{ "counts": { "adjacent": 16, "core": 3, "errors": 0, "negative": 70, "total": 89 }, "date": "2026-09-19", "errors": [], "fresh_content_days": 21, "generated_at": "2026-09-19T18:10:40Z", "items": [ { "age_days": 4, "arxiv_id": "2609.16991", "authors": [ "Xin Quan", "Reto Gubelmann", "André Freitas" ], "content_date": "2026-09-15", "freshness": "fresh", "id": "arxiv:2609.16991", "kind": "paper", "label": "core", "matched_signals": [ "autoformalization", "verifier_guided_reasoning" ], "published": "2026-09-15", "score": 9.5, "source": "arxiv-ai4math-core", "summary": "Natural language arguments are compelling before they are formally explicit. A premise supports a claim through defeasible warrants, background commitments, and exception conditions that the text leaves implicit. However, formal verification requires the opposite. Making such arguments machine-checkable requires constructing the missing commitments, not only translating given sentences into logic. Construction, however, carries a risk that translation does not: a system free to add premises can make any claim provable, and a formally valid proof may assert the claim outright, prove it without the original premise, or establish more than the claim itself. We address this problem by formulating autoformalization for argumentative material inference as guard completion, in which non-monotonic material support is turned into monotonic formal inference relative to an explicitly constructed guard set. A completion is accepted only when its proof both passes the theorem prover and survives contrastive tests of premise dependence and claim selectivity. We implement this formulation in GUARD, a neuro-symbolic framework in which LLMs construct and formalize candidate guards, Isabelle/HOL verifies the resulting theories and returns step-level feedback for iterative refinement, and the system abstains when no faithful completion can be reached. Our empirical results on Debatepedia and ARCT using different LLMs demonstrate that GUARD yields significant improvements in verified-faithful (+35.3, +32.9 points) and substantial reductions in leakage (-25.9, -21.9 points) over the state-of-the-art LLM-driven theorem proving approach. Moreover, we show that the symbolic soft critique and the explicit assumption layer account for most of these gains, with the soft critique also improving the initial validity of the elicited context and reducing the number of iterations required for successful verification.", "title": "Autoformalizing Argumentative Material Inferences", "updated": "2026-09-15", "url": "https://arxiv.org/abs/2609.16991" }, { "age_days": 2, "arxiv_id": "2609.19814", "authors": [ "Sirui Lu", "Ruixuan Deng", "Yanqiao Zhu", "Zhengfeng Ji" ], "content_date": "2026-09-17", "freshness": "fresh", "id": "arxiv:2609.19814", "kind": "paper", "label": "core", "matched_signals": [ "autoformalization", "lean_formal_proving_agents" ], "published": "2026-09-17", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Landmark mathematical formalizations have taken specialist teams years to complete. We present FormalFlow, a system that coordinates AI proving agents under human supervision to address statement drift and proof composition in long-horizon formalization. Drawing on software engineering principles and practices, it uses a shared blueprint to guide nested planning, proving and review loops. Agents strengthen verification and review throughout formalization. We completed a machine-checked Lean 4 proof of the quantum soundness of the classical low individual-degree test, a core theorem underlying MIP* = RE. Developing the proof took 63 days; greater parallelism could further reduce this time. The final library contains 126,367 lines of Lean code, all generated by agents. The formalization corrects side conditions and intermediate errors while preserving the published final error bound under corrected assumptions. This work provides a verified foundation for quantum complexity and demonstrates a route to affordable verification of major research proofs by small teams.", "title": "Long-horizon autoformalization of a core theorem underlying MIP* = RE", "updated": "2026-09-17", "url": "https://arxiv.org/abs/2609.19814" }, { "age_days": 3, "arxiv_id": "2609.19352", "authors": [ "Robert Joseph George", "Will Adkisson", "Anima Anandkumar" ], "content_date": "2026-09-16", "freshness": "fresh", "id": "arxiv:2609.19352", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "mathlib_retrieval" ], "published": "2026-09-16", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "We present FloatLib, a verified arbitrary-precision floating-point arithmetic library in Lean 4 that combines broad format coverage, machine-checked correctness, and efficient certified execution. To our knowledge, FloatLib is the first Lean library to unify IEEE binary and decimal arithmetic, arbitrary-width posits, P3109, and user-defined formats and rounding rules behind interchangeable certified software backends. Every certified backend is proved equal to a complete encoded specification, preserving signed zeros and exceptional values, while numerical theorems connect execution to real rounding, error bounds, and exactness. FloatLib combines exhaustive certified tables for small formats with verified word and limb kernels based on guard-and-sticky invariants and independently checked quotient candidates. Its posit development additionally proves standard rounding thresholds and exact quire accumulation within capacity for arbitrary widths. Across matched workloads, FloatLib achieves speedups of up to 1.46x over FLoPS and 116x over Universal, while remaining slower in some regimes such as binary arithmetic against MPFR. Independent conformance testing includes more than 102 million TestFloat evaluations with zero differences under the tested relation. We release the library, proofs, benchmarks, evaluation data, and guide as open source.", "title": "FloatLib: Verified Floating-Point Arithmetic in Lean", "updated": "2026-09-16", "url": "https://arxiv.org/abs/2609.19352" }, { "age_days": 3, "arxiv_id": "2609.19126", "authors": [ "Teng Zhang" ], "content_date": "2026-09-16", "freshness": "fresh", "id": "arxiv:2609.19126", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-09-16", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "Very recently, Lech Mazur proved the celebrated Sendov conjecture, and Terence Tao subsequently distilled the main ideas of the proof in a blog post. In this paper, we establish a quantitative strengthening of Sendov's conjecture, namely the quadratic Tang--Zhang inequality. Let $p$ be a polynomial of degree $n\\ge2$ whose zeros lie in the closed unit disk, and let $ζ_1,\\ldots,ζ_{n-1}$ denote its critical points, counted with multiplicity. We prove that, for every zero $a$ of $p$, $$ \\sum_{j=1}^{n-1}\\frac{1}{|a-ζ_j|^2}\\ge n-1. $$ Moreover, equality holds if and only if $p(z)=c(z^n-ω)$ for some $c\\in\\mathbb C\\setminus\\{0\\}$ and $|ω|=1$. We also provide a Lean 4 formalization of the main results.", "title": "Beyond Sendov's conjecture: the quadratic Tang--Zhang inequality", "updated": "2026-09-16", "url": "https://arxiv.org/abs/2609.19126" }, { "age_days": 4, "arxiv_id": "2609.19195", "authors": [ "Cameron Beeley" ], "content_date": "2026-09-15", "freshness": "fresh", "id": "arxiv:2609.19195", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-09-15", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "For a finite simple graph G, let alpha(G) denote its independence number and let l_avg(G) = (1 / |V(G)|) sum_{v in V(G)} alpha(G[N_G(v)]) be the average independence number of its open neighbourhoods. Written on the Wall II Conjecture 194 asserts that every simple connected graph on n > 1 vertices satisfying alpha(G) <= 1 + l_avg(G) has a Hamiltonian path. We give a four-parameter family of counterexamples. Its principal two-parameter subfamily satisfies the proposed inequality with equality: for every pair of integers s >= 1 and t >= 3 it has (s + 1)t^2 vertices, independence number t + 1, l_avg(G) = t, and minimum degree s, but has no Hamiltonian path. This entire infinite subfamily is machine-checked in Lean 4: one universally quantified theorem certifies its order, connectivity, independence number, average neighbourhood independence, minimum degree, conjecture hypothesis, and failure of traceability. Thus no fixed lower bound on the minimum degree repairs the conjecture. The case (s,t) = (1,3) has 18 vertices, but the formal certificate is parametric rather than a verification of that one graph alone.", "title": "Lean-Certified Infinite Counterexamples to Written on the Wall II Conjecture 194", "updated": "2026-09-15", "url": "https://arxiv.org/abs/2609.19195" }, { "age_days": 2, "arxiv_id": "2609.20492", "authors": [ "Maseeh Ghodsi" ], "content_date": "2026-09-17", "freshness": "fresh", "id": "arxiv:2609.20492", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-17", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "A Leech tree of order $n$ is a tree with positive integral edge weights whose $n(n-1)/2$ pairwise weighted distances are precisely $1,2,\\ldots,n(n-1)/2$. This paper gives a computer-assisted proof that no Leech tree of order $18$ exists. The argument has three layers. First, a development in Lean 4 verifies the structural facts used in the paper. These facts reduce every putative example to one of eight local configurations and justify several necessary conditions. Second, conventional mathematical arguments prove a component-pair whole-block exact-cover condition and the completeness of a recursive search. Third, exhaustive computations close all eight configurations. The computation records exact coverage, source and input hashes, terminal receipts, and checked exact-zero results. The structural layer is kernel-checked, but the search program, its execution, and the certificate checker have not been formalized in Lean. The result is therefore a computer-assisted proof, not an end-to-end Lean proof.", "title": "Nonexistence of a Leech Tree of Order 18: A Computer-Assisted Proof", "updated": "2026-09-17", "url": "https://arxiv.org/abs/2609.20492" }, { "age_days": 2, "arxiv_id": "2609.19583", "authors": [ "Chunhao Liao", "Hongxu Xu", "Xintong Zhou", "Yizhou Zhang", "Chengnian Sun" ], "content_date": "2026-09-17", "freshness": "fresh", "id": "arxiv:2609.19583", "kind": "paper", "label": "adjacent", "matched_signals": [ "verifier_guided_reasoning" ], "published": "2026-09-17", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "LLVM is the cornerstone of modern compilers, but its subtle intermediate representation (IR) semantics make transformations error-prone and necessitate formal verification. Alive2, a state-of-the-art translation validator based on satisfiability modulo theories, has achieved substantial success in automating the validation of LLVM transformations. However, it still faces scalability limitations, does not support symbolic bitwidths, and offers only bounded guarantees for loops. In contrast, interactive theorem provers such as Lean can address these cases but require substantial proof engineering. In this paper, we present Trivet, a framework combining large language models (LLMs) and Lean for automated translation validation of LLVM transformations. Trivet generates structured proof scaffolds based on source and target functions, automatically discharges obligations amenable to deterministic reasoning, and delegates transformationspecific obligations to LLMs. It produces refinement proofs or counterexample-based refutations, with every successful verdict checked by the Lean kernel. On 148 LLVM transformations, Trivet verifies or refutes 147, leaving one invalid case unresolved. Successful cases include 60 loop-free transformations with symbolic bitwidths, 27 cases from a restricted class of loop-containing transformations, and 10 complex valid fixed-bitwidth cases on which Alive2 times out. Compared with an unscaffolded baseline, scaffolding enables 26 additional proofs. On cases solved by both configurations, it reduces mean proof time by 75.9% and mean monetary cost by 88%.", "title": "LLVM Translation Validation Automated with Large Language Models and Lean", "updated": "2026-09-17", "url": "https://arxiv.org/abs/2609.19583" }, { "age_days": 3, "arxiv_id": "2609.18261", "authors": [ "Pierre Senellart", "Anton Gnatenko" ], "content_date": "2026-09-16", "freshness": "fresh", "id": "arxiv:2609.18261", "kind": "paper", "label": "adjacent", "matched_signals": [ "mathlib_retrieval" ], "published": "2026-09-16", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We show that descriptive complexity can serve as a foundation for formalizing computational complexity results in a proof assistant, by constructing a Lean library centered around the following concepts: decision problems are isomorphism-invariant predicates on finite structures; complexity classes are defined by their logical characterization; membership is shown by definability witnesses; hardness is shown by first-order reductions from a known hard problem. We also establish bridges to traditional machine models such as (non)deterministic Turing machines. The library proves 73 completeness results, on 68 problems or problem families, over 14 different classes; relations between the classes established inside the logic and not by machine simulation, among them NL = coNL and the Abiteboul-Vianu theorem; and unconditional lower bounds, among them $\\mathrm{FO}(\\leq) \\subsetneq \\mathrm{FO}(\\leq, \\mathrm{TC})$ and the failure of order-free FO(IFP) to capture PTIME.", "title": "Descriptive Complexity in Lean: Completeness by First-Order Reductions", "updated": "2026-09-16", "url": "https://arxiv.org/abs/2609.18261" }, { "age_days": 3, "arxiv_id": "2609.19214", "authors": [ "Ioannis Tsiokos" ], "content_date": "2026-09-16", "freshness": "fresh", "id": "arxiv:2609.19214", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-16", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Socolar and Taylor asked for a single, simply connected three-dimensional prototile that forces nonperiodicity by shape alone, admitting no weakly nonperiodic tiling; the Schmitt-Conway-Danzer biprism and the three-dimensional Socolar-Taylor tile admit screw motions or a periodic stacking direction. We exhibit a rational polyhedral $3$-ball $Q$, which we call Chair44 (R44): a seven-cube chair whose $24$ exposed unit panels carry tiny square-pyramid features, and prove, as a proof submission, that $Q$ admits tilings of $\\mathbb{R}^3$ by congruent copies, reflections allowed, and that every such tiling has no translational period and a symmetry group of order at most $24$; every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles. The solid was designed to a reading of the aperiodic-monotile phenomenon reached with the Six Birds emergence calculus (Section 3.3), and the construction turns on a single finite test, checked by machine: the tile's own contact rule survives coarsening, so that the decoded parent tiling obeys the tile's rule and no other. The proof combines a written geometric argument, that the features force every tiling onto a registered lattice, with exhaustive finite enumerations; the companion census is replayed by two independent implementations, every finite gate is kernel-checked in Lean 4 (modulo a named compiler hook per native_decide theorem), and the written geometric lemmas and the logical assembly are Lean theorems as well, so that the theorem is kernel-checked modulo the named compiler hooks; the written proofs remain as exposition.", "title": "A Strongly Aperiodic Monotile in Three Dimensions", "updated": "2026-09-16", "url": "https://arxiv.org/abs/2609.19214" }, { "age_days": 4, "arxiv_id": "2609.17027", "authors": [ "Junjie Liao" ], "content_date": "2026-09-15", "freshness": "fresh", "id": "arxiv:2609.17027", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-15", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Let $F=F(X)$ be a free group of finite rank, with palindromic length taken with respect to the fixed basis $X$. We embed $F$ as the index-two subgroup of the universal Coxeter group $W=F\\rtimes_θ\\langle t\\mid t^2=1\\rangle$, where $θ(x)=x^{-1}$ for $x\\in X$, and prove $\\mathrm{pl}(g)=\\min{\\ell_T(g),\\ell_T(gt)}$. Dyer's deletion theorem then identifies reflection length with the minimum number of unmatched positions in a noncrossing equal-label partial matching on a reduced Coxeter word. This gives an $O(n^3)$-time, $O(n^2)$-space algorithm for palindromic length, together with recovery of an optimal palindromic factorization. The matching model also gives a structural characterization. For every ordered full binary tree with $k$ leaves we define a literal word template whose leaves are palindromes and whose internal vertices carry arbitrary words. A reduced word $w$ represents an element of palindromic length at most $k$ if and only if $w$ is a literal instance of one of these templates. Hence the $C_{k-1}$ ordered binary-tree shapes give a complete finite family for each fixed $k$. For $k=4$ the five templates are exactly the five forms proposed by Frid, proving the completeness of that list. A companion Lean 4 development verifies the four-palindrome classification end to end for every finite rank, including the ordinary reduced-word formulation and the literal five-form conclusion.", "title": "Palindromic Length in Free Groups: Reflections, Noncrossing Matchings, and Catalan Forms", "updated": "2026-09-15", "url": "https://arxiv.org/abs/2609.17027" }, { "age_days": 4, "arxiv_id": "2609.17283", "authors": [ "Antonio Acuaviva" ], "content_date": "2026-09-15", "freshness": "fresh", "id": "arxiv:2609.17283", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-15", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We construct two complemented subspaces of $L_1[0,1]$. The first has the Schur property but fails the Radon--Nikodým property. The second contains a copy of $\\ell_2$ but no copy of $L_1[0,1]$. Neither space is isomorphic to a Banach lattice. This gives a negative answer to the complemented-subspace question of Lindenstrauss and Rosenthal. A Lean 4 formalisation of the main results accompanies the paper.", "title": "On complemented subspaces of $L_1[0,1]$", "updated": "2026-09-15", "url": "https://arxiv.org/abs/2609.17283" }, { "age_days": 4, "arxiv_id": "2609.16801", "authors": [ "Jacob Bedrossian" ], "content_date": "2026-09-15", "freshness": "fresh", "id": "arxiv:2609.16801", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-15", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We present a formalization in Lean 4 of Mouhot and Villani's theorem on nonlinear Landau damping in $\\mathbb T^d$ in all Gevrey regularity indices $s > 1/3$ for small backgrounds.", "title": "Formalization of Landau damping in the Vlasov--Poisson equations in Lean", "updated": "2026-09-15", "url": "https://arxiv.org/abs/2609.16801" }, { "age_days": 2, "arxiv_id": "2609.20567", "authors": [ "Yifeng Huang", "Kenny Lau", "Ken Ono" ], "content_date": "2026-09-17", "freshness": "fresh", "id": "arxiv:2609.20567", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning" ], "published": "2026-09-17", "score": 1.9, "source": "arxiv-ai4math-core", "summary": "We prove the conjecture of Huang, Jiang, and Oblomkov (HJO) giving a geometric extension of the Rogers--Ramanujan and Andrews--Gordon identities for every torus-knot singularity $X^a=Y^b$ with coprime $1 \\frac12$ for the existence of uniformly discrete uniqueness sets for spectra with periodic weak gaps, considered by Olevskii and Ulanovskii, is sharp. Our findings admit extensions to higher dimensions and are verified in Lean.", "title": "Universal completeness of exponentials", "updated": "2026-09-17", "url": "https://arxiv.org/abs/2609.20805" }, { "age_days": 2, "arxiv_id": "2609.20789", "authors": [ "Zachary Robertson" ], "content_date": "2026-09-17", "freshness": "fresh", "id": "arxiv:2609.20789", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-17", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "This article introduces mutual evaluation of a replicable task worker and a critic that incentivizes truthful reporting, both modeled as strategic agents. The critic chooses a finite-valued rule that induces an evaluation score on joint report laws. Their common payoff is analyzed through regret relative to the unrestricted critic envelope. The critic rule is distinct from the evaluation score. This class enables a peer-free information elicitation mechanism using conditionally independent replications of a worker on the same task. This replication-loop mechanism implements a type-agreement payoff using same-task replications and new-task samples. In contrast to the peer-prediction and scoring-rule literature, implementations are shown that produce unbiased Pearson and Shannon information scores without requiring peers, a ground-truth reference, or likelihood-ratio estimation. A valid binary critic also can be represented by shared finite type annotations of worker returns. One runtime restriction is that the number of required replicas is random and can depend on the critic rule. Other timing effects, such as commitment and reoptimization, yield distinct incentives, connecting the framework to variational peer prediction. This mechanism class illustrates why strategic considerations matter for both critic and worker agents.", "title": "Mutual Evaluation and Supervision without Peers", "updated": "2026-09-17", "url": "https://arxiv.org/abs/2609.20789" }, { "age_days": 2, "arxiv_id": "2609.20005", "authors": [ "Maxim Chupilkin" ], "content_date": "2026-09-17", "freshness": "fresh", "id": "arxiv:2609.20005", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-17", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "People increasingly turn to AI chatbots for news and explanations of world events. But do they receive the same political answers when they ask in different languages? Here we show that the language of a question can change how the same AI systems assess the war in Ukraine. We ask GPT, Claude and Gemini to evaluate twenty statements about the war in 112 languages, collecting 67,200 responses. The balance between Russia-leaning and Ukraine-leaning responses differs across languages. When we group responses by countries' official languages, they follow a pattern resembling worldwide political divisions: relatively more Russia-leaning answers correspond to more favourable public views of Russia, less support for Ukraine in United Nations votes, and less aid to Ukraine. The broad pattern recurs across all three models and remains when individual statement pairs are removed. Our findings suggest a possible route through which information warfare may shape the text used to train AI models, which may in turn spread geopolitical biases.", "title": "Geopolitical Divisions Across Languages in Large Language Models", "updated": "2026-09-17", "url": "https://arxiv.org/abs/2609.20005" }, { "age_days": 2, "arxiv_id": "2609.19740", "authors": [ "Najiyya Younas", "Omar Abdulkader", "Yaser Ali Shah", "Muhammad Jawad Ikram", "Jebran Khan", "Amaad Khalil" ], "content_date": "2026-09-17", "freshness": "fresh", "id": "arxiv:2609.19740", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-17", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Recent innovations in deep learning have significantly enhanced the diagnosis of medical images, although they are based on the use of centralized data storage that pose severe threats to patient privacy and medical data security. To address this issue, this research proposes a Federated Learning (FL) model that is coupled with an optimized YOLOv8 network to detect the kidney stones on a computed tomography (CT) image and at the same time, protect privacy of the patients. The suggested system can help various medical organizations to jointly train a common model without exchanging the information about the patients. This is to ensure that data protection laws like GDPR and HIPAA are adhered to. The residual feature fusion and DropBlock regularization among other architectural improvements are also included in YOLOv8 to enhance detection robustness and minimize overfitting. Experimental analysis carried out on a distributed CT dataset demonstrated that the federated YOLOv8 model has a mAP at 50 of 0.733 and is able to keep the data confidential. Moreover, its lean design facilitates fast edge deployment and real-time inference across a clinical setting. Altogether, these findings indicate that Federated Learning is a safe and efficient solution to AI-assisted diagnosis in contemporary healthcare when combined with the use of sophisticated object detection models.", "title": "Federated Learning Framework for Privacy-Preserving Kidney Stone Detection", "updated": "2026-09-17", "url": "https://arxiv.org/abs/2609.19740" }, { "age_days": 3, "arxiv_id": "2609.19234", "authors": [ "Julian Alfredo Mendez", "Timotheus Kampik" ], "content_date": "2026-09-16", "freshness": "fresh", "id": "arxiv:2609.19234", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-16", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "This paper presents the AR fairness metamodel, a framework designed to represent, analyze, and compare different fairness scenarios. The metamodel considers key elements, such as agents, resources, and their attributes, and enables the systematic definition and comparison of various fairness measures. We provide examples involving both discrete and continuous measures, including equality, equity, group fairness, individual fairness, the Gini index, the Theil index, Jain's fairness index, and a detailed fairness measure for Australia's Child Care Subsidy. We also explore relationships among group fairness, individual fairness, and envy-freeness, supported by formal proofs. At the conceptual modeling level, our approach builds on the Tiles framework, which offers modular components that can be connected to capture diverse fairness definitions. The goal is to make AR-based fairness definitions practical and adaptable across contexts, providing a clear way to define, compare, and evaluate them. An implementation of the Tiles framework is available as an open-source tool, and can support fairness modeling and evaluation across a wide range of applications.", "title": "The AR Fairness Metamodel: A Structured Framework for Fairness Measures", "updated": "2026-09-16", "url": "https://arxiv.org/abs/2609.19234" }, { "age_days": 3, "arxiv_id": "2609.19285", "authors": [ "Marco Tonin", "Michele Tizzoni", "Bruno Lepri", "Esteban Moro" ], "content_date": "2026-09-16", "freshness": "fresh", "id": "arxiv:2609.19285", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-16", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Residential and social sorting along political lines structure Americans' daily interactions, often limiting encounters with opposing viewpoints. Here, we combine high-resolution anonymized mobility data from 11 major U.S. metropolitan areas with precinct-level voting results to quantify and examine individuals' experienced partisan segregation in urban areas. We find that residential context provides a strong baseline for political exposure, but experienced partisan segregation is also shaped by everyday mobility behavior, since individuals living in similar residential environments can experience different levels of partisan segregation depending on their daily routines. In particular, individuals who travel farther encounter more politically diverse environments, especially in highly segregated areas, indicating that mobility range moderates the relationship between residential and experienced segregation, while exploring a wider range of places does not reduce partisan segregation and may even reinforce it. Moreover, place segregation largely reflects political geography, yet categories of points of interest have distinct segregation patterns, with community places and social businesses tending to attract more left-leaning visitors, reflecting visitation patterns rather than local accessibility alone. Our findings show that experienced partisan segregation extends beyond residential geography and is also associated with the mobility behavior and visitation patterns that structure people's daily lives.", "title": "Experienced partisan segregation across patterns of mobility behavior in US cities", "updated": "2026-09-16", "url": "https://arxiv.org/abs/2609.19285" }, { "age_days": 3, "arxiv_id": "2609.19479", "authors": [ "Daniel O. Martínez-Rivillas", "Arthur F. Ramos", "Ruy J. G. B. de Queiroz" ], "content_date": "2026-09-16", "freshness": "fresh", "id": "arxiv:2609.19479", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-16", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We present a typed two-dimensional $λ$-calculus whose equality evidence is \\emph{computational}: a path between two terms is an explicit finite sequence of one-step conversions (the $β$- and $η$-contractions, the congruences, and the structural rules), and every property of paths is proved \\emph{by recursion over that sequence}, with any step as a base case --- in deliberate contrast with Martin-Löf type theory, where identity is generated by reflexivity alone and all properties go through the non-computational $J$-eliminator. The higher structure is imported from the $2β$- and $2η$-conversions of the theory of an arbitrary higher $λ$-model: we obtain 2-dimensional coherence laws, computable naturality of homotopies (via inductive homotopies and their explicit evaluations), a 2-dimensional path type with transport, and a parity invariant that proves the system consistent and \\emph{really intensional}: the $β$- and $η$-contractions are provably distinct evidence, while in the native syntax of Idris (core MLTT) they are identified by definitional equality. Commutative diagrams accompany the main constructions, and the theory is fully formalized in Idris 2. A parallel Lean formalization is published in the Palomar registry \\cite{palomar2026lean}. A philosophical reading closes the paper: constructivism in the BHK sense, proof-relevant intensionality, and the boundary between syntax and semantics, drawn relative to MLTT and HoTT.}", "title": "A Theory of a Two-Dimensional Typed Lambda Calculus", "updated": "2026-09-16", "url": "https://arxiv.org/abs/2609.19479" }, { "age_days": 3, "arxiv_id": "2609.18722", "authors": [ "Shuxing Yang", "Rui Zhao", "Junyao Wu", "Yize Wang", "Wenhao Li", "Fujia Chen", "Taowen Deng", "Shenzhan Hong", "Yaqi Li", "Zichen Li", "Jincheng Mi", "Yuang Pan", "Kaihao Zhu", "Junjie Yang", "Hongsheng Chen", "Yihao Yang" ], "content_date": "2026-09-16", "freshness": "fresh", "id": "arxiv:2609.18722", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-16", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We prove that the tensor rank of $3\\times3$ matrix multiplication over $\\mathbb F_2$ is at least $21$. The structural proof, independently developed by Qiushi Engine, converts occupation constraints on a single tensor factor into algebraic relations coupling all three factors. Certified quotient-rank bounds and finite geometry force any hypothetical $20$-term decomposition to have first-factor matrix-rank profile $(16,1,3)$. The ranks of the corresponding split-flattened summands therefore sum to $27$, exactly the rank of the full split flattening. Equality in rank subadditivity forces their images to form a direct sum; normalization by the inverse flattening then makes the summands pairwise annihilating idempotents. An explicit product identity for matrix multiplication implies that at most one first factor can be invertible, contradicting the three forced by the profile. The same obstruction constrains $22$-term decompositions attaining the split-rank bound. The complete proof, including the finite quotient bounds, is formalized in Lean. The accompanying research trajectory records Qiushi Engine's long-horizon autonomous research, from numerical experiments and quotient constructions to the structural proof.", "title": "A Structural Proof of the Lower Bound 21 for $3\\times3$ Matrix Multiplication over $\\mathbb F_2$", "updated": "2026-09-16", "url": "https://arxiv.org/abs/2609.18722" }, { "age_days": 4, "arxiv_id": "2609.17643", "authors": [ "Saurabh Verma", "Manish Yadav", "Archana Dixit", "Anirudh Pradhan", "M. S. Barak" ], "content_date": "2026-09-15", "freshness": "fresh", "id": "arxiv:2609.17643", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-15", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Late-time cosmic acceleration is conventionally ascribed to a cosmological constant, though $Λ$CDM continues to face several theoretical difficulties that keep alternative gravity models under active consideration. This work examines the Hu-Sawicki $f(R)$ model in a spatially non-flat background, with the modified-gravity parameter $b$ and the curvature density $Ω_k$ both treated as free parameters, constrained using DESI-DR2 BAO, BBN, and four Type Ia supernova compilations -- PantheonPlus, PantheonPlus+SH0ES, Union3, and DESY5yr. Across all four combinations, $H_0$ and $Ω_m$ stay close to their $Λ$CDM values, with a noticeable shift in $H_0$ appearing only for the SH0ES-calibrated dataset. The parameter $b$ departs from zero at better than $2σ$ in three of the four fits, most prominently for DESY5yr, whereas the SH0ES-calibrated combination alone prefers $b<0$. A strong positive correlation among $b$, $H_0$, and $Ω_k$ underlies this shift, suggesting that curvature signatures obtained under $Λ$CDM can be reabsorbed into the modified-gravity sector once this additional freedom is allowed. Statistical model comparison via AIC and BIC gives a mixed picture: AIC leans toward the Hu-Sawicki model in three of the four combinations, while BIC's heavier penalty on the extra parameter favors $Λ$CDM in most cases, and only DESY5yr is preferred under both criteria. These findings show that the Hu-Sawicki $f(R)$ scenario with unconstrained curvature is nonetheless a statistically feasible, if not obviously preferred, alternative to $Λ$CDM, and that curvature restrictions generated inside $Λ$CDM cannot be viewed as independent of the underlying gravity model.", "title": "Hu-Sawicki $f(R)$ Gravity in a Non-Flat Universe: Constraints from DESI-DR2, BBN, and Type Ia Supernovae", "updated": "2026-09-15", "url": "https://arxiv.org/abs/2609.17643" }, { "age_days": 4, "arxiv_id": "2609.16603", "authors": [ "Jeng Wen Joshua Lean", "Ting-Yu Yen", "Wei-Fang Sun", "Simon See", "Hung-Kuo Chu", "Shih-Hsuan Hung" ], "content_date": "2026-09-15", "freshness": "fresh", "id": "arxiv:2609.16603", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-15", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Full-context neural visual geometry is impractical for thousands of images, while sequence-based chunking poorly captures irregular non-local overlap in multi-sequence aerial collections. We present Graph-Guided Neural Visual Geometry for Aerial Registration (G3AR), a graph-guided framework for scalable dense neural geometry. Before local inference, G3AR builds a geometrically verified image-proximity graph that guides bounded overlapping chunks and induces a chunk graph whose maximum spanning tree defines alignment topology. Compatible backbones process chunks independently; shared-image predictions then estimate three-dimensional similarity (Sim(3)) transforms that register local cameras and geometry in a common frame. Across four real aerial scenes, G3AR improves pose error and runtime in matched VGGT- and Pi3-backed comparisons, while its DA3 variant achieves the lowest pose error among evaluated neural-geometry methods.", "title": "G3AR: Graph-Guided Neural Visual Geometry for Scalable Multi-Sequence Aerial Registration", "updated": "2026-09-17", "url": "https://arxiv.org/abs/2609.16603" }, { "age_days": 4, "arxiv_id": "2609.16555", "authors": [ "Nikita Lebedev" ], "content_date": "2026-09-15", "freshness": "fresh", "id": "arxiv:2609.16555", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-15", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We study the augmented Zarankiewicz problem, in which disjoint pairs of cells are added to a binary matrix with no all-one $2\\times2$ submatrix. The pairs must satisfy compatibility conditions, and the objective counts each original occupied cell and each added pair once. We show that starting with a maximum $C_4$-free matrix can lower the final optimum, answering a question of Qi, Cui, and Xu. Let ${z_A}(m,n)$ be the optimum over all $C_4$-free initial matrices, and ${z_L}(m,n)$ the optimum when the initial matrix must have the maximum number of occupied cells. As $n\\to\\infty$ with $n\\le m=o(n^2)$, we prove \\[ {z_A}(m,n)-{z_L}(m,n)\\ge\\left(\\frac1{30}-o(1)\\right)mn \\] and determine the sharp second-order term: \\[ {z_A}(m,n)=\\frac{mn}{3}+\\left(\\frac1{\\sqrt6}+o(1)\\right)n\\sqrt m. \\] An explicit construction gives a separation at $m=n=1893$. We also find a sharp density threshold: when $n\\to\\infty$ and $m/n^2\\to c>0$, the limited density ${z_L}(m,n)/(mn)$ tends to $1/3$ if and only if $c\\ge1/12$. The proofs combine density and stability estimates, combinatorial constructions, and an exact polynomial certificate.", "title": "Density and separation for augmented Zarankiewicz numbers", "updated": "2026-09-15", "url": "https://arxiv.org/abs/2609.16555" } ], "lookback_days": 21, "schema": "ai4math-radar-run-v1", "timezone": "America/Los_Angeles" }