Tootfinder

Nonrational varieties with unirational parametrizations of coprime degrees
Song Yang, Xun Yu, Zigang Zhu
https://arxiv.org/abs/2508.03623 https://arxiv.org…

Nonrational varieties with unirational parametrizations of coprime degrees
We show that there exists a $2$-dimensional family of smooth cubic threefolds admitting unirational parametrizations of coprime degrees. This together with Clemens--Griffiths' work solves the long standing open problem whether there exists a nonrational variety with unirational parametrizations of coprime degrees. Our proof uses a new approach, called the Noether--Cremona method, for determining the rationality of quotients of hypersurfaces.

Factorization of periods, construction of automorphic motives and Deligne's conjecture over CM-fields
Harald Grobner, Michael Harris, Lin Jie
https://arxiv.org/abs/2509.02303

Factorization of periods, construction of automorphic motives and Deligne's conjecture over CM-fields
The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. As an application of our main result, we establish Deligne's conjecture for a class of CM-automorphic motives, which we construct in this paper. Our proof uses the results of our recent joint work with Raghuram in combination with the Ichino--Ikeda--Neal-Harris (II…

All Proof of Work But No Proof of Play
Hayder Tirmazi
https://arxiv.org/abs/2506.23435 https://arxiv.org/pdf/2506.23435

All Proof of Work But No Proof of Play
Speedrunning is a competition that emerged from communities of early video games such as Doom (1993). Speedrunners try to finish a game in minimal time. Provably verifying the authenticity of submitted speedruns is an open problem. Traditionally, best-effort speedrun verification is conducted by on-site human observers, forensic audio analysis, or a rigorous mathematical analysis of the game mechanics. Such methods are tedious, fallible, and, perhaps worst of all, not cryptographic. Motivated b…

Uniform semiclassical observable error bound of Trotterization without the Egorov theorem: a simple algebraic proof
Di Fang, Conrad Qu
https://arxiv.org/abs/2507.02783

Uniform semiclassical observable error bound of Trotterization without the Egorov theorem: a simple algebraic proof
Efficient simulation of the semiclassical Schrödinger equation has garnered significant attention in the numerical analysis community. While controlling the error in the unitary evolution or the wavefunction typically requires the time step size to shrink as the semiclassical parameter $h$ decreases, it has been observed -- and proved for first- and second-order Trotterization schemes -- that the error in certain classes of observables admits a time step size independent of $h$. In this work, …

Derived Stratifications and Arithmetic Intersection Theory for Varieties with Isolated Singularities
Jiaming Luo, Shirong Li
https://arxiv.org/abs/2508.01679 https://

Derived Stratifications and Arithmetic Intersection Theory for Varieties with Isolated Singularities
In this paper, We develop the stratified de Rham theory on singular spaces using modern tools including derived geometry and stratified structures. This work unifies and extends the de Rham theory, Hodge theory, and deformation theory of singular spaces into the frameworks of stratified geometry, $p$-adic geometry, and derived geometry. Additionally, we close a gap in Ohsawa's original proof, concerning the convergence of $L^2$ harmonic forms in the Cheeger-Goresky-MacPherson conjecture for var…

Proof-of-Behavior: Behavior-Driven Consensus for Trustworthy Decentralized Finance
Ailiya Borjigin, Wei Zhou, Cong He
https://arxiv.org/abs/2506.22171 http…

Proof-of-Behavior: Behavior-Driven Consensus for Trustworthy Decentralized Finance
Current blockchain protocols (e.g., Proof-of-Work and Proof-of-Stake) secure the ledger yet cannot measure validator trustworthiness, allowing subtle misconduct that is especially damaging in decentralized-finance (DeFi) settings. We introduce Proof-of-Behavior (PoB), a consensus model that (i) gives each action a layered utility score -- covering motivation and outcome, (ii) adapts validator weights using recent scores, and (iii) applies decentralized verification with proportional slashing. T…

On Existence and Uniqueness of the Solution of a Two-Surfaces Contact Problem Using a Fixed Point Approach
Abdelkrim Atailia, Frekh Taallah
https://arxiv.org/abs/2506.23763

On Existence and Uniqueness of the Solution of a Two-Surfaces Contact Problem Using a Fixed Point Approach
In this work, we give the proof of the existence and uniqueness of the solution to the weak form of a two-surfaces contact problem using fixed point approach. We begin by modeling the evolution of a two deformable surfaces contact problem with a general viscoplastic law, the contact is considered frictionless and governed by the Signorini-type condition with an initial gap. Then, we derive the variational formulation of the classical problem. Finally, we conclude our work by establishing an exi…

The Persson--Stepanov theorem revisited
Amiran Gogatishvili, Lubo\v{s} Pick, Hana Tur\v{c}inov\'a, Tu\u{g}\c{c}e \"Unver
https://arxiv.org/abs/2507.00867

The Persson--Stepanov theorem revisited
We develop a new proof of the result of L.-E.~Persson and V.D.~Stepanov \cite[Theorems 1 and 3]{Per:02}, which provides a characterization of a Hardy integral inequality involving two weights, and which can be applied to an effective treatment of the geometric mean operator. Our approach enables us to extend their result to the full range of parameters, in particular involving the critical case $p=1$, which was excluded in the original work. Our proof avoids all duality steps and discretization…

Combinatorial proof of a permuted basement Macdonald polynomial identity
Daniel Orr, Johnny Rivera Jr
https://arxiv.org/abs/2508.20337 https://arxiv.org/pd…

Combinatorial proof of a permuted basement Macdonald polynomial identity
A well-known and fundamental property of the Macdonald polynomials $P_λ(x;q,t)$ is their invariance under the transformation sending $(q,t)$ to $(q^{-1},t^{-1})$. Recently, Concha and Lapointe showed that this property extends in an interesting, nontrivial way to an identity for partially symmetric Macdonald polynomials. Their identity played a key role in the work of Bechtloff Weising and Orr linking partially symmetric Macdonald polynomials to parabolic flag Hilbert schemes. In this paper, w…

2D CFT and efficient Bethe ansatz for exactly solvable Richardson-Gaudin models
Grzegorz Biskowski, Franco Ferrari, Marcin R. Piatek
https://arxiv.org/abs/2507.22734 https://

2D CFT and efficient Bethe ansatz for exactly solvable Richardson-Gaudin models
This work inaugurates a series of complementary studies on Richardson-Gaudin integrable models. We begin by reviewing the foundations of classical and quantum integrability, recalling the algebraic Bethe ansatz solution of the Richardson (reduced BCS) and Gaudin (central spin) models, and presenting a proof of their integrability based on the Knizhnik-Zamolodchikov equations and their generalizations to perturbed affine conformal blocks. Building on this foundation, we then describe an alternat…

Chiral-odd generalized parton distributions in the large-$N_{c}$ limit of QCD: Next-to-leading-order contributions
June-Young Kim
https://arxiv.org/abs/2506.21013

Chiral-odd generalized parton distributions in the large-$N_{c}$ limit of QCD: Next-to-leading-order contributions
We investigate the nucleon's chiral-odd generalized parton distribution functions (GPDs) in the large-$N_c$ limit of QCD. Extending previous work on the leading-order contribution in the $1/N_c$ expansion, we focus on the next-to-leading-order contributions and provide a complete set of flavor-singlet and flavor-non-singlet chiral-odd GPDs. This study includes the derivation of the spin-flavor structure of the baryon matrix element of the chiral-odd operator, the proof of the polynomiality prop…

The Balmer spectrum of pseudo-coherent complexes over a discrete valuation ring
Beren Sanders, Yufei Zhang
https://arxiv.org/abs/2508.17603 https://arxiv.o…

The Balmer spectrum of pseudo-coherent complexes over a discrete valuation ring
We study the derived category of pseudo-coherent complexes over a noetherian commutative ring, building on prior work by Matsui-Takahashi. Our main theorem is a computation of the Balmer spectrum of this category in the case of a discrete valuation ring. We prove that it coincides with the spectral space associated to a bounded distributive lattice of asymptotic equivalence classes of monotonic sequences of natural numbers. The proof of this theorem involves an extensive study of generation beh…

New Bounds for the Ideal Proof System in Positive Characteristic
Amik Raj Behera, Nutan Limaye, Varun Ramanathan, Srikanth Srinivasan
https://arxiv.org/abs/2506.16397

New Bounds for the Ideal Proof System in Positive Characteristic
In this work, we prove upper and lower bounds over fields of positive characteristics for several fragments of the Ideal Proof System (IPS), an algebraic proof system introduced by Grochow and Pitassi (J. ACM 2018). Our results extend the works of Forbes, Shpilka, Tzameret, and Wigderson (Theory of Computing 2021) and also of Govindasamy, Hakoniemi, and Tzameret (FOCS 2022). These works primarily focused on proof systems over fields of characteristic $0$, and we are able to extend these results…

Reading McComas’ article from 1996, and I have to say it’s very refreshing.

TEN MYTHS OF SCIENCE: REEXAMINING WHAT WE THINK WE KNOW...

Some examples:

* Myth 3: A General and Universal Scientific Method Exists
* Myth 5: Science and its Methods Provide Absolute Proof
* Myth 6: Science Is Procedural More Than Creative
* Myth 10: All Work in Science is Reviewed to Keep the Process Honest

Resolving an error with path-tracing and a 2-to-1 mapping in a work of Jang, So, and Marotta
Murali Meyer, Daniel Stoertz, Mike Wang
https://arxiv.org/abs/2508.18543 https://

Resolving an error with path-tracing and a 2-to-1 mapping in a work of Jang, So, and Marotta
We identify two nontrivial errors in the proof of the main results of the work \emph{generalized baby Mandelbrot sets adorned with halos in families of rational maps} by Jang, So, and Marotta. We correct these errors, showing that the main results remain true.

A Formalization of the Ionescu-Tulcea Theorem in Mathlib
Etienne Marion (ENS de Lyon)
https://arxiv.org/abs/2506.18616 https://arxiv.…

A Formalization of the Ionescu-Tulcea Theorem in Mathlib
We describe the formalization of the Ionescu-Tulcea theorem, showing the existence of a probability measure on the space of trajectories of a Markov chain, in the proof assistant Lean using the integrated library Mathlib. We first present a mathematical proof before exposing the difficulties which arise when trying to formalize it, and how they were overcome. We then build on this work to formalize the construction of the product of an arbitrary family of probability measures.

FaultLine: Automated Proof-of-Vulnerability Generation Using LLM Agents
Vikram Nitin, Baishakhi Ray, Roshanak Zilouchian Moghaddam
https://arxiv.org/abs/2507.15241

FaultLine: Automated Proof-of-Vulnerability Generation Using LLM Agents
Despite the critical threat posed by software security vulnerabilities, reports are often incomplete, lacking the proof-of-vulnerability (PoV) tests needed to validate fixes and prevent regressions. These tests are crucial not only for ensuring patches work, but also for helping developers understand how vulnerabilities can be exploited. Generating PoV tests is a challenging problem, requiring reasoning about the flow of control and data through deeply nested levels of a program. We present F…

Towards an Analysis of Proofs in Arithmetic
Alexander Leitsch, Anela Loli\'c, Stella Mahler
https://arxiv.org/abs/2506.05837 https://

Towards an Analysis of Proofs in Arithmetic
Inductive proofs can be represented as proof schemata, i.e. as parameterized sequences of proofs defined in a primitive recursive way. Applications of proof schemata can be found in the area of automated proof analysis where the schemata admit (schematic) cut-elimination and the construction of Herbrand systems. This work focuses on the expressivity of proof schemata. We show that proof schemata can simulate primitive recursive arithmetic. The translation of proofs in arithmetic to proof schema…

Density of integral points in the Betti moduli of quasi-projective varieties
Simone Coccia, Daniel Litt
https://arxiv.org/abs/2507.00167 https://

Density of integral points in the Betti moduli of quasi-projective varieties
Let $Y$ be a smooth quasi-projective complex variety equipped with a simple normal crossings compactification. We show that integral points are potentially dense in the (relative) character varieties parametrizing $SL_2$-local systems on $Y$ with fixed algebraic integer traces along the boundary components. The proof proceeds by using work of Corlette-Simpson to reduce to the case of Riemann surfaces, where we produce an integral point with Zariski-dense orbit under the mapping class group.

Linear stability of Kerr black holes in the full subextremal range
Dietrich H\"afner, Peter Hintz, Andr\'as Vasy
https://arxiv.org/abs/2506.21183 …

Linear stability of Kerr black holes in the full subextremal range
We prove, unconditionally, the linear stability of the Kerr family in the full subextremal range. On an analytic level, our proof is the same as that of our earlier paper in the slowly rotating case. The additional ingredients we use are, firstly, the mode stability result proved by Andersson, Whiting, and the first author and, secondly, computations related to the zero energy behavior of the linearized gauge-fixed Einstein equation in work by the second author.

Quantum-inspired algorithm for simulating viral response
D. O. Konina, D. I. Korbashov, I. V. Kovalchuk, A. A. Nizamieva, D. A. Chermoshentsev, A. K. Fedorov
https://arxiv.org/abs/2506.15671

Quantum-inspired algorithm for simulating viral response
Understanding the properties of biological systems is an exciting avenue for applying advanced approaches to solving corresponding computational tasks. A specific class of problems that arises in the resolution of biological challenges is optimization. In this work, we present the results of a proof-of-concept study that applies a quantum-inspired optimization algorithm to simulate a viral response. We formulate an Ising-type model to describe the patterns of gene activity in host responses. Re…

Understanding Tool-Integrated Reasoning
Heng Lin, Zhongwen Xu
https://arxiv.org/abs/2508.19201 https://arxiv.org/pdf/2508.19201

Understanding Tool-Integrated Reasoning
We study why Tool-Integrated Reasoning (TIR) makes Large Language Models (LLMs) more capable. While LLMs integrated with tools like Python code interpreters show great promise, a principled theory explaining why this paradigm is effective has been missing. This work provides the first formal proof that TIR fundamentally expands an LLM's capabilities. We demonstrate that tools enable a strict expansion of the model's empirical and feasible support, breaking the capability ceiling of pure-text mo…

Local perturbations of block Toeplitz matrices
Lars Koekenbier, Hermann Schulz-Baldes
https://arxiv.org/abs/2506.12757 https://arxiv.…

Local perturbations of block Toeplitz matrices
This work is about the asymptotic spectral theory of tridiagonal Toeplitz matrices with matrix entries, with periodicity broken on a finite number of entries. Varying the ranks of these perturbations allow to interpolate between open boundary and circulant Toeplitz matrices. While the continuous part of the limit spectrum only depends on these ranks and no other aspect of the perturbation, the outliers of the spectrum depend continuously on the local perturbation. The proof is essentially based…

A short proof of the existence of the injective envelope of an operator space
Thomas Sinclair
https://arxiv.org/abs/2507.10931 https://

A short proof of the existence of the injective envelope of an operator space
We use Ellis' lemma to give a simple proof of the existence of the injective envelope of an operator space first shown by work of Hamana and Ruan.

Combinatorial proof of a congruence for partitions into two sizes of part
Eli R. DeWitt, William J. Keith
https://arxiv.org/abs/2507.13566 https://

Combinatorial proof of a congruence for partitions into two sizes of part
Previous work showed that, for $ν_2(n)$ the number of partitions of $n$ into exactly two part sizes, one has $ν_2(16n + 14) \equiv 0 \pmod{4}$. The earlier proof required the technology of modular forms, and a combinatorial proof was desired. This article provides the requested proof, in the process refining divisibility to finer subclasses. Some of these subclasses have counts closely related to the divisor function $d(16n + 14)$, and we offer a conjecture on a potential rank statistic.

Improving Student-AI Interaction Through Pedagogical Prompting: An Example in Computer Science Education
Ruiwei Xiao, Xinying Hou, Runlong Ye, Majeed Kazemitabaar, Nicholas Diana, Michael Liut, John Stamper
https://arxiv.org/abs/2506.19107

Improving Student-AI Interaction Through Pedagogical Prompting: An Example in Computer Science Education
With the proliferation of large language model (LLM) applications since 2022, their use in education has sparked both excitement and concern. Recent studies consistently highlight students' (mis)use of LLMs can hinder learning outcomes. This work aims to teach students how to effectively prompt LLMs to improve their learning. We first proposed pedagogical prompting, a theoretically-grounded new concept to elicit learning-oriented responses from LLMs. To move from concept design to a proof-of-co…

On Effective Banach-Mazur Games and an application to the Poincar\'e Recurrence Theorem for Category
Prajval Koul, Satyadev Nandakumar
https://arxiv.org/abs/2506.11118

On Effective Banach-Mazur Games and an application to the Poincaré Recurrence Theorem for Category
The classical Banach-Mazur game characterizes sets of first category in a topological space. In this work, we show that an effectivized version of the game yields a characterization of sets of effective first category. Using this, we give a proof for the effective Banach Category Theorem. Further, we provide a game-theoretic proof of an effective theorem in dynamical systems, namely the category version of Poincaré Recurrence. The Poincaré Recurrence Theorem for category states that for a hom…

A proof of the Brian\c{c}on-Iarrobino Conjecture in three dimensions
Owen Mackenzie, Fatemeh Rezaee
https://arxiv.org/abs/2508.21717 https://arxiv.org/pdf/…

A proof of the Briançon-Iarrobino Conjecture in three dimensions
We resolve the 1978 Briançon-Iarrobino Conjecture regarding the maximum singularity of $\mathcal{H}=\mathrm{Hilb}^{l}(\mathbb{A}^3)$, where $l$ is a tetrahedral number, by refining the work of Ramkumar-Sammartano in \cite{Ramkumar-Sammartano}. This also immediately implies the conjectural necessary condition for a point of $\mathcal{H}$ to have the maximal singularity, suggested by the second-named author in \cite{Rezaee-23-Conjectures}. In a sequel to this article, \cite{Mackenzie-Rezaee2}, w…

I made some progress on the Beat Bike!
The magnet mounted on the wheel passes a reed switch which is sensed by the control box (with a Raspberry Pi Pico) and then triggers the drum machine to step through the programmed sequence.
Still a lot of work to do, mostly mounting everything onto the bike, and some more code to write, but the proof of concept is all good.
#bikeTooter

Lower Bounds against the Ideal Proof System in Finite Fields
Tal Elbaz, Nashlen Govindasamy, Jiaqi Lu, Iddo Tzameret
https://arxiv.org/abs/2506.17210 https…

Lower Bounds against the Ideal Proof System in Finite Fields
Lower bounds against strong algebraic proof systems and specifically fragments of the Ideal Proof System (IPS), have been obtained in an ongoing line of work. All of these bounds, however, are proved only over large (or characteristic $0$) fields, yet finite fields are the more natural setting for propositional proof complexity, especially for progress toward lower bounds for Frege systems such as $AC^0[p]$-Frege. This work establishes lower bounds against fragments of IPS over fixed finite fie…

Privacy-preserving and reward-based mechanisms of proof of engagement
Matteo Marco Montanari, Alessandro Aldini
https://arxiv.org/abs/2506.12523 https://…

Privacy-preserving and reward-based mechanisms of proof of engagement
Proof-of-Attendance (PoA) mechanisms are typically employed to demonstrate a specific user's participation in an event, whether virtual or in-person. The goal of this study is to extend such mechanisms to broader contexts where the user wishes to digitally demonstrate her involvement in a specific activity (Proof-of-Engagement, PoE). This work explores different solutions, including DLTs as well as established technologies based on centralized systems. The main aspects we consider include the l…

On the squares functor and the Gaitsgory-Rozenblyum conjectures
F\'elix Loubaton, Jaco Ruit
https://arxiv.org/abs/2507.07807 https://

On the squares functor and the Gaitsgory-Rozenblyum conjectures
In the seminal work of Gaitsgory and Rozenblyum on derived algebraic geometry, eight conjectures regarding the theory of $(\infty,2)$-categories are stated. This paper aims to clarify the status of these claims, and to provide a proof for the last remaining open one. Along the way, we demonstrate the universal property of the so-called squares functor, a construction that plays an important role in the $(\infty,2)$-categorical foundations of Gaitsgory-Rozenblyum.

Differentiating Through a Quadratic Cone Program
Quill Healey, Parth Nobel, Stephen Boyd
https://arxiv.org/abs/2508.17522 https://arxiv.org/pdf/2508.17522

Differentiating Through a Quadratic Cone Program
Quadratic cone programs are rapidly becoming the standard canonical form for convex optimization problems. In this paper we address the question of differentiating the solution map for such problems, generalizing previous work for linear cone programs. We follow a similar path, using the implicit function theorem applied to the optimality conditions for a homogenous primal-dual embedding. Along with our proof of differentiability, we present methods for efficiently evaluating the derivative ope…

Pencils of projective hypersurfaces, Griffiths heights and geometric invariant theory. II Hypersurfaces with semihomogeneous singularities
Thomas Mordant
https://arxiv.org/abs/2506.22126

Pencils of projective hypersurfaces, Griffiths heights and geometric invariant theory. II Hypersurfaces with semihomogeneous singularities
This paper establishes the formula for the stable Griffiths height of the middle-dimensional cohomology of a pencil of projective hypersurfaces $H$, with semihomogeneous singularities, over some smooth projective curve $C$, that appears as Theorem 5.1 in the first part of this paper (arxiv:2506.15334). The proof of this formula relies on the strategy developed in my previous work (arxiv:2212.11019v3) to derive an expression for this Griffiths height when the only singularities of the fibers o…

Medians, Oscillations, and Distance Functions
Marcus Pasquariello, Ignacio Uriarte-Tuero
https://arxiv.org/abs/2507.21020 https://arxiv.org/pdf/2507.21020

Medians, Oscillations, and Distance Functions
We establish a geometric characterization of the sets $E \subset \mathbb{R}^n$ so that the corresponding log-distance function $\log \text{dist}(\cdot, E)$ is in $BMO$ (or equivalently $\text{dist}(\cdot, E)^{-α} \in A_\infty$ for some $α> 0$), generalizing the $A_1$ characterization in the work by Anderson, Lehrbäck, Mudarra, and Vähäkangas. The proof relies on a new median-value characterization of $BMO$: For a real-valued measurable function on $\mathbb{R}^n$ and constants $0 < s < t < …

To see the forest for the trees: On the infinite divisibility of unlabeled forests
Michal Bassan, Serte Donderwinkel, Brett Kolesnik
https://arxiv.org/abs/2507.16650

To see the forest for the trees: On the infinite divisibility of unlabeled forests
Inspired by Stufler's recent probabilistic proof of Otter's asymptotic number of unlabeled trees, we revisit work of Palmer and Schwenk, and study unlabeled forests from a probabilistic point of view. We show that the number of trees in a random forest converges, with all of its moments, to a shifted compound Poisson. We also find the asymptotic proportion of forests that are trees. The key fact is that the number of trees $t_n$ and forests $f_n$ are related by a Lévy process. As such, the res…

Short Proof: Exact Solution to the Finite Frobenius Coin Problem
Lorenzo De Gaspari, Marco Ronzani
https://arxiv.org/abs/2508.08464 https://arxiv.org/pdf/2…

Short Proof: Exact Solution to the Finite Frobenius Coin Problem
The Frobenius Coin Problem is a classic question in mathematics: given coins of specified denominations, what is the largest amount that cannot be formed using only those coins? This brief work covers a variation of such question, posing a limit on the number of coins available for each denomination. Thus, the new problem becomes finding the count of distinct values that can be represented, and those that cannot, within the finite set of integers ranging from zero to the sum of all coins. We re…

Efficient dataset construction using active learning and uncertainty-aware neural networks for plasma turbulent transport surrogate models
Aaron Ho (MIT Plasma Science and Fusion Center, Cambridge, USA), Lorenzo Zanisi (UKAEA Culham Centre for Fusion Energy, Abingdon, UK), Bram de Leeuw (Radboud University, Nijmegen, Netherlands), Vincent Galvan (MIT Plasma Science and Fusion Center, Cambridge, USA), Pablo Rodriguez-Fernandez (MIT Plasma Science and Fusion Center, Cambridge, USA), Nath…

Efficient dataset construction using active learning and uncertainty-aware neural networks for plasma turbulent transport surrogate models
This work demonstrates a proof-of-principle for using uncertainty-aware architectures, in combination with active learning techniques and an in-the-loop physics simulation code as a data labeller, to construct efficient datasets for data-driven surrogate model generation. Building off of a previous proof-of-principle successfully demonstrating training set reduction on static pre-labelled datasets, using the ADEPT framework, this strategy was applied again to the plasma turbulent transport prob…

Convergence of the Birkhoff spectrum for nonintegrable observables
Godofredo Iommi, Mike Todd, Boyuan Zhao
https://arxiv.org/abs/2506.21390 https://…

Convergence of the Birkhoff spectrum for nonintegrable observables
We consider interval maps with countably many full branches and observables with polynomial tails. We show that the Birkhoff spectrum is real analytic and that its convergence to the Hausdorff dimension of the repeller is governed by the polynomial tail exponent. This result extends previous work by Arima on more regular observables and demonstrates how the tail behaviour influences the structure of the Birkhoff spectrum. Our proof relies on techniques from thermodynamic formalism and tail esti…

This https://arxiv.org/abs/2411.12206 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csRO_…

Safe Navigation in Dynamic Environments using Density Functions
This work presents a density-based framework for safe navigation in dynamic environments characterized by time-varying obstacle sets and time-varying target regions. We propose an analytical construction of time-varying density functions that enables the synthesis of a feedback controller defined as the positive gradient of the resulting density field. The primary contribution of this paper is a rigorous convergence proof demonstrating almost-everywhere safe navigation under the proposed framew…

Time-Dependent Hamiltonian Simulation via Time-Independent Dynamics in a Larger Space
Zecheng Li, Chunhao Wang
https://arxiv.org/abs/2507.19345 https://arx…

Time-Dependent Hamiltonian Simulation via Time-Independent Dynamics in a Larger Space
In this paper, we present a proof-of-concept quantum algorithm for simulating time-dependent Hamiltonian evolution by reducing the problem to simulating a time-independent Hamiltonian in a larger space using a discrete clock Hamiltonian construction. A similar construction was first explored for this simulation problem by Watkins, Wiebe, Roggero, and Lee [PRX Quantum, 2024]. Our algorithm improves upon their work in terms of the dependence on evolution time and precision. In addition, the compl…

A note on the boundary Dehn twist of $K3$ surfaces
Yujie Lin
https://arxiv.org/abs/2506.10444 https://arxiv.org/pdf/2506.10444…

A note on the boundary Dehn twist of $K3$ surfaces
By the work of Baraglia-Konno and Kronheimer-Mrowka, the boundary Dehn twist on punctured $K3$ surfaces is nontrivial in the smooth mapping class group relative to boundary. In this short note, we prove that it becomes trivial after abelianization. The proof is based on an obstruction for $\mathrm{Spin}^\mathbb{C}$ families due to Baraglia-Konno and the global Torelli theorem of $K3$ surfaces.

Expansion creates spin-glass order in finite-connectivity models: a rigorous and intuitive approach from the theory of LDPC codes
Benedikt Placke, Grace M. Sommers, Nikolas P. Breuckmann, Tibor Rakovszky, Vedika Khemani
https://arxiv.org/abs/2507.13342

Expansion creates spin-glass order in finite-connectivity models: a rigorous and intuitive approach from the theory of LDPC codes
Complex free-energy landscapes with many local minima separated by large barriers are believed to underlie glassy behavior across diverse physical systems. This is the heuristic picture associated with replica symmetry breaking (RSB) in spin glasses, but RSB has only been rigorously verified for certain mean-field models with all-to-all connectivity. In this work, we give a rigorous proof of finite temperature spin glass order for a family of models with local interactions on finite-connectivit…

Collaborative Beamforming for Communication Applications Using a Two-Element Fully-Wireless Open-Loop Coherent Distributed Array
Jason M. Merlo, Jeffrey A. Nanzer
https://arxiv.org/abs/2506.13014

Collaborative Beamforming for Communication Applications Using a Two-Element Fully-Wireless Open-Loop Coherent Distributed Array
In this work we demonstrate a proof of concept of a fully-wireless two-node open-loop coherent distributed communication system and evaluate its performance by transmitting QPSK , 64-, and 256-QAM constellations at a symbol rate of 2 MBd over a 58 m link in an urban environment. The system is implemented in a distributed manner with on-node processing using software-defined radios (SDRs) and wireless internode communication to share coordination information and does not rely on external time or…

Substituting Proof of Work in Blockchain with Training-Verified Collaborative Model Computation
Mohammad Ishzaz Asif Rafid, Morsalin Sakib
https://arxiv.org/abs/2508.12138 https…

Substituting Proof of Work in Blockchain with Training-Verified Collaborative Model Computation
Bitcoin's Proof of Work (PoW) mechanism, while central to achieving decentralized consensus, has long been criticized for excessive energy use and hardware inefficiencies \cite{devries2018bitcoin, truby2018decarbonizing}. This paper introduces a hybrid architecture that replaces Bitcoin's traditional PoW with a centralized, cloud-based collaborative training framework. In this model, miners contribute computing resources to train segments of horizontally scaled machine learning models on prepro…

Towards Operation Proof Obligation Generation for VDM
Nick Battle, Peter Gorm Larsen
https://arxiv.org/abs/2506.12858 https://arxiv.o…

Towards Operation Proof Obligation Generation for VDM
All formalisms have the ability to ensure that their models are internally consistent. Potential inconsistencies are generally highlighted by assertions called proof obligations, and the generation of these obligations is an important role of the tools that support the method. This capability has been available for VDM tools for many years. However, support for obligation generation for explicit operation bodies has always been limited. This work describes the current state of work to address t…

Intersections of longest cycles in vertex-transitive and highly connected graphs
Jie Ma, Ziyuan Zhao
https://arxiv.org/abs/2508.17438 https://arxiv.org/pdf…

Intersections of longest cycles in vertex-transitive and highly connected graphs
Motivated by the classical conjectures of Lovász, Thomassen, and Smith, recent work has renewed interest in the study of longest cycles in important graph families, such as vertex-transitive and highly connected graphs. In particular, Groenland et al.\ proved that if two longest cycles and in a graph share $m$ vertices, then there exists a vertex cut of size $O(m^{8/5})$ separating them, yielding improved bounds toward these conjectures. Their proof combines Turán-type arguments with computer…

Decay Properties of Invariant Measure and Application to Elliptic Homogenization of Non-divergence Form with an Interface
Pengxiu Yu, Yiping Zhang
https://arxiv.org/abs/2507.15536

Decay Properties of Invariant Measure and Application to Elliptic Homogenization of Non-divergence Form with an Interface
Using the self-contained PDE analysis, this paper investigates the existence and the decay properties of the invariant measure in elliptic homogenization of non-divergence form with an interface assumptions on the leading coefficient $A$ and the drift $b$ for $b_1\equiv 0$, which partially provides an alternative proof of the previous work by Hairer and Manson [Ann. Probab. 39(2011) 648-682]. Moreover, as a direct application after using the analysis by the second author [Calc. Var. Partial Dif…

Implicitly and Differentiably Representing Protein Surfaces and Interfaces
Cory B. Scott, Charlie Rothschild, Benjamin Nye
https://arxiv.org/abs/2508.11641 https://

Implicitly and Differentiably Representing Protein Surfaces and Interfaces
We introduce a pipeline for representing a protein, or protein complex, as the union of signed distance functions (SDFs) by representing each atom as a sphere with the appropriate radius. While this idea has been used previously as a way to render images of proteins, it has not, to our knowledge, been widely adopted in a machine learning setting. Mirroring recent successful work applying SDFs to represent 3D geometry, we present a proof of concept that this representation of proteins could be u…

How to use Linux on iPad to convert PNG to GIF
This is a proof of concept. When you create animations in PostScript on SQL Notebook, you can download the PNG images as ZIP file.
Now, If I installed Linux and ffmpeg on iPad, would that work? As a matter of fact, I will show you that it is possible, but it is very slow.
https://www.belle-nuit.com/sql-notebook/index.html?url=../site/files/how-to-use-linux-on-ipad-to-convert-png-to-gif.json

The Asai--Flach Euler system in $p$-adic families
David Loeffler, Arshay Sheth
https://arxiv.org/abs/2506.19673 https://arxiv.org/pdf…

The Asai--Flach Euler system in $p$-adic families
We show that the Euler system for the Asai representation corresponding to a Hilbert modular eigenform over a real quadratic field, constructed by Lei, Loeffler and Zerbes (2018), can be interpolated $p$-adically as the Hilbert modular form varies in a Hida family. This work is used as an important input in recent work of Grossi, Loeffler and Zerbes (2025) on the proof of the Bloch--Kato conjecture in analytic rank zero for the Asai representation.

The Hiraga-Ichino-Ikeda conjecture on formal degrees for classical groups
Rapha\"el Beuzart-Plessis
https://arxiv.org/abs/2508.08470 https://arxiv.org…

The Hiraga-Ichino-Ikeda conjecture on formal degrees for classical groups
We prove a conjecture of Hiraga-Ichino-Ikeda relating formal degrees of square-integrable representations to adjoint gamma factors for symplectic and even orthogonal groups over characteristic zero non-Archimedean local fields. The proof is based on the twisted endoscopic characterization of the local Langlands correspondence for such groups and extends an approach already appearing in the original paper of Hiraga-Ichino-Ikeda itself inspired from earlier work of Shahidi. Namely, the argument c…

Convergence of empirical Gromov-Wasserstein distance
Kengo Kato, Boyu Wang
https://arxiv.org/abs/2508.03985 https://arxiv.org/pdf/2508.03985

Convergence of empirical Gromov-Wasserstein distance
We study rates of convergence for estimation of the Gromov-Wasserstein distance. For two marginals supported on compact subsets of $\R^{d_x}$ and $\R^{d_y}$, respectively, with $\min \{ d_x,d_y \} > 4$, prior work established the rate $n^{-\frac{2}{\min\{d_x,d_y\}}}$ for the plug-in empirical estimator based on $n$ i.i.d. samples. We extend this fundamental result to marginals with unbounded supports, assuming only finite polynomial moments. Our proof techniques for the upper bounds can be adap…

Reinforcement Learning-based Adaptive Path Selection for Programmable Networks
Jos\'e Eduardo Zerna Torres, Marios Avgeris, Chrysa Papagianni, Gergely Pongr\'acz, Istv\'an G\'odor, Paola Grosso
https://arxiv.org/abs/2508.13806

Reinforcement Learning-based Adaptive Path Selection for Programmable Networks
This work presents a proof-of-concept implementation of a distributed, in-network reinforcement learning (IN-RL) framework for adaptive path selection in programmable networks. By combining Stochastic Learning Automata (SLA) with real-time telemetry data collected via In-Band Network Telemetry (INT), the proposed system enables local, data-driven forwarding decisions that adapt dynamically to congestion conditions. The system is evaluated on a Mininet-based testbed using P4-programmable BMv2 sw…

Opened up YouTube, got nerd sniped with a math problem. After a bad morning, it was the perfect thing to take a break working on. Still haven't watched the actual video, just the thumbnail...will do that once I get off of work to see if we got to the answer the same way (well, the proof that is). #math

Growing a Modular Framework for Modal Systems- HOLMS: a HOL Light Library
Antonella Bilotta
https://arxiv.org/abs/2506.10048 https://…

Growing a Modular Framework for Modal Systems- HOLMS: a HOL Light Library
The present dissertation introduces the research project on HOLMS (\textbf{HOL} Light Library for \textbf{M}odal \textbf{S}ystems), a growing modular framework for modal reasoning within the HOL Light proof assistant. To provide an accessible introduction to the library, the fundamentals of modal logic are outlined first, followed by a concise manual for the proof assistant itself. The core contribution of this work on HOLMS is the development of a unified and modular strategy for proving adequ…

Voting-Based Semi-Parallel Proof-of-Work Protocol
Mustafa Doger, Sennur Ulukus
https://arxiv.org/abs/2508.06489 https://arxiv.org/pdf/2508.06489

Voting-Based Semi-Parallel Proof-of-Work Protocol
Parallel Proof-of-Work (PoW) protocols are suggested to improve the safety guarantees, transaction throughput and confirmation latencies of Nakamoto consensus. In this work, we first consider the existing parallel PoW protocols and develop hard-coded incentive attack structures. Our theoretical results and simulations show that the existing parallel PoW protocols are more vulnerable to incentive attacks than the Nakamoto consensus, e.g., attacks have smaller profitability threshold and they res…

The 2R-Conjecture for the Hegselmann--Krause Model: A Proof in Expectation and New Directions
Partha S. Dey, S. Rasoul Etesami, Aditya S. Gopalan
https://arxiv.org/abs/2508.08299

The 2R-Conjecture for the Hegselmann--Krause Model: A Proof in Expectation and New Directions
Hegselmann--Krause models are localized, distributed averaging dynamics on spatial data. A key aspect of these dynamics is that they lead to cluster formation, which has important applications in geographic information systems, dynamic clustering algorithms, opinion dynamics, and social networks. For these models, the key questions are whether a fixed point exists and, if so, characterizing it. In this work, we establish new results towards the "2R-Conjecture" for the Hegselmann--Krause model, …

Outline of the Wang-Zahl proof of the Kakeya conjecture in $\mathbb{R}^3$
Larry Guth
https://arxiv.org/abs/2508.05475 https://arxiv.org/pdf/2508.05475

Outline of the Wang-Zahl proof of the Kakeya conjecture in $\mathbb{R}^3$
We give a detailed outline of the proof that the Kakeya conjecture follows from the sticky case. This proof is due to Wang and Zahl and appears in a recent paper. The sticky case was proven in earlier work of Wang-Zahl, building on an approach suggested by Katz-Tao.

Green functions and a positive mass theorem for asymptotically hyperbolic $3$-manifolds
Klaus Kroencke, Francesca Oronzio, Alan Pinoy
https://arxiv.org/abs/2506.07108

Green functions and a positive mass theorem for asymptotically hyperbolic $3$-manifolds
We prove a new positive mass theorem for three-dimensional manifolds which are asymptotically hyperboloidal of order greater than $1$. The mass quantity under consideration is the volume-renormalized mass recently introduced in a paper by Dahl, McCormick and the first author. The proof is based on a monotonicity formula holding along the level sets of the Green function for the Laplace operator centered at an arbitrary point. In order for this argument to work out, we require that the second ho…

Error Rates for Large Deviations in the domain of an $\alpha=1$ stable law
Jonny Imbierski, Dalia Terhesiu
https://arxiv.org/abs/2506.12443 https://…

Error Rates for Large Deviations in the domain of an $α=1$ stable law
We obtain error rates for large deviations of sums of i.i.d. random variables in, a particular case, of the domain of a non-symmetric infinite mean $α=1$-stable law. The focus of this work is on the method of proof via analytic techniques rather than the particular result.

Volume Polynomials and Log-concavity of the Characteristic Polynomials of Matroids
Eric Katz
https://arxiv.org/abs/2508.08391 https://arxiv.org/pdf/2508.08…

Volume Polynomials and Log-concavity of the Characteristic Polynomials of Matroids
The Rota--Heron--Welsh conjecture (now a theorem of Adiprasito, Huh, and the author) asserts the log-concavity of the characteristic polynomial of matroids. We give an exposition of the Lorentzian polynomial proof following the work of Branden and Leake aimed at undergraduate and beginning graduate students.

Deformations of pseudocharacters and Mazur's finiteness condition
Vytautas Pa\v{s}k\=unas, Julian Quast
https://arxiv.org/abs/2506.10901 https://

Deformations of pseudocharacters and Mazur's finiteness condition
We show that deformation rings $R^{\mathrm{ps}}$ of $G$-pseudocharacters of a profinite group $Γ$ are noetherian, when $Γ$ satisfies Mazur's finiteness condition. The proof proceeds by reduction to the case when $Γ$ is finitely generated, where the result was previously established by the second author. This enables us to extend our work on moduli spaces of $R^{\mathrm{ps}}$-condensed representations of a finitely generated profinite group $Γ$, to the groups satisfying Mazur's finiteness co…

This https://arxiv.org/abs/2505.01323 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csNE_…

Proven Approximation Guarantees in Multi-Objective Optimization: SPEA2 Beats NSGA-II
Together with the NSGA-II and SMS-EMOA, the strength Pareto evolutionary algorithm 2 (SPEA2) is one of the most prominent dominance-based multi-objective evolutionary algorithms (MOEAs). Different from the NSGA-II, it does not employ the crowding distance (essentially the distance to neighboring solutions) to compare pairwise non-dominating solutions but a complex system of $σ$-distances that builds on the distances to all other solutions. In this work, we give a first mathematical proof showi…

This https://arxiv.org/abs/2412.21057 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The rectifiable rectangular peg problem
We give an affirmative answer to the rectangular peg problem for a large class of continuous Jordan curves that contains all rectifiable curves and Stromquist's locally monotone curves. Our proof is based on microlocal sheaf theory and inspired by recent work of Greene and Lobb.

On localization for the alloy-type Anderson-Bernoulli model with long-range hopping
Shihe Liu, Yunfeng Shi, Zhifei Zhang
https://arxiv.org/abs/2508.12714 https://

On localization for the alloy-type Anderson-Bernoulli model with long-range hopping
In this paper, we prove the Anderson localization near the spectral edge for some alloy-type Anderson-Bernoulli model on $\mathbb{Z}^d$ with exponential long-range hopping. This extends the work of Bourgain [Geometric Aspects of Functional Analysis, LNM 1850: 77--99, 2004], in which he pioneered a novel multi-scale analysis to treat Bernoulli random variables. Our proof is mainly based on Bourgain's method. However, to establish the initial scales Green's function estimates, we adapt the approa…

Some concerns on the border rank of Kronecker products of the Coppersmith-Winograd tensor
Daiki Kawabe
https://arxiv.org/abs/2507.13126 https://

Some concerns on the border rank of Kronecker products of the Coppersmith-Winograd tensor
This note provides a detailed proof of Conner--Gesmundo--Landsberg--Ventura's result that the border rank of the Kronecker square of the little Coppersmith--Winograd tensor is $(q+2)^{2}$.We also indicate how the same ideas seem to extend to the case of the Kronecker cube, pointing toward the conjectural value $(q+2)^{m}$ for $m\ge 4$, although a full proof is left for future work.

How to Beat Nakamoto in the Race
Shu-Jie Cao, Dongning Guo
https://arxiv.org/abs/2508.16202 https://arxiv.org/pdf/2508.16202…

How to Beat Nakamoto in the Race
This paper studies proof-of-work Nakamoto consensus under bounded network delays, settling two long-standing questions in blockchain security: How can an adversary most effectively attack block safety under a given block confirmation latency? And what is the resulting probability of safety violation? A Markov decision process (MDP) framework is introduced to precise characterize the system state (including the tree and timings of all blocks mined), the adversary's potential actions, and the sta…

Derivatives of padded Schubert polynomials through pipe dreams
Hugh Dennin
https://arxiv.org/abs/2508.11879 https://arxiv.org/pdf/2508.11879

Derivatives of padded Schubert polynomials through pipe dreams
In recent work, Hamaker, Pechenik, Speyer, and Weigandt showed that a certain differential operator $\nabla$ expands positively in the basis of Schubert polynomials. For $π$ a dominant permutation, Gaetz and Tung showed an analogous positivity result holds for a dual differential operator $Δ$ acting on $π$-padded Schubert polynomials. In this paper, we provide a new proof of the result of Gaetz and Tung using the combinatorics of pipe dreams.

On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic
Shachar Weinbaum, Elyasheev Leibtag, Rotem Kalisch, Michael Shalyt, Ido Kaminer
https://arxiv.org/abs/2507.08138

On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic
Ratios of D-finite sequences and their limits -- known as Apéry limits -- have driven much of the work on irrationality proofs since Apéry's 1979 breakthrough proof of the irrationality of $ζ(3)$. We extend ratios of D-finite sequences to a high-dimensional setting by introducing the Conservative Matrix Field (CMF). We demonstrate how classical Apéry limits are included by this object as special cases. A useful construction of CMFs is provided, drawing a connection to gauge transformations …

Optimized projection-free algorithms for online learning: construction and worst-case analysis
Julien Weibel (SIERRA), Pierre Gaillard (Thoth), Wouter M. Koolen (CWI), Adrien Taylor (SIERRA)
https://arxiv.org/abs/2506.05855

Optimized projection-free algorithms for online learning: construction and worst-case analysis
This work studies and develop projection-free algorithms for online learning with linear optimization oracles (a.k.a. Frank-Wolfe) for handling the constraint set. More precisely, this work (i) provides an improved (optimized) variant of an online Frank-Wolfe algorithm along with its conceptually simple potential-based proof, and (ii) shows how to leverage semidefinite programming to jointly design and analyze online Frank-Wolfe-type algorithms numerically in a variety of settings-that include …

Layer Cake Representations for Quantum Divergences
Po-Chieh Liu, Christoph Hirche, Hao-Chung Cheng
https://arxiv.org/abs/2507.07065 https://

Layer Cake Representations for Quantum Divergences
Defining suitable quantum extensions of classical divergences often poses a challenge due to the non-commutative nature of quantum information. In this work, we propose a new approach via what we call the layer cake representation. The resulting quantum Rényi and $f$-divergences are then proven to be equivalent to those recently defined via integral representations. Nevertheless, the approach can provide several insights. We give an alternative proof of the integral representation of the relat…

Unconditional energy dissipation of Strang splitting for the matrix-valued Allen-Cahn equation
Chaoyu Quan, Tao Tang, Dong Wang
https://arxiv.org/abs/2508.03992 https://

Unconditional energy dissipation of Strang splitting for the matrix-valued Allen-Cahn equation
The energy dissipation property of the Strang splitting method was first demonstrated for the matrix-valued Allen-Cahn (MAC) equation under restrictive time-step constraints [J. Comput. Phys. 454, 110985, 2022]. In this work, we eliminate this limitation through a refined stability analysis framework, rigorously proving that the Strang splitting method preserves the energy dissipation law unconditionally for arbitrary time steps. The refined proof hinges on a precise estimation of the double-we…

This https://arxiv.org/abs/2412.02003 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Polynomial algorithm for alternating link equivalence
Link equivalence up to isotopy in a 3-space is the problem that lies at the root of knot theory, and is important in 3-dimensional topology and geometry. We consider its restriction to alternating links, given by two alternating diagrams with $n_1$ and $n_2$ crossings, and show that this problem has polynomial algorithm in terms of $max\{n_1, n_2\}$. For the proof, we use Tait flyping conjectures, observations stemming from the work of Lackenby, Menasco, Sundberg and Thistlethwaite on alternati…

There and Back Again: Revisiting the Failure of Concatenation in Mittag-Leffler Functions
Paulo M. Carvalho-Neto, Cesar E. T. Ledesma
https://arxiv.org/abs/2508.05788 https://…

There and Back Again: Revisiting the Failure of Concatenation in Mittag-Leffler Functions
The function $t \mapsto E_α(λt^α)$ is widely regarded as the fractional analogue of the exponential function, yet its algebraic properties remain poorly understood. In particular, standard references lack a rigorous proof of the failure of the semigroup property. In this work, we fill this gap by providing an analytical proof that the semigroup property holds if and only if either $α= 1$ or $λ= 0$. Our result establishes a precise criterion for when Mittag-Leffler dynamics are compatible w…

This https://arxiv.org/abs/2502.20327 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Parabolic bundles and the intersection cohomology of moduli spaces of vector bundles on curves
The study of the intersection cohomology of moduli spaces of semistable bundles was initiated by Frances Kirwan in the 1980's. In this paper, we give a complete geometric proof of a recursive formula, which reduces the calculation of the intersection Betti numbers of the moduli spaces of semistable bundles on Riemann surfaces in degree-0 and arbitrary rank to the known formulas of the Betti numbers of the smooth, degree-1 moduli spaces. Our formula was motivated by the work of Mozgovoy and Rein…

Gr\"unbaum's inequality for probability measures with concavity properties
Matthieu Fradelizi, Dylan Langharst, Jiaqian Liu, Francisco Mar\'in Sola, Shengyu Tang
https://arxiv.org/abs/2507.06759

Grünbaum's inequality for probability measures with concavity properties
A celebrated result in convex geometry is Grünbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Grünbaum-type inequalities - with equality characterizations - for probability measures under certain concavity assumptions. Of particular interest is the case of $s$-concave Radon measures. For such measures, we provide a simpler proof of known results and, more importantl…

Generating Hodges' Graviton MHV Formula with an $Lw_{1 \infty}$ Ward Identity
Alfredo Guevara, Elizabeth Himwich, Noah Miller
https://arxiv.org/abs/2506.05460

Generating Hodges' Graviton MHV Formula with an $Lw_{1+\infty}$ Ward Identity
Hodges' formula expresses the tree-level all-multiplicity Einstein gravity MHV amplitude as a matrix determinant. In this work, we prove that Hodges' determinant is generated by an $Lw_{1+\infty}$ Ward identity on the celestial sphere. The Ward identity takes the form of a recursion relation that has not previously appeared in the literature and is unrelated to BCFW. The proof makes use of the matrix-tree theorem.

Quaternionic Kolyvagin systems and Iwasawa theory for Hida families
Francesco Zerman
https://arxiv.org/abs/2507.04980 https://arxiv.o…

Quaternionic Kolyvagin systems and Iwasawa theory for Hida families
We build a modified universal Kolyvagin system for the Galois representation attached to a Hida family of modular forms, starting from the big Heegner point Euler system of Longo--Vigni built in towers of Shimura curves. We generalize the work of Büyükboduk to a quaternionic setting, relaxing the classical \emph{Heegner hypothesis} on the tame conductor of the family. As a byproduct of this construction, we give a proof of one divisibility of the anticyclotomic Iwasawa main conjecture for Hid…

Composable Quantum Fault-Tolerance
Zhiyang He, Quynh T. Nguyen, Christopher A. Pattison
https://arxiv.org/abs/2508.08246 https://arxiv.org/pdf/2508.08246…

Composable Quantum Fault-Tolerance
Proving threshold theorems for fault-tolerant quantum computation is a burdensome endeavor with many moving parts that come together in relatively formulaic but lengthy ways. It is difficult and rare to combine elements from multiple papers into a single formal threshold proof, due to the use of different measures of fault-tolerance. In this work, we introduce composable fault-tolerance, a framework that decouples the probabilistic analysis of the noise distribution from the combinatorial analy…

On K3 surfaces with hyperbolic automorphism groups
Koji Fujiwara, Keiji Oguiso, Xun Yu
https://arxiv.org/abs/2507.13726 https://arxiv…

On K3 surfaces with hyperbolic automorphism groups
We show the finiteness of the Néron-Severi lattices of complex projective K3 surfaces whose automorphism groups are non-elementary hyperbolic with explicit descriptions, under the assumption that the Picard number $\ge 6$ which is optimal to ensure the finiteness. Our proof of finiteness is based on the study of genus one fibrations on K3 surfaces and recent work of Kikuta and Takatsu.

Doubly nonlinear parabolic equation with perturbation
Shun Uchida
https://arxiv.org/abs/2506.11503 https://arxiv.org/pdf/2506.11503…

Doubly nonlinear parabolic equation with perturbation
In this paper, we consider the initial boundary value problem of a doubly nonlinear parabolic equation with nonlinear perturbation. We impose the homogeneous Dirichlet condition on this problem. We aim to reduce the growth condition of the nonlinear term and the largeness of exponent as possible so that we can apply our result to both singular and degenerate type parabolic equations. We use in our proof the $L ^{\infty }$-estimate of the time-discrete equation derived in the previous work and a…

Tatuzawa's theorem for Rankin-Selberg $L$-functions
Gergely Harcos, Jesse Thorner
https://arxiv.org/abs/2508.10844 https://arxiv.org/pdf/2508.10844

Tatuzawa's theorem for Rankin-Selberg $L$-functions
Let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ with unitary central characters. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,π\timesπ')$, generalizing Tatuzawa's refinement of Siegel's work on Dirichlet $L$-functions. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,π\times\widetildeπ)$ by an idele class character $χ$ apart from a pos…

Refuting Perfect Matchings in Spectral Expanders is Hard
Ari Biswas, Rajko Nenadov
https://arxiv.org/abs/2506.07700 https://arxiv.org…

Refuting Perfect Matchings in Spectral Expanders is Hard
This work studies the complexity of refuting the existence of a perfect matching in spectral expanders with an odd number of vertices, in the Polynomial Calculus (PC) and Sum of Squares (SoS) proof system. Austrin and Risse [SODA, 2021] showed that refuting perfect matchings in sparse $d$-regular \emph{random} graphs, in the above proof systems, with high probability requires proofs with degree $Ω(n/\log n)$. We extend their result by showing the same lower bound holds for \emph{all} $d$-regul…

A Gaussian integral formula for the Hermite polynomials: Combinatorics, Asymptotics and Applications
Mihai Nica, Janosch Ortmann
https://arxiv.org/abs/2508.13910 https://…

A Gaussian integral formula for the Hermite polynomials: Combinatorics, Asymptotics and Applications
The Hermite polynomials are ubiquitous but can be difficult to work with due to their unwieldy definition in terms of derivatives. To remedy this, we showcase an underappreciated Gaussian integral formula for the Hermite polynomials, which is especially useful for generalizing to multivariable Hermite polynomials. Taking this as our definition, we prove many useful consequences, including: 1. Combinatorial interpretations for the Hermite polynomials, including a proof of orthogonality. 2. A mor…

ZKTorch: Compiling ML Inference to Zero-Knowledge Proofs via Parallel Proof Accumulation
Bing-Jyue Chen, Lilia Tang, Daniel Kang
https://arxiv.org/abs/2507.07031

ZKTorch: Compiling ML Inference to Zero-Knowledge Proofs via Parallel Proof Accumulation
As AI models become ubiquitous in our daily lives, there has been an increasing demand for transparency in ML services. However, the model owner does not want to reveal the weights, as they are considered trade secrets. To solve this problem, researchers have turned to zero-knowledge proofs of ML model inference. These proofs convince the user that the ML model output is correct, without revealing the weights of the model to the user. Past work on these provers can be placed into two categories…

Higher dimensional Sacks-Uhlenbeck-type functionals and applications
Gianmichele Di Matteo, Tobias Lamm
https://arxiv.org/abs/2506.17166 https://

Higher dimensional Sacks-Uhlenbeck-type functionals and applications
In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new null-homotopic such maps. The proof follows a similar perturbative argument, which in high dimensions leads to a degenerate and double-phase-type Euler-Lagrange system, making the uniform regularity needed to formalize the bubbling harder to achieve. Then, we develop a r…

Finding Partite Hypergraphs Efficiently
Ferran Espu\~na
https://arxiv.org/abs/2508.10641 https://arxiv.org/pdf/2508.10641

Finding Partite Hypergraphs Efficiently
We provide a deterministic polynomial-time algorithm that, for a given $k$-uniform hypergraph $H$ with $n$ vertices and edge density $d$, finds a complete $k$-partite subgraph of $H$ with parts of size at least ${c(d, k)(\log n)^{1/(k-1)}}$. This generalizes work by Mubayi and Turán on bipartite graphs. The value we obtain for the part size matches the order of magnitude guaranteed by the non-constructive proof due to Erdős and is tight up to a constant factor.

Almost a Complete Proof of the Generalized Erd\H{o}s-Straus Conjecture: ${5}/{a} = {1}/{b} {1}/{c} {1}/{d}$
Bilal Ghermoul
https://arxiv.org/abs/2508.07367 https://

Almost a Complete Proof of the Generalized Erdős-Straus Conjecture: ${5}/{a} = {1}/{b} + {1}/{c} + {1}/{d}$
The generalized Erdős-Straus conjecture, proposed by Wacław Sierpiński in 1956, asks whether the Diophantine equation \[ \frac{5}{a} = \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \] admits positive integer solutions $b,c,d \in \mathbb{N}$ for every integer $a \ge 2$. In this work we present explicit solutions for all integers $a \ge 2$. We begin with the simplest known cases where $a \equiv i \pmod{5}$ for $i \in \{0,2,3,4\}$, providing direct decompositions. The remaining open case, $a \equiv 1…

Abelian motives and Shimura varieties in nonzero characteristic
James S. Milne
https://arxiv.org/abs/2508.09972 https://arxiv.org/pdf/2508.09972

Abelian motives and Shimura varieties in nonzero characteristic
Much of the work on Shimura varieties over the last thirty years has been devoted to constructing the theory that would follow from a good notion of motives, one incorporating the Hodge, Tate, and standard conjectures. These conjectures are believed to be beyond reach, and may not even be correct as stated. I argue in this article that there exists a theory of motives, accessible to proof, weaker than Grothendieck's, but with many of the same consequences.

Willchain: Decentralized, Privacy-Preserving, Self-Executing, Digital Wills
Jovonni L. PHarr
https://arxiv.org/abs/2507.03694 https://

Willchain: Decentralized, Privacy-Preserving, Self-Executing, Digital Wills
This work presents a novel decentralized protocol for digital estate planning that integrates advances distributed computing, and cryptography. The original proof-of-concept was constructed using purely solidity contracts. Since then, we have enhanced the implementation into a layer-1 protocol that uses modern interchain communication to connect several heterogeneous chain types. A key contribution of this research is the implementation of several modern cryptographic primitives to support vari…

Single photon emission from lithographically-positioned engineered nanodiamonds for cryogenic applications
Vivekanand Tiwari, Zhaojin Liu, Hao-Cheng Weng, Krishna C Balram, John G Rarity, Soumen Mandal, Oliver A Williams, Gavin W Morley, Joe A Smith
https://arxiv.org/abs/2508.06424

Single photon emission from lithographically-positioned engineered nanodiamonds for cryogenic applications
Nitrogen-vacancy centres in nanodiamonds (NDs) provide a promising resource for quantum photonic systems. However, developing a technology beyond proof-of-principle physics requires optimally engineering its component parts. In this work, we present a hybrid materials platform by photolithographically positioning ball-milled isotopically-enriched NDs on broadband metal reflectors. The structure enhances the photonic collection efficiency, enabling cryogenic characterisation despite the limited …

Blockchain-Based Decentralized Domain Name System
Guang Yang, Peter Trinh, Alma Nkemla, Amuru Serikyaku, Edward Tatchim, Osman Sharaf
https://arxiv.org/abs/2508.05655 https://…

Blockchain-Based Decentralized Domain Name System
The current Domain Name System (DNS) infrastructure faces critical vulnerabilities including poisoning attacks, censorship mechanisms, and centralized points of failure that compromise internet freedom and security. Recent incidents such as DNS poisoning attacks on ISP customers highlight the urgent need for resilient alternatives. This paper presents a novel blockchain-based Decentralized Domain Name System (DDNS). We designed a specialized Proof-of-Work blockchain to maximize support for DNS-…

Character varieties, hexagonal 3-webs and Hess connection
Adjaratou Arame Diaw (UCAD), Karamoko Diarra (USTTB), Frank Loray (IRMAR), Tangue Ndawa Bertuel
https://arxiv.org/abs/2507.07542

Character varieties, hexagonal 3-webs and Hess connection
We investigate the existence of some affine structure on SL 2 (C)-character varieties on the four-punctured sphere through web and symplectic geometry. This work provides a new proof of non integrability (i.e. irreducibility) of Painlev{é} VI equations, previously proved by S. Cantat and the third author.

Replaced article(s) found for cs.CR. https://arxiv.org/list/cs.CR/new
[1/1]:
- Proof-of-Useful-Work Blockchain for Trustworthy Biomedical Hyperdimensional Computing
Jinghao Wen, Dongning Ma, Sizhe Zhang, Hasshi Sudler, Xun Jiao

Wall-crossing for Calabi-Yau fourfolds: framework, tools, and applications
Arkadij Bojko
https://arxiv.org/abs/2507.05922 https://arx…

Wall-crossing for Calabi-Yau fourfolds: framework, tools, and applications
This work develops new ideas and tools to establish wall-crossing in Calabi-Yau four categories as originally conjectured by Gross-Joyce-Tanaka. In the process, I set up some necessary new language, including a natural refinement of Joyce's vertex algebras to equivariant homology. The proof is then given for Calabi-Yau four dg-quivers and local CY fourfolds. A crucial part of the problem is showing that the generalized invariants counting stable objects are well-defined. Using a conceptual argu…