Papers

The van der Waerden number $w(k)$ is the smallest positive integer $N$ such that every two-coloring of $\{1,2,\ldots,N\}$ contains a monochromatic $k$-term arithmetic progression. We prove that $w(k) \geq (1-o(1))k2^{k-1}$ holds for all positive integers $k$. This verifies a conjecture of Erdős. In 1968, Berlekamp proved the same result when $k-1$ is prime. The coloring for general $k$ can be viewed as a product of Berlekamp's colorings for various primes. It was found by ChatGPT 5.6 Sol Pro.

Longo and Vigni extended a result of Greenberg by giving a relation between the cardinality of the Selmer group and the characteristic power series of its Pontryagin dual over the cyclotomic $\mathbb{Z}_p$-extension for a $p$-adic Galois representation. In this paper, we extend this result to the case of Artin representations, following the framework of Greenberg and Vatsal.

Let $p\ge 5$ be a prime with $p\equiv -1\pmod 3$, let $q=p^r$, and consider \[ \cC:\qquad y^3=x^{(q^2+1)/2}+x \] over $\F_{q^6}$. We prove the exact formula \[ \#\cC(\F_{q^6})=q^6+1+(-1)^{r+1}(q^2-1)q^3. \] Since $g(\cC)=(q^2-1)/2$, the curve is maximal when $r$ is odd and minimal when $r$ is even. The proof uses a birational Kummer model and an explicit Jacobi-sum point count. A congruence together with Frobenius invariance reduces the relevant Jacobi sums to cubic Gauss sums, whose sign is determined from the Fermat cubic. In particular, the maximality of $y^3=x^{13}+x$ over $\F_{5^6}$ appears as the first case of an infinite family.

We consider systems of homogeneous multivariate polynomials equivariant under the complex reflection group $G(r,n) = (\mathbb{Z}/r\mathbb{Z})^n \rtimes S_n$. Using divided differences indexed by partitions of $n$, we establish a decomposition formula expressing the resultant of such a system as a product of resultants of smaller, partition-indexed subsystems. Combining this with the classical resultant–discriminant relation, we show that the discriminant of a $G(r,n)$-invariant homogeneous polynomial splits explicitly into a product of resultants of smaller subsystems, considerably easier to compute; we illustrate both decompositions with worked examples.

We study polylogarithmic Chabauty–Kim loci for $S$-integral points on $\mathbb{P}^1\smallsetminus\{0,1,\infty\}$ over number fields. We compare motivic and étale Selmer schemes, describe Galois actions on Selmer schemes and period rings, and give an explicit formula for the localisation map on the polylogarithmic Selmer scheme. For imaginary and real quadratic fields, we derive equations and determine several polylogarithmic and full Chabauty–Kim loci and show that Kim's Conjecture holds in several new cases. In other cases, we show that the polylogarithmic Chabauty–Kim method is insufficient to cut out precisely the $S$-integral points, even when combined with $S_3$-symmetrisation, due to additional points arising from $p$-adic roots of unity. As a further application, we give a Chabauty–Kim theoretic proof of the $S$-Selmer Section Conjecture for imaginary quadratic fields when $S$ is empty or $S$ consists of a single prime not fixed by complex conjugation.

A sequence $a_1,\ldots,a_n$ of integers is nearly consecutive if $a_{i+1}-a_i \in \{1,2\}$ for $1 \leq i \leq n-1$. We prove that there are nearly consecutive sequences of length $\Omega(2^k/k^2)$ that contain no $k$-term arithmetic progression. This improves on the previous best known bound of Alon and Zaks from 1998. We also prove a generalization for sequences with bounded gaps.

The least primitive root $g(p)$ modulo a prime $p$ is conjectured by Grosswald to satisfy $g(p)<\sqrt{p}-2$ for any $p>409$. We make progress towards the conjecture by proving $g(p)<\sqrt{p}-2$ when $p > 1.77 \times 10^{54}$.

Let $\text{F}$ be a non-archimedean local field of characteristic zero with residual characteristic $p$. In this paper we give an explicit description of the local Langlands correspondence for middle supercuspidal representations of $\text{GL}(2n,\text{F})$, under the tameness condition $p\nmid 2n$, in terms of the maximal simple types that define them. We achieve this by explicitly computing and comparing the gamma factors on the automorphic and Galois sides of the local Langlands correspondence. The computation on the automorphic side requires neither the tameness condition nor the characteristic zero condition.

We define and study the arithmetic de Rham stack $X^{\operatorname{arith}}$ of a scheme $X$ over a field of characteristic $p$, and analyze its relation with related stacks such as the Hyodo–Kato stack. We show that $X^{\operatorname{arith}}$ gives a stack-theoretic approach to rigid cohomology and its coefficients, known as overconvergent isocrystals and arithmetic $D$-modules, which avoids the long-standing problem of frame-choosing in earlier approaches. We finally give some arithmetic applications of our formalism, such as a proof of Berthelot's conjecture on the preservation of overconvergent isocrystals by smooth proper pushforward.

The refined Diophantine exponent, recently introduced by the author, is a quantity that measures the periodicity of an infinite word. In this article, we study this exponent from combinatorial and topological viewpoints. First, we show that, over a ternary alphabet, the spectrum of the refined Diophantine exponent is $[1,\infty]$. Second, we show that this exponent has topological properties similar to those of the set of Liouville numbers. Finally, we provide concrete examples with the Champernowne, Rudin–Shapiro, and Thue–Morse words, words coming from coding a rotation by intervals, and bracket words.

Let $K$ be a number field and $p$ be an odd prime. Greenberg introduced a natural topology on the space of all $\mathbb{Z}_p$-extensions of $K$ and established several boundedness results for the classical Iwasawa invariants. We extend this framework to compare the Iwasawa invariants of Selmer groups attached to an ordinary $p$-adic representation across $\mathbb{Z}_p$-extensions lying in a Greenberg neighbourhood in the sense of Greenberg. We also establish analogous results for the fine Selmer groups. Finally, in a neighbourhood of the cyclotomic $\mathbb{Z}_p$-extension, we provide evidence for the expected connection between the characteristic ideal of the Selmer group and the conjectural $p$-adic $L$-function introduced by Disegni.

Let $p, N \geq 5$ be primes such that $N \equiv 1 \bmod p$. We prove modularity theorems at levels $N$ and $N^2$, showing that suitable Eisenstein localizations of the weight-$2$ $p$-adic Hecke algebra at these levels are isomorphic to certain natural quotients of a universal pseudodeformation ring. This universal ring parametrizes pseudorepresentations that are residually Eisenstein, unramified outside $Np$ and finite-flat at $p$, and satisfy appropriate conditions at $N$ depending on the level.

Let $B_{m}$ and $D_{n}$ be the denominators of the $m$th and $n$th convergent of the real numbers $\alpha$ and $\beta$, respectively. We introduce an abnormal method in the theory of simultaneous Diophantine approximation to the pair $\alpha,\beta$. Namely, the question of finding simultaneous approximation is turned into study of small solutions of a linear Diophantine equation $xB_{m} + yB_{m+1} = zD_{n} + vD_{n+1}$. The Thue-Siegel's lemma guarantees the existence of a non-zero integer vector $(x,y,z,v)$ in such a way that $|x|,|y|,|z|,|v|$ are bounded above by $\big( B_{m} + B_{m+1} + D_{n} + D_{n+1} \big)^{1/3}$. Thereby we can construct an integer $q:=xB_{m} + yB_{m+1} = zD_{n} + vD_{n+1}\ge 1$, a common denominator, which by the theory of continued fractions gives simultaneous approximations to $\alpha$ and $\beta$. We give also a variety of explicit constructions without the Thue-Siegel's lemma. Let $\|\alpha\|:=\underset{k\in\mathbb{Z}}\min\{|\alpha-k|\}$. For a class of numbers, including particular equivalent numbers $\alpha$ and $\beta$, we show there exist a real number $\kappa=\kappa(\alpha,\beta)>1/2$ and infinitely many explicitly constructible positive integers $q$ such that $\|q\alpha\| \le \frac{1}{q^{\kappa}}$ and $\|q\beta\| \le \frac{1}{q^{\kappa}}$. As the result improves Dirichlet's theorem on simultaneous approximation it also confirms the classical Littlewood conjecture for such a pair $\alpha,\beta$. In addition, we present general criteria for the classical Littlewood conjecture as well as for its $p$-adic counterpart.

Let \(\psi(n)\), \(\phi(n)\), and \(\sigma(n)\) denote Dedekind's arithmetic function, Euler's totient function, and the sum-of-divisors function, respectively. We study exceptional sets associated with the compositions \(\phi(\psi(n))\), \(\phi(\sigma(n))\), \(\psi(\psi(n))\), and \(\psi(\sigma(n))\). For every fixed \(c>0\), we obtain quantitative upper bounds for the numbers of integers \(n\le x\) satisfying \(\phi(\psi(n))\ge cn\) and \(\phi(\sigma(n))\ge cn\), thereby refining density results of Sándor and Dixit and Bhattacharjee, respectively. We further establish quantitative density-zero estimates for the sets of integers \(n\le x\) for which \(\psi(\psi(n))\le cn\) or \(\psi(\sigma(n))\le cn\). More generally, we allow the fixed thresholds in the first two problems to be replaced by thresholds involving non-decreasing functions subject to mild growth conditions.

Let $F_1=F_2=1$ and $F_{t+2}=F_{t+1}+F_t$ for $t\geq1$. For every $n\geq2$, there are unique integers $a,b,t$ such that $n=aF_t+bF_{t-1}$ with $t\geq2$ and $1\leq a\leq b\leq F_t$. The Fibonacci walk with initial pair $(b,a)$ reaches $n$ as late as possible, and the term following $n$ in this walk is $\lfloor\phi n\rfloor$ when $t$ is even and $\lceil\phi n\rceil$ when $t$ is odd, where $\phi=(1+\sqrt5)/2$. Let $D=\{d_1<d_2<\cdots\}$ and $U=\{u_1<u_2<\cdots\}$ be the sets corresponding to even and odd $t$, respectively. For $\ell,m\geq1$, define $D_\ell=\{d_{k+\ell}-d_k:k\geq1\}$, $U_\ell=\{u_{k+\ell}-u_k:k\geq1\}$, $D_\ell(m)=\{d_k:d_{k+\ell}-d_k=m\}$ and $U_\ell(m)=\{u_k:u_{k+\ell}-u_k=m\}$. Chung, Graham and Spiro conjectured that $D_\ell=U_\ell$ for all $\ell$, and asked for the densities of $D_\ell(m)$ and $U_\ell(m)$, especially when $\ell=1$. In this paper, we determine the third and fourth order gap spectra, and show that the conjecture holds for $\ell=3$ but fails for $\ell=4$. We also answer their density question by characterizing when $D_\ell(m)$ and $U_\ell(m)$ have natural densities and proving that their logarithmic densities always exist and are equal. For $\ell=1$, we give the exact logarithmic densities.

This paper establishes a new upper bound for the irrationality measure of arctan(1/2), namely 8.585166... Following the strategy and proof techniques of Zudilin and Zeilberger (who improved the bound for pi), we use a family of complex contour integrals with symmetric integrands. The integrality and divisibility of the partial-fraction coefficients are derived via p-adic valuations and a specially defined prime set P_n. Combined with saddle-point asymptotics, the growth rates of the integral sequence and its rational/logarithmic components yield the desired irrationality measure.

Silverman conjectured that the geometric divisibility sequence attached to a Zariski-dense point on an irreducible commutative algebraic group of dimension at least two, with no unipotent part, returns to its initial value infinitely often. We construct, for the first time, unconditional examples of this phenomenon on geometrically nonsplit semiabelian varieties over number fields using torsion translates of normalized Ribet points. More precisely, let $A/K$ be a positive-dimensional abelian variety over number field $K$, let \[ 1\longrightarrow\mathbf G_m\xrightarrow{\iota}G_q\xrightarrow{\pi}A \longrightarrow0 \] be the extension represented by $q\in A^\vee(K)$, and let $R_\beta(q)\in G_q(K)$ be the normalized Ribet point associated with a homomorphism $\beta:A^\vee\to A$. We set $\delta:=\beta-\widehat\beta$ and assume that $\delta$ is an isogeny and that $\mathbf Z(\delta q)$ is Zariski dense in $A$. For a torsion point $t\in\mathbf G_m(K)_{\mathrm{tors}}$, identify $t$ with $\iota(t)$ and set $P=R_\beta(q)+t$. Then $P$ has Zariski-dense cyclic orbit in the geometrically nonsplit extension $G_q$. There exists an explicit integer $N_{\delta,t}$ such that if $N_{\delta,t}>1$, then $N_{\delta,t}\mid \mathrm{ord}(\overline P_v)$ at all but finitely many places $v$. Consequently, there is a squarefree integer $Q_P>1$ such that \[ (n,Q_P)=1 \quad\Longrightarrow\quad \mathfrak d_{\mathcal N}(nP)=\mathfrak d_{\mathcal N}(P), \] where $\mathfrak d_{\mathcal N}$ denotes the full denominator ideal on the Néron lft-model $\mathcal N$. In particular, we construct explicitly a geometrically nonsplit semiabelian surface $G/\mathbf Q$ and a semiabelian threefold over $\mathbf Q$ satisfying the Silverman conjecture. It follows that we can construct instances satisfying the Silverman conjecture for every dimension at least two.

The four orders of the quaternion algebra B6 carry four reflective wall data on lattices of signature (3,2); three integrate to Borcherds denominators and one is obstructed. The failed denominator survives as a weakly harmonic Maass form, and we prove its shadow is a Hecke eigenform on the line of the newform 6.4.a.a, with zero twist component. The mechanism is invariance selection: the obstruction functional is invariant under the discriminant isometry group, whose invariants in S_{5/2} are one-dimensional; the same mechanism, verified at quaternion discriminants 10 and 22, places the shadows there on 10.4.a.a and 22.4.a.c. On the weight-1/2 layer we prove a determination theorem: the canonical form exists and is unique precisely when the obstruction space vanishes, and among the 71 discriminants below 230 this happens exactly for D in {6, 10, 22}, the genus-zero compact Shimura curves, whose maximal-order ternaries are reflective with integral Weyl chambers of ranks 3, 4, 4; completeness beyond that range is reduced to an estimate on a quadratic Dedekind-type sum, given a bound on the Gauss-sum term. Parity confines this layer to odd channels; on the obstructed orientation the section layer is obstructed outside an explicit 40-element locus of orientations (a double shadow: CM, 36.2.a.a, at weight 3/2 and newform at weight 5/2), while the deck-symmetric directions instead carry a unique canonical weight-1/2 form. The defect invariant satisfies ||Xi||^2 <v_+,v_+> = 144 exactly and equals L(f,2)/48 pi^2 <f,f> to 31 digits. On the section layer the Petersson geometry is rigid: the Gram matrix of S_{3/2}(rho_4) is a single transcendental multiple of an exact rational form, and that transcendental is identified, to 40 digits, as 3 Gamma(1/3)^3 / (2^{7/3} pi^2): the weight-3/2 shadow norms lie in the Chowla-Selberg ring; the weight-5/2 norm is numerically excluded from it.

The notion of $p$-derivation as introduced by Buium has a rich history of applications in arithmetic geometry. Working over $\ZZ_p$, Buium-Ralph-Simanca showed that arithmetic differential operators built from these determine $p$-adic analytic functions and vice versa. Notably, over $\ZZ_p$, there is only one $p$-derivation. In this article, we consider the same question for ramified extensions $A_\pi$ with uniformizer $\pi$. This has the effect of passing from ordinary to partial differential operators, since such extensions can enjoy multiple $\pi$-derivations. We introduce a notion of analytic functions which are naturally determined by these operators in a way analogous to that in the unramified case, but with a significantly richer structure. We show that under mild conditions, analytic functions and partial differential operators determine each other in this setting.

Suppose $F$ is a nonarchimedean local field whose residue field is a proper extension of $\mathbb{F}_p$ with $p > 3$. Generalizing results of Ghate–Le–Sheth, we show that any split, connected, reductive group $G$ over $F$ which is not a torus admits a smooth, irreducible, non-admissible mod $p$ representation. We use this to show that the functor of pro-$p$-Iwahori invariants does not induce an equivalence between the category of smooth mod $p$ $G$-representations generated by their pro-$p$-Iwahori invariant vectors and modules over the pro-$p$-Iwahori–Hecke algebra, contrary to what happens for $\textrm{GL}_2(\mathbb{Q}_p)$.

We prove quantitative polynomial Szemerédi-type theorems involving polynomial progressions with shift parameter restricted to the set of shifted primes $\mathbb{P}-1$. The types of configurations covered are distinct degree progressions and progressions involving integer multiples of a fixed polynomial. For nonlinear configurations of length at least three, these results provide the first quantitative versions of such theorems. In the linear case, our results improve on work by the last two authors. Our density bounds are strongest in the case of distinct degree polynomials, where they give polylogarithmic bounds, of the same shape as recent bounds by Shao and Wang with integer shifts. The proofs combine recent quantitative results for polynomial configurations in the integers with quantitative Gowers uniformity bounds of the primes. For multiples of a fixed polynomial, we adapt a comparison argument of Altman and Sawhney to obtain uniformity over the polynomial families produced by the $W$-trick. For distinct degree progressions, we establish a comparison between prime-weighted and unweighted polynomial counts that is uniform throughout the density increment argument and accounts for a possible Siegel zero.

We show that the shifted log-sine integrals with odd entries, introduced by Umezawa, are multiple zeta values. In particular, they define an algebra homomorphism from the f-alphabet algebra to the algebra of multiple zeta values. The proof uses a motivic realization at sixth roots of unity and the Galois descent from level six to level one studied by Glanois. Finally, we give an interpretation of our results using Zinbiel algebras and introduce iterated cotangent integrals.

For any $m, s \in \mathbb{N}$, we study the number $N_{m\times s,q}(\kappa, \beta)$ of solutions $(x_1,\ldots,x_s) \in (\mathbb{F}_q)^s$ of the monic system of diagonal equations $$ X_{1}^{k_i} + \cdots + X_{s}^{k_i}= \beta_i, \qquad (1\le i \le m), $$ with $\kappa=(k_1,\ldots,k_m) \in \mathbb{N}^m$ and $\beta=(\beta_1,\ldots,\beta_m) \in (\mathbb{F}_q)^m$. We show that this number can be obtained in terms of some data of diagonal GP-graphs $\Gamma(\kappa,q)$. This is a new family of graphs that we introduce here, $\Gamma(\kappa,q)$, with $\kappa = (k_1,\ldots,k_m) \in \mathbb{N}^{m}$, is the directed graph with vertex set the finite field $\mathbb{F}_q$ and there is an arc from $u$ to $v$ if and only if $v-u \in R_{\kappa} = \{ (x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_{q}^*\}$. In particular, we give three different expressions for $N_{m\times s,q}(\kappa, \beta)$: one in terms of walks, another in terms of adjacency matrices of $\Gamma(\kappa,q)$ and the last one in terms of the spectrum of $\Gamma(\kappa,q)$. Finally, we explicitly derive combinatorial formulas for the number of solutions $N_{m}(s,q) = N_{m\times s,q}(\kappa_\ell, 0)$ of monic homogeneous systems of diagonal equations of the form $$ X_1^{q^{\ell_i}+1} + \cdots + X_s^{q^{\ell_i}+1} = 0 \qquad (1\le i \le m),$$ with $\kappa_\ell=(\ell_1,\ldots,\ell_m)=(1,3,\ldots,2m-1)$ and $m\ge 2$, via the known spectrum of Hermitian-form graphs, which can be viewed as diagonal GP-graphs. For any $m,s \in \mathbb{N}$, we give general summation and recursive formulas for $N_m(s,q) \in \mathbb{Z}[q]$. For the small cases $N_{1}(s,q)$, $N_{2}(s,q)$ and $N_{m}(s,q)$, with $1\le s \le 5$, we give explicit expressions.

Characterizing isolated points on the modular curve $X_1(N)$ is a key obstruction to classifying all points of a fixed degree. These points do not lie in infinite parameterized families, making them difficult to obtain through geometric constructions. In this paper, we focus on the collection of "isolated $j$-invariants" for $X_1(N)$, which are the values obtained by mapping isolated points to the $j$-line. Prior work of the author in collaboration with Ejder, Liu, Odumodu, and Viray asks whether there are only finitely many isolated $j$-invariants lying in extensions of bounded degree. Here, we explore how this question relates to other uniformity problems in the field and give new finiteness results for isolated $j$-invariants in $\mathbb{Q}$. As an application, we show similar methods give sharpened polynomial bounds on torsion for non-CM elliptic curves having rational $j$-invariant.

The kernel of a random symmetric p-adic matrix is a random abelian group, equipped with a symmetric pairing. If we consider not only the matrix but also its top-left corners, we get a process valued in isomorphism classes of abelian groups, equipped with such a pairing. We show that when the matrix is Haar random, this process is a Markov chain, generated by an operator that we explicitly describe. We will also prove that this operator is reversible with respect to a Cohen-Lenstra type measure.

We consider the spherical Borel Eisenstein series induced from the trivial representation for a split semisimple linear algebraic group over a number field. We prove that its regularization at the special point corresponding to half the weighted marking of a distinguished coadjoint nilpotent orbit in the Langlands dual Lie algebra is nonzero and square-integrable. Our proof follows the philosophy of Kazhdan and Okounkov. We give a geometric interpretation of Langlands' square-integrability criterion in this setting and, using the equivariant integration formula, prove that the regularization satisfies this criterion. As an immediate consequence, we obtain a simple and uniform proof of Arthur's unitarity conjecture, without case-by-case analysis or machine computation.

Let $G$ be a nontrivial, finite abelian group. Then $\nu (G)$ is the smallest integer $\ell$ such that every zero-sum free sequence $T$ over $G$ of length at least $\ell$ has the following property: all nonzero elements of $G$ that do not occur as a subsequence sum of $T$ lie in a proper coset of some subgroup of $G$. We study the invariant $\nu (G)$, which was introduced in Zero-Sum Theory in the 1960s.

We state and prove the Theorem: for primes $p < 10^n$ with base-$10$ expansion $p = \sum_{k=0}^{n(p)-1} d_k(p) 10^k$, the positional digit probabilities $P_n(d \mid k)$ satisfy \[ \lim_{n \to \infty} P_n(d \mid k) = \begin{cases} 1/10, & k \ge 1,\ d \in \{0,\dots,9\}, \\[4pt] 1/9, & k = \mathrm{lead},\ d \in \{1,\dots,9\}. \end{cases} \] The limiting behavior splits cleanly into two distinct mechanisms: an arithmetic regime for interior digits and an Archimedean regime for the leading digit. For fixed interior positions ($k \ge 1$), digit extraction modulo $10^{k+1}$ reduces the problem to prime counts in reduced residue classes, where uniform distribution follows from Siegel–Walfisz (with Bombieri–Vinogradov allowing $k$ to grow with $n$). For the leading digit, the $1/9$ limit is not a Benford-type scale invariance, but arises from the local near-constancy of the prime density $1/\log t$ within single decades combined with a Toeplitz-type error averaging. Explicit classical and conditional error bounds are recorded for both regimes.

We present Eureka, a task-conditioned Meta-Agent architecture that compiles long-horizon tasks into dynamic obligation graphs with explicit acceptance semantics. During execution, Eureka forms Macro-Agents with specialized state, memory, operators, tools, verifiers, and local topology via receding-horizon planning, architecture promotion, and minimal-sufficient compilation. When bottlenecks recur, cost-benefit-gated evolution updates the local architecture under constraints. Theoretically, we establish results on regret, planning invalidation, amortization, subtree interfaces, serializability, and verification. Experimentally, Eureka completes 170/170 recursive tasks and generates 3,948 certificates with no false acceptances. Active context compresses median input from 9,490 to 4,005 tokens; incremental processing avoids 65.38% recomputation across 12,000 tasks; 16,000 concurrent executions serialize consistently. The same Meta-Agent instantiates a Theory-Discovery Agent and a Math/Conjecture Agent. The former yields structural results in quantum-process and spacetime theory. The latter identifies bottlenecks in Riemann Hypothesis research and advances a positivity certificate for Suzuki's localized Weil quadratic form to 0 < a <= 69/200 = 0.345, reaching ~99.55% of (log 2)/2. These results suggest that scientific-agent capability depends not only on the base model but on whether an architecture can be formed to match the task's cognitive structure.

We develop additive-energy estimates and weighted Young inequalities for finite sets on bounded-degree real algebraic varieties. For an irreducible $m$-dimensional variety $V$, let $\sigma(V)=2m-\dim\overline{V-V}^{\mathrm{Zar}}$ and $\alpha(V)=\max{2,1+\frac{2\sigma(V)}{m}}$. For every $a\in[\alpha(V),3)$ we define a finite-degree translation-partition flag parameter $\Lambda_{a,R}(X;V)$ and prove $E(X)\ll \Lambda_{a,R}(X;V)^{3-a}|X|^{a+\varepsilon}$. This recovers the line-concentration theorem of Jing and Wu for algebraic surfaces in $\mathbb{R}^3$. For codimension-two quadratic threefolds ${(u,Q_1(u),Q_2(u))\in\mathbb{R}^3}\subset\mathbb{R}^5$ with positive-definite $Q_1$ and simple generalized spectrum, we prove the sharp estimate $E(X)\ll_{\varepsilon}|X|^{2+\varepsilon}$ without a flag loss. Hereditary versions of these estimates imply weighted $L^4$ restriction bounds and off-diagonal Young inequalities; at the near-diagonal threshold the sharp region is $1\le p,q\le 2$ and $p^{-1}+q^{-1}\ge 1$. We also prove a sharp turning-complexity extension of the Cushman-Demeter-Wu theorem: $J_3(P)\ll_{\varepsilon}\kappa(P)^2|P|^{3+\varepsilon}$, with matching examples at every power scale.