined with finite inequalities covering every allowed input configuration. Two constructions control the unobserved input beyond a finite window: one uses a common outside survivor sequence and bounds omitted deletion patterns, while the other cancels an entropy term to make outside probabilities enter linearly. For the lower bound, finite-state inputs combine output-entropy estimates with selected disjoint counts of compatible deletion masks; independent-run inputs retain additional uncertainty about output-run boundaries. Directed numerical checks establish the finite inequalities. An analytic comparison between deletion probabilities then extends the pointwise bounds over the entire parameter range. The lower endpoint supplies rates within $0.019$ bits of capacity in the asymptotic coding sense. We give the derivations, recorded computational costs, and complete numerical inputs and programs needed to verify the result.

This paper studies axially symmetric Abrikosov-Nielsen-Olesen vortices in the quartic Abelian Higgs model as the magnetic flux becomes arbitrarily large. The emphasis is on developing the rigorous asymptotic analysis behind the formal large-flux expansions of Hakim, Lemaitre, and Mallick '01 and Dumitrescu, Gaikwad '25. For every coupling constant less than 4 we prove that the radius of the magnetic core is of the size predicted by the physics literature. We identify the interface between then normal and super-conducting regimes with the classical normal-to-superconducting transition layer in Ginzburg-Landau theory. The theorem of Chapman-Howison-McLeod-Ockendon '91 proves its existence, uniqueness and monotonicity. The connection of the Gaussian tail of the scalar field on the left with the interface is made directly with the Weber equation. The core Higgs equation is treated as a semiclassical Sturm-Liouville problem in the left overlap region, where the Higgs field decays like a Gaussian. The exterior region is described perturbatively around modified Bessel functions. A key step is to show that the matching map between the admissible Cauchy data at the two ends of the linearized interface equations has maximal rank.

Consider a sequence of i.i.d.~trials $X=\{X_1, X_2, \ldots, X_n\}$ where $p(X_i=j)={1}/{d}; j=1,2,\ldots, d$, or more generally $p(X_i=j)=p_j; \sum_{1\le j\le d}p_j=1$. We consider the variable $D$ that counts the number of distinct substrings of all lengths,$1\le k\le n$ in $X$ and prove results concerning $E(D)$.

We prove that exact block recovery in the square Gaussian binary MIMO model can be achieved in polynomial time at the same first-order SNR threshold as exhaustive maximum-likelihood detection. Specifically, for $y=\sqrt{\rho/N}Hx^\star+w, \; x^\star\in\{\pm1\}^N,$ and independent standard Gaussian $H\in\mathbb R^{N\times N}$ and $w$, rounded linear MMSE followed by steepest single-bit descent recovers $x^\star$ with failure probability tending to zero, uniformly over every transmitted word and every $\rho\ge2\log N$, using $O(N^3)$ unit-cost exact-real arithmetic operations. The model is a special case of Gaussian random linear estimation, for which AMP state evolution and replica/MMSE formulas rigorously characterize fixed-parameter normalized performance. Those results predict the same $2\log N$ scale, but do not by themselves yield an all-coordinate guarantee in the dimension-dependent regime considered here. To the best of our knowledge, no prior work gives polynomial-time exact block recovery at the ML boundary for this setting; the closest prior square-system theorem, for the box relaxation, has first-order threshold $4\log N$. The proof places the rounded LMMSE estimate at sublinear Hamming distance from the truth, and then establishes, uniformly over every error set the local search can visit, that some wrong bit offers a quantified cost decrease while an objective barrier confines the search path. Conversely, if $0<\rho\le2\log N-\log\log N-s_N$ with $s_N\to\infty$ and $s_N=o(\log N)$, then a one-bit neighbor beats the transmitted word with probability tending to one, so even ML detection fails. Therefore, the statistical and polynomial-time exact-recovery thresholds coincide to first order in the stated arithmetic model.

In [SX25], we introduced the notion of nondegenerate cylindrical singularity using the ideas from dynamical systems. In this paper, we further study the properties of these singularities, showing that they are stable under small perturbations. Furthermore, we show that we can perturb some mean curvature flow with degenerate singularities to produce nondegenerate singularities.

We provide a tractable sufficient condition for strict comparison for solutions of the one-dimensional stochastic differential equation \[ d X_t^x=\sigma(X_t^x)\,d B_t, \quad X_0^x=x\in I=(\ell,r), \] for $\sigma>0$ and continuous. Our proof is based on a two-dimensional Lyapunov argument, which allows us to prove strict comparison for some coefficients in $W^{1,p}_{\text{loc}}(I)$, $1\le p<2$. We illustrate using $\sigma(x):=1+|x|^\beta$ for $x\in\mathbb{R}$, $\beta \in (0,1)$, and show that strict comparison holds if and only if $\beta\in[\frac12,1)$. We give examples showing that neither $\sigma\in W^{1,p}_{\text{loc}}(\mathbb R)$, $p\in(1,2)$, nor $\sigma\in C^\beta(\mathbb R)$, $\beta\in[\frac12,1)$, is sufficient for strict comparison, even when combined with boundedness, uniform ellipticity, global strong existence, and pathwise uniqueness.

In projective space $\mathbb{P}^r$, the normal bundle of a general rational curve of any degree $d \geq r$ is balanced except over fields of characteristic 2. In arXiv:2404.08102, Coskun, Larson, and Vogt found counterexamples to the naive generalization of this statement to general rational curves in Grassmannians and proposed a conjecture for when the normal bundle is balanced. In this paper, we prove their conjecture for the Grassmannians $G(2, 4)$, $G(2, 5)$, and $G(2, 6)$.

The Pachter-Sturmfels phylogenetic rank of a graph G is the minimal number of metric trees needed to embed G isometrically. Here, all edges of G are of length one and the product of metric trees is endowed with the supremum norm. We develop both a greedy and an exact algorithm for computing phylogenetic ranks. Using our algorithms, we construct a database of phylogenetic ranks which includes all graphs on 6 and 7 vertices. In particular, we exhibit examples disproving that the phylogenetic rank is hereditary, bounded by $\lceil \frac{n}{2} \rceil$, and a generalised 4-point conjecture by Pachter and Sturmfels. In addition, we show that the phylogenetic rank is subadditive under 1-sums and certain 2-vertex-sums, and that it is trivially upper bounded by n-1, where n denotes the number of vertices. We also provide a complete classification of graphs with phylogenetic rank 1 and construct several infinite families with phylogenetic rank $\lceil \frac{n}{2} \rceil$.

We use metric functionals to build a weak topology associated with $d$-weak convergence on metric spaces. We also show the existence of a sequence in the space $C[0,1]$ that is unbounded in the sup-norm yet converges $d$-weakly. This points out an error in an earlier work.

The query and key projections $\WQ,\WK$ in attention are almost always trained by Euclidean optimizers with no geometric constraint. We constrain them to the Stiefel manifold and optimize with a Riemannian Adam carrying one scalar second moment per frame—the form of \citet{becigneul2019}, here extended to the compact, non-Hadamard $\St(d,r)$ with a tangent projector, step-norm cap, and polar retraction. Four propositions prove steepest descent in the embedded metric, gradient-scale independence, well-conditioning, and exact $\mathrm{O}(d)$-equivariance. A fifth records that weight decay has identically zero Riemannian gradient on $\St(d,r)$ ($W{=}WI_r$ lies in the normal space), so decay cannot act on the constrained frames. On a CIFAR-10 patch benchmark at $n{=}10\mathrm{k}$ this rule gains $\mathbf{+6.79}$\,pp over AdamW across 12 paired starts ($t{=}38.33$, $12/12$); earlier fixed-step Riemannian SGD gains $+1.97$\,pp, of which $+1.69$\,pp comes from frozen orthonormal initialization alone. The corrected Adam's lead grows with data: $+1.9$\,pp at $n{=}1\mathrm{k}$ to $+6.7$\,pp at $n{=}50\mathrm{k}$. A 12-seed ablation credits all gain to the scale-free step ($+4.63$\,pp, $12/12$), nothing to the projector or equivariance; a targeted $\varepsilon$-sweep causally confirms the mechanism ($-2.6$\,pp at $\varepsilon{=}0.1$, $p{<}0.001$). Two five-seed grokking studies confirm the constrained arm does not grok better than the baseline ($p{=}0.019$, A2 wins): the weight-decay exemption has no grokking consequence. A single-seed pilot exploiting this localization achieves the first stable grokking under slingshot conditions—Stiefel + targeted circuit regularization keeps routing-frame isometry error $10^6\times$ lower than the unconstrained ablation through every collapse.

We study high-order safety-critical control of quadrotor teams under bounded inputs and pairwise collision-avoidance constraints. Squared-distance barriers may lose thrust effectiveness when the relative displacement is orthogonal to the available thrust directions, while nonsingular constraints may still be jointly infeasible under shared bounds. We characterize both phenomena through pairwise effectiveness and aggregate feasibility measures. A torque-aware dynamic extension exposes attitude torques in a fourth-order barrier and prevents the extended-input row from vanishing under positive thrust. Gaussian processes directly learn the fourth-order HOCBF residual, providing robust margins without differentiating unknown perturbations. Under residual-bound and persistent-feasibility assumptions, the resulting QP guarantees collision avoidance and recovers the nominal input whenever it satisfies the robust safety and actuator constraints.

Assuming Jensen's diamond principle, we construct a separable crowded compact zero-dimensional space $K$ of weight $\aleph_1$ and cardinality $2^{\aleph_1}$ such that $C_p(K)$ has no infinite-dimensional Hausdorff separable linear quotient. In particular, $K$ is an Efimov space: it contains neither a nontrivial convergent sequence nor a copy of $\beta\mathbb N$. We also prove in ZFC that the existence of a separable quotient of this kind is equivalent to the existence of a metrisable one. On the positive side, established finite-support random-sign methods give a split quotient of $C_p(K)$ onto $(c_0)_p$ whenever $K$ is infinite compact and $w(K)<\operatorname{cov}\mathcal N$, with explicit norm-one operators. Consequently, the universal quotient assertion for compacta of weight $\aleph_1$ is independent relative to the consistency of ZFC alone. A real-valued measurable continuum gives the positive result below the continuum and reduces the remaining problem to separable compacta of weight exactly the continuum. The counterexample uses direct diamond prediction, a simultaneous successor lemma, and a compactness argument converting localised measure limits into differences of point masses.

Designing road networks for wind farms in complex terrain is a challenging task, especially under tight construction budgets. Traditional manual methods are time-consuming and may yield suboptimal results. We propose a structured three-phase optimization framework, TriPhase, to automate and minimize road construction costs from corridor selection to earthwork. Phase one formulates corridor selection as a Steiner minimum tree problem over a terrain-aware graph, incorporating slope and curvature constraints. In phase two, each road segment undergoes horizontal alignment optimization using a bilevel model, where a mixed-integer program evaluates vertical alignment costs. To ensure solver compatibility and performance, the model is reformulated explicitly for Gurobi. The final phase applies a network-wide convex optimization model to refine vertical alignment. Numerical experiments on real-world sites demonstrate up to 14% cost savings compared to industry-standard manual designs, validating the framework's effectiveness and practical relevance.

According to the classical Skitovich–Darmois theorem, the Gaussian distribution on the real line is characterized by the independence of two linear forms of a finite number of independent random variables $\xi_j$. This theorem has been extended in various directions. In particular, S.G. Ghurye and I. Olkin established an analogous result for the case where $\xi_j$ are $n$-dimensional independent random vectors and the coefficients of the linear forms are invertible $n\times n$ matrices. Subsequently, A.A. Zinger and later I.A. Ibragimov investigated linear forms of an infinite sequence of $n$-dimensional independent random vectors. In the present paper, we investigate, for the first time, linear forms of a sequence of independent random variables taking values in a second-countable locally compact Abelian group $X$ under either of the following conditions: $X$ contains no nontrivial compact subgroups, or $X$ contains no subgroups topologically isomorphic to the circle group and has a finite topological automorphism group. Furthermore, we investigate the case where the independent random variables take values in an $\mathbf{a}$-adic solenoid $\Sigma_{\mathbf{a}}$. In all these settings, the coefficients of the linear forms are topological automorphisms of the corresponding group.

We compute the convex envelope of (nonconvex) bivariate piecewise linear-quadratic (PLQ) functions (bivariate quadratic functions defined on a polyhedral subdivision). Our algorithm is composed of the following steps: (1) compute the convex envelope of each quadratic piece obtaining piecewise rational functions (quadratic divided by linear function) defined over a polyhedral subdivision; (2) compute the (Legendre-Fenchel) conjugate of each resulting piece to obtain piecewise quadratic functions defined over a parabolic subdivision; (3) compute the maximum of all those functions to obtain the conjugate of the original PLQ function as a piecewise quadratic function defined on a parabolic subdivision; (4) compute the conjugate of each resulting piece; and finally (5) compute the maximum over all those functions to obtain the convex envelope (biconjugate) as rational functions (quadratic divided by linear function) defined over a polyhedral subdivision. Our contribution includes a practical algorithm running in linear time, and proving that the convex envelope is a piecewise rational function.

We compare two mixed Hodge structures attached to a unipotent variation $\mathbb V$ of mixed Hodge–Tate structures on $X=\mathbb P^1\setminus{0,1,\infty}$: the cohomology of its extension by $*$ at $0$ and by $!$ at $\infty$, and the vanishing cycles at $1$. The semi-holonomy isomorphism constructed in [arXiv:2604.26357] identifies these two structures on their underlying rational structures, compatibly with their weight filtrations, but need not respect their Hodge filtrations. We compute the resulting Hodge defect. Its universal noncommutative series is given by a specific component of the inverse Drinfeld associator multiplied by an explicit local monodromy factor. We conclude with a formal $A_\infty$ interpretation of the coefficients of the Hodge defect in terms of weight-framed Tate extensions.

We propose a deep learning framework for the numerical approximation of weakly singular semilinear elliptic equations subject to homogeneous Dirichlet boundary conditions and positivity constraints. These problems are challenging because the nonlinear source term becomes unbounded as the positive solution approaches zero near the boundary. The proposed approach combines three components. First, the Dirichlet boundary conditions are embedded directly into the neural representation through a hard-constraint formulation. Second, a Softplus transformation preserves positivity in the interior of the computational domain, ensuring that the singular nonlinear term remains well-defined during training. Third, a singularity-aware weighted residual loss emphasizes regions where the predicted solution becomes small and the singular behavior is more pronounced. The method is evaluated in the weak singularity regime through numerical reference solutions and controlled verification tests. Comparisons with the standard residual formulation are used to assess approximation accuracy and convergence behavior. The numerical experiments illustrate the potential of combining hard boundary enforcement, positivity preservation, and singularity-aware residual weighting for weakly singular elliptic problems.

Mixed-variable blackbox optimization arises in simulation-based engineering and machine learning, where objective and constraint functions are expensive to evaluate and variables may be continuous, integer, binary, or categorical. Several software libraries are available for derivative-free optimization, but few combine ease of use, computational efficiency, flexibility, and theoretical convergence guarantees. This work presents NomadBBO, a user-friendly Python library for inequality-constrained mixed-variable blackbox optimization built around Nomad. Its core optimization framework is CatADS, a new direct-search method that extends Adaptive Direct Search (ADS) to mixed-variable problems. CatADS combines the mechanisms of ADS for quantitative variables with neighborhoods for handling categorical variables. These neighborhoods use Gaussian process-based or empirical Wasserstein distances derived from available data. Inequality constraints are handled through the progressive barrier. The mixed-variable and surrogate-based mechanisms are implemented in Python and connected to the efficient C++ backend of Nomad via Cython. This integration allows Nomad to handle categorical variables. It also enables the use of external libraries and hybrid optimization strategies, including Gaussian process models from SMT 2.0 and Bayesian optimization. Numerical experiments compare NomadBBO with other solvers on constrained and unconstrained mixed-variable problems from the Cat-Suite benchmark collection. The beta release is available at https://test.pypi.org/project/NomadBBO/

We stabilize the linearized 2-D Poiseuille channel flow at arbitrary Reynolds numbers, and assign the decay of every wavenumber to any user-prescribed rate. The actuation is as in Vazquez–Krstic (IEEE TAC, 2007), VK'07 below: the two velocity components at only one wall. This is the first Fredholm backstepping design for a Navier–Stokes system. In VK'07 the wall-normal input renders the plant spatially causal, after which the tangential input closes a Volterra backstepping loop. Here the normal input carries a finite-rank feedback of arbitrarily small gain, whose sole role, in the sense of Heymann's lemma (TAC, 1968), is to give the plant simple spectrum and make every mode controllable from the tangential velocity. The tangential input then applies a stabilizing feedback, obtained through an invertible Fredholm transform to a Stokes target PDE, shifted left by an arbitrary decay. No simplicity of eigenvalues of the Orr–Sommerfeld operator is assumed and the exclusions of plant coefficient values that have accompanied Fredholm backstepping are removed. We state the comparison with the Volterra design without embellishment: the Fredholm architecture is powerful, but its gain is spectral where VK'07 is closed-form, the simpler of the two, and one from which the Fredholm approach takes its model, its actuation, and its target. What is demonstrated here is the reach of the Fredholm method on its most demanding plant to date. The feedback is tested at Reynolds number $30000$, with the dominant part of the closed-loop spectrum assigned so that it amounts to slowing the flow about twentyfold.