{ "counts": { "adjacent": 15, "core": 1, "errors": 0, "negative": 74, "total": 90 }, "date": "2026-09-10", "errors": [], "fresh_content_days": 21, "generated_at": "2026-09-10T18:38:24Z", "items": [ { "age_days": 2, "arxiv_id": "2609.09264", "authors": [ "Idan Davidovich", "Debargha Ganguly", "Vikash Singh", "Vipin Chaudhary" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "arxiv:2609.09264", "kind": "paper", "label": "core", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-09-08", "score": 7.9, "source": "arxiv-ai4math-core", "summary": "Leading benchmarks for formal theorem proving with large language models are small collections drawn from competition math, such as the IMO and Putnam, that poorly represent field-specific applications. We introduce StochBench, a Lean 4 benchmark of 450 graduate stochastic-processes problems at varying abstraction levels, each paired with its natural-language source. Addressing a field underrepresented in Mathlib, it covers finite and countable Markov chains, renewal processes, random walks, martingales, stopping times, queues, Brownian motion, stochastic calculus, weak convergence, and Poisson and continuous-time Markov processes. Our Opus 4.8-based agent achieves a 34.9% proof rate (157/450) under a 15-minute per-problem limit. StochBench better represents domain-specific applied mathematics while remaining challenging for advanced provers.", "title": "StochBench: A Domain-Specific Benchmark for Stochastic Processes in Lean", "updated": "2026-09-08", "url": "https://arxiv.org/abs/2609.09264" }, { "age_days": 2, "arxiv_id": "2609.08921", "authors": [ "Antonio Acuaviva" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "arxiv:2609.08921", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-09-08", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "We give a negative solution to the complemented subspace problem for Banach spaces with unconditional bases over both the real and complex fields. For every $ρ>0$, we construct a projection $P_ρ$ of norm less than $1+ρ$ on a separable superreflexive space \\begin{equation*} X_ρ=\\left(\\bigoplus_{j=1}^{\\infty}\\ell_{p_j}^{N_j}\\right)_2, \\qquad p_j\\downarrow2, \\end{equation*} such that $Z_ρ=P_ρ(X_ρ)$ and its dual $Z_ρ^*$ have Schauder bases but admit no unconditional bases. Over the real field, both spaces have Gordon--Lewis local unconditional structure (GL-lust) but fail Dubinsky--Pełczyński--Rosenthal local unconditional structure (DPR-lust), disproving a conjecture of Figiel, Johnson and Tzafriri. In particular, neither is isomorphic to a Banach lattice, giving a negative solution to the separable Banach-lattice complemented subspace problem. A modification of the construction also shows that the class of separable real Banach lattices is not primary. A Lean 4 formalisation of the main results accompanies the paper.", "title": "A negative solution to the complemented subspace problem for Banach spaces with unconditional bases", "updated": "2026-09-08", "url": "https://arxiv.org/abs/2609.08921" }, { "age_days": 4, "arxiv_id": "2609.06477", "authors": [ "Junyu Guo", "Hao Shen", "Junqi Liu", "Lihong Zhi" ], "content_date": "2026-09-06", "freshness": "fresh", "id": "arxiv:2609.06477", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents", "seed_author:Junqi Liu" ], "published": "2026-09-06", "score": 4.7, "source": "arxiv-ai4math-core", "summary": "Iima and Yoshino asked for an ideal $I$ in $S=k[x_1,x_2,\\ldots]$, with $\\operatorname{deg} x_i=i$, and a monomial order such that $S/I\\cong k[x_i:i\\equiv\\pm1\\pmod5], \\operatorname{in}(I)=(x_i^2,x_ix_{i+1}:i\\geq1).$ We construct such an ideal and monomial order over every field $k$ of characteristic different from $5$ containing an element $c$ with $c^2+c=1$. The ideal has an explicit infinite homogeneous reduced Gröbner basis. A five-periodic syzygy derived from a pentagon identity proves that all basis relations belong to $I$ and supplies standard representations for the non-coprime critical pairs. Triangular elimination establishes the graded quotient isomorphism. Together, the quotient and initial ideal descriptions yield the partition form of the first Rogers-Ramanujan identity. In each weighted degree, a perfect matching in the support of the normal-form matrix gives a bijection between the two partition classes. We formalize the complex specialization in Lean 4 using Mathlib and our set-based theory of infinite Gröbner bases, including the reduced basis, the graded quotient isomorphism, and the partition-matching theorem.", "title": "A Solution to Iima--Yoshino Problem 2.3", "updated": "2026-09-06", "url": "https://arxiv.org/abs/2609.06477" }, { "age_days": 1, "arxiv_id": "2609.10520", "authors": [ "Chengkai Zhu", "Xin Wang" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "arxiv:2609.10520", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-09", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Private communication over a noisy quantum channel requires reliable transmission to the receiver and secrecy from the environment. Whether two channels with zero private capacity can jointly enable private communication is a longstanding open problem in quantum information theory. Here we resolve this problem by exhibiting a four-level channel and a qubit erasure channel with half erasure probability, each with zero private capacity, whose joint use achieves more than 0.0001903 private bits per product use. The encoding gives the receiver a linear information gain with at most quadratic environmental leakage, enabling privacy through a fixed joint measurement and classical coding. This superactivation, impossible for independent classical memoryless wiretap channels, shows that a channel's private capacity alone does not determine its value for secure communication. The initial activation example was identified through interactions with large language models, and the result has been formalized in Lean 4.", "title": "Private communication via zero-private-capacity quantum channels", "updated": "2026-09-09", "url": "https://arxiv.org/abs/2609.10520" }, { "age_days": 1, "arxiv_id": "2609.10281", "authors": [ "Achyuth Jayadevan" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "arxiv:2609.10281", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-09", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "For an ordinary finite presentation $P=\\langle x_1,\\ldots,x_n\\mid R\\rangle$, put $G(P)=F_n/\\langle\\langle R\\rangle\\rangle$. We construct a primitive-recursive predicate $V$ with $G(P)''=1 \\Longleftrightarrow \\exists c\\in\\mathbb{N}: V(P,c)=1$. Thus finite presentations of metabelian groups are recursively enumerable, answering Kourovka Problem 17.124. An effective form of the Bieri-Strebel covering construction, using signed Laurent relations and rational separation, gives a family of finitely presented metabelian groups cofinal under epimorphisms. Products of conjugates of defining relators witness these epimorphisms. The construction and enumeration theorem are formalized in Lean 4.", "title": "Finite presentations of metabelian groups: effective enumeration via Laurent relations", "updated": "2026-09-09", "url": "https://arxiv.org/abs/2609.10281" }, { "age_days": 1, "arxiv_id": "2609.09938", "authors": [ "Hao Shen", "Jiaqi Wang", "Lihong Zhi" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "arxiv:2609.09938", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-09", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We prove that every trace-zero matrix $A\\in M_n(\\mathbb{C})$ admits a representation $A=BC-CB$ with $B,C\\in M_n(\\mathbb{C})$ and $\\lVert B\\rVert\\lVert C\\rVert\\le K\\lVert A\\rVert$, where $K$ is an absolute constant independent of $n$, and $\\lVert\\cdot\\rVert$ denotes the operator norm. For a fixed $t>0$, the proof splits according to whether $\\lVert\\operatorname{Re}(e^{\\mathrm{i}θ}A)\\rVert_1\\ge tn\\lVert A\\rVert$ holds for all $θ\\in\\mathbb{R}$, where $\\lVert\\cdot\\rVert_1$ denotes the trace norm. When this lower bound holds, we construct a commutator representation directly. Otherwise, the vector-selection theorem of Marcus, Spielman, and Srivastava yields smaller trace-zero compressions whose norms are small enough for the induction to close. We also construct an explicit family of zero-diagonal Hermitian unitaries that forces a lower bound of order $\\sqrt{\\log n}$ for $\\lVert B\\rVert\\lVert C\\rVert$ when either factor is required to be diagonal in the prescribed basis. The same family admits $\\varepsilon$-pavings with fewer than $2\\varepsilon^{-2}$ blocks and representations by two normal factors with optimal norm product $1/2$. This establishes a distinction between unrestricted commutator bounds and bounds under a prescribed diagonal restriction. The main results and their essential inputs are formalized in Lean 4 using Mathlib. The development also includes a formal derivation of the Kadison-Singer state-extension theorem from the same vector-selection theorem.", "title": "A Dimension-Independent Commutator Bound", "updated": "2026-09-09", "url": "https://arxiv.org/abs/2609.09938" }, { "age_days": 2, "arxiv_id": "2609.08431", "authors": [ "Barinder S. Banwait" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "arxiv:2609.08431", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-08", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "For an elliptic curve $E/\\mathbb{Q}$ of rank two with complex multiplication, Coates, Liang and Sujatha gave a criterion for the vanishing of $Sha(E/\\mathbb{Q})[p^\\infty]$ at a good ordinary prime $p$ and applied it to five such curves for $p < 30{,}000$. We prove a cyclotomic criterion of the same kind: outside an explicit set of primes, the normalised second Taylor coefficient of the Mazur-Tate-Teitelbaum $p$-adic $L$-function at the central point is a $p$-adic unit if and only if the cyclotomic $p$-adic regulator is a unit and $Sha(E/\\mathbb{Q})[p^\\infty] = 0$, and, by the theory of Bannai and Kobayashi, if and only if an explicit combination of three critical Hecke $L$-values of weight $2p - 1$ has valuation exactly two. Following the algorithm of Stein and Wuthrich, we compute the regulator for the same five curves at every good ordinary prime below $30{,}000$: it is a unit at all but three of the $8{,}050$ primes outside the excluded set. The one case that the criterion of Coates, Liang and Sujatha left open, $p = 577$ for $y^2 = x^3 + 34x$, is settled by the new criterion. A Lean 4 formalisation of the first equivalence, assuming stated results from the literature, is provided.", "title": "Second derivatives of $p$-adic $L$-functions and the Shafarevich--Tate group of rank-two CM elliptic curves", "updated": "2026-09-08", "url": "https://arxiv.org/abs/2609.08431" }, { "age_days": 2, "arxiv_id": "2609.08319", "authors": [ "Kay Akiyama" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "arxiv:2609.08319", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-08", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We prove that no strongly regular graph with parameters $(266, 45, 0, 9)$ exists. The proof is formalized in Lean 4 and Mathlib without external infeasibility certificates or assumed classification theorems. A hypothetical graph gives a rank-$12$ integral Gram lattice with an integral centroid. A Lorentzian change of form, a marked $D_7$ gluing, and an explicit rank-six complement produce a positive-definite even unimodular lattice of rank $24$, together with the original indexed family of $220$ vectors. Harmonic theta identities and a root-isolation inequality force the root system $A_{11} \\perp D_7 \\perp E_6$. First and second moments then exclude the possible complements: the final case reduces to an impossible binary projection identity $4x + 4y - 2z = 50$. A type-$A$ subcase is closed by a separate classification-free proof of the known nonexistence of a quasi-symmetric $2$-$(56, 12, 9)$ design with intersections $0, 3$. That argument constructs a Krein graph and forces a Steiner $3$-$(12, 4, 1)$ design, contradicting its replication equation. The formal theorem depends only on the three standard Lean axioms and has also been checked independently with nanoda. The archived formalization is release v2.0.0.", "title": "Nonexistence of a Strongly Regular Graph with Parameters (266,45,0,9): A Certificate-Free Lean Proof", "updated": "2026-09-08", "url": "https://arxiv.org/abs/2609.08319" }, { "age_days": 2, "arxiv_id": "2609.08160", "authors": [ "Andreas Andreakis" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "arxiv:2609.08160", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-08", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Change-data capture (CDC) feeds downstream systems like caches, search indexes, and data warehouses from a database's log of committed row changes. When bootstrapping, adding a table, or repairing downstream data, a pipeline must also copy existing rows. Merging this copy with the active log introduces the copy-to-log handoff problem. Changes must not fall through a gap, and older copied state must not overwrite a newer logged update or resurrect a deleted row. DBLog, developed at Netflix, addressed this problem by reading tables in chunks and interleaving those reads with the live log. Watermarks identify the changes that overlap each read, and the log wins when a copied row is stale. Debezium and Flink CDC have since adapted this design. Earlier work proved that applying the original algorithm's copied rows and logged changes in their emitted order reconstructs the source's rows, including the effect of every logged insert, update, and delete processed. Generalized DBLog asks when the same result holds for variants of that design. We state the conditions the source and capture implementation must satisfy. Once copying and reconciliation are complete, we prove that the result holds across all selected tables and key ranges even when their rows were read at different times. A single database snapshot is not required for the copy. Further logged changes advance the reconstructed state one event at a time. We establish these guarantees for classic watermarking, Debezium's signal-table and read-only modes, Flink CDC's parallel chunks, reads and dumps tied to exact log positions, and engine-consistent backups whose log position lies within known bounds. The complete theory is machine-checked in Isabelle/HOL, its core independently verified in Lean 4, and the protocols are also examined by bounded model checking in TLA+.", "title": "Generalized DBLog: A Verified Contract for Interleaving Database Rows with a Change Log", "updated": "2026-09-08", "url": "https://arxiv.org/abs/2609.08160" }, { "age_days": 2, "arxiv_id": "2609.08528", "authors": [ "Mario Raso", "Daniele Venturi" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "arxiv:2609.08528", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-08", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "For each integer $n\\geq 1$, let $N(n)$ denote the number of distinct subsets of $\\{2,\\ldots,n+1\\}$ obtained by choosing one forbidden residue class modulo each integer from $2$ to $n$; this is OEIS sequence A396595 (https://oeis.org/A396595). Equivalently, $N(n)$ is the initial-restriction complexity of the family of global residue-profile survivor sequences. We derive a closed formula, depending on the parity of $n$, for the number of locally distinct residue profiles, and an exact inclusion--exclusion formula for profiles realizing a prescribed survivor set. We prove that $\\log N(n)$ has order $n/\\log n$, with any possible leading constant between $\\log 2$ and $2\\log 2$. For prime traces, the logarithm of their number is asymptotic to $(\\log 2)n/\\log n$. Our main comparison theorem shows that $N(n)$ is exponentially equivalent to the block complexity of prime-admissible subsets of an interval of length $n$. The combinatorial component of the private composite coordinates argument used in the comparison theorem is formalized in Lean 4/Mathlib. We also establish an exact structural recurrence, characterize extendibility by a residue-class covering criterion, and give a dynamic enumeration algorithm. As further illustrations of the model, we exhibit purely periodic global profiles generating prime-valued survivor sequences for which we have not identified corresponding OEIS entries.", "title": "Counting Survivor Sets: Exponential Equivalence with Prime-Admissible Sets", "updated": "2026-09-08", "url": "https://arxiv.org/abs/2609.08528" }, { "age_days": 3, "arxiv_id": "2609.08007", "authors": [ "Jeffrey S. Baggett" ], "content_date": "2026-09-07", "freshness": "fresh", "id": "arxiv:2609.08007", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-07", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We prove that the realization graph of every graphical degree sequence is maximally Hamiltonian: it is Hamilton-laceable when bipartite on more than one vertex, and Hamilton-connected otherwise. This answers Problem P59 of Mütze's survey of combinatorial Gray codes, and the Hamiltonicity question recorded as open by Barrus, in the strongest form either admits. The argument is an induction on the number of ground vertices, cutting the realization graph at a single ground vertex into fibers and the quotient they lie over. The proof is formalized in Lean 4 and checked by its kernel, with seven results cited from the literature and nothing else assumed. Its engine is a classification. The realizable neighborhoods of a ground vertex form a shifted family -- one closed under replacing an element by a smaller one -- and the quotient is the Johnson graph of that family. Such a Johnson graph can fail to be Hamilton-connected, and we determine exactly when: the failures are one explicit family of examples, the Y-families, and each of them fails between a single pair of its members. A shifted family with a greatest member never fails, and those families are exactly the shifted matroids, where the conclusion already follows from the theorem of Naddef and Pulleyblank on the graphs of 0/1-polytopes. The obstruction lives entirely outside the matroid case, which is why it has not been met before.", "title": "Maximal Hamiltonicity of realization graphs of degree sequences", "updated": "2026-09-07", "url": "https://arxiv.org/abs/2609.08007" }, { "age_days": 3, "arxiv_id": "2609.07750", "authors": [ "Yury N. Berdinsky" ], "content_date": "2026-09-07", "freshness": "fresh", "id": "arxiv:2609.07750", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-07", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We classify all natural, conformally covariant, symmetric (0,2)-tensors of conformal weight -2 and differential order less than or equal to 4 in dimension 4, built from the metric, the Schouten tensor, covariant derivatives, and the Kulkarni-Nomizu product. We prove that the space of such tensors is exactly 2-dimensional, spanned by the Bach tensor (order 2) and the Eastwood-Singer tensor (order 4). The algebraic core is formally verified using Lean 4 with Mathlib. The 8x11 constraint matrix and divergence-free condition are verified computationally with exact rational arithmetic. Both English and Russian versions provided; Russian version available as ancillary.", "title": "Classification of Conformally Covariant 2-Tensors from the Kulkarni--Nomizu Product in Dimension 4", "updated": "2026-09-07", "url": "https://arxiv.org/abs/2609.07750" }, { "age_days": 3, "arxiv_id": "2609.07745", "authors": [ "James Alexander Schreib" ], "content_date": "2026-09-07", "freshness": "fresh", "id": "arxiv:2609.07745", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-07", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "For a fixed graph F, the F-degree of a vertex v in a host graph H is the number of subgraphs of H isomorphic to F that contain v, and H is F-irregular if its F-degrees are pairwise distinct. We show that every finite connected graph F on at least three vertices admits a finite connected F-irregular host. For noncomplete F, the proof builds the host from a threshold graph with one deleted edge; when the minimum degree is at least two, a small incidence gadget with distinct weighted column sums separates the remaining exceptional vertices. The construction also yields infinitely many pairwise non-isomorphic finite connected F-irregular hosts for every noncomplete F. The complete-pattern case follows from a theorem of Chartrand, Holbert, Oellermann and Swart. A Lean 4 formalization of Theorem 1.1 is described, taking the published complete-pattern theorem as its sole custom axiom.", "title": "A construction of F-irregular graphs", "updated": "2026-09-07", "url": "https://arxiv.org/abs/2609.07745" }, { "age_days": 4, "arxiv_id": "2609.06730", "authors": [ "Zhipeng Lu" ], "content_date": "2026-09-06", "freshness": "fresh", "id": "arxiv:2609.06730", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-06", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "For every finite $P\\subset\\mathbb{R}^2$ we prove $|\\{p\\cdot q: p,q\\in P\\}|\\gg |P|^{199/295}$, with an absolute constant and no logarithmic loss, where $199/295 = 2/3+7/885$. This improves the bound $2/3+7/1425$ of Kokkinos (arXiv:2502.12727), which in turn had improved the first superthreshold bound of Hanson, Roche-Newton, and Senger. The main new ingredient is the inequality $|F^{(2)}G^{(2)}/(F^{(2)}G)| \\le |AB|(|AF|/|A|)^4(|BG|/|B|)^3$: each radial profile retains its own Petridis growth-minimizing subset, and the product of the two subsets serves as a common base for both growth operators; a prime-power construction shows this is sharp under its hypotheses. The second ingredient is a weighted-median replacement for the dyadic selection in the squeezing argument of Roche-Newton and Wong, upgrading their seven-factor expander to a logarithm-free form; an appendix gives the complete proof from the Solymosi-Zahl incidence theorem. We complement the lower bound with a third-moment structure theorem for near-extremal configurations and a construction showing that the pinned problem admits no linear lower bound: $\\max_{p\\in P}|p\\cdot P| = O(|P|/\\sqrt{\\log |P|})$ is attainable. The algebraic core, the median lemma, the surface identities, and the pinned construction are formally verified in Lean 4 and included as ancillary files.", "title": "A dot-product bound from separate growth-minimizing bases", "updated": "2026-09-06", "url": "https://arxiv.org/abs/2609.06730" }, { "age_days": 5, "arxiv_id": "2609.06226", "authors": [ "Yupeng Ren", "Zhaoxuan Li", "Rui Zhang" ], "content_date": "2026-09-05", "freshness": "fresh", "id": "arxiv:2609.06226", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-05", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "Large language models (LLMs) have recently made substantial progress in formal proof generation, yet presenting distinctive challenges in cryptographic area. Computational security arguments posit that a valid proof must coordinate probability, adversarial games, invariants, assumptions and bounds, which can be provided by a machine-checked framework named EasyCrypt. Although all objects may appear in available context, LLMs still struggle because proof-theoretic dependencies are typically implicit in a linear representation and distributed across multiple programs. So, this paper presents Scratchy, a visual-scratchpad approach that exposes these dependencies for multimodal generation. Given the natural-language security description, with formal context and target propositions, the proof objects can be normalized into a typed proof-relation graph. Then a structure-preserving visual compiler transforms the graph into the formula-rich visual proof state that guides a multimodal model in generating the EasyCrypt proof. Also, the Scratchy-eval, a 114-task dataset derived from reliable official EasyCrypt files, has been introduced. It contains 64 security-form proof generations and 50 multiple-choice knowledge tests. After a series of evaluations, covering semantic grounding, relational invariants, and game reductions, classical LLMs like GPT-5.6-Sol and Claude-Opus-5 have gained a clear advantage from Scratchy's structured visual proof states. This contrast suggests that explicit proof structure can make the improvement and multimodal proof-state representation as a promising direction for computer-aided cryptography.", "title": "Scratchy: Visual-Scratchpad Multimodal Reasoning for Cryptographic Proof Generation in EasyCrypt", "updated": "2026-09-05", "url": "https://arxiv.org/abs/2609.06226" }, { "age_days": 3, "arxiv_id": "2609.07238", "authors": [ "Ahaan Kallat" ], "content_date": "2026-09-07", "freshness": "fresh", "id": "arxiv:2609.07238", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning" ], "published": "2026-09-07", "score": 1.9, "source": "arxiv-ai4math-core", "summary": "Let $a(n)$ be the sequence A158690 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function $\\sum_{n\\ge0} a(n)t^n/n! = 1+\\sum_{m\\ge1}\\prod_{j=1}^m(1-e^{-(2j-1)t})$. We prove two congruence conjectures of Peter Bala. The first states that, for every integer $k\\ge1$, the sequence $a(n)$ modulo $k$ is eventually periodic with period dividing $\\varphi(k)$. We prove the stronger statement that the Carmichael function $λ(k)$ is an eventual period. The second conjecture asserts the shifted Gauss congruences $a(np^r+i)\\equiv a(np^{r-1}+i)\\pmod{p^r}$ for every $i\\ge0$, every prime $p$, and all $n,r\\ge1$. Both results follow from a general theorem for exponential generating functions of the form $G(e^t-1)$ with $G\\in\\mathbb Z[[y]]$, together with the standard power-sum formula for Stirling numbers of the second kind.", "title": "A Proof of Bala's Congruence Conjectures for A158690", "updated": "2026-09-07", "url": "https://arxiv.org/abs/2609.07238" }, { "age_days": 0, "authors": [ "Joe Hendrix" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:3f6d20e5da82", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: use CMAKE_COMMAND instead of bare cmake in build (#15001)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/3f6d20e5da82ec7facb56a45c165d1c5c76ec282" }, { "age_days": 0, "authors": [ "Sebastian Ullrich" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:645266e0c0ad", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: update to mimalloc 3.4.5 (#15060)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/645266e0c0ade78b8a108b05902d73f5d04619cd" }, { "age_days": 0, "authors": [ "Sebastian Ullrich" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:843c8318e5f4", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: restore trailing whitespace of nested tactic blocks in the info tree (#15093)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/843c8318e5f44a17dd89b1861a4c2d560bd45e1a" }, { "age_days": 0, "authors": [ "Kim Morrison" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:9f1f954f5384", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: expose Vector DecidableEq for kernel reduction (#14988)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/9f1f954f53846c91df98549658fd4ca35bdc47fe" }, { "age_days": 0, "authors": [ "Kim Morrison" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:59731777e11c", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: expose Array.zipWith for kernel reduction (#15078)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/59731777e11cf6694e8b9da2d6ad7b19a5ab1f4f" }, { "age_days": 0, "authors": [ "Kim Morrison" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:1e5d993bbf55", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: expose Array.ofFn for kernel reduction (#14989)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/1e5d993bbf552c2ec408c1324187ad6832e55acc" }, { "age_days": 0, "authors": [ "Kim Morrison" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:587587f23e88", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: expose Array.modify for kernel reduction (#15079)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/587587f23e88af64e9059e400b771b0db49f2a4b" }, { "age_days": 0, "authors": [ "Kim Morrison" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:682d26ad9537", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: expose Array.mapM for kernel reduction (#14996)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/682d26ad95371f156ecbf2a2caebc3c3c041ad00" }, { "age_days": 0, "authors": [ "Kim Morrison" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:575c84496d7e", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: expose Array equality for kernel reduction (#14270)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/575c84496d7e97dde5a7fff205840f9df3e51efc" }, { "age_days": 0, "authors": [ "Henrik Böving" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:1116ca037000", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: the compiler may remove all arguments of a function if they are unused (#15107)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/1116ca03700074381b2d95346bc15fd533e761b7" }, { "age_days": 0, "authors": [ "Sebastian Ullrich" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:53823a47ae0f", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: bundle the `nanoda` external checker with release toolchains (#15099)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/53823a47ae0f4293bda3de80e99d7c12d3c2a78f" }, { "age_days": 0, "authors": [ "Sebastian Ullrich" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:a5886a0f8d2d", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: bundle the `lean4lean` external checker with release toolchains (#15048)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/a5886a0f8d2d8a0265e83ac11c452e37c4fc7c59" }, { "age_days": 0, "authors": [ "Garmelon" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover/lean4:c1f270e0bc1e", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: stop creating lean-pr-testing-* branches in mathlib4 (#15098)", "updated": "2026-09-10", "url": "https://github.com/leanprover/lean4/commit/c1f270e0bc1e3fe5fe4e23c231937ed91cabde98" }, { "age_days": 0, "authors": [ "Michael Rothgang" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:1192d6246b46", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: refactor: make several definitions take concrete morphism classes (#43596)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/1192d6246b462d5d423cccde4066d15b18718ca9" }, { "age_days": 0, "authors": [ "Thomas Browning" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:403547feec13", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: refactor(RingTheory/Multiplicity): switch `multiplicity` to have junk value of `0` (#43573)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/403547feec13239abab39e599f1b6a55841fd23e" }, { "age_days": 0, "authors": [ "Thomas Browning" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:edbb3ee7f5f8", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: refactor(Data/Nat/Factorization/Defs): redefine `Nat.factorization` in terms of `primeFactorsList` instead of `padicValNat` (#43581)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/edbb3ee7f5f844d6e04064810ae9e39aec737819" }, { "age_days": 0, "authors": [ "Eric Wieser" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:a5c1af9396da", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: fix: `macro_inline` `DFunLike` instances to make `Multiset.toFinsupp` computable (#43620)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/a5c1af9396da72739b771223b8d82aa3b3361a63" }, { "age_days": 0, "authors": [ "Stefan Kebekus" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:83f60fb9a0ec", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat: separation lemma of Value Distribution Theory (#42020)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/83f60fb9a0ec4746f89f2d25620e92db3f1c6f7c" }, { "age_days": 0, "authors": [ "Kim Morrison" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:7e195f2805ab", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat: injectivity of Function.extend for disjoint ranges (#43325)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/7e195f2805abfa785cd944303a388be37aaccf0b" }, { "age_days": 0, "authors": [ "Yaël Dillies" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:00586edcc007", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat: constructors for `IsOrderedMonoid` from `MulLeftMono`/`MulRightMono` (#43663)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/00586edcc007f633e66c0a0b958af7c5517847e2" }, { "age_days": 0, "authors": [ "FMLJohn" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:c02174567c98", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(RingTheory/Adjoin/Polynomial/Basic): `Algebra.adjoin_mem_exists_aeval'` and `Algebra.adjoin_eq_exists_aeval'` (#40344)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/c02174567c98cac8b9a8e2ee34ee8e2ffa98df94" }, { "age_days": 0, "authors": [ "Snir Broshi" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:b8d3b9bdfe72", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Logic/Relation): `Map r f g ≤ s ↔ r ≤ s.bicompl f g` (#38432)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/b8d3b9bdfe72207e7b368f4644272a0685a836ab" }, { "age_days": 0, "authors": [ "Marko Herkaliuk" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:c4dc2c944453", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Analysis/SpecialFunctions/Integrals): the integral of `x / (c ^ 2 + x ^ 2)` (#43640)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/c4dc2c9444530979b651972c37e6bf9f105bc3d9" }, { "age_days": 0, "authors": [ "Bhavik Mehta" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:83abb3e776bd", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Algebra/Regular/SMul): add IsVAddRegular (#43647)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/83abb3e776bdefcbc447a1e44d0debe4010039e5" }, { "age_days": 0, "authors": [ "Vlad Tsyrklevich" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:a429856c734d", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Algebra): generalize Baer's injective criterion to non-commutative rings (#43513)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/a429856c734d2d0d38a9efe32ae3a1f2aa067961" }, { "age_days": 0, "authors": [ "Jeremy Tan Jie Rui" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:f6734f8e476d", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: doc: correct reference to Carmichael numbers (#43651)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/f6734f8e476df9b1afe56d16abcb5c7ec6541042" }, { "age_days": 0, "authors": [ "Carlo Perassi" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:217ba069a556", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: weaken unused hypotheses (Data and Order: thread on zulip) (#43660)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/217ba069a556ad9de837366b35c2ec1bec1de384" }, { "age_days": 0, "authors": [ "Kim Morrison" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:86020b6cf1c3", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: move `liftReflToEq` and `rel_of_eq_and_refl` from core (#43601)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/86020b6cf1c3562f73895bfc2d7f12dff250e200" }, { "age_days": 0, "authors": [ "Michael Rothgang" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:2631d1cc8c2a", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore: deprecate Subring.closure_preimage_le (#43674)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/2631d1cc8c2ace6c6a900425d6e5d2b5963966e9" }, { "age_days": 0, "authors": [ "Thomas Browning" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:1f5affdb20dc", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Algebra/Group/Equiv/TypeTags): generalize to `Add` and `Mul` and deprecate duplicates (#43661)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/1f5affdb20dc9e17a56aa90b9df6f0d8e1fae620" }, { "age_days": 0, "authors": [ "Harald Husum" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:f6d82fe9b990", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-10", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Algebra): tidy markdown headers (#42594)", "updated": "2026-09-10", "url": "https://github.com/leanprover-community/mathlib4/commit/f6d82fe9b99071d4c27024afa1babe17757df082" }, { "age_days": 1, "authors": [ "Henrik Böving" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover/lean4:2d2ed324c3d1", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: perf: never materialize the export string in comparator (#15096)", "updated": "2026-09-09", "url": "https://github.com/leanprover/lean4/commit/2d2ed324c3d1ec3b4503d8dbd5b343074b1d91b0" }, { "age_days": 1, "authors": [ "Sebastian Ullrich" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover/lean4:11636539d415", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: show goal of empty nested `by` block at positions indented past the enclosing tactic (#15095)", "updated": "2026-09-09", "url": "https://github.com/leanprover/lean4/commit/11636539d4150daeabe43cbc5cc5826b06f1e3ce" }, { "age_days": 1, "authors": [ "Henrik Böving" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover/lean4:cb43d9d86fd5", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: floatLetIn should be pessimistic in the presence of side effects (#15089)", "updated": "2026-09-09", "url": "https://github.com/leanprover/lean4/commit/cb43d9d86fd55ac6c2f483b7c19cad91a8958996" }, { "age_days": 1, "authors": [ "Sebastian Ullrich" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover/lean4:0243d9f4bc04", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: sandbox `lake challenge` with `bwrap` instead of `landrun` (#15005)", "updated": "2026-09-09", "url": "https://github.com/leanprover/lean4/commit/0243d9f4bc04983c037caa738a29ab5e78b5459f" }, { "age_days": 1, "authors": [ "Julia Markus Himmel" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover/lean4:061006998a1c", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: missing order instances for `Fin` (#15092)", "updated": "2026-09-09", "url": "https://github.com/leanprover/lean4/commit/061006998a1c7bce18143c6614c6baad875c389a" }, { "age_days": 1, "authors": [ "Henrik Böving" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover/lean4:eea0213293ea", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: linearity marker for HashMaps (#15049)", "updated": "2026-09-09", "url": "https://github.com/leanprover/lean4/commit/eea0213293eaeb2c355603ba7d7bf41326081244" }, { "age_days": 1, "authors": [ "Garmelon" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover/lean4:3029c5976054", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: stop creating lean-pr-testing-* branches in reference manual (#15074)", "updated": "2026-09-09", "url": "https://github.com/leanprover/lean4/commit/3029c59760543da5bc820008d6d182eb30d07b10" }, { "age_days": 1, "authors": [ "Henrik Böving" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover/lean4:fe53f21e0598", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: fixup riscv-ast bench (#15084)", "updated": "2026-09-09", "url": "https://github.com/leanprover/lean4/commit/fe53f21e05980b97d7bbb6717cf7566e3700e576" }, { "age_days": 1, "authors": [ "Kim Morrison" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover/lean4:d7918b742bbf", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: deprecate `liftReflToEq` and `rel_of_eq_and_refl` (#15080)", "updated": "2026-09-09", "url": "https://github.com/leanprover/lean4/commit/d7918b742bbfbf2a3fb94ba224ae4cac863586b9" }, { "age_days": 1, "authors": [ "Julia Markus Himmel" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover/lean4:e3f528649739", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: `LinearOrderPackage Int` (#15088)", "updated": "2026-09-09", "url": "https://github.com/leanprover/lean4/commit/e3f52864973926391a27067e1a52bd45a4ef54de" }, { "age_days": 1, "authors": [ "Julia Markus Himmel" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover/lean4:c1b4faef289a", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: `LawfulOrderBEq Nat` (#15071)", "updated": "2026-09-09", "url": "https://github.com/leanprover/lean4/commit/c1b4faef289a314d00d5c9a15d3f12a57adf67ad" }, { "age_days": 1, "authors": [ "Michael Rothgang" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:5614ff5c2f31", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: refactor: make several NonUnitalAlgHom definitions take a particular algebra homomorphism (#43579)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/5614ff5c2f3198252fee17490df1ce69fcd2abf4" }, { "age_days": 1, "authors": [ "Anne Baanen" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:ac7917d6a517", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat: solve equalities of instances in `convert` (#40663)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/ac7917d6a517b3b710987e9f8037bdcac522a461" }, { "age_days": 1, "authors": [ "Anatole Dedecker" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:aac9be8f596b", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat: a locally closed subgroup is closed (#42501)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/aac9be8f596bb4bfd8576e93d7a140b2e9e7e179" }, { "age_days": 1, "authors": [ "Yi.Yuan" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:ce3179490c5a", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Topology/Closure): add frontier inclusion for finite unions (#43468)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/ce3179490c5a6005e5731666b00c9583f090263c" }, { "age_days": 1, "authors": [ "Yaël Dillies" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:2e52d79f8ad4", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Data/Finsupp): support of `mapDomain` is image of support if nonneg (#43610)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/2e52d79f8ad4a3f152ebefc37070df2cb177f97e" }, { "age_days": 1, "authors": [ "Marko Herkaliuk" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:42a3845c6d7e", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Analysis/SpecialFunctions/Trigonometric/Bounds): bounds for `arctan` against the identity (#43535)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/42a3845c6d7ec6866eefa4cc327a306a0c4a7d3c" }, { "age_days": 1, "authors": [ "Moritz Doll" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:01c8d16a8c31", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(Algebra/Module): use `IsApply` for `LieHom` (#42132)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/01c8d16a8c31eed40a54a38a8c8006c4ee001152" }, { "age_days": 1, "authors": [ "mathlib-splicebot[bot]" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:f5e908760f36", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(RingTheory/Nilpotent/Basic): automated extraction from #43343 (#43619)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/f5e908760f367cc0ad66f64fb3b3a689965bc7a1" }, { "age_days": 1, "authors": [ "Monica Omar" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:2cc1b8858bfc", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(LinearAlgebra/Matrix/PosDef): generalize `PosSemidef.dotProduct_mulVec_zero_iff` (#43617)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/2cc1b8858bfcd37cb2eb130dd59d4a63f3cf69a2" }, { "age_days": 1, "authors": [ "Gabe" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:d83c546b2429", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Combinatorics/SimpleGraph): golf `Coloring.even_length_iff_congr` (#43632)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/d83c546b2429983f8a5195f21e05f5926ba2a767" }, { "age_days": 1, "authors": [ "mathlib-splicebot[bot]" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:e37d88a26f37", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Analysis/Calculus/ContDiff/Basic): automated extraction from #43046 (#43630)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/e37d88a26f3791ed5a93daa1f949af1021b8d103" }, { "age_days": 1, "authors": [ "mathlib-splicebot[bot]" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:398cec6a9732", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-09", "repo": "leanprover-community/mathlib4", "score": 0.8, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Algebra/Polynomial/Monic): automated extraction from #43343 (#43618)", "updated": "2026-09-09", "url": "https://github.com/leanprover-community/mathlib4/commit/398cec6a9732219765ad9df511d7a7932a44379c" }, { "age_days": 2, "authors": [ "Henrik Böving" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "github:leanprover/lean4:9e3f6c6c8671", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-08", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: fix: ctor size limit checking (#15075)", "updated": "2026-09-08", "url": "https://github.com/leanprover/lean4/commit/9e3f6c6c8671b3270684c4ce46b40449929f93f2" }, { "age_days": 2, "authors": [ "Henrik Böving" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "github:leanprover/lean4:541b04729ad9", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-08", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: linearity markers for Vector (#15069)", "updated": "2026-09-08", "url": "https://github.com/leanprover/lean4/commit/541b04729ad996e732e9ef5cb3024b4aca1bcd98" }, { "age_days": 2, "authors": [ "Henrik Böving" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "github:leanprover/lean4:00e8d4f76ff4", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-08", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: dynamic non-linearity tracking primitives (#15052)", "updated": "2026-09-08", "url": "https://github.com/leanprover/lean4/commit/00e8d4f76ff45935419c77e14e28a7ca19ce7cbf" }, { "age_days": 2, "authors": [ "Sebastian Graf" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "github:leanprover/lean4:0b8b09612966", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-08", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: feat: add `ext` theorems for `ULift`, `PULift`, `PLift`, and `MProd` (#15062)", "updated": "2026-09-08", "url": "https://github.com/leanprover/lean4/commit/0b8b096129661e42d901e5734ed59461e2a3fd87" }, { "age_days": 2, "authors": [ "Lean stage0 autoupdater" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "github:leanprover/lean4:235b7598586a", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-08", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: update stage0", "updated": "2026-09-08", "url": "https://github.com/leanprover/lean4/commit/235b7598586a37909478143129fed0af93049a24" }, { "age_days": 2, "authors": [ "Sebastian Ullrich" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "github:leanprover/lean4:0ec0b26ebf24", "kind": "github_update", "label": "negative", "matched_signals": [], "published": "2026-09-08", "repo": "leanprover/lean4", "score": 0.8, "source": "lean4-github", "summary": "Recent commit on leanprover/lean4.", "title": "leanprover/lean4: chore: CI: ignore OOMing RelWithAssert test", "updated": "2026-09-08", "url": "https://github.com/leanprover/lean4/commit/0ec0b26ebf2412e1792c8a07a577c8a9bccec868" }, { "age_days": 1, "arxiv_id": "2609.10461", "authors": [ "Marc Kegel", "Shana Yunsheng Li", "Qiuyu Ren" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "arxiv:2609.10461", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-09", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We construct a $3$-generator $9$-relator group with unsolvable word problem. We use the group to construct two fixed-size Adian--Rabin families of group presentations, one with $4$ generators and $11$ relators, and another with $2$ generators and $10$ relators. As a consequence, $\\#_7(S^2\\times S^2)$ is topologically unrecognizable and $\\#_9(S^2\\times S^2)$ is smoothly unrecognizable. These algebraic and topological results improve the previous best known bounds by Borisov, Tancer, and Gordon. The construction of the group builds upon an example of Borisov and uses additional HNN extensions and Tietze eliminations to reduce the size of the presentation. We also provide a machine-checked Lean~4 formalization of the algebraic results.", "title": "Small undecidable groups and unrecognizable 4-manifolds", "updated": "2026-09-09", "url": "https://arxiv.org/abs/2609.10461" }, { "age_days": 1, "arxiv_id": "2609.10525", "authors": [ "Xiaoyu Li", "Andi Han", "Jiaojiao Jiang", "Junbin Gao" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "arxiv:2609.10525", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-09", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Language generation in the limit asks for valid unseen elements from every exhaustive positive presentation of an unknown infinite language. We characterize this task for arbitrary families over a countable universe. Generation is possible exactly when each target can be assigned a finite positive witness so that the targets activated by any finite sample have an infinite common intersection. The necessary direction follows from a universal normalization: a search through unconfirmed histories converts any successful generator into one depending only on the observed set. We then ask how large compatible witnesses must be. Positive separation width records the smallest uniform size bound, with two further levels for unbounded finite witnesses and the absence of any compatible finite-witness assignment. Every level occurs. Countable families admit singleton witnesses, explicit families realize every finite width, and a union of two families with infinite common cores requires unbounded finite witnesses. Finally, countable-support and finite-profile obstructions explain why local combinatorial data cannot determine generation in the limit. The characterization and full width hierarchy are checked in Lean, including the simplified normalization and a direct diagonal capture lemma. The accompanying Lean development is maintained at https://github.com/xiaoyulics/language-generation-characterization", "title": "Characterizing Language Generation in the Limit: Finite Witnesses and a Separation-Width Hierarch", "updated": "2026-09-09", "url": "https://arxiv.org/abs/2609.10525" }, { "age_days": 1, "arxiv_id": "2609.09687", "authors": [ "Hazem Ibrahim", "Yasir Zaki" ], "content_date": "2026-09-09", "freshness": "fresh", "id": "arxiv:2609.09687", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-09", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Studies of careers in science, film, music, and books report a common pattern. When a person's most successful work arrives is close to a random draw over the works they produce. How large their successes tend to be, in contrast, follows a stable, person-specific factor. We test whether this pattern holds for open-source software careers and whether it changed when generative AI coding tools arrived. From the complete public record of GitHub push events (2015-2025), we reconstruct 102.2M career works by 6.15M contributors, and for the 908k contributors whose repositories publish packages, we measure each work's impact by how many downstream packages come to depend on it. First, we find that the timing of a career's biggest hit is close to a lottery over their works, as in science and the arts, with a small, replicable lean toward early career that grows as careers get longer. Second, some coders reliably produce higher-impact work than others, but this lasting personal factor accounts for only part of why impact persists (about a fifth in our primary specification); the rest behaves like momentum, success feeding on itself for a period of time. Third, within the same contributors, this structure did not change after ChatGPT's release. The stable factor's weight grew by about as much as it grew for an earlier cohort that simply aged, and subtracting the effect of aging from the effect of generative AI puts the shift at +0.03 (95% CI [-0.22, +0.23]), indistinguishable from zero. The success pattern documented in science and the arts therefore describes open-source careers too, and it shows no detectable break across the arrival of generative AI. These results have implications for how track records on open platforms should be read and on what to expect from generative AI for the careers built on them.", "title": "Chance, Persistent Advantage, and the Generative-AI Era in Open-Source Package Careers", "updated": "2026-09-09", "url": "https://arxiv.org/abs/2609.09687" }, { "age_days": 2, "arxiv_id": "2609.09033", "authors": [ "Sevag Gharibian", "Carsten Hecht", "Dorian Rudolph" ], "content_date": "2026-09-08", "freshness": "fresh", "id": "arxiv:2609.09033", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-08", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We prove quantitative lower bounds on the semidefinite extension complexity of the set of separable quantum states on $\\mathbb{C}^d\\otimes\\mathbb{C}^d$. We consider semidefinite programs (SDPs) that approximate the maximum acceptance probability of a measurement over separable states, the optimization problem underlying QMA(2). In the extended-formulation model of Harrow, Natarajan, and Wu (HNW), all measurements share a common feasible region and an objective-independent embedding of product states that exactly reproduces their acceptance probabilities. For every $0<θ<2/7$, there are constants $c_θ,a_θ>0$ such that, for sufficiently large $d$, any such SDP with uniform additive error $01 DSP48 per butterfly and -50% stored twiddle bits at a whole-core Fmax cost of about 4% (best of three seeds; within the seed-to-seed spread), measured on the released datapath driven by a controller we reconstructed (the reference FSM was never released) and validated by full-core simulation; a sequential core with our own controller builds to a timing-gated Basys-3 bitstream.", "title": "FoldNTT: A Multiplier- and Twiddle-Lean NTT Core with Formally Verified Arithmetic for Proth Primes", "updated": "2026-09-06", "url": "https://arxiv.org/abs/2609.06412" }, { "age_days": 4, "arxiv_id": "2609.06428", "authors": [ "Elena Caviglia", "Luca Mesiti", "Tim Van der Linden" ], "content_date": "2026-09-06", "freshness": "fresh", "id": "arxiv:2609.06428", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-06", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We prove a Snake Lemma for 2-categories. Working in a 2-category with a strong bizero object, we develop 2-kernels and 2-cokernels, 2-monomorphisms and 2-epimorphisms as fully faithful and cofully faithful 1-cells, short 2-exact sequences and normal image factorisations, and we prove a two-dimensional Normal Short Five Lemma. We then introduce the dinversion of an antinormal pair and the notion of a homologically self-dual 2-category, characterised equally by the self-duality of homology, by a Pure Snake Lemma and by a Third Isomorphism Property. Two-dimensional di-exactness implies homological self-duality. Our main result is the Snake Lemma in a 2-di-exact 2-category: a ladder of 2-exact rows with normal verticals induces a 2-exact six-term sequence, with a connecting 1-cell that is 2-natural in the ladder. Di-exactness can be traded for two hypotheses that are not self-dual: that dinversion preserve normality, and that normal 2-epimorphisms compose. Everything specialises, on passing to a locally discrete 2-category, to its classical counterpart. We close by exhibiting three models: a 2-di-exact 2-category of abelian categories containing $\\mathsf{Coh}(X)$ for every noetherian scheme $X$; the locally ordered 2-category of complete modular lattices, in which 2-di-exactness amounts to Dedekind's transposition principle; and the 2-category of Hilbert lattices, which is not 2-di-exact but satisfies the non-self-dual hypotheses, by a theorem of Mackey on pairs of closed subspaces.", "title": "A two-categorical Snake Lemma", "updated": "2026-09-06", "url": "https://arxiv.org/abs/2609.06428" } ], "lookback_days": 21, "schema": "ai4math-radar-run-v1", "timezone": "America/Los_Angeles" }