Tootfinder

The commutative algebra of congruence ideals and applications to number theory
Srikanth B. Iyengar, Chandrashekhar B. Khare, Jeffrey Manning
https://arxiv.org/abs/2510.05418 htt…

The commutative algebra of congruence ideals and applications to number theory
In his proof of Fermat's Last Theorem, Wiles deployed a commutative algebra technique, namely a numerical criterion for detecting isomorphisms of rings. In our recent work we pick up on Wiles' work and generalize the numerical criterion to ``higher codimension''. A critical ingredient is a notion of congruence module in higher codimension: this has turned out to be a key definition whose utility extends beyond the role it plays in the numerical criterion. In this paper we trace the origin of so…

The classification of quasi-elliptic fibrations and unexpected plane cubics in characteristics 2 and 3
Jake Kettinger
https://arxiv.org/abs/2510.06365 https://

The classification of quasi-elliptic fibrations and unexpected plane cubics in characteristics 2 and 3
In this paper, we categorize all isomorphism classes of quasi-elliptic surfaces over a field $k$ of characteristic 2 or 3. For every quasi-elliptic surface $X$, we classify all possible sequences of blow-downs from $X$ to the projective plane $\mathbb{P}^2_k$. We then use these categorizations to identify all unexpected plane cubic curves in characteristic 2 and present a proof of the lack of unexpected cubics in characteristic 3. Before the work in this paper -- based partly on the author's th…

Non-Termination Proving: 100 Million LoC and Beyond
Julien Vanegue, Jules Villard, Peter O'Hearn, Azalea Raad
https://arxiv.org/abs/2509.05293 https://…

Non-Termination Proving: 100 Million LoC and Beyond
We report on our tool, Pulse Infinite, that uses proof techniques to show non-termination (divergence) in large programs. Pulse Infinite works compositionally and under-approximately: the former supports scale, and the latter ensures soundness for proving divergence. Prior work focused on small benchmarks in the tens or hundreds of lines of code (LoC), and scale limits their practicality: a single company may have tens of millions, or even hundreds of millions of LoC or more. We report on apply…

@… I will probably also post on the original thread when it reopens, but I just wanted to post my thoughts about this thread here as well: https://comm…

Framework supporting far-right racists?
So evidence and proof is the opinion of a known activist with specific political leanings? An activist that has said and done many controversial things as well, I might add.

Saturation-Driven Dataset Generation for LLM Mathematical Reasoning in the TPTP Ecosystem
Valentin Quesnel, Damien Sileo
https://arxiv.org/abs/2509.06809 https://

Saturation-Driven Dataset Generation for LLM Mathematical Reasoning in the TPTP Ecosystem
The scarcity of high-quality, logically sound data is a critical bottleneck for advancing the mathematical reasoning of Large Language Models (LLMs). Our work confronts this challenge by turning decades of automated theorem proving research into a scalable data engine. Rather than relying on error-prone LLMs or complex proof-assistant syntax like Lean and Isabelle, our framework leverages E-prover's saturation capabilities on the vast TPTP axiom library to derive a massive, guaranteed-valid cor…

Verifying Sampling Algorithms via Distributional Invariants
Kevin Batz, Joost-Pieter Katoen, Tobias Winkler, Daniel Zilken
https://arxiv.org/abs/2509.06410 https://

Verifying Sampling Algorithms via Distributional Invariants
This paper develops a verification framework aimed at establishing the correctness of discrete sampling algorithms. We do so by considering probabilistic programs as distribution transformers. Inspired by recent work on distributional verification of Markov models, we introduce the notion of (inductive) distributional loop invariants for discrete probabilistic programs. These invariants are embedded in a Hoare-like verification framework that includes proof rules for total and partial correctne…

PoS-CoPOR: Proof-of-Stake Consensus Protocol with Native Onion Routing Providing Scalability and DoS-Resistance
Ivan Homoliak, Martin Pere\v{s}\'ini, Marek Tama\v{s}kovi\v{c}, Timotej Ponek, Luk\'a\v{s} Hellebrandt, Kamil Malinka
https://arxiv.org/abs/2510.04619

PoS-CoPOR: Proof-of-Stake Consensus Protocol with Native Onion Routing Providing Scalability and DoS-Resistance
Proof-of-Stake (PoS) consensus protocols often face a trade-off between performance and security. Protocols that pre-elect leaders for subsequent rounds are vulnerable to Denial-of-Service (DoS) attacks, which can disrupt the network and compromise liveness. In this work, we present PoS-CoPOR, a single-chain PoS consensus protocol that mitigates this vulnerability by integrating a native onion routing mechanism into the consensus protocol itself. PoS-CoPOR combines stake-weighted probabilistic …

Eigenvalues of Universal Covers and the Matching Polynomial
Thom\'as Jung Spier
https://arxiv.org/abs/2510.05041 https://arxiv.org/pdf/2510.05041

Eigenvalues of Universal Covers and the Matching Polynomial
In this work, we prove that the universal and maximal abelian covers of a finite multi-graph have the same eigenvalues. This strengthens a recent result of Li, Magee, Sabri, and Thomas (2025), and answers one of their questions. Our proof relies on the theory of the matching polynomial of a graph, and we further prove new results about its Gallai--Edmonds decomposition.

A Small Collusion is All You Need
Yotam Gafni
https://arxiv.org/abs/2510.05986 https://arxiv.org/pdf/2510.05986

A Small Collusion is All You Need
Transaction Fee Mechanisms (TFMs) study auction design in the Blockchain context, and emphasize robustness against miner and user collusion, moreso than traditional auction theory. \cite{chung2023foundations} introduce the notion of a mechanism being $c$-Side-Contract-Proof ($c$-SCP), i.e., robust to a collusion of the miner and $c$ users. Later work \cite{chung2024collusion,welfareIncreasingCollusion} shows a gap between the $1$-SCP and $2$-SCP classes. We show that the class of $2$-SCP mechan…

Electron-beam-induced Contactless Manipulation of Interlayer Twist in van der Waals Heterostructures
Nicola Curreli, Tero S. Kulmala, Riya Sebait, Nicol\`o Petrini, Matteo Bruno Lodi, Roman Furrer, Alessandro Fanti, Michel Calame, Ilka Kriegel
https://arxiv.org/abs/2510.03347

Electron-beam-induced Contactless Manipulation of Interlayer Twist in van der Waals Heterostructures
The ability to dynamically control the relative orientation of layers in two dimensional (2D) van der Waals (vdW) heterostructures represents a critical step toward the realization of reconfigurable nanoscale devices. Existing actuation methods often rely on mechanical contact, complex architectures, or extreme operating conditions, which limit their applicability and scalability. In this work, we present a proof-of-concept demonstration of contactless electrostatic actuation based on electron-…

Combining relatively hyperbolic groups over a complex of groups
Darius Alizadeh
https://arxiv.org/abs/2510.02602 https://arxiv.org/pdf/2510.02602

Combining relatively hyperbolic groups over a complex of groups
Given a complex of groups $G(\mathcal{Y}) = (G_σ, ψ_a, g_{a,b})$ where all $G_σ$ are relatively hyperbolic, the $ψ_a$ are inclusions of full relatively quasiconvex subgroups, and the universal cover $X$ is CAT$(0)$ and $δ$--hyperbolic, we show $π_1(G(\mathcal{Y}))$ is relatively hyperbolic. The proof extends the work of Dahmani and Martin by constructing a model for the Bowditch boundary of $π_1(G(\mathcal{Y}))$. We prove the model is a compact metrizable space on which $G$ acts as a geo…

Two Calabi-Yau theorems for degenerations of compact K\"ahler manifolds
David Witt Nystr\"om
https://arxiv.org/abs/2510.03044 https://arxiv.org/p…

Two Calabi-Yau theorems for degenerations of compact Kähler manifolds
We discuss two closely related Calabi-Yau theorems for degenerations of compact Kähler manifolds. The first is a Calabi-Yau theorem for big test configurations, that generalizes a result in [WN24]. It follows from recent joint work with Mesquita-Piccione [MW25], but is here given a more direct proof. The second result is a Calabi-Yau theorem for a wider class of degenerations, formulated in the language of non-Archimedean Kähler geometry. It was first proved in the algebraic setting by Boucks…

Cloud-Assisted Remote Control for Aerial Robots: From Theory to Proof-of-Concept Implementation
Achilleas Santi Seisa, Viswa Narayanan Sankaranarayanan, Gerasimos Damigos, Sumeet Gajanan Satpute, George Nikolakopoulos
https://arxiv.org/abs/2509.04095

Cloud-Assisted Remote Control for Aerial Robots: From Theory to Proof-of-Concept Implementation
Cloud robotics has emerged as a promising technology for robotics applications due to its advantages of offloading computationally intensive tasks, facilitating data sharing, and enhancing robot coordination. However, integrating cloud computing with robotics remains a complex challenge due to network latency, security concerns, and the need for efficient resource management. In this work, we present a scalable and intuitive framework for testing cloud and edge robotic systems. The framework co…

On convergence of normal form transformations
Valery G. Romanovski, Sebastian Walcher
https://arxiv.org/abs/2510.00925 https://arxiv.org/pdf/2510.00925

On convergence of normal form transformations
We discuss various aspects concerning transformations of local analytic, or formal, vector fields to Poincare-Dulac normal form, and the convergence of such transformations. We first review A.D. Bruno's approach to formal normalization, as well as convergence results in presence of certain (simplified) versions of Bruno's ``Condition A'', and along the way we also identify a large class of systems that satisfy Bruno's diophantine ``Condition omega''. We retrace the proof steps in Bruno's work, …

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…

A novel approach through spherical functions in the characterization of invariant polynomials
Roc\'io D\'iaz Mar\'in, Linda Saal
https://arxiv.org/abs/2510.01043 htt…

A novel approach through spherical functions in the characterization of invariant polynomials
Given a compact subgroup K of the orthogonal group acting on the Euclidean space Rn, Gerald Schwarz proved that every smooth K-invariant function on Rn can be expressed as a smooth function of a generating set of $K$-invariant polynomials on n variables. The goal of this work is to provide an alternative and more straightforward proof of this result, based on Gelfand theory, with a particular focus on spherical functions.

Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$
Mirna D\v{z}amonja
https://arxiv.org/abs/2509.19519 …

Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$
We prove a theorem on iterated forcing that can be used for preservation of $\aleph_2$ and $\aleph_1$ in iterations with supports of size $\aleph_1$ of forcings that have amalgamation properties similar to those present in the perfect set forcing. The work is modelled after Baumgartner's Axiom A and his proof that iterations with countable support of the same preserve $\aleph_1$. In honour of James E. Baumgartner, the property introduced here is called Property B$(κ)$. The known additional dif…

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…

Atypical Hodge Loci
Phillip Griffiths
https://arxiv.org/abs/2510.01155 https://arxiv.org/pdf/2510.01155

Atypical Hodge Loci
In the recent works of a number of people there has emerged a beautiful new perspective on the arithmetic properties of Hodge structures. A central result in that development appears in a paper by Baldi, Klingler, and Ullmo. In this expository work we will explain that result and give a proof. The main conceptual step is to formulate Noether-Lefschetz loci in terms of intersections of period images with Mumford-Tate subdomains of period domains. The main technical step is to use the alignment o…

Purification of quantum trajectories in infinite dimensions
Federico Girotti, Alessandro Vitale
https://arxiv.org/abs/2509.13377 https://arxiv.org/pdf/2509…

Purification of quantum trajectories in infinite dimensions
In this work we exhibit a class of examples that show that the characterization of purification of quantum trajectories in terms of 'dark' subspaces that was proved for finite dimensional systems fails to hold in infinite dimensional ones. Moreover, we prove that the new phenomenon emerging in our class of models and preventing purification to happen is the only new possibility that emerges in infinite dimensional systems. Our proof strategy points out that the emergence of new phenomena in inf…

On the equivalence and optimality of transformations of diffusive systems
Davide Gabrielli, Giovanni Jona-Lasinio
https://arxiv.org/abs/2509.14450 https://…

On the equivalence and optimality of transformations of diffusive systems
In this paper we introduce, inspired by Clausius and developing the ideas of \cite{pre}, the concept of equivalence of transformations in non equilibrium theory of diffusive systems within the framework of macroscopic fluctuation theory. Besides providing a new proof of a formula derived in \cite{mft,qc}, which is the basis of the equivalence, we show that equivalent quasistatic transformations can be distinguished in finite terms, by the renormalized work introduced in \cite{45,46,mft,qc}. Thi…

"However, international justice can seem like a faith-based community. We believe in it, but proof of its existence is rare, and almost miraculous when it happens" -- @…
Let's all pray for a miracle, then.

Justice vs Peace
As the world watches the shaky ceasefire in Gaza, the issues of peace and justice intermingle, strain, and collide. For those who work on conflicts, these are familiar tensions, but the underlying debates rarely get wider attention. The basic issue is this: to get any kind of peace deal in

The Stein-Weiss inequality in variable exponent Morrey spaces
David Cruz-Uribe, Arash Ghorbanalizadeh, Durvudkhan Suragan
https://arxiv.org/abs/2510.02235 https://

The Stein-Weiss inequality in variable exponent Morrey spaces
In this paper we prove the Stein-Weiss inequality in variable exponent Morrey spaces over a bounded domain. Our work extends earlier results in the variable exponent Lebesgue and Morrey settings, and utilizes new proof techniques applicable to Morrey spaces. We build on the foundational paper by Almeida, Hasanov, and Samko, which introduced Morrey spaces of variable exponents. As an application of our main result, we prove Poincaré-type inequalities using the approach of a recent paper by the …

Pose Estimation of a Thruster-Driven Bioinspired Multi-Link Robot
Nicholas B. Andrews, Yanhao Yang, Sofya Akhetova, Kristi A. Morgansen, Ross L. Hatton
https://arxiv.org/abs/2510.01485

Pose Estimation of a Thruster-Driven Bioinspired Multi-Link Robot
This work demonstrates pose (position and shape) estimation for a free-floating, bioinspired multi-link robot with unactuated joints, link-mounted thrusters for control, and a single gyroscope per link, resulting in an underactuated, minimally sensed platform. Through a proof-of-concept hardware experiment and offline Kalman filter analysis, we show that the robot's pose can be reliably estimated. State estimation is performed using an unscented Kalman filter augmented with Gaussian process res…

Secret Leader Election in Ethereum PoS: An Empirical Security Analysis of Whisk and Homomorphic Sortition under DoS on the Leader and Censorship Attacks
Tereza Burianov\'a, Martin Pere\v{s}\'ini, Ivan Homoliak
https://arxiv.org/abs/2509.24955

Secret Leader Election in Ethereum PoS: An Empirical Security Analysis of Whisk and Homomorphic Sortition under DoS on the Leader and Censorship Attacks
Proposer anonymity in Proof-of-Stake (PoS) blockchains is a critical concern due to the risk of targeted attacks such as malicious denial-of-service (DoS) and censorship attacks. While several Secret Single Leader Election (SSLE) mechanisms have been proposed to address these threats, their practical impact and trade-offs remain insufficiently explored. In this work, we present a unified experimental framework for evaluating SSLE mechanisms under adversarial conditions, grounded in a simplified…

Composition Direction of Seymour's Theorem for Regular Matroids -- Formally Verified
Martin Dvorak, Tristan Figueroa-Reid, Rida Hamadani, Byung-Hak Hwang, Evgenia Karunus, Vladimir Kolmogorov, Alexander Meiburg, Alexander Nelson, Peter Nelson, Mark Sandey, Ivan Sergeev
https://arxiv.org/abs/2509.20539

Composition Direction of Seymour's Theorem for Regular Matroids -- Formally Verified
Seymour's decomposition theorem is a hallmark result in matroid theory presenting a structural characterization of the class of regular matroids. Formalization of matroid theory faces many challenges, most importantly that only a limited number of notions and results have been implemented so far. In this work, we formalize the proof of the forward (composition) direction of Seymour's theorem for regular matroids. To this end, we develop a library in Lean 4 that implements definitions and result…

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…

@… Buttt, proof of work! Years of academic (cryptocurency) research has shown this is the new kid on the block.

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

Higher tensor categories and their extensions: notes from the Scottish Talbot On Algebra and Topology
Diogo Andrade, Julia Bierent, Jennifer Brown, Tudor Caba, Matthew Cellot, Jonathan Davies, Adrien DeLazzer Meunier, Jannik Gr\"one, Benjamin Ha\"ioun, Alea Hofstetter, Theo Johnson-Freyd, David Jordan, Tessa Kammermeier, Patrick Kinnear, Cameron Krulewski, Theodoros Lagiotis, Leon Liu, Adri\`a Mar\'in Salvador, Nivedita, David Reutter, Lorenzo Riva, Iordanis Romaidis, Ja…

Higher tensor categories and their extensions: notes from the Scottish Talbot On Algebra and Topology
These lecture notes are the product of a week-long learning workshop on the work of Johnson-Freyd and Reutter on the problem of the existence of minimal nondegenerate extensions of braided fusion categories (arXiv:2105.15167). They recount the mathematical arguments of the original paper from an expository angle, with background material covering the algebra and homotopy theory required to understand the statement and follow the proof. The notes are aimed at newcomers to the field of (braided) …

Es ist mir ein Rätsel warum gängige CAPTCHA/Anti-Bot Dienste wie Anubis, ALTCHA etc. auf Proof-of-Work (PoW) setzen. Das (allein) verplempert doch nur Rechenleistung und lässt sich leicht umgehen, oder verstehe ich das nur nicht richtig?

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.

Mass-Centered GCM Framework in Perturbations of Kerr(-Newman)
Allen Juntao Fang, Elena Giorgi, Jingbo Wan
https://arxiv.org/abs/2510.10811 https://arxiv.or…

Mass-Centered GCM Framework in Perturbations of Kerr(-Newman)
The nonlinear stability problem for black hole solutions of the Einstein equations critically depends on choosing an appropriate geometric gauge. In the vacuum setting, the use of Generally Covariant Modulated (GCM) spheres and hypersurfaces has played a central role in the proof of stability for slowly rotating Kerr spacetime. In this work, we develop an alternative GCM framework, that we call mass-centered, designed to overcome the breakdown of the standard GCM construction in the charged cas…

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…

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…

On the quadratic complexity of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC_{2}}$-dimension
C. Terry, J. Wolf
https://arxiv.org/abs/2510.12767 https://

On the quadratic complexity of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC_{2}}$-dimension
In prior work, we showed that subsets of $\mathbb{F}_{p}^{n}$ of $\mathrm{VC_{2}}$-dimension at most $k$ are well approximated by a union of atoms of a quadratic factor of complexity $(\ell,q)$, where the complexity $\ell$ of the linear part and the complexity $q$ of the quadratic part are both bounded in terms of $k$, $p$, and the desired level of approximation $μ$. A key tool in the proof of this result was an arithmetic regularity lemma for the Gowers $U^3$-norm by Green and Tao, which resu…

"McTT: A Verified Kernel for a Proof Assistant"
Presentation by Junyoung Jang, joint work with Antoine Gaulin, Jason Z. S. Hu, Brigitte Pientka.
Repo: https://github.com/Beluga-lang/McTT/tree/icfp25

Green's functions for magnetic Dirac operators and bulk-edge correspondence
Jean-Marie Barbaroux, Horia D. Cornean, Lo\"ic Le Treust, Nicolas Raymond, Edgardo Stockmeyer
https://arxiv.org/abs/2509.17151

Green's functions for magnetic Dirac operators and bulk-edge correspondence
We study the resolvent kernel (Green's function) of magnetic Dirac operators on a half-plane with boundary conditions interpolating between infinite mass and zigzag cases, excluding the latter. We show that these kernels have all the required properties so that the proof of bulk-edge correspondence from the infinite mass case shown recently can be repeated ad litteram. The zigzag case exhibits qualitatively different behavior and will be addressed in future work.

Proof of the $p$-adic Kazhdan-Lusztig hypothesis for $\mathrm{GL}(n)$
Kristaps John Balodis
https://arxiv.org/abs/2510.09788 https://arxiv.org/pdf/2510.097…

Proof of the $p$-adic Kazhdan-Lusztig hypothesis for $\mathrm{GL}(n)$
In this article, we prove the $p$-adic Kazhdan-Lusztig hypothesis for $\mathrm{GL}_n(F)$. While the approach via graded affine Hecke algebras due to recent work of Solleveld leads to more general results, this article serves to completes and clarifies the approach via affine Hecke algebras of Chriss and Ginzburg. In particular, this article serves as an opportunity to articulate several results which are undoubtedly known to experts, but have not been formally recorded in the literature.

Zk-SNARK Marketplace with Proof of Useful Work
Samuel Oleksak, Richard Gazdik, Martin Peresini, Ivan Homoliak
https://arxiv.org/abs/2510.09729 https://arxi…

Zk-SNARK Marketplace with Proof of Useful Work
Proof of Work (PoW) is widely regarded as the most secure permissionless blockchain consensus protocol. However, its reliance on computationally intensive yet externally useless puzzles results in excessive electric energy wasting. To alleviate this, Proof of Useful Work (PoUW) has been explored as an alternative to secure blockchain platforms while also producing real-world value. Despite this promise, existing PoUW proposals often fail to embed the integrity of the chain and identity of the m…

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…

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…

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…

Algorithmic Perspective on Toda's Theorem
Dror Fried, Etay Segal, Gad E. Yaron
https://arxiv.org/abs/2509.13871 https://arxiv.org/pdf/2509.13871…

Algorithmic Perspective on Toda's Theorem
Toda's Theorem is a fundamental result in computational complexity theory, whose proof relies on a reduction from a QBF problem with a constant number of quantifiers to a model counting problem. While this reduction, henceforth called Toda's reduction, is of a purely theoretical nature, the recent progress of model counting tools raises the question of whether the reduction can be utilized to an efficient algorithm for solving QBF. In this work, we address this question by looking at Toda's red…

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…

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…

$\mathsf{P} \neq \mathsf{NP}$: A Non-Relativizing Proof via Quantale Weakness and Geometric Complexity
Ben Goertzel
https://arxiv.org/abs/2510.08814 https://

$\mathsf{P} \neq \mathsf{NP}$: A Non-Relativizing Proof via Quantale Weakness and Geometric Complexity
We give a compositional, information-theoretic framework that turns short programs into locality on many independent blocks, and combine it with symmetry and sparsity of masked random Unique-SAT to obtain distributional lower bounds that contradict the self-reduction upper bound under $\mathsf{P}=\mathsf{NP}$. We work in the weakness quantale $w_Q=K_{\mathrm{poly}}(\cdot\mid\cdot)$. For an efficiently samplable ensemble $D_m$ made by masking random $3$-CNFs with fresh $S_m\ltimes(\mathbb{Z}_2)^…

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…

Mechanizing Synthetic Tait Computability in Istari
Runming Li, Yue Yao, Robert Harper
https://arxiv.org/abs/2509.11418 https://arxiv.org/pdf/2509.11418

Mechanizing Synthetic Tait Computability in Istari
Categorical gluing is a powerful technique for proving meta-theorems of type theories such as canonicity and normalization. Synthetic Tait Computability (STC) provides an abstract treatment of the complex gluing models by internalizing the gluing category into a modal dependent type theory with a phase distinction. This work presents a mechanization of STC in the Istari proof assistant. Istari is a Martin-Löf-style extensional type theory with equality reflection. Equality reflection eliminate…

Charting GLOBs in Asymptotically Safe Gravity
Francesco Del Porro, Jonas Pfeiffer, Alessia Platania, Samuele Silveravalle
https://arxiv.org/abs/2509.14309 https://

Charting GLOBs in Asymptotically Safe Gravity
Computing the gravitational effective action provides a direct route to charting the landscape of admissible black hole spacetimes and their alternatives, which we will collectively call "gravitationally localized objects" (GLOBs). In this work, we provide a proof of principle of this idea within the framework of asymptotically safe gravity. Focusing on the Einstein-Weyl truncation, we identify the unique ultraviolet-complete trajectory emanating from the asymptotically safe fixed point and use…

An efficient iteration method to reconstruct the drift term from the final measurement
Dakang Cen, Wenlong Zhang, Zhidong Zhang
https://arxiv.org/abs/2510.10940 https://

An efficient iteration method to reconstruct the drift term from the final measurement
This work investigates the inverse drift problem in the one-dimensional parabolic equation with the final time data. The authors construct an operator first, whose fixed points are the unknown drift, and then apply it to prove the uniqueness. The proof of uniqueness contains an iteration converging to the drift, which inspires the numerical algorithm. To handle the ill-posedness of the inverse problem, the authors add the mollification on the data first in the iterative algorithm, and then prov…

$C^1$ perturbations of a continuum of critical points
Rafael Ortega, Antonio J. Urena
https://arxiv.org/abs/2509.10457 https://arxiv.org/pdf/2509.10457

$C^1$ perturbations of a continuum of critical points
Given a real valued function having a nondegenerate compact manifold of critical points, some of these points survive under small $C^2$ perturbations. This is a well-known result in critical point theory. In 1986 Weinstein obtained the analogous conclusions when the perturbation is only $C^2$ and the ambient space is a finite dimensional manifold. In this work we present a complete proof for $C^1$ perturbations in infinite dimensional Hilbert spaces.

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, …

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…

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…

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…

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…

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…

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…

A new proof on quasilinear Schr\"{o}dinger equations with prescribed mass and combined nonlinearity
Jianhua Chen, Jijiang Sun, Chenggui Yuan, Jian Zhang
https://arxiv.org/abs/2509.11073

A new proof on quasilinear Schrödinger equations with prescribed mass and combined nonlinearity
In this work, we study the quasilinear Schrödinger equation \begin{equation*} \aligned -Δu-Δ(u^2)u=|u|^{p-2}u+|u|^{q-2}u+λu,\,\, x\in\R^N, \endaligned \end{equation*} under the mass constraint \begin{equation*} \int_{\R^N}|u|^2\text{d}x=a, \end{equation*} where $N\geq2$, $20$ is a given mass and $λ$ is a Lagrange multiplier. As a continuation of our previous work (Chen et al., 2025, arXiv:2506.07346v1), we establish s…

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.

Man-in-the-Middle Proof-of-Concept via Krontiris' Ephemeral Diffie-Hellman Over COSE (EDHOC) in C
Daniel Hennig, Joaquin Garcia-Alfaro
https://arxiv.org/abs/2510.10574 https…

Man-in-the-Middle Proof-of-Concept via Krontiris' Ephemeral Diffie-Hellman Over COSE (EDHOC) in C
This report presents some technical details on the authentication process of a lightweight key exchange protocol, paying attention on how Man-in-the-Middle (MitM) attacks could undermine its security, e.g., under the scope of lawful interception and its risk to facilitate mass surveillance. We focus only on some technical aspects associated to the attack scenario. Perspectives for future work are also discussed. Other specific aspects of the work, mainly focusing on the security implications of…

Counting $2\times 2$ matrices with fixed determinant and bounded coefficients
Kavita Dhanda, Alan Haynes, Silmi Prasala
https://arxiv.org/abs/2509.16890 https://

Counting $2\times 2$ matrices with fixed determinant and bounded coefficients
Recent work by M. Afifurrahman established the first asymptotic estimates with error terms for the number of $2\times 2$ matrices with fixed non-zero determinant $n\in\mathbb{N}$, and with coefficients bounded in absolute value by $X$. In this paper we present a new proof of this result, which also gives an improved error term as $X\rightarrow\infty$. Similar to Afifurrahman's result, our error term is uniform in both $n$ and $X$, and our estimates are significant for $X$ as small as $n^{1/2+δ…

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.

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.

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…

Entire Large Solutions for Competitive Semilinear Elliptic Systems with General Nonlinearities Satisfying Keller--Osserman Conditions
Dragos-Patru Covei
https://arxiv.org/abs/2509.11933

Entire Large Solutions for Competitive Semilinear Elliptic Systems with General Nonlinearities Satisfying Keller--Osserman Conditions
We generalize a theorem of Lair concerning the existence of positive entire large solutions to competitive semilinear elliptic systems. While Lair's original result \cite{Lair2025} was established for power-type nonlinearities, our work extends the theory to a broad class of general nonlinearities satisfying a Keller--Osserman-type growth condition. The proof follows the same conceptual framework monotone iteration to construct global positive solutions, reduction to a scalar inequality for the…

A Continuum Beck-type Theorem for Hyperplanes
Paige Bright, Alexander Ortiz, Dmitrii Zakharov
https://arxiv.org/abs/2510.10907 https://arxiv.org/pdf/2510.1…

A Continuum Beck-type Theorem for Hyperplanes
We prove a sharp continuum Beck-type theorem for hyperplanes. Our work is inspired by foundational work of Beck on the discrete problem, as well as refinements due to Do and Lund. The inductive proof uses recent breakthrough results in projection theory by Orponen--Shmerkin--Wang and Ren, who proved continuum Beck-type theorems for lines in $\mathbb{R}^2$ and $\mathbb{R}^n$.

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…

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 …

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.

Unaligned Incentives: Pricing Attacks Against Blockchain Rollups
Stefanos Chaliasos, Conner Swann, Sina Pilehchiha, Nicolas Mohnblatt, Benjamin Livshits, Assimakis Kattis
https://arxiv.org/abs/2509.17126

Unaligned Incentives: Pricing Attacks Against Blockchain Rollups
Rollups have become the de facto scalability solution for Ethereum, securing more than $55B in assets. They achieve scale by executing transactions on a Layer 2 ledger, while periodically posting data and finalizing state on the Layer 1, either optimistically or via validity proofs. Their fees must simultaneously reflect the pricing of three resources: L2 costs (e.g., execution), L1 DA, and underlying L1 gas costs for batch settlement and proof verification. In this work, we identify critical m…

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-…