Tootfinder

from my link log —
Arend: a theorem prover based on Homotopy Type Theory.
https://arend-lang.github.io/
saved 2019-08-07 https://dotat.at/:/AR6J2.html

The Theory of Strategic Evolution: Games with Endogenous Players and Strategic Replicators
Kevin Vallier
https://arxiv.org/abs/2512.07901 https://arxiv.org/pdf/2512.07901 https://arxiv.org/html/2512.07901
arXiv:2512.07901v1 Announce Type: new
Abstract: This paper develops the Theory of Strategic Evolution, a general model for systems in which the population of players, strategies, and institutional rules evolve together. The theory extends replicator dynamics to settings with endogenous players, multi level selection, innovation, constitutional change, and meta governance. The central mathematical object is a Poiesis stack: a hierarchy of strategic layers linked by cross level gain matrices. Under small gain conditions, the system admits a global Lyapunov function and satisfies selection, tracking, and stochastic stability results at every finite depth. We prove that the class is closed under block extension, innovation events, heterogeneous utilities, continuous strategy spaces, and constitutional evolution. The closure theorem shows that no new dynamics arise at higher levels and that unrestricted self modification cannot preserve Lyapunov structure. The theory unifies results from evolutionary game theory, institutional design, innovation dynamics, and constitutional political economy, providing a general mathematical model of long run strategic adaptation.
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Pedro Abreu aka #TypeTheoryForall had an epic conversation with me about all things programming languages, out now on the podcast.
https://www.typetheoryforall.com/…

Dimensionality reduction and width of deep neural networks based on topological degree theory
Xiao-Song Yang
https://arxiv.org/abs/2511.06821 https://arxiv.org/pdf/2511.06821 https://arxiv.org/html/2511.06821
arXiv:2511.06821v1 Announce Type: new
Abstract: In this paper we present a mathematical framework on linking of embeddings of compact topological spaces into Euclidean spaces and separability of linked embeddings under a specific class of dimension reduction maps. As applications of the established theory, we provide some fascinating insights into classification and approximation problems in deep learning theory in the setting of deep neural networks.
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Topological Structure of Infrared QCD
J. Gamboa
https://arxiv.org/abs/2511.07455 https://arxiv.org/pdf/2511.07455 https://arxiv.org/html/2511.07455
arXiv:2511.07455v1 Announce Type: new
Abstract: We investigate the infrared structure of QCD within the adiabatic approximation, where soft gluon configurations evolve slowly compared to the fermionic modes. In this formulation, the functional space of gauge connections replaces spacetime as the natural arena for the theory, and the long-distance behavior is encoded in quantized Berry phases associated with the infrared clouds. Our results suggest that the infrared sector of QCD exhibits features reminiscent of a \emph{topological phase}, similar to those encountered in condensed-matter systems, where topological protection replaces dynamical confinement at low energies. In this geometric framework, color-neutral composites such as quark--gluon and gluon--gluon clouds arise as topological bound states described by functional holonomies. Illustrative applications to hadronic excitations are discussed within this approach, including mesonic and baryonic examples. This perspective provides a unified picture of infrared dressing and topological quantization, establishing a natural bridge between non-Abelian gauge theory, adiabatic Berry phases, and the topology of the space of gauge configurations.
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Natural transformations between braiding functors in the Fukaya category
Yujin Tong
https://arxiv.org/abs/2511.10462 https://arxiv.org/pdf/2511.10462 https://arxiv.org/html/2511.10462
arXiv:2511.10462v1 Announce Type: new
Abstract: We study the space of $A_\infty$-natural transformations between braiding functors acting on the Fukaya category associated to the Coulomb branch $\mathcal{M}(\bullet,1)$ of the $\mathfrak{sl}_2$ quiver gauge theory. We compute all cohomologically distinct $A_\infty$-natural transformations $\mathrm{Nat}(\mathrm{id}, \mathrm{id})$ and $\mathrm{Nat}(\mathrm{id}, \beta_i^-)$, where $\beta_i^-$ denotes the negative braiding functor. Our computation is carried out in a diagrammatic framework compatible with the established embedding of the KLRW category into this Fukaya category. We then compute the Hochschild cohomology of the Fukaya category using an explicit projective resolution of the diagonal bimodule obtained via the Chouhy-Solotar reduction system, and use this to classify all cohomologically distinct natural transformations. These results determine the higher $A_\infty$-data encoded in the braiding functors and their natural transformations, and provide the first step toward a categorical formulation of braid cobordism actions on Fukaya categories.
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Global Convergence of Four-Layer Matrix Factorization under Random Initialization
Minrui Luo, Weihang Xu, Xiang Gao, Maryam Fazel, Simon Shaolei Du
https://arxiv.org/abs/2511.09925 https://arxiv.org/pdf/2511.09925 https://arxiv.org/html/2511.09925
arXiv:2511.09925v1 Announce Type: new
Abstract: Gradient descent dynamics on the deep matrix factorization problem is extensively studied as a simplified theoretical model for deep neural networks. Although the convergence theory for two-layer matrix factorization is well-established, no global convergence guarantee for general deep matrix factorization under random initialization has been established to date. To address this gap, we provide a polynomial-time global convergence guarantee for randomly initialized gradient descent on four-layer matrix factorization, given certain conditions on the target matrix and a standard balanced regularization term. Our analysis employs new techniques to show saddle-avoidance properties of gradient decent dynamics, and extends previous theories to characterize the change in eigenvalues of layer weights.
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from my link log —
Homotopy type theory for dummies.
http://www.chriswarbo.net/blog/2015-09-11-hott_for_dummies.html
saved 2025-11-03 …

S-D-RSM: Stochastic Distributed Regularized Splitting Method for Large-Scale Convex Optimization Problems
Maoran Wang, Xingju Cai, Yongxin Chen
https://arxiv.org/abs/2511.10133 https://arxiv.org/pdf/2511.10133 https://arxiv.org/html/2511.10133
arXiv:2511.10133v1 Announce Type: new
Abstract: This paper investigates the problems large-scale distributed composite convex optimization, with motivations from a broad range of applications, including multi-agent systems, federated learning, smart grids, wireless sensor networks, compressed sensing, and so on. Stochastic gradient descent (SGD) and its variants are commonly employed to solve such problems. However, existing algorithms often rely on vanishing step sizes, strong convexity assumptions, or entail substantial computational overhead to ensure convergence or obtain favorable complexity. To bridge the gap between theory and practice, we integrate consensus optimization and operator splitting techniques (see Problem Reformulation) to develop a novel stochastic splitting algorithm, termed the \emph{stochastic distributed regularized splitting method} (S-D-RSM). In practice, S-D-RSM performs parallel updates of proximal mappings and gradient information for only a randomly selected subset of agents at each iteration. By introducing regularization terms, it effectively mitigates consensus discrepancies among distributed nodes. In contrast to conventional stochastic methods, our theoretical analysis establishes that S-D-RSM achieves global convergence without requiring diminishing step sizes or strong convexity assumptions. Furthermore, it achieves an iteration complexity of $\mathcal{O}(1/\epsilon)$ with respect to both the objective function value and the consensus error. Numerical experiments show that S-D-RSM achieves up to 2--3$\times$ speedup compared to state-of-the-art baselines, while maintaining comparable or better accuracy. These results not only validate the algorithm's theoretical guarantees but also demonstrate its effectiveness in practical tasks such as compressed sensing and empirical risk minimization.
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Infinite-dimensional Lagrange-Dirac systems with boundary energy flow II: Field theories with bundle-valued forms
Fran\c{c}ois Gay-Balmaz, \'Alvaro Rodr\'iguez Abella, Hiroaki Yoshimura
https://arxiv.org/abs/2511.05687 https://arxiv.org/pdf/2511.05687 https://arxiv.org/html/2511.05687
arXiv:2511.05687v1 Announce Type: new
Abstract: Part I of this paper introduced the infinite dimensional Lagrange--Dirac theory for physical systems on the space of differential forms over a smooth manifold with boundary. This approach is particularly well-suited for systems involving energy exchange through the boundary, as it is built upon a restricted dual space -a vector subspace of the topological dual of the configuration space- that captures information about both the interior dynamics and boundary interactions. Consequently, the resulting dynamical equations naturally incorporate boundary energy flow. In this second part, the theory is extended to encompass vector-bundle-valued differential forms and non-Abelian gauge theories. To account for two commonly used forms of energy flux and boundary power densities, we introduce two distinct but equivalent formulations of the restricted dual. The results are derived from both geometric and variational viewpoints and are illustrated through applications to matter and gauge field theories. The interaction between gauge and matter fields is also addressed, along with the associated boundary conditions, applied to the case of the Yang-Mills-Higgs equations.
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After a few intense weeks — first at #icfpsplash25 in Singapore, then four days of interviewing candidates for tenure-track positions as part of the faculty appointment committee (Lärarförslagsnämnd) at the University of Gothenburg — I’m now in Budapest for (mostly) vacation.
Tomorrow (2025-10-27) I’ll give a talk at the local Type Theory Seminar (invited by Ambrus Kaposi). Looking f…

I just realised there's a really nice and simple description of ℝPⁿ as a cubical cell complex: one can construct ℝPⁿ (for a known external n) as a HIT in cubical type theory with reversals like this:
This is nice because it doesn't involve composition, so the eliminator is easy to define (in fact my macro can do it automatically). This makes it very easy to define coherent involutions in n-types as maps out of Bℤ₂, defined as ℝPⁿ⁺² an n-truncation constructor.
ping @… I think you were thinking about this at one point

On Dynamic Programming Theory for Leader-Follower Stochastic Games
Jilles Steeve Dibangoye, Thibaut Le Marre, Ocan Sankur, Fran\c{c}ois Schwarzentruber
https://arxiv.org/abs/2512.05667 https://arxiv.org/pdf/2512.05667 https://arxiv.org/html/2512.05667
arXiv:2512.05667v1 Announce Type: new
Abstract: Leader-follower general-sum stochastic games (LF-GSSGs) model sequential decision-making under asymmetric commitment, where a leader commits to a policy and a follower best responds, yielding a strong Stackelberg equilibrium (SSE) with leader-favourable tie-breaking. This paper introduces a dynamic programming (DP) framework that applies Bellman recursion over credible sets-state abstractions formally representing all rational follower best responses under partial leader commitments-to compute SSEs. We first prove that any LF-GSSG admits a lossless reduction to a Markov decision process (MDP) over credible sets. We further establish that synthesising an optimal memoryless deterministic leader policy is NP-hard, motivating the development of {\epsilon}-optimal DP algorithms with provable guarantees on leader exploitability. Experiments on standard mixed-motive benchmarks-including security games, resource allocation, and adversarial planning-demonstrate empirical gains in leader value and runtime scalability over state-of-the-art methods.
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Hamiltonian flow between standard module Lagrangians
Yujin Tong
https://arxiv.org/abs/2511.06431 https://arxiv.org/pdf/2511.06431 https://arxiv.org/html/2511.06431
arXiv:2511.06431v1 Announce Type: new
Abstract: In Aganagic's Fukaya category of the Coulomb branch of quiver gauge theory, the $T_\theta$-brane algebra gives a symplectic realization of the Khovanov-Lauda-Rouquier-Webster (KLRW) algebra, where each standard module is known to admit two Lagrangian realizations: the 'U'-shaped $T$-brane and the step $I$-brane. We show that the latter arises as the infinite-time limit of the Hamiltonian evolution of the former, thus serving as a generalized thimble. This provides a geometric realization of the categorical isomorphism previously established through holomorphic disc counting.
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… zwischen zwei Saunagängen ne Idee gehabt wie man Martin-Löf dependent type theory im @… erklärt und mir jetzt was Schickes angezogen. Jetzt geht’s noch auf nen 80. Geburtstag (nicht meiner).
What a Life. 🤗
*so lange Star Wars Raumschiffe sagen bis einer keine neuen mehr weiß.
2/2

Polyharmonic Cascade
Yuriy N. Bakhvalov
https://arxiv.org/abs/2512.17671 https://arxiv.org/pdf/2512.17671 https://arxiv.org/html/2512.17671
arXiv:2512.17671v1 Announce Type: new
Abstract: This paper presents a deep machine learning architecture, the "polyharmonic cascade" -- a sequence of packages of polyharmonic splines, where each layer is rigorously derived from the theory of random functions and the principles of indifference. This makes it possible to approximate nonlinear functions of arbitrary complexity while preserving global smoothness and a probabilistic interpretation. For the polyharmonic cascade, a training method alternative to gradient descent is proposed: instead of directly optimizing the coefficients, one solves a single global linear system on each batch with respect to the function values at fixed "constellations" of nodes. This yields synchronized updates of all layers, preserves the probabilistic interpretation of individual layers and theoretical consistency with the original model, and scales well: all computations reduce to 2D matrix operations efficiently executed on a GPU. Fast learning without overfitting on MNIST is demonstrated.
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Invariant Price of Anarchy: a Metric for Welfarist Traffic Control
Ilia Shilov, Mingjia He, Heinrich H. Nax, Emilio Frazzoli, Gioele Zardini, Saverio Bolognani
https://arxiv.org/abs/2512.05843 https://arxiv.org/pdf/2512.05843 https://arxiv.org/html/2512.05843
arXiv:2512.05843v1 Announce Type: new
Abstract: The Price of Anarchy (PoA) is a standard metric for quantifying inefficiency in socio-technical systems, widely used to guide policies like traffic tolling. Conventional PoA analysis relies on exact numerical costs. However, in many settings, costs represent agents' preferences and may be defined only up to possibly arbitrary scaling and shifting, representing informational and modeling ambiguities. We observe that while such transformations preserve equilibrium and optimal outcomes, they change the PoA value. To resolve this issue, we rely on results from Social Choice Theory and define the Invariant PoA. By connecting admissible transformations to degrees of comparability of agents' costs, we derive the specific social welfare functions which ensure that efficiency evaluations do not depend on arbitrary rescalings or translations of individual costs. Case studies on a toy example and the Zurich network demonstrate that identical tolling strategies can lead to substantially different efficiency estimates depending on the assumed comparability. Our framework thus demonstrates that explicit axiomatic foundations are necessary in order to define efficiency metrics and to appropriately guide policy in large-scale infrastructure design robustly and effectively.
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Can You Hear Me Now? A Benchmark for Long-Range Graph Propagation
Luca Miglior, Matteo Tolloso, Alessio Gravina, Davide Bacciu
https://arxiv.org/abs/2512.17762 https://arxiv.org/pdf/2512.17762 https://arxiv.org/html/2512.17762
arXiv:2512.17762v1 Announce Type: new
Abstract: Effectively capturing long-range interactions remains a fundamental yet unresolved challenge in graph neural network (GNN) research, critical for applications across diverse fields of science. To systematically address this, we introduce ECHO (Evaluating Communication over long HOps), a novel benchmark specifically designed to rigorously assess the capabilities of GNNs in handling very long-range graph propagation. ECHO includes three synthetic graph tasks, namely single-source shortest paths, node eccentricity, and graph diameter, each constructed over diverse and structurally challenging topologies intentionally designed to introduce significant information bottlenecks. ECHO also includes two real-world datasets, ECHO-Charge and ECHO-Energy, which define chemically grounded benchmarks for predicting atomic partial charges and molecular total energies, respectively, with reference computations obtained at the density functional theory (DFT) level. Both tasks inherently depend on capturing complex long-range molecular interactions. Our extensive benchmarking of popular GNN architectures reveals clear performance gaps, emphasizing the difficulty of true long-range propagation and highlighting design choices capable of overcoming inherent limitations. ECHO thereby sets a new standard for evaluating long-range information propagation, also providing a compelling example for its need in AI for science.
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