Tootfinder

The Fractal Logic of $\Phi$-adic Recursion
Milan Rosko
https://arxiv.org/abs/2510.08934 https://arxiv.org/pdf/2510.08934

The Fractal Logic of $Φ$-adic Recursion
We establish that valid $Σ_1$ propositional inference admits reduction to Fibonacci-indexed witness equations. Specifically, modus ponens verification reduces to solving a linear Diophantine equation in $O(M(\log n))$ time, where $M$ denotes integer multiplication complexity. This reduction is transitive: tautology verification proceeds via Fibonacci index arithmetic, bypassing semantic evaluation entirely. The core discovery is a transitive closure principle in $Φ$-scaled space (Hausdorff di…

Quasisymmetric mappings on two variants of fractal percolation
Roope Anttila, Sylvester Eriksson-Bique, Aleksi Py\"or\"al\"a
https://arxiv.org/abs/2510.06900 http…

Quasisymmetric mappings on two variants of fractal percolation
We study quasisymmetric maps on two variants of the classical fractal percolation model: the fat and dense fractal percolations. We show that, almost surely conditioned on non-extinction, the Hausdorff dimension of the fat fractal percolation cannot be lowered with a quasisymmetry and the Hausdorff dimension of the dense fractal percolation cannot be lowered with a power quasisymmetry.

Crosslisted article(s) found for cond-mat.dis-nn. https://arxiv.org/list/cond-mat.dis-nn/new
[1/1]:
- Fractal structure of multipartite entanglement in monitored quantum circuits
Vaibhav Sharma, Erich J Mueller

Can Fractal Dimension Distinguish between Grand-design and Flocculent Spiral Arms? #galaxies

Box dimension of the graphs of recurrent fractal interpolation functions
Lai Jiang, Xiao-Hui Li, Zhen Liang, Huo-Jun Ruan
https://arxiv.org/abs/2510.02754 https://

Box dimension of the graphs of recurrent fractal interpolation functions
Let $f$ be a generalized affine recurrent fractal interpolation function with vertical scaling functions. In this paper, by introducing underlying local iterated function systems of $f$, we define restricted vertical scaling matrices. Then we prove the monotonicity of spectral radii of these matrices without additional conditions. We also prove the irreducibility of these matrices under the assumption that vertical scaling functions are positive. With these results, we estimate the upper and lo…

BOFH excuse #303:
fractal radiation jamming the backbone

Taking notes from the successes and failures of the Russian revolution, a group of anarchists (including Nestor Makhno, a Ukrainian anarchist militant who was critical in defeating the Tzar's army and who later also fought the Red Army) wrote a document titled the "Organizational Platform of the Libertarian Communists." This document came to be known as "The Platform." It remains one of the most important first-hand revolutionary documents, outlining a clear revolutionary plan.
I've taken this, the Viable System Model from cybernetics, and my own organizing experience, to describe an organization to confront the current set of crises.
This continues to build on the stuff I have been writing, but it's a lot less high level theory and a lot more specific.
https://anarchoccultism.org/building-zion/a-solarpunk-fractal-microservices
As always, editing notes (typos, grammar, spelling, etc) are always welcome, as are any questions. My ADHD brain tends to go a lot faster than anything else, so I have a tendency to drop words and have a lot of trouble catching them later. Between my ADHD and mild dyslexia, it can be pretty hard for me to catch when autocorrect gives me the wrong word.
A lot of folks have already been super helpful in offering their editing support, and I'm really grateful. Writing this has felt collaborative, and it should. On the one hand this comes from my own experience and research, but on the other I'm also voicing things that have come from conversations here. This has all been a bit of my voice and a bit of the federated world, and I'm really appreciating that.

(Yet another) 1-minute excerpt of an #Umbra variation, one of my recent algorithmic/generative realtime animation art pieces, constantly evolving and exhibited exclusively on Layer.
Very much love this color theme, especially after ~15 seconds... 🤌
(See above hashtag for other clips)
Made with ❤️ and my opensource

Hausdorff dimension of the Wedding cake type surfaces
Bal\'azs B\'ar\'any, Manuj Verma
https://arxiv.org/abs/2510.04613 https://arxiv.org/pdf/2…

Hausdorff dimension of the Wedding cake type surfaces
In this paper, we study the Hausdorff dimension of fractal interpolation surfaces (FISs) over a triangular domain. These FISs are known as `wedding cake surfaces'. These surfaces are the attractor of some deterministic self-affine iterated function systems (IFS) on $\mathbb{R}^3$ generated by a fractal interpolation algorithm. Due to the recent seminal result of Rapaport (Adv. Math. 449 (2024) 109734), the dimension theory of self-affine IFS on $\mathbb{R}^3$ is known whenever the IFS is strong…

Localization Transition on Random Graphs with Chiral and Bogoliubov-de Gennes Symmetry Classes
Daniil Kochergin
https://arxiv.org/abs/2510.10255 https://ar…

Localization Transition on Random Graphs with Chiral and Bogoliubov-de Gennes Symmetry Classes
We studied single-particle Anderson localization in ensembles of graphs that correspond to chiral and Bogoliubov-de Gennes (BdG) symmetry classes. For a random biregular bipartite graph with chiral symmetry, the density of states was found using the cavity approach. Calculating the fractal dimension shows the effects of disordered zero modes. For Bogoliubov-de Gennes ensembles with an underlying random regular graph (RRG), the density of states was calculated both numerically and analytically. …

Riddled basin geometry sets fundamental limits to predictability and reproducibility in deep learning
Andrew Ly, Pulin Gong
https://arxiv.org/abs/2510.05606 https://

Riddled basin geometry sets fundamental limits to predictability and reproducibility in deep learning
Fundamental limits to predictability are central to our understanding of many physical and computational systems. Here we show that, despite its remarkable capabilities, deep learning exhibits such fundamental limits rooted in the fractal, riddled geometry of its basins of attraction: any initialization that leads to one solution lies arbitrarily close to another that leads to a different one. We derive sufficient conditions for the emergence of riddled basins by analytically linking features w…

Semi-Rings, Semi-Vector Spaces, and Fractal Interpolation
Peter R. Massopust
https://arxiv.org/abs/2509.24512 https://arxiv.org/pdf/2509.24512

Semi-Rings, Semi-Vector Spaces, and Fractal Interpolation
In this paper, we introduce fractal interpolation on complete semi-vector spaces. This approach is motivated by the requirements of preservation of positivity or monotonicity of functions for some models in approximation and interpolation theory. The setting in complete semi-vector spaces does not requite additional assumptions but is intrinsically built into the frame work. For the purposes of this paper, fractal interpolation in the complete semi-vector spaces $C^+$ and $L_p^+$ is considered.

Dimension dependence of critical phenomena in long-range percolation
Tom Hutchcroft
https://arxiv.org/abs/2510.03951 https://arxiv.org/pdf/2510.03951

Dimension dependence of critical phenomena in long-range percolation
Statistical mechanical systems at and near their points of phase transition are expected to exhibit rich, fractal-like behaviour that is independent of the small-scale details of the system but depends strongly on the dimension in which the model is defined. Moreover, many models are conjectured to have an upper critical dimension with important quantitative and qualitative differences between critical behaviour at, above, and below the upper critical dimension. For models with long-range inter…

Replaced article(s) found for stat.ML. https://arxiv.org/list/stat.ML/new
[2/2]:
- Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region
Shuang Liang, Guido Mont\'ufar

🇺🇦 #NowPlaying on KEXP's #Early
Brian Eno:
🎵 Blank Frank
#BrianEno
https://frankreich1.bandcamp.com/track/fractal-zoom-brian-eno-cover

Another 1-minute excerpt of an #Umbra variation, one of my recent algorithmic/generative realtime animation art pieces, constantly evolving and exhibited exclusively on Layer.
(See above hashtag for other clips)
Made with ❤️ and my opensource https://th…

Hitting times of shrinking targets: Transversality and an ergodic theorem
Vicente Saavedra-Araya
https://arxiv.org/abs/2510.07450 https://arxiv.org/pdf/251…

Hitting times of shrinking targets: Transversality and an ergodic theorem
In this paper, we investigate ergodic and fractal properties of the sets $$Λ_y:=\Big\{n\in\mathbb{N}:\ \{u_ny\}\in I_n\Big\},$$ where $\{\cdot\}$ denotes the fractional part function, $(u_n)_{n\in\mathbb{N}}$ is an increasing sequence of real numbers, $y\in [0,1]$ and each $I_n$ is a finite union of intervals with decreasing Lebesgue measure. Our main result shows that, under suitable conditions, the set $Λ_y$ is good for pointwise convergence of ergodic averages for Lebesgue almost every $y\…

Fractal failures of Ratner rigidity in higher rank geometry
Subhadip Dey, Hee Oh
https://arxiv.org/abs/2509.17915 https://arxiv.org/pdf/2509.17915

Fractal failures of Ratner rigidity in higher rank geometry
Ratner's theorem shows that in a locally symmetric space of noncompact type and finite volume, every immersed totally geodesic subspace of noncompact type is topologically rigid: its closure is an immersed submanifold. We construct the first explicit higher-rank, infinite-volume examples in which this rigidity fails, via floating geodesic planes. Specifically, we exhibit a Zariski-dense Hitchin surface group $Γ<\mathrm{SL}_3(\mathbb{R})$ such that $Γ\backslash \mathrm{SL}_3(\mathbb{R})/ \ma…

Fractal growth of higher-order topological insulators
Yutaro Tanaka, Shuai Zhang, Tiantian Zhang, Shuichi Murakami
https://arxiv.org/abs/2509.09875 https://

Fractal growth of higher-order topological insulators
Understanding crystal growth and morphology is a fundamental issue in condensed matter physics. In this work, we reveal the fractal morphology of growing crystals of higher-order topological insulators and show that the corners of the crystals grow preferentially compared to the edges in the presence of the corner states. We further demonstrate that when we compare the crystal shape of the higher-order topological insulator with that of the trivial insulator with the same value of the fractal d…

Fractal Rhetorics: Scaling Small and Subtle Feminist Rhetorical Actions
https://ift.tt/sLgBrbp
updated: Monday, November 3, 2025 - 3:29pmfull name / name of organization: Jessi Thomsen, Western…
via Input 4 RELCFP …

Bilinear spherical maximal function with fractal dilations
Surjeet Singh Choudhary, Chun-Yen Shen, Saurabh Shrivastava
https://arxiv.org/abs/2510.03094 https://

Bilinear spherical maximal function with fractal dilations
In this paper, we investigate $L^p-$boundedness of the bilinear spherical maximal function associated with a general dilation set $E\subset\R_+$. We quantify the range of $L^p-$boundedness in terms of a dilation-invariant notion of upper Minkowski dimension of the set $E$. A particular case of this study settles an open question of $L^p-$boundedness of the lacunary bilinear spherical maximal function at borderline cases $p_1=1$ or $p_2=1$ in dimension $d\geq4$.

Scaling limit for Brownian motions on the $l$-level Sierpinski gaskets: The fractal to Euclidean crossover
David A. Croydon, Ben Hambly, Takashi Kumagai
https://arxiv.org/abs/2509.20657

Scaling limit for Brownian motions on the $l$-level Sierpinski gaskets: The fractal to Euclidean crossover
In two dimensions, the $l$-level Sierpinski gasket $\mathrm{SG}(l)$ is obtained by splitting an equilateral triangle into a collection of $l^2$ equilateral triangles of equal size and with the same total area, retaining only the $l(l+1)/2$ triangles with the same orientation as the original triangle, and then iterating this procedure indefinitely. We show that the canonical diffusions on the spaces $\mathrm{SG}(l)$, $l\geq2$, can be rescaled to yield Brownian motion on the initial triangle. Our…

Some Structures arising from the Farey Fractal
Shai Haran
https://arxiv.org/abs/2509.10959 https://arxiv.org/pdf/2509.10959

Some Structures arising from the Farey Fractal
This work explores new arithmetic and combinatorial structures arising from the interplay between Farey-type graphs, Fibonacci expansions, and operadic constructions. We introduce Fibonadic numbers, defined as an inverse limit under the Zeckendorf shift, equipped with a metric, order, and commutative rig structure. A normalization lemma provides canonical representatives, while quotients of the non-zero Fibonadic numbers under shifts and it's fundamental domain covering the circle via phi-value…

Holographic Turbulence and the Fractal Dimension of the Turbulent Horizon
Jia Du, Yu Tian, Hongbao Zhang
https://arxiv.org/abs/2510.12198 https://arxiv.org…

Holographic Turbulence and the Fractal Dimension of the Turbulent Horizon
We study two-dimensional turbulence driven by a scalar operator within the framework of the AdS/CFT correspondence, where the external driving source is used to sustain a quasi-steady turbulent state. We numerically construct dynamical and spatially inhomogeneous turbulent black holes in the asymptotically $\mathrm{AdS}_4$ spacetime by solving the full nonlinear equations of motion in the Bondi-Sachs formalism. The inverse energy cascade and the corresponding energy spectrum of both decaying an…

4K psytrance visuals feel like riding cosmic waves through fractal dimensions, perfect for melting into the trip!
https://www.youtube.com/watch?v=Y3lKTBWwr6Y
Especially 8K!

Been working on some additions/adjustments for #Umbra, one of my recent algorithmic/generative art pieces, currently shown exclusively on Layer.
Here's a 3 minute segment (of an infinite realtime animation). Give it time to develop/change! Fullscreen playback probably best...
Made with

I was just thinking about how the fact that #Musk named his AI "Grok" is evidence that he "reads sci-fi" in the same way he "plays video games." Like, he claims to do it but when it comes time to show the evidence it's clear he does not actually "grok" it.
Like... To grok something is to have a layer deeper than simply knowledge, but mathematically encoding statistical relationships between words is pretty obviously not even understanding much less qualifying as "groking" it. In the book, the ability to grok something is also the ability to annihilate that thing with a thought. Just pretending that an LLM actually *was* something that could become AGI (which it's not), this name would imply the AI would have the power to annihilate reality. That's bad. That's a bad name for an AI.
And why would a greedy fascist name something of his after something an anarchist communist space Jesus taught to the hippie cult he started? There are so many layers of facepalm to this. It's some kind of php-esque fractal of incompetence.
Like, there's no reason to talk about this but my brain does this to me sometimes and now it's your problem.

Generalized Thue-Morse measures: spectral and fractal analysis
Philipp Gohlke, Marc Kesseb\"ohmer, Tanja I. Schindler
https://arxiv.org/abs/2509.22109 https://

Generalized Thue-Morse measures: spectral and fractal analysis
We investigate a family of Riesz products and show that they can be regarded as diffraction measures of generalized Thue-Morse sequences, possibly over an infinite alphabet. These measures are closely related to the dynamical system arising from the doubling map together with an observable exhibiting a logarithmic singularity. For this system, we develop a generalized thermodynamic formalism beyond the standard setting, which yields explicit formulas for Birkhoff and dimension spectra. A furthe…

Replaced article(s) found for cond-mat.str-el. https://arxiv.org/list/cond-mat.str-el/new
[1/1]:
- Fractal dimension and the counting rule of the Goldstone modes
Qian-Qian Shi, Yan-Wei Dai, Huan-Qiang Zhou, Ian P. McCulloch

From Quasiperiodicity to a Complete Coloring of the Kohmoto Butterfly
Ram Band, Siegfried Beckus
https://arxiv.org/abs/2509.24025 https://arxiv.org/pdf/250…

From Quasiperiodicity to a Complete Coloring of the Kohmoto Butterfly
The spectra of the Kohmoto model give rise to a fractal phase diagram, known as the Kohmoto butterfly. The butterfly encapsulates the spectra of all periodic Kohmoto Hamiltonians, whose index invariants are sought after. Topological methods - such as Chern numbers - are ill defined due to the discontinuous potential, and hence fail to provide index invariants. This Letter overcomes that obstacle and provides a complete classification of the Kohmoto model indices. Our approach encodes the Kohmot…

What does the tree of life look like as it grows? Evolution and the multifractality of time
Kevin Hudnall, Raissa D'Souza
https://arxiv.org/abs/2509.25615 https://

What does the tree of life look like as it grows? Evolution and the multifractality of time
By unifying three foundational principles of modern biology, we develop a mathematical framework to analyze the growing tree of life. Contrary to the static case, where the analogy between phylogenetic trees and the tree that grows in soil is drawn, our framework shows that the living tree of life is analogous to a Cantor dust where each branch is a distinct fractal curve. The system as a whole is therefore multifractal in the sense that it consists of many unique fractals. The three foundation…

Distinct dimensions for attractors of bi-Lipschitz iterated function systems
Simon Baker, Amlan Banaji, De-Jun Feng, Chun-Kit Lai, Ying Xiong
https://arxiv.org/abs/2509.22084 ht…

Distinct dimensions for attractors of bi-Lipschitz iterated function systems
In this paper, we construct an iterated function system on the line consisting of two bi-Lipschitz contractions whose attractor has distinct lower, Hausdorff, lower box, upper box, and Assouad dimensions, thereby providing negative answers to certain folklore questions. Furthermore, as a by-product of our study of bi-Lipschitz IFSs, we construct IFSs within this family that exhibit interesting fractal behaviour. In particular, we prove the following two statements: (i) There exists a bi-Lipsc…

Log-Hausdorff multifractality of the absolutely continuous spectral measure of the almost Mathieu operator
Jie Cao, Xianzhe Li, Baowei Wang, Qi Zhou
https://arxiv.org/abs/2509.09945

Log-Hausdorff multifractality of the absolutely continuous spectral measure of the almost Mathieu operator
This paper focuses on the fractal characteristics of the absolutely continuous spectral measure of the subcritical almost Mathieu operator (AMO) and Diophantine frequency. In particular, we give a complete description of the (classical) multifractal spectrum and a finer description in the logarithmic gauge. The proof combines continued$-$fraction$/$metric Diophantine techniques and refined covering arguments. These results rigorously substantiate (and quantify in a refined gauge) the physicists…

Delocalization Induced by Enhanced Hyperuniformity in One-Dimensional Disordered Systems
Junmo Jeon, Harukuni Ikeda, Shiro Sakai
https://arxiv.org/abs/2509.23036 https://…

Delocalization Induced by Enhanced Hyperuniformity in One-Dimensional Disordered Systems
In one dimension, any disorder is traditionally believed to localize all states. We show that this paradigm breaks down under hyperuniform disorder, which suppresses long-wavelength fluctuations and interpolates between random and periodic potentials. In tight-binding chains, strong hyperuniformity induces a sharp delocalization transition and the emergence of mobility edges. The transition is identified by the generalized fractal dimension and corroborated by the scaling of localization length…

Nonlinear Fractal Histopolation Function
Aiswarya T, Srijanani Anurag Prasad
https://arxiv.org/abs/2509.18657 https://arxiv.org/pdf/2509.18657