Papers

We introduce and systematically study the weak diametral diameter two property (weak-DD2P), a new geometric property that lies strictly between the diametral diameter two property and both the diameter two property and the convex diametral local diameter two property. A necessary condition for the weak-DD2P is obtained through the structure of the extreme points of the dual unit ball, which leads to characterisations for several classical classes of spaces, including $C(K)$ spaces, $L_1$-preduals, unital uniform algebras, and (vector-valued) function algebras. We establish stability results under standard constructions such as absolute sums, Köthe–Bochner spaces, and projective (symmetric) tensor products. Moreover, we provide complete descriptions of the weak-DD2P for vector-valued spaces of the form $L_1(\mu,X)$, $L_\infty(\mu,X)$, and $C(K,X)$. These results yield a wide range of new examples and show that the weak-DD2P exhibits a behaviour genuinely different from that of other diameter two properties.

Addressing a question posed by Erdős and Hajnal, Chen and Ma proved that, for all $n \ge 600$, the complete bipartite graph $K_{n,n+1}$ is the unique graph on $2n+1$ vertices with at least $n^2+n$ edges that contains no two vertices of equal degree joined by a path of length three. In this paper, we extend this result and prove that for every fixed integer \(\ell\ge 2\) and sufficiently large \(n\), the unique \(2n\)-vertex graph with at least \((n^2+n)/2\) edges that contains no two vertices of equal degree joined by a path of length \(2\ell\) is the half graph \(H_n\). This resolves the problem posed by Chen and Ma, as well as a related question of Attwa, Azócar Carvajal, Boyadzhiyska, Pierron, and Taraz concerning paths of even length with equal-degree endpoints.

Let $p_1, \dotsc, p_k$ be primes not exceeding $x$ ($k \geqslant 2$), and define the additive Mertens sum \[ S_k(x) = \sum_{p_1 \leqslant x} \cdots \sum_{p_k \leqslant x} \frac{1}{p_1 + \dotsm + p_k}. \] In contrast to Tenenbaum's generalized (multiplicative) Mertens sum, whose leading term has order $(\log \log x)^k$, the sum $S_k(x)$ has leading term of order $x^{k-1}/\log^k x$. We establish the complete asymptotic expansion \[ S_k(x) = \frac{x^{k-1}}{\log^k x} \sum_{n=0}^{N} \frac{E_{k,n}}{\log^n x} + O_{k,N}\left(\frac{x^{k-1}}{\log^{k+N+1} x}\right) \quad (\forall\, N \geqslant 0), \] where the coefficients are given by absolutely convergent multiple logarithmic integrals \[ E_{k,n} = (-1)^n \int_{(0,1]^k} \frac{h_n(\log t_1, \dotsc, \log t_k)}{t_1 + \dotsm + t_k}\, \mathrm{d}\mathbf{t}, \] with $h_n$ the complete homogeneous symmetric polynomial of degree $n$. We derive closed-form expressions for the first two coefficients $E_{k,0}$ and $E_{k,1}$ for all $k$, and obtain the closed form for the diagonal part of the third coefficient $E_{k,2}$ (with $E_{k,2}$ fully explicit for $k \leqslant 7$), together with a uniform closed form for the coefficients of the new constants $G(-1/m)$, where $G(z)=\int_0^z \mathrm{Li}_2(t)/(1+t)\,\mathrm dt$, whose genuinely new instances first occur for $k \geqslant 5$ (Proposition 5.4); consequently, the first three terms of the expansions of $S_2(x)$ and $S_3(x)$ are fully explicit. For $k = 2$, we further obtain a closed-form expression for the entire sequence $\{E_{2,n}\}_{n \geqslant 0}$, whose values are explicit $\mathbb{Q}$-linear combinations of $\log 2$ and zeta values $\zeta(j)$. The proofs rely on the real-variable form of the prime number theorem, variable rescaling, and multivariate Taylor remainder estimates.

For an odd perfect number $N$, write $q=\min\{p:p\mid N\}$ for its smallest prime divisor. This paper proves a certified branch-closure theorem for the five minimal-prime branches $q=5,7,11,13,17$. The proof combines the exact $q$-adic valuation balance for $\sigma(N)=2N$ with lower-prime avoidance: primes below $q$ cannot occur in the support and therefore cannot divide any divisor-sum factor. These constraints reduce each branch to a finite first-input coverage split and then to terminal forced-or-pure cofactor records. The terminal records are checked by the frozen certificate release C-small-2026-07, consisting of JSONL certificate bundles, Python verifier scripts, expected terminal outputs, and SHA256 hashes. The $q=5$ branch is presented as the detailed audit model, while the branches $q=7,11,13,17$ are closed by the same forced-or-pure mechanism. The result is scoped: it does not prove nonexistence of odd perfect numbers, and the branches $q=3$ and $q\ge 19$ remain outside the paper.

Volterra series feedback linearizes a class of nonlinear hyperbolic PDEs but produces a controller that, even after truncation, demands solving a tower of plant-specific kernel PDEs and evaluating nested integrals. We prove the truncated controller is jointly Lipschitz in plant and state, and learn it as a single neural operator from plant nonlinearity and state to boundary control. Once trained, no kernel is ever solved again, for any plant in the trained class. The closed loop is practically stable in class-$\mathcal{KL}$ form, with a residual ball scaling linearly with training accuracy.

Backstepping for nonlinear PDEs yields exact feedback linearizing laws in the form of infinite Volterra series – elegant in theory, but with challenges for implementation. This paper shows that even very low-order truncations of such controllers, no longer exactly linearizing, retain the stabilizing power. The key insight is that higher-order terms become negligible near the origin, so stability is recovered for any fixed truncation order by restricting the initial condition size. We establish spatial sup-norm results: finite-time practical stability and asymptotic stability characterized by a class-$\mathcal{KL}$ estimate. The region-of-attraction estimate grows with the truncation order and shrinks with the growth rate of the nonlinearity. The analysis overcomes the lack of pointwise kernel bounds and resolves well-posedness of the nonlinear closed loop, showing that surprisingly simple approximations already capture the essence of nonlinear PDE feedback linearization.

For an integer $L$, write $C_{\ge L}$ for the family of cycles of length at least $L$. For $L=2a$ let $H(n,L)=K_a+\overline K_{n-a}$, and for $L=2a+1$ let $H(n,L)$ be obtained from $K_a+\overline K_{n-a}$ by adding one edge inside the independent part. We prove sharp results for two even target graphs, namely even cycles $C_{2s}$ and even paths $P_{2r+1}$. For even cycles, with $s\ge3$ and $L\ge2s$, we have \[ \mathrm{ex}(n,C_{2s},C_{\ge L+1})=C_{2s}(H(n,L)) \] for all sufficiently large $n$. Together with the known $C_4$ case of Zhu, Győri, He, Lv, Salia and Xiao~[Bull. Lond. Math. Soc. 55 (2023)], this verifies the even-cycle case of their conjecture on $\mathrm{ex}(n,C_k,C_{\ge L+1})$. For even paths, with $r\ge2$ and $L\ge2r$, we have \[ \mathrm{ex}(n,P_{2r+1},C_{\ge L+1})=N(P_{2r+1},H(n,L)) \] for all sufficiently large $n$. We also derive the corresponding exact results when the forbidden graph is a path $P_{p+1}$, sharpening the relevant even-cycle and even-path asymptotic results of Győri, Salia, Tompkins and Zamora~[Discrete Math. Theor. Comput. Sci. 21 no. 1 (2019)].

We study one-centre point interactions for the Dirichlet Laplacian on unbounded domains in dimensions two and three, with emphasis on exterior domains and special Lipschitz domains. These operators are singular perturbations constructed as self-adjoint extensions of the Dirichlet Laplacian restricted to functions vanishing at the interaction centre, and their resolvents are given by an explicit Kreĭn formula with a single extension parameter $\alpha$. The negative spectrum is completely characterized by a scalar equation and the critical coupling $\alpha$ separating binding from non-binding is the threshold limit of the Weyl function appearing in the Kreĭn formula. We establish domain monotonicity of the Weyl function, of the critical coupling, and of the unique negative eigenvalue when existing, and we derive sharp near-boundary asymptotics of the critical coupling in uniformly $C^{1,1}$ geometries. These estimates imply that, for every fixed coupling, nonpositive spectrum disappears when the interaction centre approaches the Dirichlet boundary. We also prove limiting absorption principles and purely absolutely continuous positive spectrum for a point-interaction in exterior domains case and in classes of special Lipschitz domains. We also analyze in depth several threshold phenomena. We show that the critical coupling is governed by the far-field behavior of the zero-energy Green function: exterior domains give threshold resonances, domains contained in a three-dimensional half-space give threshold eigenvalues, the half-plane gives a $p$-wave resonance, and planar wedges exhibit types of threshold states that are aperture-dependent. Finally, low-energy resolvent expansions are computed in the model cases and persistence or disappearance of eigenvalues at threshold are studied. The present paper seems to be the first systematic work on the subject.

Starting from the stability theorem of Erdős and Simonovits, stability problems for graphs forbidding a fixed subgraph have been studied in terms of edge numbers, spectral radii and subgraph counts. Let $\mathcal{N}(F,G)$ denote the number of unlabeled copies of $F$ in $G$. It is known that, for every fixed path $P_t$ and even cycle $C_{2a}$, the maximum number of copies in an $n$-vertex $C_{2\ell+1}$-free graph is attained by the bipartite Turán graph $T_{n,2}$. In this paper we obtain strong structural stability for $C_{2\ell+1}$-free graphs in terms of copies of paths and even cycles. For fixed $\ell\ge2$ and $3\le r\le2\ell-1$, we show that if an $n$-vertex $C_{2\ell+1}$-free graph contains at least as many copies of $P_t$ or $C_{2a}$ as the corresponding suspended extremal construction, then it has the corresponding suspension structure. This gives exact high-chromatic extremal theorems for paths and even cycles. We also prove a counting theorem for nearly complete bipartite graphs. It shows that, for every fixed matching-admissible connected bipartite graph $F$, both imbalance between the two parts and missing cross-edges decrease the number of copies of $F$ by a term with a specified main coefficient. This theorem is independent of the forbidden odd cycle and converts subgraph-count assumptions into the edge bounds needed for the structural theorem.

Balancing the efficiency of an estimator under ideal conditions against its robustness under contamination remains a central challenge in robust statistics. While minimum divergence methods offer a flexible alternative to traditional M-estimation, choosing the appropriate discrepancy measure has historically relied on heuristic or empirical justifications. This manuscript introduces a rigorous optimality criterion for this selection process. By investigating the comprehensive Generalized Alpha-Beta Divergence (GABD) family, we explicitly characterize the Pareto frontier dictating the lowest possible asymptotic variance for any strictly enforced asymptotic breakdown point. Our main theoretical results establish that the estimator achieving this mathematical optimum invariably falls within the extended $(\phi, \gamma)$-divergence class. Crucially, the derived optimal tuning parameter, $\phi^*$, given other parameters, depends solely on the desired breakdown threshold and is entirely invariant to both the assumed parametric model and the exact nature of the data contamination. Supported by comprehensive derivations of asymptotic normality, influence functions, and breakdown thresholds for both continuous and discrete settings, this work offers a unified, theoretical resolution to the long-standing problem of optimal divergence selection in robust inference.

In this paper, the universal KZB connection on the configuration space of points on a closed Riemann surface of an arbitrary genus, introduced by Enriquez, is lifted to the configuration space of points with tangent vectors. This lifted connection is then used to prove the existence of a higher genus version of a Drinfeld associator in the sense of Gonzalez.

This paper counts Frobenius-periodic vector bundles on elliptic curves over an algebraically closed field of characteristic $p>0$. By translating the problem into continuous representations of the étale fundamental group, it derives explicit generating functions and exact-period formulas, with separate treatments of the ordinary and supersingular cases.

Using the Hadamard-Weierstrass factorization theorem for Riemann's {\xi} function, we discuss and prove Riemann's hypothesis.

We prove that fundamental groups of graph 3-manifolds are virtually poly-free and lie in the family Lex. As a consequence, we prove that all finitely generated 3-manifold groups also have these two properties. The first property is a purely group-theoretical concept, and the second is related to the left-exactness property of bounded cohomology of groups. Both properties are proved by constructing sequences of covers of graph 3-manifolds.

A dissociation set in a graph is a subset of vertices that induces a subgraph of maximum degree at most one, which is a natural generalization of the notion of an independent set. The dissociation number of a graph is defined as the maximum cardinality of a dissociation set. This paper studies the minimum spectral radius of connected graphs with a given order $n$ and a given dissociation number $\psi$. For $\psi=n-k$ with $k\ge 4$ fixed and $n$ sufficiently large, we establish both upper and lower bounds for this minimum spectral radius and prove the extremal graphs must belong to a specific graph class.

A classical theorem of Andrásfai, Erdős, and Sós states that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\delta(G)>\frac{2n}{2k+1}$ is bipartite (i.e., homomorphic to $K_2$). Messuti and Schacht proved that the same odd girth condition with $\delta(G)>\frac{3n}{4k}$ forces a homomorphism to $C_{2k+1}$. In this paper, we strengthen the above results by showing that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\delta(G)>\frac{4n}{6k-1}$ is homomorphic to the Möbius ladder on $4k$ vertices. This answers a question of Messuti and Schacht and generalizes a result of Brandt and Ribe-Baumann.

We analytically determine the ultimate limits of state preparation in two-level open quantum systems driven by coherent control. For a dissipative qubit governed by a GKSL master equation, we give an exact characterization of the reachable set in the Bloch ball. Dissipation excludes a region of states in the Bloch ball which cannot be approached even under arbitrarily strong coherent driving, and we prove that this region has a nontrivial geometry whose boundary is a surface of revolution around the $x$-axis which is analytic except for two conical singularities. We derive a closed-form control protocol for moving on this boundary, and construct an explicit protocol that steers the system arbitrarily close to any prescribed boundary state. These results provide a complete geometric constructive description of reachable qubit states in the standard dissipative environment, establishing fundamental bounds on controllability and state-preparation fidelity for open two-level quantum systems.

We study algebraic conjugacy of linear flows on connected Lie groups. To each linear flow we associate a derivation of the Lie algebra, and we analyze how the conjugacy problem can be transferred to the infinitesimal level. In the connected and simply connected case, algebraic conjugacy is characterized by the conjugacy of the associated derivations. For non-simply connected Lie groups, we show that this condition must be supplemented by global restrictions arising from the universal covering group. More precisely, only projectable derivations define linear flows on the quotient, and only admissible Lie algebra isomorphisms descend to conjugacies between the quotient groups. This provides a framework for the algebraic classification of linear flows by computing projectable derivations and admissible automorphisms. We also describe fixed point subgroups and kernels of derivations as obstructions to algebraic conjugacy. Finally, we apply the general results to the three-dimensional Heisenberg group and to its central quotients, showing explicitly how the quotient topology restricts both the admissible derivations and the conjugating automorphisms.

We introduce a Gaussian–Perron prime-force defect that compares a smoothed prime-side logarithmic force with the logarithmic derivative of the Riemann zeta function. The construction turns the explicit formula into a local diagnostic for zero geometry. Its kernel produces an error-function prime weight and an anisotropic zero-side damping law, with an explicit boundary separating amplified and suppressed nonlocal zero contributions. We prove an exact zero-side formula, derive a universal selected-zero profile on the logarithmic scale, and formulate a finite-window damping certificate for non-selected residues. Under explicit damping, pole, and contour-regularity hypotheses, these ingredients localize the full defect near a selected zero. Assuming the Riemann Hypothesis and the stated pole-damping condition, the full defect near each fixed simple critical-line zero has the selected-zero profile up to an exponentially small nonlocal remainder. The framework provides a local diagnostic for zero geometry associated with the Riemann zeta function.

Presolve for mixed integer programming (MIP) problems aims to eliminate redundant information, strengthen the formulation, and extract useful structural information for the subsequent branch-and-cut process. An important type of such structural information is the variable implications (VIs), which describe how a bound on a variable depends on a bound of a binary variable. In this paper, we develop two new presolve techniques that exploit VIs to derive reductions for MIP problems. The first technique, called VI aggregation, aggregates multiple VIs into a single inequality by using implications between a variable and a set of binary variables that form a clique. This aggregation can reduce the number of constraints and tighten the linear programming relaxation. The second technique, called VI-aware linear constraint propagation (LCP), builds on the standard LCP but incorporates VIs associated with the variable being tightened to derive more reductions and can derive tighter variable bounds. We show that although VI information is additionally considered, the tightest lower or upper bound of a variable can still be derived in linear time. Moreover, compared with a state-of-the-art approach in the literature, the proposed VI-aware LCP can derive tighter variable bounds. Computational results on MIPLIB 2017 benchmark instances demonstrate the effectiveness of VI aggregation and VI-aware LCP in improving the performance of the open-source MIP solver HiGHS. In particular, using the two proposed presolve techniques, a reduction of 4% in solving time and 6% in node number on HiGHS can be achieved.

We study information design when a designer with commitment shapes the information of strategically interacting, far-sighted agents whose actions drive a persistent, controlled Markov state. We introduce the Markov Bayes correlated equilibrium (Markov BCE), the controlled-Markov generalisation of the BCE of Bergemann and Morris (2016), characterised by a dynamic obedience condition that adds a continuation-value term to the static one and reduces to it when actions cannot move the state. Recommending actions is without loss; the designer's problem is recursive in the agents' promised continuation utilities and is solved by a set-valued backward-induction algorithm whose optimum exists and lies between the no-disclosure and first-best values. For linear-quadratic-Gaussian payoffs the obedience condition becomes a covariance condition with a modified interaction matrix, and the stationary case reduces to an algebraic Riccati equation. When agents instead learn the transition, we identify the rent an agent earns from a model of the dynamics sharper than the designer anticipates: it is non-negative, zero at the known-dynamics benchmark, and deterred only by building slack into obedience. Under persistent excitation the cumulative rent grows logarithmically as heterogeneous agents' estimates converge. Two worked examples, in congestion and resource coordination, together with a numerical study illustrate the theory.

We consider an elliptic problem with slightly subcritical nonlinearities at the interior and on the boundary; the nonlinearity at the interior is also depending on a gradient term. For any $u$ weak solution, we provide explicit $L^\infty$ a priori estimates depending only on both nonlinearities, on the $H^1(\Omega)$ norm of $u,$ and on the domain $\Omega$. To obtain our results, we combine De Giorgi-Nash-Moser iteration procedure, elliptic regularity when the gradient term appears, Lebesgue interpolation and the Gagliardo-Nirenberg interpolation inequalities.

We investigate the existence and (in)stability of standing waves that are discontinuous at the vertex for the one-dimensional nonlinear Schrödinger equation (NLS) with a power nonlinearity, posed on a star graph consisting of a finite number $N$ of half-lines and endowed with a delta-prime interaction. Our non-variational approach characterizes all possible configurations of such standing wave profiles and shows, rather surprisingly, that the components of any discontinuous-at-the-vertex stationary solution split into two groups, each determined by one of exactly two distinct types of shifted soliton for the NLS on a half-line. More precisely, for every $N\geqq 2$ and every $n\in\{1,\dots,N-1\}$, there are two groups consisting of exactly $n$ and $N-n$ components, with each group sharing a common shift. The stability properties of these discontinuous-at-the-vertex stationary states are analyzed via the Grillakis-Shatah-Strauss framework and the Grillakis-Jones Instability Theorem, where non-standard techniques are required to establish the necessary spectral properties of the linearization operators, in particular their Morse and deficiency indices. Our approach yields a complete description of these indices. The results of this manuscript optimally extend previous findings in the literature concerning delta-prime type interactions on star graphs. Moreover, the methods developed here have the prospect of being adapted to study the stability of other discontinuous-at-the-vertex stationary solutions of the NLS on non-compact metric graphs, such as looping-edge graphs.

We establish stable rank one for the reduced group C$^*$-algebras of the C$^*$-simple topological full groups and dynamical alternating groups constructed by Kerr and Tucker-Drob. The proof relies on both type II$_1$ and type III phenomena, in the first case via the use of Følner towers and in the second via Ozawa's recent results on selflessness as applied to direct products of free products.

In this paper, we prove a spectral upper-pinching theorem for complete properly immersed self-shrinking hypersurfaces. Our argument is inspired by the second author's recent work\cite{Zhao2025}. If \(\lambda_\rho(\Sigma)\geq\lambda>0\) and \(S=|A|^2<1+\lambda\), then \(\Sigma\) is either a hyperplane or a generalized round cylinder. In the properly embedded case, the Ding–Xin and Brendle–Tsiamis weighted Poincaré estimate gives \(\lambda_\rho(\Sigma)\geq1/2\). Consequently, the pointwise upper pinching \(S<3/2\) forces \(\Sigma\) to be a hyperplane or a generalized round cylinder. For embedded self-shrinking surfaces in \(\mathbb R^3\), we also obtain the endpoint case \(S\leq3/2\). These results remove the lower pointwise pinching assumption in the corresponding embedded upper-pinching range and improve the ranges in earlier work of Ding–Xin~\cite{DingXin2014}, Cheng–Wei~\cite{ChengWei2015}, and Lei–Xu–Xu~\cite{LeiXuXu2020}.

We extend the concept of microlocal defect functionals to test functions belonging to the space $\mathrm{L}^\infty\cap\mathrm{VMO}_c$. Following L.~Tartar's remark that an extension of such concepts to $\mathrm{VMO}$ spaces should be possible, we establish a functional-analytic framework for this extension within the $\mathrm{L}^p-\mathrm{L}^q$ setting. Because the topological dual of $\mathrm{VMO}$ is the Hardy space $\mathcal{H}^1$, the resulting object takes the form of an H-distribution rather than a non-negative Radon measure. By assuming strict Hölder conjugate inequalities, we use the John-Nirenberg inequality over localized domains to construct these functionals. We show that the functional acts as a distribution on the inductive limit topology of the test spaces, giving geometric localisation principles for equations with rough $\mathrm{VMO}$ coefficients, that is, coefficients admitting sharp transitions of vanishing mean oscillation. We illustrate the framework in three settings: stratified transport, zero-order non-local cross-phase energies, and sub-critical acoustic scattering in high-contrast media. These differ in the nonlinearity generating the companion sequence but share a common geometric conclusion, since in each case the macroscopic energy defect is confined to the characteristic variety of the underlying flow, being supported where $\sum_j a_j({\bf x})\xi_j=0$.

Intertwined complete projective experiments share outcomes, motivating the search for a common spectral labeling across contexts. Strong coloring tests whether complete contexts admit one context-independent nondegenerate spectrum; when that labeling exists, chromatic separability tests which event distinctions it preserves. We distinguish these questions from ordinary Kochen–Specker statelessness, geometric realizability, event-order representation, and Born-probability models. A state extends to a strong d-coloring exactly when deletion of its support leaves a (d-1)-colorable graph. Faithful real 17- and 21-ray restrictions of completed Yu–Oh have separating ordinary states but forced color identities; one quantum preparation requires exactly 1/9 nonextendable weight despite admitting an ordinary-state model. Independent Hermitian projection arguments correct earlier inferences from a chromatic gap to statelessness or nonrealizability. Hesse's complex 37-ray example separates real and complex geometry and has exact state-family minima 6/7/18 for three representation requirements. A rational two-plane construction gives a linear real 65-ray example with 393,216 states and explicit event-order and probability failures. General and specified ancestry results distinguish geometric KS extensions from abstract chromatic-to-stateless constructions. Detailed enumerations, regenerated state certificates, and extended proofs are supplementary.

We propose two numerical domain decomposition methods (DDMs) for elliptic equations on compact Riemannian manifolds, based on physics-informed neural networks (PINNs). Our approach incorporates the DDM technique for manifolds with the advantages of neural networks in high-dimensional settings. The proposed methods are validated through numerical experiments on various manifolds, both with and without boundary, in dimensions ranging from $5$ to $10$.

The extension of microlocal defect measures to the $L^p$-$L^q$ framework necessitates the use of distributions which possess strictly positive finite order. A topological $\eps$-bounding definition of orthogonality for such distributions was recently introduced by Antonić, Mitrović, and Perić. Building on their framework, we systematically map the obstructions that arise when this natural approach is transported to strictly positive order, and we provide a complementary algebraic resolution. We first establish that while smooth spatial multipliers provide a robust framework for separating order-zero Radon measures, the partition-of-unity route to phase-space geometric separation breaks down for higher-order distributions: the separating frequency multiplier suffers an $\mathcal{O}(\delta^{-\kappa})$ derivative divergence when frequency supports touch, which we prove defeats the a priori bound at every order $\kappa \ge 1$ and produces an explicit non-orthogonality already at order one. Transitioning to algebraic phase-space projectors reveals a second severe obstruction: it is mathematically impossible to separate scalar distributions algebraically, as one-dimensional smooth projectors enforce topological triviality. We demonstrate that the resolution is achieved exclusively within vector-valued PDE systems. Rather than relying on arbitrarily constructed, non-physical geometric scalar cut-offs $\chi(x,\xi)$ to separate distinct modes, we redefine orthogonality canonically via zero-order algebraic matrix projectors. This replaces artificial geometric separation with the intrinsic spectral polarization of the governing system, yielding a coordinate-free microlocal orthogonality for systems of higher-order distributions.

We prove that the number of primes in an interval of length $N$ is at most $2N/(\log N+3.53)$ when $N$ is large enough. This is obtained through a sieving process which can be seen as a hybrid between the large sieve and the Selberg sieve, and draws on what we call "local models". This paper has been published in Acta Arithmetica, 2008, vol. 131, nb. 4, pages 351–366.