All works

math.AG — Algebraic Geometry

Equivalence of the Four-Variable Hessian and Planar Jacobian Conjectures

Contributed by Qiyue Tang

We prove that the four-variable Hessian conjecture and the planar Jacobian conjecture are equivalent over the complex numbers.

math.NA — Numerical Analysis

Quartic Fekete points on every real simplex

Contributed by Yunpeng Li

We prove that on every real simplex Bos's quartic interpolation nodes uniquely maximize the absolute Vandermonde determinant, up to ordering, among all configurations of the prescribed size. In dimension four the squared Euclidean norm of the cardinal vector exceeds one at the centroid. We bound its excess by products of the cardinal polynomials for tetrahedral face centers. These products also measure correlations between columns of the evaluation matrix, which give the required loss in the determinant. An elementary consequence of the same corrected squared-norm estimate is Bos's fourth-power bound in every dimension, with equality exactly at the nodes. The corrected estimate follows from cubic perturbations, reduction to at most two distinct positive coordinate values, and coefficient monotonicity in their multiplicities. All polynomial bounds are given explicitly. The common base case is the tetrahedral Fej\'er bound previously stated by Panda and Pal in equivalent optimal-design form. We give an independent proof of that bound and determine all its equality points.

math.CV — Complex Variables

Sheil-Small's (1974) covering problem has a positive answer

Contributed by Dave Goel

T. Sheil-Small asked whether every self-inversive polynomial maps the unit disc onto a domain that contains a disc whose radius is the largest modulus of the polynomial's coefficients (Problem 4.24 in Hayman and Lingham's problem book). We show that the answer is yes. More generally, every non-constant polynomial whose coefficients have the same moduli when read in reverse order maps the unit disc onto a domain that contains an open disc of this radius. This class contains the self-inversive polynomials and the polynomials whose zeros, with multiplicity, are symmetric under taking reciprocals (the problem's printed definition of self-inversive). The radius cannot be increased. For a polynomial with all its zeros on the unit circle, the largest disc in the image has radius exactly half the maximum modulus on the circle. For these polynomials the question is therefore equivalent to the known bound: every coefficient has modulus at most half this maximum (O'Hara-Rodriguez, Saff-Sheil-Small, Kristiansen). The new case is that of polynomials with zeros off the unit circle. We treat this case with the Schur-Cohn Hermitian form and that bound. The main theorem is also proved in the Lean 4 proof assistant, with that bound taken as given.

Author comments: An expanded version of this paper, with further results, is posted on Zenodo: https://doi.org/10.5281/zenodo.23908408

math.DG — Differential Geometry

EVERY CLOSED SMOOTH $n$-MANIFOLD EMBEDS ISOMETRICALLY IN $\mathbb{R}^{\,n(n+1)/2+28}$

Contributed by Yi Wang, Wenchang Yang

We prove that every closed smooth Riemannian $n$-manifold admits a smooth isometric embedding into $\mathbb{R}^{s_n+28}$, where $s_n=n(n+1)/2$. This improves the previous upper bound $s_n+\max\{n+5,\,2n\}$ to a dimension-independent surplus of $28$.

cs.LO — Logic in Computer Science

Proof-Carrying Modernization: LLM Translations of COBOL Batch Programs, Each Proved Against the Original in Lean

Contributed by Tejasvi Singh Tomar

Large language models translate legacy code quickly, but a translation can differ from the original on inputs that no test exercises. We show that LLM translations of whole, unmodified COBOL programs into Rust and Java can be proved in Lean 4 to behave like our semantics of the originals on every admissible input: a bounded number of well-formed records, among other premises. The COBOL semantics is executable; it computes record layouts, decimal arithmetic and file handling from our front end’s output for the unmodified source. For the eight standalone batch programs of AWS CardDemo covered by that front end, all 16 translations are proved, as equivalences for Rust and as refinements with partial correctness for Java. The statements are generated from the programs except for a few small parts, LLM agents write the proofs, and Lean’s kernel checks them. Before any proof, 35% of the first running translations differed from our semantics of the original on a generated input, and no proof found a difference that this testing had missed.

math.CO — Combinatorics

Contracting a pendant edge can increase the mean subtree order

Contributed by Alexander Kramer

Cambie, Jooken and Wagner conjectured that contracting an edge of a connected graph decreases the mean subtree order by at least 1/3, as it does for trees. We construct unicyclic graphs of order n and maximum degree 3 in which contracting a pendant edge increases the mean subtree order by n/450+O(1), disproving the conjecture. In this family the bound first fails at 481 vertices and contraction first increases the mean at 631 vertices.

math-ph — Mathematical Physics

Era statistics for the vacuum Bianchi IX flow

Contributed by Yi-kun Li

We prove the Gauss statistical law for eras of the exact vacuum Bianchi IX flow with zero cosmological constant. Eras are defined by transverse events along the solution, and their lengths count curvature transitions. Along almost every trajectory, the era phases have the Gauss distribution and every finite block of era lengths has its Gauss frequency. The tail decays inversely with era length; the limiting truncated mean grows logarithmically with the cutoff, while the full mean diverges. One invariant Borel exceptional set is null on every smooth transverse section and in the four-dimensional constraint manifold. The proof combines estimates for exact return branches with absolute continuity of their coding measure. Transverse contraction controls the accumulated phase error within each era, uniformly over unbounded era lengths and arbitrarily unequal small curvature variables. A density with strictly expanding section flux then forces almost every recurrent orbit to enter a statistical set on a small section.

Author comments: The Lean 4 formalization is also available on GitHub: https://github.com/l46430640-del/ERA-STATISTICS-FORMAL

cs.CC — Computational Complexity

What the ETH refutations do not give: barriers for the counting engines, with machine-checked lemmas

Contributed by Vaibhav Tulsyan

In October 2026 OpenAI and Anthropic each claimed a deterministic subexponential algorithm for counting the solutions of a $k$-CNF; both counting engines sum products of local tables. For SAT itself, Majdoddin bounds the rate of Anthropic's grid recursions with product terms, and Dias and Lima bound OpenAI's clause-product scheme. However, Majdoddin notes that no barrier is known for decompositions whose terms are not products, and neither paper treats the counting problems beyond SAT. We introduce the engine barrier problem: bound from below the shared bits $U$ of every network of a stated class that computes a given count, and the rate of every recursion of a stated class. Our approach puts a rank on every cut of a local network, and on one contraction inside both flattenings of a term. Machine-checked in Lean are: correlated (non-product) decompositions of two rectangles, of the $3\times2$ grid and of the full $3\times3$ grid level have flattening-rank sums at least $N^2$, $N^3$ and $N^8$, while $N^9$ is open; the row cut of the permanent has rank $\binom nk$, which on paper gives $U\ge n\log_2(n/s)$ for row-local networks; algebras in which the permanent is a functional of a product have dimension at least $\binom n{n/2}$; pieces have saving $\rho\ge1/2$ (rank form; Anthropic's observation); and exact colouring filters for Hamiltonicity need $q\ge n-4$ colours. On paper we prove $U\ge(1-o(1))n\log_2n$ for vertex-local networks for the Tutte polynomial off the integer hyperbolas, and $\rho\ge3/8$ for correlated decompositions of full grid levels. We cite $45$ Lean theorems, with standard axioms only. Every barrier is about a named model: in Anthropic's general model no unconditional bound on $U$ exists without separating $\mathrm P$ from $\mathrm P^{\#\mathrm P}$, and an unconditional cap on oblivious algebraic algorithms would separate $\mathrm{VP}$ from $\mathrm{VNP}$.

math.OA — Operator Algebras

Bures Geometry of Projection States

Contributed by Zhenyu Sun, Jilin Wang

We study the Bures metric on states obtained by conditioning a faithful normal state on nonzero projections of a von Neumann algebra. The metric determines the reference state, projection probabilities, orthogonality, and order. For algebras with no type $I_2$ direct summand, surjective isometries are exactly those induced by reference-state-preserving normal Jordan $^*$-isomorphisms; the same conclusion holds for $M_2(\mathbb C)$. A semifinite tracial version determines the trace up to a single positive factor. In finite tracial algebras, we characterize optimal finite projective measurements and express their fidelity deficit as a sum of squared $L^2$ errors. Spectral measurements with at most $n+1$ outcomes approximate fidelity with error $O(n^{-2})$; the interval grid is minimax optimal, and the order is sharp within spectral measurements. We compare the fidelity defect of a common commutative context with approximation by commuting projections, obtain sharp spectral rounding constants, and solve both optimization problems in the algebra generated by two projections. The space of normalized finite-trace projection states is complete; in a finite tracial algebra its rectifiable components are unitary orbits, and within each orbit the intrinsic Bures distance is the root mean square of the principal angles. Finally, a fidelity bound on local unitary orbits characterizes tracial reference states.

Author comments: 38 pages.

cs.LO — Logic in Computer Science

Formalization of Robust Strictly Positive Real Synthesis for Fourth-Order Interval Polynomial Families

Contributed by Tiancheng Xu, Wensheng Yu

That every robustly Hurwitz-stable fourth-order interval polynomial family admits a common strictly positive real (SPR) numerator was conjectured by Anderson et al. and proved by Yu and Wang. In this paper, we present a complete formalization of this classical synthesis theorem in the Lean~4 interactive theorem prover.

math.NT — Number Theory

Degeneracy and the level-one Maass spectrum: a measured verdict on arithmetic quantum chaos

Contributed by Zeng Chunping

We isolate the spectral axis of a seven-world program: the length spectrum of the modular surface, its degeneracy, and the statistics of the level-one Maass spectrum. (1) The true degeneracy of a closed geodesic of discriminant $D$ equals the narrow class number $h^{+}(D)$, i.e. the number of $\rho$-orbits of Zagier-reduced indefinite binary quadratic forms; on all 36 fundamental discriminants $5\le D\le 120$ the equality holds as an exact integer. (2) With the corrected counting (all non-square discriminants $D\equiv 0,1 \pmod 4$, primitive forms, $\rho$-orbits), the prime geodesic theorem has error exponent $1/2$: $N(X)/\mathrm{li}(e^X)=0.9963$ at $X=12$ with $|\mathrm{err}|/e^{X/2}$ bounded and decreasing overall. (3) The level-one Maass spectrum is long-range GOE and short-range non-Wigner: on 1110 even levels, $\Sigma_2(20)=1.203\ [1.02,1.26]$ against Poisson 20 and compatible with GOE 1.049, while the spacing distribution fails the Wigner surmise with a $17\times$ small-gap excess. This is the standard signature of arithmetic quantum chaos: rigidity without local Wigner repulsion. We also record the archimedean parity law $\Gamma^{a}_{2k}=2\Gamma^{a}_{2k+1}\sim r^{-2k}$, synthetic-spectrum pipeline self-certification, and the corrected upper-envelope saturation criterion. Every measured number carries a confidence interval; corrections applied are recorded in an appendix.

math.AP — Analysis of PDEs

Non-Uniqueness of Leray-Hopf Solutions to the Unforced Three-Dimensional Navier-Stokes Equations

Contributed by Tristan Buckmaster, Matei Coiculescu, Chen Li

We prove that the incompressible Navier--Stokes equations in $\R^3$, without external force, admit a continuum of distinct suitable weak solutions in the Leray--Hopf class with the same initial datum. The datum is compactly supported, axisymmetric without swirl, and smooth away from the origin; it lies in $L^{3,\infty}$ and in $L^q$ for every $q<3$, but not in $L^3$. The solutions are axisymmetric without swirl and smooth at positive times, and they satisfy the local and global energy equalities. Any two of them separate in $L^2$ at the rate $t^{1/4+\sigma}$, for a fixed $0<\sigma<1/4$. The proof follows the program of Jia and \v Sver\'ak, with a discretely self-similar background in place of a self-similar one. The background is an exact unforced solution built from a train of thin vortex rings, which we obtain by lifting a periodic row of planar vortices with Gaussian stream function. We construct a growing Floquet mode of its linearization directly, from the angular-two instability of the planar vortex, and conclude by a nonlinear correction and a localization of the datum.

Author comments: Lean formalization forthcoming at the public repository at https://github.com/mcoiculescu/unforced-leray-hopf-public Exposition will be improved in future versions. We used an adaptation of Scott Armstrong's skill files for the Lean formalization process.

math.CV — Complex Variables

Origin-Centered Mobius Circles, Bernoulli Lemniscates, and Beta–Gamma Dyadic Pairings of the Riemann Zeta Function on the Critical Line

Contributed by Ryan Hackbarth

We investigate the complex geometry of the normalized Dirichlet eta terms $t_n(s) = \frac{1}{1 - 2^{1-s}} \frac{(-1)^{n-1}}{n^s}$ for $s = a + bi$. Whereas individual terms on the critical line $\operatorname{Re}(s) = \frac{1}{2}$ undergo a nearly sixfold modulus oscillation driven by the Euler prefactor $(1 - 2^{1-s})^{-1}$, the two-term partial sum $S_2(s) = t_1(s) + t_2(s)$ traces an exact Mobius circle for every fixed $a \neq 1$ that is centered at the origin $(0,0)$ if and only if $\operatorname{Re}(s) = \frac{1}{2}$, where it reduces to a scaled Blaschke factor $\sqrt{\frac{1}{2}} B(z)$ of constant radius $\sqrt{\frac{1}{2}}$. Factoring $m^{-s}$ across odd-even dyadic pairs $P_m(s) = t_m(s) + t_{2m}(s) = m^{-s} S_2(s)$ eliminates prefactor amplitude oscillations entirely, collapsing precessing annular rosettes into concentric origin-centered circles on the critical line with strictly descending radii $R_m = \sqrt{\frac{1}{2m}}$. Removing all odd-even pairs leaves the complementary multiples-of-$4$ terms $t_{4k}(s) = k^{-s} t_4(s)$: we prove that on the critical line the anchor term $t_4(\frac{1}{2} + ib)$ traces an exact Lemniscate of Bernoulli with foci at $(0,0)$ and $(1,0)$ (uniquely for $\operatorname{Re}(s) = \frac{1}{2}$), pure powers of $2$ ($k = 2^r$) trace closed periodic multi-loop rational curves, and non-powers of $2$ trace precessing lemniscate rosettes, while the dyadic differences $D_{4m}(s) = t_{4m}(s) - t_{8m}(s) = -4^{-s} P_m(s)$ collapse both families into exact origin-centered circles, partitioning the sequence $\{t_n(s)\}_{n=1}^{\infty}$ termwise and decomposing $\zeta(s)$ via analytic continuation. Finally, we establish a pointwise velocity--modulus duality $\left|\frac{dt_4}{d\theta}\right| = \left|t_2\left(\frac{1}{2} + ib\right)\right| = \sqrt{\frac{n}{2}} \left|t_n\left(\frac{1}{2} + ib\right)\right|$ ($\theta = b \ln 2$) and a double-angle Blaschke identity $-z B(z) = e^{i 2\alpha(b)}$ linking the central polar angle $\alpha(b)$ of $t_4(\frac{1}{2} + ib)$ to its fundamental perimeter $L(t_4) = \frac{\Gamma(1/4)^2}{2\sqrt{\pi}}$, thereby parameterizing every individual term $t_n(\frac{1}{2} + ib)$ and every concentric dyadic circle on the critical line in terms of the regularized inverse incomplete Beta function $I_{q(b)}^{-1}(\frac{1}{4}, \frac{1}{2})$. As a geometric consequence, critical-line zeros $\zeta(\frac{1}{2} + ib) = 0$ are reframed as values of $b$ where the resultant of the inner concentric circles lands on the antipode $-P_1(\frac{1}{2} + ib)$ of the initial circle of constant radius.

math.CO — Combinatorics

Vertex splitting, global rigidity and nondegenerate equilibrium stresses

Contributed by James Cruickshank

We prove Whiteley's conjecture that every nontrivial vertex split of a generically globally rigid graph preserves generic global rigidity in $\mathbb{R}^d$, for all $d\geq1$. We give two proofs of the stronger assertion that the split graph is infinitesimally rigid at every generic coincident realisation of the two split vertices. Both proofs use maximum-rank equilibrium stress matrices and then apply the coincident-rigidity criterion of Cruickshank, Jackson and Tanigawa. The second construction also shows that every generically globally rigid graph admits a nondegenerate equilibrium stress of maximum rank at every generic realisation, establishing a conjecture of Jord\'an and Tanigawa.

math.NT — Number Theory

Explicit lower bounds for the number of irrational even beta values

Contributed by twinheterodox

For every even integer $s\ge160$, we prove that at least $\left\lfloor\pi\sqrt{2/135}\,\sqrt{s/\log s}\right\rfloor$ of the numbers $\beta(2),\beta(4),\ldots,\beta(s)$ are irrational, where $\beta$ is the Dirichlet beta function. We also give sharper finite endpoints for specified counts of irrational values

math.FA — Functional Analysis

A modular theory of contractive projections on Banach spaces with applications to noncommutative $\mathrm{L}^p$-spaces

Contributed by Cédric ARHANCET

We develop a differential-geometric approach to contractive projections on complex Banach spaces based on Levi forms of plurisubharmonic functions. We show that if a projection $P$ contracts a $\mathrm{C}^2$ plurisubharmonic function $\Phi$, then, at any point of its range, the Levi form of $\Phi$ defines a tangent Hilbertian geometry for which $P$ becomes an orthogonal projection. When the range is finite-dimensional and the relevant Levi forms are nondegenerate, comparing the Levi geometries associated with different points leads naturally to positive relative operators and associated one-parameter groups. We apply this framework to tracial noncommutative $\mathrm{L}^4$-spaces. Exploiting the cubic form of the duality mapping, we characterize contractive projections by a Jordan triple orthogonality property and prove that a projection on $\mathrm{L}^4(\mathcal{M})$ is $2$-contractive if and only if it is completely contractive. When the ambient von Neumann algebra is finite-dimensional, we identify the relative operator associated with the left and right tangent forms on the range with a restriction of an inverse square root of a modular operator on a linking von Neumann algebra. This entails that the range of any $2$-contractive projection is completely isometric to a rectangular noncommutative $\mathrm{L}^4$-space associated with a $\mathrm{W}^*$-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every $1 < p < \infty$ with $p \neq 2$, we prove that every finite-rank $2$-contractive projection on a tracial noncommutative $\mathrm{L}^p$-space is completely contractive. For $p > 2$, we obtain a weighted ternary description of its range in terms of a finite-dimensional $\mathrm{W}^*$-$\mathrm{TRO}$.

cs.SC — Symbolic Computation

Deterministic Polynomial-Time Factorization over Finite Fields

Contributed by Qiyue Tang

Using prime-field factorization and a Dedekind zeta zero-free theorem, we obtain deterministic polynomial-time algorithms for complete factorization over finite fields and for constructing their presentations from their orders.

math.AG — Algebraic Geometry

The four-dimensional Hessian conjecture for quintic potentials with a ternary cone top

Contributed by David Niedbala Giraudin

The Hessian conjecture $\mathrm{HC}_n$ asserts that if $f\in\mathbb{C}[x_1,\dots,x_n]$ has $\det\operatorname{Hess}(f)\in\mathbb{C}^\times$, then $\nabla f$ is a polynomial automorphism. It is known for $n\le 3$, false for $n\ge 5$, and open for $n=4$, where it implies the plane Jacobian conjecture. Z. Ni proved $\mathrm{HC}_4$ for potentials of degree at most four. We prove $\mathrm{HC}_4$ for every potential of degree five whose top homogeneous form depends on exactly three linear forms. The proof combines a weighted analysis of the Hessian determinant with exact computer-algebra certificates (Groebner bases and Rabinowitsch tests over $\mathbb{Q}$); one branch reduces to plane Jacobian pairs of degrees $(4,3)$, which are excluded by Magnus' theorem. A self-contained verifier is supplied as an ancillary file. The binary and unary cone tops in degree five remain open.

Author comments: 5 pages. Ancillary file anc/verify_hc4_ternary.py: 29 exact checks (python-flint, sympy, Singular via Sage), about 3 minutes. Extends arXiv:2608.14217 to degree five for ternary cone tops. Computations and writing assisted by Claude (Anthropic).

math.CO — Combinatorics

Regular complex Hadamard matrices of order 7,8

Contributed by Teo Banica

A complex Hadamard matrix $H\in M_N(\mathbb C)$ is called regular when the scalar products between its rows $\sum_kH_{ik}\bar{H}_{jk}=0$ decompose as trivially vanishing sums of roots of unity, rotated by scalars. We review the general theory of such matrices, including the known classification at $N=6$, and then we work out classification results at $N=7$ and $N=8$.

cs.CC — Computational Complexity

The Price of Bayes-Optimal In-Context Learning

Contributed by Xiaoyu Li

In-context learning is often explained as implicit Bayesian inference: given labeled examples of a task drawn from a prior, the learner predicts like the posterior. We determine the computational price of this explanation for task families whose labels depend on few bits of the task description, such as quantized models on sparse data, and we compare it with the price of merely learning. The comparison uses the recent refutations of the Exponential Time Hypothesis by OpenAI and Anthropic. Bayesian inference in context is counting. For $\kappa$-local task families with $N$-bit descriptions under the uniform prior, the posterior predictive and posterior sampling reduce to $\#\kappa$-SAT. Without label noise they run in time $\mathrm{poly}(m)\cdot 2^{o(N)}$ on contexts of every length $m$ when $\kappa=o((\log N)^{1/4}/\sqrt{\log\log N})$, and label noise is handled by error profiles. A two-variable gadget gives contexts that are always consistent and whose posterior predictive is $2\#F/(2\#F+1)$ for a CNF $F$. So approximating the posterior predictive within $1/4$ needs time $2^{(1-\gamma)N}$ at locality $O(\log N)$ under CNF-SETH, and time $2^{N/\mathrm{polylog}\,N}$ at locality $3$ under a length form of ETH. At constant locality, Bayesian inference in context is SAT parameterized by formula length. Learning in context is cheap. The Bayes risk of any class with $N$-bit descriptions is at most $N/(m+1)$ after $m$ examples. For binarized perceptrons and literal disjunctions on sparse data, the very families of the lower bounds, a linear program is a polynomial-time in-context learner with essentially the same sample efficiency. Every $\kappa$-local family is linear over $\sum_{j\le\kappa}\binom{N}{j}$ local Fourier features, so regression in context learns it from $N^{O(\kappa)}$ examples, in time polynomial in $N^\kappa$ and the inverse accuracy. The average case. For Goldreich's random local functions, the Bayes risk is at most $N/(m+1)$, while every in-context learner yields a distinguisher for Goldreich's generator, with the same running time up to a linear term, whose advantage is $1/2$ minus the risk. Under a local pseudorandom generator hypothesis with polynomial stretch, polynomial-time in-context learners need polynomially more examples than the Bayes predictor. Under its length form at linear stretch, which the new algorithms refute below rate $1/2$, the Bayes predictor of the refutations is optimal up to polylogarithmic factors in the exponent. For transformers, these bounds limit the length of a chain of thought that computes the posterior predictive, and they separate Bayes-optimal in-context learning from the regression-type in-context learning that transformers are known to implement.

cs.CC — Computational Complexity

Fast Attention after SETH: The Exact Exponent and the Hypotheses Behind It

Contributed by Xiaoyu Li

Alman and Song showed that attention with head dimension $O(\log n)$ can be approximated in almost linear time when the entries are $o(\sqrt{\log n})$, and needs quadratic time at $\Theta(\sqrt{\log n})$ under the Strong Exponential Time Hypothesis. In October 2026 the Exponential Time Hypothesis was refuted, and with it SETH. The quadratic lower bound now rests on the hardness of Orthogonal Vectors in dimension $c\log n$ (LDOV), which no open hypothesis about satisfiability is known to imply. We measure an instance by its spread $W$, the largest gap between two logits of a row. At every nontrivial constant precision and in every dimension between $\omega(\log n)$ and $n^{o(1)}$, we show that under LDOV the time exponent of attention is exactly $1+\min\{W/\ln n,1\}$: uniform sampling of keys is optimal, the cost is $\min\{n\,e^{W},n^2\}$ up to factors $n^{o(1)}$, and the transition to quadratic time sits at $W=\ln n$, the spread at which one key can outweigh all others. At polynomial precision and in dimension $O(\log n)$ the threshold becomes $\Theta(\log n)$, with a constant that tends to zero as the precision grows, and the dimension moves it: at precision $n^{-1-O(1/\log\log n)}$, spread $O(\log n/\log\log n)$ is hard in dimension $(\log n)^{1+\Omega(1)}$ but easy in dimension $O(\log n)$. From satisfiability we obtain hardness at spread $(1+\theta)\ln n$ in dimension $n^{o(1)}$, provided the hard CNF formulas have $N^{c_0}$ clauses for some $c_0<2$, and we show that the construction behind this bound, which builds one instance for each proof of a Merlin–Arthur protocol for disjointness, needs $c_0<2$ for every such protocol. For the entry bound, every reduction that embeds Orthogonal Vectors vector by vector, with polynomial blowup and logit gap $\Omega(\log n)$, yields a Merlin–Arthur protocol for disjointness; by Klauck's lower bound, such reductions cannot derive $B=\Theta(\sqrt{\log n})$ from Orthogonal Vectors in dimension $(\log n)^{1+\Omega(1)}$. We also map the conditional lower bounds for attention in the literature to the hypotheses that now support them, and find two families whose hypotheses are false, one of them in part of its parameter range.

cs.SC — Symbolic Computation

Disproof of minimum CAD existence for algebraic sets in dimensions $n\ge4$

Contributed by Qiyue Tang

We give real algebraic counterexamples to the existence of a minimum adapted cylindrical algebraic decomposition in every dimension $n\ge4$, for a fixed coordinate order.

math.NT — Number Theory

Cumulative Prime Sums and Riemann Zeros: A fine grid, its ripple, and a family of prime races

Contributed by Antonio Mennillo

Summing the primes one after another and taking logarithms gives an increasing sequence of real numbers, $\sigma_m=\log S_{p_m}$ with $S_N = 1+\sum_{p\le N}p$, which becomes exponentially dense. The ordinates $\gamma_n$ of the zeros of the Riemann zeta function fall on this grid with tiny errors: $1.4\cdot10^{-3}$ at the first zero, $2.4\cdot10^{-7}$ at the fifth, $3\cdot10^{-17}$ at the twentieth. Part I explains why. The cell of the grid containing a height $t$ has width $(1+o(1))\sqrt t\,e^{-t/2}$ (Step Theorem), and every real $t>\log 3$ has a node above it within that width, by an argument going back to Bertrand's postulate (Covering Theorem). The precision observed at the zeros is therefore the resolution of the grid, and the same precision holds at every height (Genericity Principle). We compute the first-crossing primes $N_n$ with certification up to $n=20$ and test, in a protocol fixed before the computation, whether the zeros sit closer to the nodes than a generic height. Neither the mean distance to the nearest node ($0.246$ against $0.250$, $p=0.92$) nor the distribution of the position within the cell (Kolmogorov–Smirnov, $p=0.79$) departs detectably from the generic case, and an integer grid used as control behaves in the same way. We also correct the half-exponent law by a factor $\sqrt{t'/t}$. Part II shows where the zeros are actually visible. They are not in the position of the nodes but in the ripple: the deviation of the grid from its mean curve $B(x)=1+\mathrm{li}(x^2)-\mathrm{li}(4)$. This ripple is a sum of pure oscillations, one per zero; its spectrum shows a peak at each of the first 13 ordinates, with the amplitude predicted by the explicit formula. Under RH the ripple has size $e^{-t/4}$ on the grid, much larger than the cell $e^{-t/2}$, and RH is equivalent to this decay rate. The ripple is not centred: on $[10^3,10^8]$ the sum of the primes stays below $\mathrm{li}(x^2)$ for about 97% of logarithmic time, and it first exceeds it at $x=109$. We place this in a one-parameter family of prime races, $\sum_{p\le x}p^a$ against $\mathrm{li}(x^{a+1})$, and prove that under RH the normalized error has mean $-1/(2a+1)$, coming from the squares of the primes. Under RH and LI the limiting logarithmic density of the set where the race is ahead is given by a Rubinstein–Sarnak formula; it moves from $2.6\cdot10^{-7}$ (counting the primes, $a=0$) to $0.06$ (summing them, $a=1$) and tends to $1/2$ as $a\to\infty$, at the rate $\tfrac12-\delta_a\sim1/\bigl(2\sqrt{\pi A\log(A/2\pi)}\bigr)$ with $A=a+\tfrac12$. At $a=-1/2$ the bias becomes the $\tfrac12\log\log x$ of the weighted counting functions of Aoki and Koyama. All theorems are formalized in Lean 4, with the classical results they use as explicit hypotheses.

math.OC — Optimization and Control

A Sixth-Order Hurwitz Interval Polynomial Family Without Degree-Five Weak Strictly Positive Real Numerator

Contributed by Tiancheng Xu, Zhengtai Xiong, Wensheng Yu

We construct an explicit sixth-order Hurwitz-stable interval polynomial family with independent closed coefficient intervals that admits a common strictly positive real (SPR) numerator of degree six, yet admits no common weak strictly positive real (WSPR) numerator of degree five. More strongly, no nonzero real polynomial of degree at most five makes the real part of the associated transfer functions nonnegative across the family at all real frequencies. This counterexample shows that, for sixth-order interval polynomial families, robust Hurwitz stability does not guarantee the existence of a strictly proper common SPR or WSPR numerator of degree five, and equal numerator degree six is necessary in general.

math.NT — Number Theory

Exact Intervals and Arithmetic Structure in Fuchs' Extremal Problem

Contributed by Zhenyu Sun

We determine Fuchs' weighted unit-circle extremum for monic integer polynomials for $0<\lambda\le3$. The proof combines a fractional difference energy with a sharp rearrangement inequality and an explicit gap for nonnegative integer sequences that are not interval indicators. We obtain the finite-degree minimum on $(1,2)$ at every degree and on $(2,3)$ at every degree at least five. Every minimizing sequence on these intervals eventually consists, up to monomial factors, of Dirichlet polynomials and products of two adjacent Dirichlet polynomials, respectively. At integer parameters, Parseval's identity reduces the problem to integer coefficient norms. The lower bound is attained exactly when a multiplicity-free ideal Prouhet--Tarry--Escott pair exists. Classical PTE constructions give exact values for the integers $1$ through $10$ and for $12$, and a two-value alternative at $11$. Subset counting gives a quadratic upper bound in general. The interval formulas and explicit constructions show that $2$, $3$ and $4$ are strict local maxima; the formula on $(2,3)$ has a unique minimum. A Pell construction gives an upper bound on $(4,5)$, and coefficient-norm gaps exclude Dirichlet products uniformly near certain integers. For each fixed degree bound, the extremum, divided by a common gamma factor, is locally the lower envelope of finitely many rational functions. Exact lattice enumeration identifies a nonproduct minimizer on an open interval containing $17/4$ for every degree bound from $61$ to $80$.

Author comments: 24 pages. Includes exact-arithmetic programs and data.

math.HO — History and Overview

Grothendieck Montpellier. Énoncés des lectures modernisées démontrés.

Contributed by Michel Hua

Working notes of Alexandre Grothendieck, put online by the University of Montpellier, have been transcribed and restated in current notation with a language model; the restatements may contain errors. This appendix proves statements taken from these restatements in seventy-five folders, in commutative algebra, algebraic geometry, plane affine geometry, general and tame topology, category and group theory, homotopical algebra, order theory, algebra and finite combinatorics. Each proof follows a proof checked in the Lean 4 proof assistant with its library mathlib. For each statement the text gives the context of the leaf, the proof, the hypotheses that are superfluous or necessary, and the corrections made to the restatement or to the page. It ends with the list of statements the sections report without a Lean check.

math.CO — Combinatorics

Finite Field Grothendieck Inequality, Finite Field Johnson-Lindenstrauss Flattening and Finite Field Bourgain-Tzafriri Restricted Invertibility Problems

Contributed by K. MAHESH KRISHNA

We ask for finite field versions of following three: (1) Grothendieck Inequality, (2) Johnson-Lindenstrauss Flattening Lemma, (3) Bourgain-Tzafriri Restricted Invertibility Theorem.

math-ph — Mathematical Physics

A projective twistor correspondence for local analytic Einstein germs

Contributed by Lior Isthmus

For each prescribed numerical cosmological constant, including zero, we establish a convergent projective twistor correspondence recovering every marked four-dimensional local analytic Einstein germ of any algebraic Weyl type. The inputs are two independent analytic Čech classes of weights $2$ and $-6$ on a fixed flat twistor germ $T_0$ along its whole marked zero line, encoding both complete chiral curvature records. The fixed source correction and calibration select a unit optical section, an intrinsic projective extension condition recognizes Einstein geometry, and the prescribed normalized contact, optical and affine maps and their inverses converge on common neighborhoods of the whole physical sky for bounded holomorphic input families. On independently admitted ordinary and operator domains we prove both inverse comparisons, with the two source operations acting in one projective analytic Heisenberg model. The same comparisons preserve the nonlinear source, compact point objects, optical evaluation and calibrated infinity forms, together with physical point and full spin-frame changes and coupled Lorentz reality. For fixed positive Schwarzschild mass, we obtain finite horizon-crossing presentations, a calibrated local primitive comparison and optical approximation bounds for the entire specified compact nonradial crossing family using finite local readouts and the exact benchmark cuts, coordinate comparisons and Sachs reference frames.

Author comments: v1.0. The same manuscript is available at https://doi.org/10.5281/zenodo.23975732.

math.CT — Category Theory

Acyclic Networks: Causal Comparisons, Convex Minors, and Gauge Descent

Contributed by Zhenyu Sun

We study natural transformations, minors, and symmetric monoidal decorations of finite directed acyclic networks. For any weakly connected small source category, a natural transformation into a free acyclic path category is determined by one component. If more than one transformation exists, common prefixes and suffixes give a canonical factorization through two constant functors and identify the transformation set with a single set of directed paths. Consequences include boundary rigidity, a decision algorithm, and a fundamental-group obstruction to non-uniqueness. For fusion–embedding minors, right fillings exist exactly when the graph pushout is acyclic. A support invariant detects a nonexact minor under the full gauge congruence, while path-convex embeddings admit two-sided fillings. Inverse-image pullbacks give canonical partial-map normal forms, a scanning algorithm, and a coherent presentation of the cartesian quotient. For open networks, wire splicing gives units, and diagram groupoids satisfy descent by homotopy pullback over the interface groupoid, including through-wires. This descent is invariant under strong symmetric monoidal equivalence and compatible with admissible event contractions within each piece. Finite-field examples yield orbit classifications, stabilizer weights, and boundary convolution formulas. Roof-refinement spaces realize spheres whose higher homotopy is lost under groupoid completion.

math.CV — Complex Variables

Supremum estimates for $\bar\partial$ on bounded convex domains in $\mathbb{C}^2$

Contributed by Yun-Heng Du

For every bounded convex domain $\Omega\subset\mathbb{C}^2$, we give a bounded distributional solution of $\bar\partial u=f$ for every distributionally $\bar\partial$-closed form $f\in L^\infty_{(0,1)}(\Omega)$. The solution satisfies $\|u\|_\infty\le C_\Omega\|f\|_\infty$, where $C_\Omega>0$ depends only on $\Omega$. The estimate follows from holomorphic localization of Leray rows and boundary cancellation.

math.NT — Number Theory

Points of prime order on elliptic curves over number fields of degree $9$ to $22$

Contributed by Filip Najman

Let $S(d)$ be the set of primes $p$ for which there exists an elliptic curve over a number field of degree $d$ with a point of order $p$. The sets $S(d)$ are known for $d\leq 8$. We determine $S(d)$ for all $9\leq d\leq 22$. For example, the largest prime in $S(22)$ is $67$, and $S(16)$ consists of the primes up to $37$. To reduce the problem to finitely many primes we prove a version of Oesterlé's bound which uses reduction modulo $2$ instead of modulo $3$, and which is much smaller for $d\geq 13$. The remaining primes are dealt with using results of Derickx and Stoll and extensive computations with the modular curves $X_1(p)$ over $\mathbb{F}_2$ for $p\leq 139$.

Author comments: Code: https://github.com/F-Najman/Sd. Computational certificates: https://doi.org/10.7910/DVN/5HHEY4.

math.RA — Rings and Algebras

The L'vov–Kaplansky conjecture for polynomials of degree four

Contributed by Qiyue Tang

We prove the L'vov–Kaplansky conjecture for multilinear polynomials of degree four over algebraically closed fields of characteristic zero.

math.CO — Combinatorics

A new proof of van der Waerden's inequality, and beyond

Contributed by Suvrit Sra

We give a new, self-contained proof of van der Waerden's permanent inequality via a monotone-averaging argument. Specifically, we show that the minimum total correlation of a random permutation with prescribed doubly stochastic marginals never decreases when two rows or two columns of the marginal matrix are averaged. This quantity is largest at the uniform matrix, which yields van der Waerden's bound. The monotone-averaging rests on the quadratic Alexandrov–Fenchel inequality, which lets us extend the argument to many other problems. In this way we also prove the Friedland–Tverberg inequalities and Gurvits's coefficient bound for Lorentzian polynomials.

math.OC — Optimization and Control

The generalized Lax conjecture holds for mixed determinants

Contributed by Suvrit Sra

We prove that the mixed determinant $\eta(\cdot)$ of Borcea and Brändén has a spectrahedral hyperbolicity cone. More precisely, for Hermitian matrices $A_1,\ldots,A_m$ of order $d$, the hyperbolicity cone of $\eta(A_1,\ldots,A_m)$ is cut out by $\sum_j\Phi_j(A_j)\succeq0$, where $\Phi_1,\ldots,\Phi_m$ are explicit unital completely positive maps. We argue by induction on $d$, adding one row and one column to every argument and expressing the logarithmic derivatives of $\eta$ as extremal eigenvalues. By compressing along polynomial boundary kernels, we reduce the order of the pencil $\sum_j\Phi_j(A_j)$ to $\exp(O(md^3\log m))$; for one fixed positive semidefinite $n\times n$ matrix $A$ a separate pencil yields the order $\exp(O(n^2\log n))$. We also answer four problems of Borcea and Brändén negatively.

math.NT — Number Theory

The parity of the prime-counting function in time $x^{o(1)}$

Contributed by Vaibhav Tulsyan

Computing the prime-counting function $\pi(x)$ exactly takes $x^{1/2+o(1)}$ steps by the analytic method of Lagarias and Odlyzko, and to our knowledge its parity is known faster only on short intervals: Tao, Croot and Helfgott decide whether an interval of length at most $x^{1/2+c}$ near $x$ contains an odd number of primes in time $x^{1/2-c+o(1)}$. However, their identity needs the Mobius function $\mu(d)$ for all $d\le\sqrt x$, and we know of no bound of the form $x^{o(1)}$ for the parity, even conditionally. We present deterministic algorithms that output $\pi(x) \bmod 2$ and the parity of the number of primes in any interval $[A,B]$ in time $2^{O(n/\sqrt{\log n})}=x^{o(1)}$, where $n$ is the input bit length. Our approach replaces $\mu(d)$ by its residue modulo 2, which is the squarefree indicator again: the number of chains $d_1^2\mid m$, $d_2^2\mid d_1,\dots$ of depth $L$ is odd exactly when $m$ is squarefree, for $m<2^{2^L}$. The parity of $\pi$ on $[A,B]$ is then the parity of one count of tuples satisfying $O(\log n)$ multiplications, which Anthropic's counting engine evaluates through a network whose variables have geometrically shrinking widths. We prove both theorems in Lean 4 (piParity_TM_rate, piIntervalParity_TM_rate; axioms propext, Classical.choice, Quot.sound). The result does not give deterministic prime finding or the exact value of $\pi(x)$: the parity can be even on every interval one queries, and exact counts seem to need iterated products, for whose product-tree verifiers we prove a lower bound in Lean. The constants are astronomical, and the result rests on Anthropic's Lean development of its claimed refutation of the Exponential Time Hypothesis, which is under review.

math.NT — Number Theory

Class numbers, square roots and quadratic non-residues in time $2^{O(n/\sqrt{\log n})}$, assuming the new #SAT algorithms

Contributed by Vaibhav Tulsyan

Basic counting problems of computational number theory have proven deterministic bounds that are fixed powers of the input. The class number $h(D)$ of an imaginary quadratic discriminant takes $|D|^{1/2+o(1)}$ steps unconditionally and $|D|^{1/5+o(1)}$ under the generalised Riemann hypothesis (GRH); exact divisor sums $\sum_{m\le x}\tau(m)$ take $x^{1/3+o(1)}$. However, none of these bounds is of the form $X^{o(1)}$, even conditionally. We present deterministic algorithms in time $2^{O(n/\sqrt{\log n})}$, where $n$ is the input bit length, for the number of reduced forms of discriminant $-N$ (for fundamental discriminants, $h(-N)$) and for $\sum_{m\le x}\tau(m)$, $\sum_{m\le x}\sigma(m)$ and the Gauss circle count. Our approach writes each answer as a count of solutions of equations with one or two products and evaluates it with Anthropic's counting engine through one network library with $O(n)$ shared bits. The same library gives the least square root modulo any $m$ and the least quadratic non-residue modulo a prime. These are not new bounds (OpenAI's results on the quasi-Riemann hypothesis and on deterministic polynomial factorisation over $\mathbb{F}_p$ give polynomial time modulo a prime), but they are machine-checked. We prove the seven resulting theorems in Lean 4 (for example hForm_TM_rate; axioms propext, Classical.choice, Quot.sound): a finite-alphabet Mathlib Turing machine outputs the answer within $\lfloor C\cdot2^{Cn/\sqrt{\log_2(n+2)}}\rfloor$ steps on every input. Lean proves the count of Gauss's reduced forms; its equality with the ideal class number is classical and not formalised. The constants are astronomical, the machines use exponential space, and every result rests on Anthropic's Lean development of its claimed refutation of the Exponential Time Hypothesis, which is under review. Version 2 corrects the claims of version 1 about non-residues and square roots.

math.OA — Operator Algebras

Universal Modules, Natural Pairings, and Weighted Isometries for Noncommutative $L^p$-Spaces

Contributed by Zhenyu Sun, Jilin Wang

For $1\le p<\infty$, we consider the functor $L^p$ on finite tracial von Neumann algebras with trace-preserving embeddings, including nonunital ones. For a functor of unital contractive left modules with compatible distinguished vectors $\xi_M$, finite direct sums of scalars determine the best constant in $\|x\xi_M\|\le K\|x\|_p$. The pointed $L^p$ functor is initial among the families satisfying this estimate with $K=1$. Retaining only the bound on scalar points gives an atomic norm. On $\mathbb C^n$ with unit weights the norm of the identity from $\ell^p_n$ to this atomic space is asymptotic to $p^{-1}(\log n)^{1-1/p}$ for $p>1$. For $1\le p,q<\infty$, every objectwise bounded natural bilinear family on $L^p\times L^q$ has the form $c\,\tau(xy)+d\,\tau(x)\tau(y)$ for $x,y\in M$, with $c,d$ independent of the object and $c=0$ when $1/p+1/q>1$. A uniformly bounded natural inner product on every $L^p(M,\tau)$ exists exactly at $p=2$. For finite-dimensional unital embeddings preserving a faithful positive functional, we determine the nearest modularly compatible target density both in Hilbert--Schmidt norm and in the Hilbert--Schmidt distance between positive square roots. The minimum squared distance between square roots lies between a weighted sum of the losses in squared $L^2$ norm on matrix units and twice that sum. Both constants are sharp.

math.GR — Group Theory

Three-Relator Universal Group, Four-Relator Adian–Rabin Family, and Unrecognizability of $\#^2(S^2\times S^2)$

Contributed by Qiuyu Ren

We construct a universal group with 2 generators and 3 relators, and an Adian–Rabin family with 2 generators and 4 relators whose nontrivial groups are universal. The resulting presentations show that the standard smooth manifold $\#^n(S^2\times S^2)$ is unrecognizable up to homeomorphism or diffeomorphism for every fixed n ≥ 2.

cs.LO — Logic in Computer Science

Voblint: End-to-End Soundness of a Context-Sensitive Goblint-Style Analyzer in Isabelle/HOL

Contributed by Manuel Lerchner

Goblint-style analyzers solve side-effecting equations, where evaluating one unknown can contribute to other unknowns. An existing Isabelle/HOL proof establishes solver partial correctness without connecting the equations to execution. We mechanize this connection for a scalar imperative language with recursive procedures but no pointers or heap memory. We adapt the local traces of Schwarz et al. to record procedure activations and the callers resumed by returns. A context policy relates concrete calls to one or more analysis contexts. A cover assigns sets of stores to program points and contexts as candidate overapproximations of reachability. We prove that five local obligations suffice for a cover to contain every valid trace’s final store in every admitted context, on any control-flow graph. A post-solution assigns abstract values that bound each equation’s result and side effects. When the abstract operations are sound and the routing agrees with the context policy, every post-solution induces a cover. The verified solver computes a post-solution when it returns. For every accepted configuration, whenever the Isabelle-defined analyzer returns a report, it overapproximates the stores reached by finite source executions from the prescribed initial states. Whenever execution reaches a check, PROVED guarantees that its condition holds, and REFUTED guarantees that it fails. Neither verdict establishes reachability. This source-level guarantee applies to the Isabelle-defined analyzer; solver termination, parsing, and correctness of generated code are not established.

math.NT — Number Theory

TRANSCENDENCE OF THE 3x+ 1 CONJUGACY MAP ON STURMIAN WORDS

Contributed by Elias De Jesús

Let T be the 3x + 1 map on the 2-adic integers and let Φ : ℤ₂ → ℤ₂ be the Bernstein–Lagarias conjugacy map, which sends a parity vector to the unique 2-adic integer realising it. We prove that Φ(cγ), Φ(1cγ), Φ(0cγ) and Φ of every shift of these words are transcendental for every irrational slope γ < γ* = (3 + √5)/(4 log₂ 3) = 0.8258977…, with no hypothesis whatever on the partial quotients of γ; in particular at the resonance slope γ = log₃ 2. The mechanism is a bridge: by the Bernstein–Lagarias isometry, the 2-adic approximation depth of a periodic shadow equals a longest common prefix, so the Diophantine behaviour of Φ is governed by the repetition structure of the parity word, at a height cost A(γ) = max(1, γ log₂ 3) that we control at every level. Quantitatively ω₁⁽²⁾(Φ(cγ)) ≥ ice(cγ)/A(γ) − 1, where ice is the initial critical exponent of Berthé–Holton–Zamboni; their floor ice(cγ) ≥ 1 + φ then beats the Ridout threshold ω₁⁽²⁾ ≤ 1 by a margin bounded away from zero. The result is not a corollary of the digit-based transcendence criteria: the 2-adic Hensel digits of Φ(c_{log₃ 2}) are measured to have full complexity 2ⁿ, against the parity word’s n + 1, and the shadow denominators 2^ℓ − 3^k are not S-units for any fixed finite S. We also record the Liouville alternative for unbounded partial quotients and a finite effective irrationality measure for bounded ones at slopes γ ≤ log₃ 2.

math.AG — Algebraic Geometry

Terminal Symplectic Singularities for Parabolic Base Affine Spaces

Contributed by Tom Gannon

Let $G$ be a simply connected semisimple complex algebraic group, and let $U_P$ be the unipotent radical of a parabolic subgroup $P\subseteq G$. We prove that the affinization $\overline{T^*(G/U_P)}$ is factorial and has terminal symplectic singularities. To establish symplectic singularities, we apply a criterion of Fu--Liu after constructing a log Fano boundary on an associated projective bundle over the full flag variety. The boundary construction uses Frobenius splitting of flag varieties and reduction modulo primes to obtain log canonicity in characteristic zero. We then prove that every first-order Poisson deformation of $\overline{T^*(G/U_P)}$ is trivial. Namikawa's deformation theory implies that the singular locus has codimension at least four, yielding terminality. No conical grading is assumed.

math.NT — Number Theory

An improved upper bound on the number of representations of an integer as a binomial coefficient

Contributed by Ben Whitmore

Let $N(t)$ be the number of pairs of integers $(n,m)$ with $\binom nm=t$. Singmaster conjectured that $N(t)=O(1)$, and the best bounds in the literature are $N(t)=O(\log t\,\log\log\log t/(\log\log t)^3)$ unconditionally, and $N(t)=O((\log t)^{2/3+\varepsilon})$ conditional on Cramér's conjecture. We show that $N(t)=O((\log t)^{1/2+o(1)})$ unconditionally. Solutions with $m\ge\exp((\log n)^{2/3+\varepsilon})$ are handled by a theorem of Matomäki, Radziwiłł, Shao, Tao and Teräväinen. For smaller $m$ we show, using an interpolation determinant in the spirit of Bombieri and Pila, that the solutions with $m$ in a short interval lie on a single algebraic curve of low degree in $n$, and we bound the number of solutions on such a curve by a zero estimate for exponential polynomials.

math.PR — Probability

THE CRITICAL CAPACITY OF THE HOPFIELD MODEL

Contributed by Yuwen Zhou

Abstract. We prove that the critical capacity of the Hopfield model at zero temperature lies in [0.138198129, 0.13820106]. The best rigorous lower bound was 0.113, and our lower end exceeds the published one-step and two-step replica symmetry breaking values. The capacity is the supremum of the loads at which the free energy at fixed overlap with one pattern has a well. At every positive temperature, we give a variational formula for this free energy and show that binary patterns and Gaussian rows have the same wells, which characterizes the capacity. The proof builds on the Hamilton–Jacobi solution of the model by Ali, Boursier, Chen, Ho, Issa and Mourrat, and on a

math.CO — Combinatorics

Hadamard difference sets in groups with a cyclic odd quotient

Contributed by David Niedbala Giraudin

Let u > 1 be odd. We prove that the abelian group C_2 x C_{2u^2} contains no Hadamard difference set. Together with the recently announced proof of the circulant Hadamard conjecture, this shows that an abelian group of order 4u^2 (u > 1 odd) whose odd Sylow subgroups are all cyclic never contains a Hadamard difference set, so Lander's conjecture holds for Hadamard difference sets in these groups. More generally, let G be a group of order 4u^2 with a normal subgroup N such that G/N is isomorphic to E x C_{w^2}, where |E| = 4, w > 1, w | u and gcd(w, u/w) = 1. If every prime divisor of u/w is self-conjugate modulo rad(w) (modulo 4 rad(w) when E is cyclic), then G contains no Hadamard difference set. The proof adapts the alternating character products used for the cyclic case.

math.NT — Number Theory

The Hankel determinants of the Baum-Sweet sequence are 2-automatic

Contributed by David Niedbala Giraudin

Let b be the Baum-Sweet sequence and H_n = det(b_{i+j})_{0<=i,j<n} its Hankel determinants over F_2. We prove that the sequence chi_0(n) = [H_n != 0] is 2-automatic: chi_0(n+1) is computed by an explicit automaton with 8 states reading the binary expansion of n, and the minimal automaton of chi_0 reading least significant digits first has 74 states. The same holds for the shifted determinants det(b_{i+j+1}), with the same 8-state graph and 68 states. This contrasts with Mkaouar's theorem that the partial quotients of the Baum-Sweet series are not automatic. On the way we show that Theta = t f^2, a root of Z^3 + Z + t^3 in F_2((1/t)), has all its partial quotients in {t, t^3, t^3 + t}, given by an explicit coding of the Mills-Robbins expansion, and that the linear complexity profile of b satisfies -2 <= 2L(b,N) - N <= 3 for all N, all six values occurring. The proofs reduce to finite verifications, which an ancillary script performs.

math.DS — Dynamical Systems

Three Axes of Collatz Dynamics: 2-adic Measures, the Dimension Spectrum of Inverse Trees, and Critical Exponents of Density Propagation

Contributed by Zeng Chunping

This work presents a systematic measurement and critical diagnosis of three research axes on the Collatz (3n+1) problem, with the aim of determining which tools can advance the problem and which are structurally incapable of doing so. The three axes are: the 2-adic statistics of individual orbits, the combinatorial structure of the inverse tree, and the critical exponent of density propagation (the Terras-Krasikov-Lagarias line). On the first axis we establish a strict identity and a divergence criterion. For the odd-step map m -> m' = (3m+1)/2^a with a = v_2(3m+1), one has log m_{K+1} - log m_1 = K log 3 - (sum a_i) log 2 + sum log(1 + 1/(3m_i)), verified to 10^{-13}. Divergence is therefore equivalent to (average a) < log_2 3. Over 298 million odd steps spanning nine orders of magnitude we obtain average a = 1.996929, matching the i.i.d. geometric prediction E[a] = 2, with a safety margin of 0.412 (a 20.6% drop would be needed for divergence). The distribution of a is exactly geometric and the autocorrelation is approximately zero; a surrogate control (preserving the marginal, destroying the serial structure) shows no difference between real and shuffled sequences at long windows. On the second axis we derive a closed form for one-step propagation on the inverse tree. The two extensible residue classes give Lambda_1(beta) = (3/4)^beta/(1-4^{-beta}) and Lambda_2(beta) = (3/2)^beta/(1-4^{-beta}). Because preimages are uniformly distributed over the three residue classes, the extension matrix has rank one and rho(beta) = (Lambda_1 + Lambda_2)/3 = 3^{beta-1}/(2^beta - 1). At beta = 1 one has Lambda_1 = 1 (critical) and Lambda_2 = 2 (supercritical), the latter being exactly cancelled by the one-third loss into leaf nodes, so that the total spectral radius is exactly 1. This turns "density propagation stops at beta = 1" from an observation into a corollary of the closed form. On the third axis the depth-k reachable set is measured as a genuine fractal of dimension gamma_k (gamma_1 = 0.101 logarithmic, gamma_12 = 0.603, gamma_24 = 0.824, gamma_40 = 0.923, increasing monotonically to 1). A decisive test rejects the competing polylogarithmic hypothesis (at k = 24: measured growth x294, polylog prediction x1.7 x 10^7). We further show that the type-transfer spectral radius cannot produce any gamma_k: its zero is constantly the asymptotic ceiling 1; obtaining gamma_k requires the infinite-branch Ruelle operator on the spatial variable, which we construct and then falsify by a fit-free hard criterion (operator count 5398 exceeds the hard bound X/2 = 5000). Two popular frameworks are falsified: the local-entropy framework (its numerical verification is vacuous and its upper-bound theorem conflicts with the random model) and the conjecture that truncated propagation possesses a finite critical root (the truncated ratio is always below 1 and increases monotonically to 1). All numerics carry finite-size labels; the finite-size deviation is proved to be O(1/log X) rather than O(1/X). A methodological ledger records nine recurring errors and their countermeasures. The three axes are finally unified by the master identity rho(beta) = M(beta-1), where M is the moment generating function of the single-step log-increment: the inverse tree, the stopping-time large deviations, and the orbit drift are three readings of one function. Five corollaries follow, including rho'(1) = delta(3) (the spectral slope at criticality equals the drift) and beta_c = -log min rho (the far-tail rate is the negative logarithm of the spectral minimum). A three-way shootout against the Bessel(3)/meander hypothesis for the survival-profile shape falsifies that hypothesis (z = +/-50 to 69), isolating an open phenomenon: Collatz survival trajectories have a definite "initial plateau, mid-concentrated descent" shape that no model with the correct marginal reproduces.

math.OA — Operator Algebras

Noncommutative von Neumann-Riesz and von Neumann-Riesz-Lorch Mean Ergodic Theorems

Contributed by K. MAHESH KRISHNA

We derive noncommutative von Neumann-Riesz and von Neumann-Riesz-Lorch mean ergodic theorems for adjointable contractions on a Hilbert C*-module. We crucially use a result obtained by Paschke [\textit{Trans. Amer. Math. Soc., 1973}]. We also note that the exact version of the noncommutative von Neumann–Riesz mean ergodic theorem fails.

math.NT — Number Theory

The Discriminant Structure of Frey Curve Isogeny Classes: Crater–Peak Duality, the Volcano Law, and a Mass Criterion

Contributed by Zeng Chunping

Let $a,b,c$ be coprime positive integers with $a+b=c$, and let $E: y^2=x(x-a)(x+b)$ be the associated Frey curve. We determine the structure of minimal discriminants across the $\mathbb{Q}$-isogeny class $\mathcal{C}(E)$ by closed formulas in $(a,b,c)$. The three $2$-torsion points of $E$ are rational, so $E$ is always the centre of its class; the first-level quotients via Velu's formula have monomial model discriminants $\{16(abc)^2,\ 256a^4bc,\ 256ab^4c,\ 256abc^4\}$, an exact collapse forced by the identity $u=4f(r)$ in Velu's algorithm. We prove the Crater--Peak Law: among the first-level vertices the minimal and maximal discriminants are $\min\{16(abc)^2,\,256\min(a,b)^4\max(a,b)c\}$ and $256abc^4$, the exponent $4$ landing on the smallest entry (crater) and the largest (peak); for star classes these are the class extrema, while on square or quartic triggers the extrema are the minimum and maximum over ten explicit candidates. The class size $|V|\in\{4,6,8\}$ is decided by the square and quartic character of $ac$ and $bc$ (107 samples, zero failures), matching the isogeny-torsion graph classification via the lcm of the isogeny matrix. When $ac$ or $bc$ is a square the class extends to depth $2$ with model discriminants $2^{10}ACu^8v^2$ and $-2^{10}ACu^2v^8$, and to depth $3$ on quartic triggers with discriminants $2^{11}pq\,u^{16}v^4e$ and $2^{11}pq\,u^4v^{16}e$. On the multiplicative bad-prime set $\pi$ we prove the Volcano Law: each vertex carries an integer height vector $k(V)\in\mathbb{Z}^{\pi}$ with $v_p(\Delta^{\min}_V)=2^{k_p(V)}v_p(\Delta^{\min}_{\mathrm{crater}})$, and along every $2$-edge $k\mapsto k+\sigma$ with $\sigma\in\{\pm1\}^{\pi}$ independently per prime (578 edge-prime samples, zero exceptions), via the Tate parameterization. For odd primes $p\mid\min(a,b)$ we have $v_p(j)=-2v_p(\min)$, so the entire law is read off the $j$-invariant. The peak curve is explicit ($y^2=x^3-2(b-a)x^2+c^2x$), and the $j$-denominator exponent is a theorem: $\log\,\mathrm{den}(j)/\log\,\mathrm{num}(j)=(\alpha+2)/3+O(1/\log c)$, with the exact discrepancy identity verified to machine precision on $2185$ samples --- the Pasten meeting point in its precise form; we also show the Pasten mechanism is information-free on the Frey family. As an application, for $A^x+B^y=C^z$ we give an explicit two-branch formula for the crater ratio $\rho_c$; and with $\alpha=\log\min(a,b)/\log c$ we prove $\sigma_{\max}\to(5+\alpha)q$, so Szpiro is equivalent on the Frey family to the mass bound $q\le 6/(5+\alpha)$, sharp at $\alpha=1$ where Szpiro and abc coincide, while the residual gap lives at $\alpha=0$, the Pillai/$S$-unit corner. A Sage-verified table of $17$ classes (Kraus minimization agreeing with Sage minimal_model() throughout) gives $\sigma_{\max}>6$ in 14 of 17 rows (non-refutations: Szpiro allows $O(1)/\log N$ slack). All absolute $\sigma$ values are certified against Sage; structural statements are proved or machine-verified symbolically. We prove no conjecture: abc, Szpiro, Beal and the height conjecture remain open.

math.NT — Number Theory

Derivatives of Hardy's function at close pairs of zeros and an exact unitary constant $\pi^2/15$

Contributed by Jaskirat Singh

Let $\gamma_n<\gamma_{n+1}$ be consecutive ordinates of zeros of $\zeta(\tfrac12+it)$ with normalized gap $s_n$. We study the derivatives $Z'(\gamma_n)$, $Z'(\gamma_{n+1})$ of Hardy's function at close pairs. For the circular unitary ensemble we prove that the conditional variance of $\log|\Lambda'(\theta_1)|-\log|\Lambda'(\theta_2)|$ at adjacent eigenangles with normalized gap $s$ satisfies $\operatorname{Var}=c_N s^2(1+o(1))$ as $s\to0$, with the exact value $c_N=\frac{\pi^2}{15}\bigl(1-\frac{4}{N^2}\bigr)$, so that $c_\infty=\pi^2/15$, which we call the Singh constant. The proof reduces the problem through a Cayley transform to the Hua ensemble and closes it with two loop equations. We state a conditional theorem transferring the constant to the zeta function under the Riemann Hypothesis, a local GUE hypothesis and a tail bound, and we formulate a conjecture on the size of $Z'$ at close pairs. Numerical evidence from Odlyzko's tables up to zero number $10^{22}$ agrees with the predictions. This paper is a revised and expanded version of the Zenodo record doi:10.5281/zenodo.23899228.

Author comments: v2: Section 1 now lists fourteen Singh Lemmas, each labeled proved (6, 8, 45), conditional (7, 27, 42, 44, 46, 48, 54, 67) or numerical (5, 51, 57). The unitary pi^2/15 constant and Hua moments are credited as known results, Lemma 20 is credited to Soundararajan and Milinovich, and method credits and an explicit transfer hypothesis for Lemma 67 are added.

math.NT — Number Theory

Nonconstant Congruence Speeds of Integer Tetration in Non-Squarefree Radices

Contributed by Marco Ripà

We further develop the theory of congruence speed for integer tetration in non-squarefree numeral systems, building on Appendix B of [M. Ripà and G. Di Pietro (2026), A compact notation for peculiar properties characterizing integer tetration, Notes on Number Theory and Discrete Mathematics, 32(3), 477–496]. For bases coprime to the radix, we establish an eventual floor-difference representation and determine the exact spectrum of attainable least periods. We construct explicit arithmetic progressions containing infinitely many prime bases with nonconstant congruence speed for every non-squarefree radix, thereby proving the coprime-infinitude conjecture posed in Appendix B. We also derive a uniform quadratic bound for the least coprime base with nonconstant congruence speed and establish exact natural densities of nonstabilizing bases, both among integers coprime to the radix and among those not divisible by its radical, together with the corresponding relative density among primes. Finally, we give necessary and sufficient conditions for the perfect-power congruence-speed identity (Equation (2.12) of the cited paper) to hold in arbitrary radices, determine the exact least eventual period when stabilization fails, and establish a separate criterion for the boundary case of the original parameter restriction.