Tootfinder

In Los Angeles wird es am Wochenende heiß und feucht. Temperaturen bis über 37 Grad werden in den Tälern und Wüsten erwartet.
Dazu kommt #tropische Feuchtigkeit durch #Sturm Alvin, was lokal zu #Gewittern

California heat wave expected to break records this weekend
Large swaths of California are under heat advisories Friday and Saturday with officials cautioning against outdoor activities in triple-digit areas.

A Finite-State Symbolic Automaton Model for the Collatz Map and Its Convergence Properties
Leonard Ben Aurel Brauer
https://arxiv.org/abs/2506.21728 https:…

A Finite-State Symbolic Automaton Model for the Collatz Map and Its Convergence Properties
We present a finite-state, deterministic automaton that emulates the Collatz function by operating on base-10 digit sequences. Each digit is represented as a symbolic triplet capturing its value, the parity of the next digit, and a local carry value, resulting in a state space of exactly 60 configurations. The transition rules are local, total, and parity-dependent, yet collectively reproduce the global behavior of the Collatz map through digitwise operations. All symbolic trajectories reduce t…

Apple reports Q3 revenue from Services, which includes the App Store, Apple TV , and Apple Music, up 13.3% YoY to $27.42B, vs. $26.8B est. (Todd Spangler/Variety)
https://variety.com/2025/digital/news/apple-june-2025-quarter-e…

Apple Services Revenue Grows 13% to New Record, Earnings Beat Street Amid Trump Tariff Uncertainty
Apple continued to drive double-digit growth in its Services business, which increased revenue 13.3% to $27.42 billion for the June 2025 quarter -- a new quarterly record

Skies cleared up just in time for solar activity to get even higher (Kp 8)...
... and the sun to come up. I grabbed a single digit number of shots but the eastern sky was already visibly bright naked eye and my long-exposure shots started getting washed out.
Forecast shows activity will be decent all day (good luck Asian/European sky watchers!) and might persist into the dark for me tonight. Fingers crossed.

Apple reports Q3 revenue from Services, which includes the App Store, Apple TV , and Apple Music, up 13.3% YoY to $27.42B, vs. $26.8B est. (Todd Spangler/Variety)
https://variety.com/2025/digital/news/apple-june-2025-quarter-e…

Apple Services Revenue Grows 13% to New Record, Earnings Beat Street Amid Trump Tariff Uncertainty
Apple continued to drive double-digit growth in its Services business, which increased revenue 13.3% to $27.42 billion for the June 2025 quarter -- a new quarterly record

Additions with unique digits: A tale of puzzling and AI
The other day, Markus Krötzsch and I were catching up when one of his kids came in. She told us about a puzzle her math teacher gave her:
How many integer sums are there where the equation uses each digit at most once?
The simplest example is 1 2=3, but 12 47=59 also works. On the other hand, 1 9=10 isn’t a solution because the digit ‘1’ appears twice (in the first number and the sum).
1/19

Polymarket raised "double-digit millions" from 1789 Capital, which Donald Trump Jr. joined as a partner in 2024; Trump Jr. will join Polymarket's advisory board (Dan Primack/Axios)
https://www.axios.com/2025/08/26/trump-jr-vc-fund-polymarket

Scoop: Donald Trump Jr.'s VC firm bets on Polymarket
1789 Capital invested double-digit millions of dollars into Polymarket

Mathematicians, a plea for understanding. Or at least I point out a curiosity.
I was playing around in ivy (high precision calculator) with Stirling's approximation and noticed something. I compute (Stirling N)/N! for successive powers of 10:
approx 10 100 1000 10000 100000 1000000
0.99170403955606148634 0.99916701656784300017 0.99991667014156998579 0.99999166670139157019 0.99999916666701389157 0.99999991666667013889
The successive approximations improve (approach 1.0) by one decimal digit of result for each digit of argument, but that's not the curious part. The curious part happens after the 9's: the result seems to be trending towards a limit of 0.9...166666...... It adds not only a 9 for each input digit, it extends the next part by a digit 6 for every factor of 10. Now 1/6 is 0.166666 and that makes me think there's something going on here.
Is this true? Is it known? Is it proven?

Obamacare Insurers Seek Double-Digit Premium Hikes Next Year (Anna Wilde Mathews/Wall Street Journal)
https://www.wsj.com/health/healthcare/obamacare-insurers-seek-double-digit-premium-hikes-next-year-454f1e1c
http://www.memeorandum.com/250718/p39#a250718p39

Treasury released a bunch of market modernisation updates this morning, including for DIGIT - the thing (cryptocurrency, but not in those words) that the press dubbed "Britcoin".
https://www.gov.uk/government/publications/digital-gilt-instrumen…

Digital Gilt Instrument (DIGIT) Pilot Update
The government are setting out features to be tested as part of the digital gilt instrument (DIGIT) pilot.

NFL initially proposed double-digit game suspension for Chiefs' Rashee Rice, per report

https://www.cbssports.com/nfl/news/nfl-initially-pro…

Structural Foundations for Leading Digit Laws: Beyond Probabilistic Mixtures
Vladimir Berman
https://arxiv.org/abs/2508.13237 https://arxiv.org/pdf/2508.13…

Structural Foundations for Leading Digit Laws: Beyond Probabilistic Mixtures
This article presents a modern deterministic framework for the study of leading significant digit distributions in numerical data. Rather than relying on traditional probabilistic or mixture-based explanations, we demonstrate that the observed frequencies of leading digits are determined by the underlying arithmetic, algorithmic, and structural properties of the data-generating process. Our approach centers on a shift-invariant functional equation, whose general solution is given by explicit af…

Benford Behavior in Stick Fragmentation Problems
Bruce Fang, Ava Irons, Ella Lippelman, Steven J. Miller
https://arxiv.org/abs/2508.17360 https://arxiv.org…

Benford Behavior in Stick Fragmentation Problems
Benford's law is the statement that in many real-world data sets, the probability of having digit \(d\) in base \(B\), where \(1 \leq d \leq B\), as the first digit is \(\log_{B}\left(\tfrac{d+1}{d}\right)\). We sometimes refer to this as weak Benford behavior, and we say that a data set exhibits strong Benford behavior in base \(B\) if the probability of having significand at most \(s\), where \(s \in [1,B)\), is \(\log_{B}(s)\). We examine Benford behaviors in the stick fragmentation model. B…

Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices
Anjelo Gabriel R. Cruz, Manuel Joseph C. Loquias, J\"org M. Thuswaldner
https://arxiv.org/abs/2507.06158

Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices
Let $A$ be an expanding $2 \times 2$ matrix with rational entries and $\mathbb{Z}^2[A]$ be the smallest $A$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^2$. Let $\mathcal{D}$ be a finite subset of $\mathbb{Z}^2[A]$ which is a complete residue system of $\mathbb{Z}^2[A]/A\mathbb{Z}^2[A]$. The pair $(A,\mathcal{D})$ is called a {\em digit system} with {\em base} $A$ and {\em digit set} $\mathcal{D}$. It is well known that every vector $x \in \mathbb{Z}^2[A]$ can be written uniquely in the…

This keyboard component is called “6-digit-verification entry pad”

Benford's Curse: Tracing Digit Bias to Numerical Hallucination in LLMs
Jiandong Shao, Yao Lu, Jianfei Yang
https://arxiv.org/abs/2506.01734 https://

Benford's Curse: Tracing Digit Bias to Numerical Hallucination in LLMs
Large Language Models (LLMs) exhibit impressive performance on complex reasoning tasks, yet they frequently fail on basic numerical problems, producing incorrect outputs. Inspired by Benford's Law -- a statistical pattern where lower digits occur more frequently as leading digits -- we hypothesize that the long-tailed digit distributions in web-collected corpora may be learned by LLMs during pretraining, leading to biased numerical generation. To investigate the hypothesis, we first examine whe…

The Anatomy of India's Industrial Interdependencies: 7-Digit Product-Level Supply-Use and Input-Output Tables from ASI Data (2016-2022) with a Case Study of the Mobile Phone Sector
Sourish Dutta
https://arxiv.org/abs/2506.13936

The Anatomy of India's Industrial Interdependencies: 7-Digit Product-Level Supply-Use and Input-Output Tables from ASI Data (2016-2022) with a Case Study of the Mobile Phone Sector
Outlines the construction of high-resolution, 7-digit product-level Supply-Use Tables (SUTs) and symmetric Input-Output Tables (IOTs) for the Indian economy, leveraging microdata from the Annual Survey of Industries (ASI) for the period 2016-2022. We delineate a robust methodology encompassing the generation of detailed input and output flows, with a particular focus on the reconciliation of data from registered and unregistered manufacturing sectors through a meticulously developed NPCMS-NIC c…

Exhaustive Generation of Pattern-Avoiding s-Words
Samuel Buick, Madeleine Goertz, Amos Lastmann, Kunal Pal, Helen Qian, Sam Tacheny, Aaron Williams, Leah Williams, Yulin Zhai
https://arxiv.org/abs/2508.16039

Exhaustive Generation of Pattern-Avoiding s-Words
The most well-known Gray code of permutations is plain changes. It was discovered in the 1600s by bell-ringers who wished to order the permutations of [n] by swaps (e.g., 123, 132, 312, 321, 231, 213 for n = 3). In other words, plain changes traces a Hamilton path in the permutohedron. In 2013 it was shown that plain changes can be generated by a greedy algorithm: swap the largest value. Algorithm J replaces the swap operation with the jump operation (which moves a larger digit past one or more…

High-Performance Pipelined NTT Accelerators with Homogeneous Digit-Serial Modulo Arithmetic
George Alexakis, Dimitrios Schoinianakis, Giorgos Dimitrakopoulos
https://arxiv.org/abs/2507.12418

High-Performance Pipelined NTT Accelerators with Homogeneous Digit-Serial Modulo Arithmetic
The Number Theoretic Transform (NTT) is a fundamental operation in privacy-preserving technologies, particularly within fully homomorphic encryption (FHE). The efficiency of NTT computation directly impacts the overall performance of FHE, making hardware acceleration a critical technology that will enable realistic FHE applications. Custom accelerators, in FPGAs or ASICs, offer significant performance advantages due to their ability to exploit massive parallelism and specialized optimizations. …

They get what they deserve. #Signal is waiting for you with open (source) arms. https://www.tagesschau.de/wirtschaft/digitales/meta-whatsapp-werbung-100.html

Meta kündigt Werbung auf WhatsApp an
Werbung auf WhatsApp: Darüber wurde schon seit Langem spekuliert. Nun soll es tatsächlich vorbei sein mit der Werbefreiheit auf dem Messenger. Das bestätigte der Mutterkonzern Meta.

The AI Action Plan suggests Trump wants to inculcate a "try-first" culture through regulatory sandboxes for AI, which have already popped up in some US states (Aaron Mak/Politico)
https://www.politico.com/newsletters/digit

💨 This 24,000-Watt Scooter Is Going for a 100 MPH Speed Record at Bonneville
https://www.thedrive.com/news/this-24000-watt-scooter-is-going-for-a-100-mph-speed-record-at-bonneville

This $30,000 Electric Scooter Is Gunning for a Bonneville Speed Record
Believe it or not, there's more than one electric scooter that claims it can break triple-digit speeds.

Source: Fox has officially closed its upfront discussions, reaching $2B for core sports properties, excluding the World Cup, for the first time (Saleah Blancaflor/Adweek)
https://www.adweek.com/convergent-tv/fox-wraps-upfront-2-billion-sports-properties/…

Fox Wraps Upfront, Reaches $2 Billion for Core Sports Properties
Fox closed its upfront talks with double-digit volume growth across its portfolio.

I had been happily reading and posting on my phone all morning with single digit battery charge.
Emboldened and reckless, I went to take a screenshot of my phone at 1% battery and it shut down as it was playing the screenshot sound effect.

The fractal Goodstein principle
David Fern\'andez-Duque, Andreas Weiermann
https://arxiv.org/abs/2508.14768 https://arxiv.org/pdf/2508.14768

The fractal Goodstein principle
The original Goodstein process is based on writing numbers in hereditary $b$-exponential normal form: that is, each number $n$ is written in some base $b\geq 2$ as $n=b^ea+r$, with $e$ and $r$ iteratively being written in hereditary $b$-exponential normal form. We define a new process which generalises the original by writing expressions in terms of a hierarchy of bases $B$, instead of a single base $b$. In particular, the `digit' $a$ may itself be written with respect to a smaller base $b'$. W…

There are no working quantum computers capable of factoring a 3-digit number. Last I heard, the record was the number 21.

That means there is no “production” environment capable of testing the critical feature. Beyond that, PQC deployment in the wild is new/small enough that we don’t even have a lot of assurance that the implementations live up to theory.

Tactile Beyond Pixels: Multisensory Touch Representations for Robot Manipulation
Carolina Higuera, Akash Sharma, Taosha Fan, Chaithanya Krishna Bodduluri, Byron Boots, Michael Kaess, Mike Lambeta, Tingfan Wu, Zixi Liu, Francois Robert Hogan, Mustafa Mukadam
https://arxiv.org/abs/2506.14754…

Tactile Beyond Pixels: Multisensory Touch Representations for Robot Manipulation
We present Sparsh-X, the first multisensory touch representations across four tactile modalities: image, audio, motion, and pressure. Trained on ~1M contact-rich interactions collected with the Digit 360 sensor, Sparsh-X captures complementary touch signals at diverse temporal and spatial scales. By leveraging self-supervised learning, Sparsh-X fuses these modalities into a unified representation that captures physical properties useful for robot manipulation tasks. We study how to effectively …

Did it turn out that the railway pass bought in KD ticket office doesn't work with the barcode scanners used in KW? Perhaps.
Did I immediately start scanning and analyzing the barcodes? Of course.
Well, KD tickets have two codes: a QR code with binary mash encoded in base64 (unnecessarily), which unfortunately I don't have time to analyze, and a code128 barcode with the ticket series and number, encoded as:
AA000000000 (0)
On the other hand, KW/PR have only code128 with the series and number, but without the digit in parentheses (perhaps a checksum?), even though it is printed on the pass:
AA000000000
#rail

My head hurts. At least they are trying.
=======
Password does not conform to policy. The current policy requirements are:
Must be at least 14 character(s) long
Must have at least 1 upper case letter(s)
Must have at least 1 lower case letter(s)
Must have at least 1 digit(s)
Must have at least 1 special character(s)
The only special characters allowed are ~@#$%^&*()-_= >?/\;:[]{}!,.
Must not match any of the last 24 password(s)

This is the header from a german book seller's website. Notice, that the sorta *wishlist* heart icon badge maxes out at single digit 9 !
Is that the real world, where people only have like nine items that they save? Is this just for “design aesthetics”? My wishlist contains 107!

Crosslisted article(s) found for stat.ME. https://arxiv.org/list/stat.ME/new
[1/1]:
- Structural Foundations for Leading Digit Laws: Beyond Probabilistic Mixtures
Vladimir Berman

NFC North betting preview: Who gets to double-digit wins? Trust Bears hype? https://www.espn.com/espn/betting/story/_/id/46017611/2025-nfc-north-division-betting-nfl-preview-favorites-odds-standings-…

Day 15 of 49ers Training Camp: Big Plays, High Heat in Joint Practice vs. Raiders https://www.49ers.com/news/day-15-of-49ers-training-camp-big-plays-high-heat-in-joint-practice-vs-raiders

Radius-zero Extended Symmetries and Irregular Fibres of $\mathbb{Z}^d$-Substitution Subshifts
\'Alvaro Bustos-Gajardo, Daniel Luz, Neil Ma\~nibo
https://arxiv.org/abs/2506.13392

Radius-zero Extended Symmetries and Irregular Fibres of $\mathbb{Z}^d$-Substitution Subshifts
In this work, we consider $\mathbb{Z}^d$-shifts generated by digit substitutions. For such a shift $\mathbb{X}$, we study the elements of the normaliser of $\mathbb{Z}^d$ in the group of self homeomorphisms (called extended symmetries) whose local maps guaranteed by the generalised Curtis--Hedlund--Lyndon theorem have radius-zero. Using the formalism of minimal sets developed by Lemańczyk, Müllner and Yassawi, we provide an algorithm to compute elements of $\mathcal{N}(\mathbb{X})$ that prese…

Crosslisted article(s) found for math.ST. https://arxiv.org/list/math.ST/new
[1/1]:
- Structural Foundations for Leading Digit Laws: Beyond Probabilistic Mixtures
Vladimir Berman

Benford behavior resulting from stick and box fragmentation processes
Bruce Fang, Steven J. Miller
https://arxiv.org/abs/2508.12915 https://arxiv.org/pdf/2…

Benford behavior resulting from stick and box fragmentation processes
Benford's law is the statement that in many real world data sets, the probability of having digit $d$ in base $B$, where $1 \leq d \leq B$, as the first digit is \log_{B}\!\left(\frac{d+1}{d}\right). We sometimes refer to this as weak Benford behavior, and we say that a data set exhibits strong Benford behavior in base $B$ if the probability of having significand at most s, where $1 \leq s < B$, is \log_{B}\!\left(s\right). We examine Benford behaviors in two different probabilistic models: sti…

Revisiting H\"otte (2025): A Companion Analysis with Extended Evidence from UK Inter-Industry Payment Data, 2017-2024
Kerstin H\"otte
https://arxiv.org/abs/2508.17695 …

Revisiting Hötte (2025): A Companion Analysis with Extended Evidence from UK Inter-Industry Payment Data, 2017-2024
In 2025, the UK Office for National Statistics released a novel dataset of monthly inter-industry payment flows during January 2017 to November 2024 at the 5-digit SIC level (ONS, 2025a), covering $>$3.1 million UK organizations. Annual aggregates amount to 490 million transactions with an aggregate value of over £3.1 trillion in 2023. Such publicly available data are unprecedented by their granularity and timeliness, providing a rich basis for economic research and real-time policy advice. H�…

SFATTI: Spiking FPGA Accelerator for Temporal Task-driven Inference -- A Case Study on MNIST
Alessio Caviglia, Filippo Marostica, Alessio Carpegna, Alessandro Savino, Stefano Di Carlo
https://arxiv.org/abs/2507.10561

SFATTI: Spiking FPGA Accelerator for Temporal Task-driven Inference -- A Case Study on MNIST
Hardware accelerators are essential for achieving low-latency, energy-efficient inference in edge applications like image recognition. Spiking Neural Networks (SNNs) are particularly promising due to their event-driven and temporally sparse nature, making them well-suited for low-power Field Programmable Gate Array (FPGA)-based deployment. This paper explores using the open-source Spiker+ framework to generate optimized SNNs accelerators for handwritten digit recognition on the MNIST dataset. S…

Probing for Arithmetic Errors in Language Models
Yucheng Sun, Alessandro Stolfo, Mrinmaya Sachan
https://arxiv.org/abs/2507.12379 https://

Probing for Arithmetic Errors in Language Models
We investigate whether internal activations in language models can be used to detect arithmetic errors. Starting with a controlled setting of 3-digit addition, we show that simple probes can accurately decode both the model's predicted output and the correct answer from hidden states, regardless of whether the model's output is correct. Building on this, we train lightweight error detectors that predict model correctness with over 90% accuracy. We then extend our analysis to structured chain-of…

"UK rooftop solar installations surge, new battery storage record set"
#UK #UnitedKigdom #Solar #SolarPower

UK rooftop solar installations surge, new battery storage record set
Latest MCS data for small-scale renewables reveal a strong month for solar installations up to 50 kW, double-digit growth for battery energy storage installations and a milestone reached despite a muted month for heat pump installations.

The Measurement Imbalance in Agentic AI Evaluation Undermines Industry Productivity Claims
Kiana Jafari Meimandi, Gabriela Ar\'anguiz-Dias, Grace Ra Kim, Lana Saadeddin, Mykel J. Kochenderfer
https://arxiv.org/abs/2506.02064

The Measurement Imbalance in Agentic AI Evaluation Undermines Industry Productivity Claims
As industry reports claim agentic AI systems deliver double-digit productivity gains and multi-trillion dollar economic potential, the validity of these claims has become critical for investment decisions, regulatory policy, and responsible technology adoption. However, this paper demonstrates that current evaluation practices for agentic AI systems exhibit a systemic imbalance that calls into question prevailing industry productivity claims. Our systematic review of 84 papers (2023--2025) reve…

Decomposition of real numbers into sums of L\"uroth sets
Maiken Gravgaard, Ying Wai Lee
https://arxiv.org/abs/2506.12513 https://

Decomposition of real numbers into sums of Lüroth sets
We study the decomposition of real numbers into sums of Lüroth sets, which are defined by numbers whose Lüroth expansions have prescribed digit constraints. We establish several results on the congruence modulo 1 of sums of Lüroth sets, including summands with digits bounded above, below, and combinations of the two. We also analyse the Hausdorff dimension of Lüroth sets and their sums. The results extend classical findings on continued fractions to Lüroth expansions.

CXL-GPU: Pushing GPU Memory Boundaries with the Integration of CXL Technologies
Donghyun Gouk, Seungkwan Kang, Seungjun Lee, Jiseon Kim, Kyungkuk Nam, Eojin Ryu, Sangwon Lee, Dongpyung Kim, Junhyeok Jang, Hanyeoreum Bae, Myoungsoo Jung
https://arxiv.org/abs/2506.15601

CXL-GPU: Pushing GPU Memory Boundaries with the Integration of CXL Technologies
This work introduces a GPU storage expansion solution utilizing CXL, featuring a novel GPU system design with multiple CXL root ports for integrating diverse storage media (DRAMs and/or SSDs). We developed and siliconized a custom CXL controller integrated at the hardware RTL level, achieving two-digit nanosecond roundtrip latency, the first in the field. This study also includes speculative read and deterministic store mechanisms to efficiently manage read and write operations to hide the endp…

Chiefs WR Rashee Rice could still be suspended prior to start of regular season https://www.nfl.com/news/chiefs-wr-rashee-rice-could-still-be-suspended-prior-to-start-of-regular-season

Sources: Elon Musk is selling $5B in debt for xAI at a double-digit interest rate, with demand already exceeding $3.5B (Bloomberg)
https://www.bloomberg.com/news/articles/2025-06-02/musk-s-xai-launches-5-billion-deb…

Sufficient digits and density estimation: A Bayesian nonparametric approach using generalized finite P\'olya trees
Mario Beraha, Jesper M{\o}ller
https://arxiv.org/abs/2506.09437

Sufficient digits and density estimation: A Bayesian nonparametric approach using generalized finite Pólya trees
This paper proposes a novel approach for statistical modelling of a continuous random variable $X$ on $[0, 1)$, based on its digit representation $X=.X_1X_2\ldots$. In general, $X$ can be coupled with a random variable $N$ so that if a prior of $N$ is imposed, $(X_1,\ldots,X_N)$ becomes a sufficient statistics and $.X_{N+1}X_{N+2}\ldots$ is uniformly distributed. In line with this fact, and focusing on binary digits for simplicity, we propose a family of generalized finite P{ó}lya trees that i…

Purmutation--invariant Niven numbers
Hui-Ling Wu, S. Y. Lou
https://arxiv.org/abs/2508.01611 https://arxiv.org/pdf/2508.01611

Purmutation--invariant Niven numbers
This paper introduces permutation-invariant Niven numbers--a novel class of Niven numbers where all digit permutations (with leading zeros automatically ignored) must retain the Niven property. We demonstrate that there exist infinitely many such numbers and that their magnitude is unbounded. Furthermore, we present an exhaustive search method for identifying permutation--invariant Niven numbers.

Solving Sudoku Using Oscillatory Neural Networks
Stefan Porfir, Bram F. Haverkort, Federico Sbravati, Aida Todri-Sanial
https://arxiv.org/abs/2508.02250 https://

Solving Sudoku Using Oscillatory Neural Networks
This paper explores the application of Oscillatory Neural Networks (ONNs) to solving Sudoku puzzles, presenting a biologically inspired approach based on phase synchronization. Each cell is represented by an oscillator whose phase encodes a digit, and the synchronization is governed by the Kuramoto model. The system dynamically evolves towards a valid solution by having the puzzle constraints encoded into the weight matrix of the network, and through a proposed novel phase mapping of the Sudoku…

How AppLovin, which lets advertisers use IP addresses to track users, began eroding Google's ad market share and pushed Google to loosen its privacy stance (Catherine Perloff/The Information)
https://www.theinformation.com/articles/applovin-scared-google

How AppLovin Scared Google
In 2023, Google ad executives noticed something alarming about one aspect of their business. Once-red hot revenue growth from selling ads promoting mobile apps, a business worth double-digit billions of dollars for Google, had slowed, a former employee said. When staffers discussed what was ...

Reason to Rote: Rethinking Memorization in Reasoning
Yupei Du, Philipp Mondorf, Silvia Casola, Yuekun Yao, Robert Litschko, Barbara Plank
https://arxiv.org/abs/2507.04782

Reason to Rote: Rethinking Memorization in Reasoning
Large language models readily memorize arbitrary training instances, such as label noise, yet they perform strikingly well on reasoning tasks. In this work, we investigate how language models memorize label noise, and why such memorization in many cases does not heavily affect generalizable reasoning capabilities. Using two controllable synthetic reasoning datasets with noisy labels, four-digit addition (FDA) and two-hop relational reasoning (THR), we discover a reliance of memorization on gene…

$p$-adic moments of $L$-functions
Daniel Kriz, Asbj{\o}rn Christian Nordentoft
https://arxiv.org/abs/2507.01836 https://arxiv.org/pdf…

$p$-adic moments of $L$-functions
We obtain a formula for the $p$-adic valuation of weighted moments of central $L$-values of holomorphic cusp forms twisted by Dirichlet characters of order $p$. In some cases we give an arithmetic interpretation of the constants in the formula. The result is obtained via the study of the digit map, turning a horizontal $p$-adic measure into a vertical one, applied to the horizontal $p$-adic $L$-functions as defined by the authors in previous work.