{
  "counts": {
    "adjacent": 15,
    "core": 1,
    "errors": 0,
    "negative": 74,
    "total": 90
  },
  "date": "2026-09-09",
  "errors": [],
  "fresh_content_days": 21,
  "generated_at": "2026-09-09T18:48:44Z",
  "items": [
    {
      "age_days": 5,
      "arxiv_id": "2609.05157",
      "authors": [
        "Weichen Winston Yin",
        "Jacob M. Taylor",
        "Dirk R. Englund",
        "Frank H. L. Koppens"
      ],
      "content_date": "2026-09-04",
      "freshness": "fresh",
      "id": "arxiv:2609.05157",
      "kind": "paper",
      "label": "core",
      "matched_signals": [
        "autoformalization",
        "lean_formal_proving_agents",
        "mathlib_retrieval"
      ],
      "published": "2026-09-04",
      "score": 9.5,
      "source": "arxiv-ai4math-core",
      "summary": "Formalizing mathematics in a proof assistant, where a machine checks every definition, statement and proof, has set a new standard of rigor. Large language models are now capable of formalizing autonomously, even at the scale of whole textbooks. We bring this standard of rigor to physics, where theoretical arguments carry idealizations that are rarely stated fully, and any logical gaps could have a cascading effect on interdependent results. Recognizing the need to evaluate autoformalization systems for physics, we release AxQM, 1,019 kernel-checkable proof-synthesis tasks over 479 items drawn from the textbook Quantum Computation and Quantum Information by Nielsen and Chuang. The tasks are stated in a custom Lean library of finite-dimensional quantum mechanics. By task count, it is the largest proof-synthesis benchmark in physics by a factor of four. AxQM is derived from a near-complete formalization of the formal portions of the textbook, so every task is guaranteed a solution, which we keep private. Grading of the benchmark is done deterministically by the Lean kernel, which checks that the proof compiles, that no sorry appears in it or in any declaration it depends on, and that it introduces no new axioms.",
      "title": "AxQM: A Textbook-Scale Benchmark for Formal Proof Synthesis in a Library of Finite-Dimensional Quantum Mechanics",
      "updated": "2026-09-04",
      "url": "https://arxiv.org/abs/2609.05157"
    },
    {
      "age_days": 1,
      "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": 3,
      "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": "2609.08431",
      "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": 1,
      "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": 1,
      "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": 1,
      "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": 2,
      "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": 2,
      "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": 2,
      "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": 3,
      "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": 4,
      "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": 4,
      "arxiv_id": "2609.05847",
      "authors": [
        "Qian Qin"
      ],
      "content_date": "2026-09-05",
      "freshness": "fresh",
      "id": "arxiv:2609.05847",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-05",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Let $π(\\mathrm{d} x)\\propto e^{-U(x)}\\,\\mathrm{d} x$ on $\\mathbb R^d$, where $0<m\\leq L<\\infty$, $mI_d\\preceq\\nabla^2U(x)\\preceq LI_d$, and $κ=L/m$. It is known that, under warm-start assumptions, fixed-step Metropolis-adjusted Langevin algorithm (MALA) with properly tuned step size has mixing time of order $κ\\sqrt{d}$ up to logarithmic factors. By contrast, when the condition number is bounded away from one, no single fixed step size yields a matching spectral-gap lower bound of order $(κ\\sqrt{d})^{-1}$ uniformly over this target class. We show that MALA with a uniformly randomized step size admits a spectral-gap lower bound of this size. At each iteration, the randomized-step MALA considered here draws $h$ uniformly from $(0,H)$ and performs one ordinary MALA transition with step size $h$. We show that, when $H$ is of order $(L\\sqrt{d})^{-1}$, the right spectral gap of randomized-step MALA admits a lower bound of order \\[ \\frac{1}{κ\\sqrt{d}\\,[1+\\log(d+1)+\\logκ]}. \\] The main new ingredient in the proof is a Cheeger-type inequality for aggregating estimates of the one-step flow of MALA out of measurable sets at various step-size scales. It allows the scale used to control the flow to depend on the set and avoids the additional loss that would result from first summing the flows and then applying the standard Cheeger inequality. This work was developed with substantial assistance from ChatGPT, which suggested the uniformly randomized-step approach, developed the principal proof arguments, and generated the simulation and Lean 4 code. The human author checked and verified the mathematical content and take full responsibility for the results.",
      "title": "A global spectral gap for Metropolis-adjusted Langevin algorithm with a uniformly randomized step size",
      "updated": "2026-09-05",
      "url": "https://arxiv.org/abs/2609.05847"
    },
    {
      "age_days": 5,
      "arxiv_id": "2609.05690",
      "authors": [
        "Jiaming Zhao",
        "Bing Wu",
        "Xu Miao"
      ],
      "content_date": "2026-09-04",
      "freshness": "fresh",
      "id": "arxiv:2609.05690",
      "kind": "paper",
      "label": "adjacent",
      "matched_signals": [
        "lean_formal_proving_agents"
      ],
      "published": "2026-09-04",
      "score": 3.5,
      "source": "arxiv-ai4math-core",
      "summary": "Finite model finders cannot witness an Austin law: an identity whose finite models are all trivial but which has a nontrivial infinite model. We introduce rank-decreasing sparse trace-tree magmas, finitely presented total operations on a countably infinite constructor-tree carrier. The default product pairs its arguments; finitely many positive Horn clauses define exceptions. Our main procedure derives clauses from symbolic evaluation traces. For every model found, it proves functionality of the exceptional relation by descent on constructor size, proves the identity by exhaustive symbolic case analysis, and emits a self-contained Lean 4 certificate. A least simultaneous fixed point gives an implementation-independent semantics, so bounded search may miss models but cannot invalidate certified results. On ETP's 96 order-five Austin candidates, we discover and Lean-verify infinite countermodels for 28 identities with no prior public classification in our audit. They form 14 duality classes and establish 28 new Austin classifications. Four ALPS-known cases bring the total to 32 certified candidates. On Canonical-4187, the deduplicated union of Order5-130 and the 4,141-row ALPS pool, a fresh trace run produces 636 certificates, all accepted by Judge v3. At equal resource limits, Vampire 5.0.1, E 3.5.1, and complete Twee 2.6.1 jointly prove implications in 94 canonical classes. Only Twee returns trusted counter-satisfiable outcomes, for 18 classes; independent finite-side certificates force 16 to be infinite. None of these ATPs emits an explicit model or Lean certificate, and none decides the 28 new classifications. To the best of our audit, this is the first automated system to synthesize this trace-tree model family, generate well-founded inversion proofs, and emit self-contained Lean 4 certificates.",
      "title": "Trace-Tree Magmas: Proof-Producing Infinite Countermodels and 28 New Order-Five Austin Classifications",
      "updated": "2026-09-04",
      "url": "https://arxiv.org/abs/2609.05690"
    },
    {
      "age_days": 2,
      "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"
    },
    {
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      "arxiv_id": "2609.05358",
      "authors": [
        "Hikaru Manabe"
      ],
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      "freshness": "fresh",
      "id": "arxiv:2609.05358",
      "kind": "paper",
      "label": "adjacent",
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        "general_ai_math_reasoning"
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      "published": "2026-09-04",
      "score": 1.9,
      "source": "arxiv-ai4math-core",
      "summary": "A subtraction game is played on a heap of tokens. The players take turns removing s tokens for some s in a fixed set S of positive integers, and the player who cannot move loses. The sequence of Sprague-Grundy values of such a game is eventually periodic. We ask when it is purely periodic, meaning periodic from the very start, for the three-move sets S={a,b,c} with 0<a<b<c. After reduction to primitive rulesets with a>=2, we give an explicit sufficient criterion for pure periodicity. For every non-additive set that satisfies it we determine the least period, all P-positions in closed form, and all nim-values, and the additive case c=a+b is treated separately. We conjecture that the criterion is also necessary, and we prove this for the period a+b whenever c>=2(a+b). The criterion is a finite test on the angle rho = c mod (a+b), built from the P-position pattern of the two-move game {a,b}, and the admissible angles form explicit unions of arcs in Z/(a+b)Z.",
      "title": "Purely Periodic Three-move Subtraction Games",
      "updated": "2026-09-04",
      "url": "https://arxiv.org/abs/2609.05358"
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    {
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        "Henrik Böving"
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      "freshness": "fresh",
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      "label": "negative",
      "matched_signals": [],
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      "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"
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        "Sebastian Ullrich"
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      "freshness": "fresh",
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      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-09",
      "repo": "leanprover/lean4",
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      "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",
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        "Henrik Böving"
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      "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"
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    {
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      "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"
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    {
      "age_days": 0,
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        "Julia Markus Himmel"
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      "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"
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    {
      "age_days": 0,
      "authors": [
        "Henrik Böving"
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      "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"
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    {
      "age_days": 0,
      "authors": [
        "Garmelon"
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      "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"
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    {
      "age_days": 0,
      "authors": [
        "Henrik Böving"
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      "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"
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    {
      "age_days": 0,
      "authors": [
        "Kim Morrison"
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      "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"
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        "Julia Markus Himmel"
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      "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"
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      "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"
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      "age_days": 0,
      "authors": [
        "Kim Morrison"
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      "content_date": "2026-09-09",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:141ad76b7125",
      "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(CategoryTheory/Limits): use compEvaluation in limitIsoFlipCompLim (#43568)",
      "updated": "2026-09-09",
      "url": "https://github.com/leanprover-community/mathlib4/commit/141ad76b7125150c3b4f5e09a3f0d0b13112cd7c"
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    {
      "age_days": 0,
      "authors": [
        "Violeta Hernández Palacios"
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      "content_date": "2026-09-09",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:d1298f10aed8",
      "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: `Sum.swap` is strictly monotonic (#43598)",
      "updated": "2026-09-09",
      "url": "https://github.com/leanprover-community/mathlib4/commit/d1298f10aed809aecc68eba075a2b95d1d850798"
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    {
      "age_days": 0,
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        "Yi.Yuan"
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      "content_date": "2026-09-09",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:4e399bf32940",
      "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/Subpath): compose consecutive subpaths up to homotopy (#43400)",
      "updated": "2026-09-09",
      "url": "https://github.com/leanprover-community/mathlib4/commit/4e399bf3294031cb7a0249083a14c0997fd04848"
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      "age_days": 0,
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        "Yi.Yuan"
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      "content_date": "2026-09-09",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:ce3179490c5a",
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      "repo": "leanprover-community/mathlib4",
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      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(Topology/Closure): add frontier inclusion for finite unions (#43468)",
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    {
      "age_days": 0,
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        "Kim Morrison"
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      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:4edb0dbaa3b3",
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      "repo": "leanprover-community/mathlib4",
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      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(Tactic/NormNum): run simp's default simprocs inside norm_num (#43003)",
      "updated": "2026-09-09",
      "url": "https://github.com/leanprover-community/mathlib4/commit/4edb0dbaa3b3cf729d86d1e2f035474d5ae2ac09"
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      "age_days": 0,
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        "Xavier Roblot"
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      "id": "github:leanprover-community/mathlib4:a4c8ef0a69f5",
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      "repo": "leanprover-community/mathlib4",
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      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(GroupTheory/SpecificGroups/Cyclic): comparison of subgroups of a cyclic group (#40597)",
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      "published": "2026-09-09",
      "repo": "leanprover-community/mathlib4",
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      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(Data/List): sublist is a partial order and subperm is a preorder (#43602)",
      "updated": "2026-09-09",
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      "repo": "leanprover-community/mathlib4",
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      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(Data/Finsupp): support of `mapDomain` is image of support if nonneg (#43610)",
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      "repo": "leanprover-community/mathlib4",
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      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(Counterexamples): the space ω₁ (#41962)",
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      "repo": "leanprover-community/mathlib4",
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      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(Combinatorics/SimpleGraph/Trails): any walk on the empty graph is Eulerian (#43511)",
      "updated": "2026-09-09",
      "url": "https://github.com/leanprover-community/mathlib4/commit/a0dd2b1fce1135d225d96d30aad544f97f045d37"
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      "id": "github:leanprover-community/mathlib4:67b1d3659da4",
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      "matched_signals": [],
      "published": "2026-09-09",
      "repo": "leanprover-community/mathlib4",
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      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(Combinatorics/SimpleGraph/Trails): an Eulerian graph without isolated vertices is connected (#43335)",
      "updated": "2026-09-09",
      "url": "https://github.com/leanprover-community/mathlib4/commit/67b1d3659da48bfa683f1189bc8b6cf6583fb135"
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      "published": "2026-09-09",
      "repo": "leanprover-community/mathlib4",
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      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(Combinatorics/SimpleGraph): star graphs (and more) are bipartite (#43514)",
      "updated": "2026-09-09",
      "url": "https://github.com/leanprover-community/mathlib4/commit/763fa9e12cc1343d35316f5cc47e65e29f63b72c"
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      "repo": "leanprover-community/mathlib4",
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      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(Analysis/SpecialFunctions/Trigonometric/Bounds): bounds for `arctan` against the identity (#43535)",
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      "repo": "leanprover-community/mathlib4",
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        "Garmelon"
      ],
      "content_date": "2026-09-08",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:d3e54696ea7c",
      "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: sync adaptation PR titles (#15061)",
      "updated": "2026-09-08",
      "url": "https://github.com/leanprover/lean4/commit/d3e54696ea7cd47c75a6dcf536bfc023b3ef6096"
    },
    {
      "age_days": 1,
      "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,
      "authors": [
        "Snir Broshi"
      ],
      "content_date": "2026-09-08",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:110cb7f03dd1",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-08",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: feat(Topology/Order/LowerUpperTopology): the order topology in a linear order is the infimum of the lower & upper topologies (#43540)",
      "updated": "2026-09-08",
      "url": "https://github.com/leanprover-community/mathlib4/commit/110cb7f03dd1c2d9edd957b937f55175c9fc3343"
    },
    {
      "age_days": 1,
      "authors": [
        "Jovan Gerbscheid"
      ],
      "content_date": "2026-09-08",
      "freshness": "fresh",
      "id": "github:leanprover-community/mathlib4:56124ab70b55",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-08",
      "repo": "leanprover-community/mathlib4",
      "score": 0.8,
      "source": "mathlib4-github",
      "summary": "Recent commit on leanprover-community/mathlib4.",
      "title": "leanprover-community/mathlib4: chore: remove imports added by `shake` (#40961)",
      "updated": "2026-09-08",
      "url": "https://github.com/leanprover-community/mathlib4/commit/56124ab70b5521a9a39b9ca6cad0ff4dbd060b03"
    },
    {
      "age_days": 2,
      "authors": [
        "Henrik Böving"
      ],
      "content_date": "2026-09-07",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:c40679c304a1",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-07",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: feat: enable checking lean4export via leanchecker (#15050)",
      "updated": "2026-09-07",
      "url": "https://github.com/leanprover/lean4/commit/c40679c304a1f8324981783c12275e109b051349"
    },
    {
      "age_days": 2,
      "authors": [
        "Sebastian Ullrich"
      ],
      "content_date": "2026-09-07",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:64906da38cdd",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-07",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: feat: add `lake check` to check a project against external checkers (#14990)",
      "updated": "2026-09-07",
      "url": "https://github.com/leanprover/lean4/commit/64906da38cdd68f83829f45f6fafe35468f06f7b"
    },
    {
      "age_days": 4,
      "authors": [
        "Mac Malone"
      ],
      "content_date": "2026-09-05",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:5549307b4a93",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-05",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: fix: lake: apply `moreServerOptions` to package modules (#15042)",
      "updated": "2026-09-05",
      "url": "https://github.com/leanprover/lean4/commit/5549307b4a93889ed1cb48d7436a2d02df8bf62a"
    },
    {
      "age_days": 4,
      "authors": [
        "Mac Malone"
      ],
      "content_date": "2026-09-05",
      "freshness": "fresh",
      "id": "github:leanprover/lean4:c155094f54ea",
      "kind": "github_update",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-05",
      "repo": "leanprover/lean4",
      "score": 0.8,
      "source": "lean4-github",
      "summary": "Recent commit on leanprover/lean4.",
      "title": "leanprover/lean4: feat: lake: `precompileLibrary` & `precompileImports` (#15015)",
      "updated": "2026-09-05",
      "url": "https://github.com/leanprover/lean4/commit/c155094f54eab345cca3da867dbd888a34fbf0d2"
    },
    {
      "age_days": 1,
      "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 $0<a\\le a_θ$ has size at least $d^{c_θ\\min\\{a^{-1/3},d^θ\\}}$. The bound applies at sufficiently small constant error, is superpolynomial in $d$ whenever $a=o(1)$, and becomes $d^{Ω(d^θ)}$ when $a\\le d^{-3θ}$, improving HNW's quasipolynomial bound at inverse- square error. The same bound holds for any SDP-representable convex set of states that contains all separable states and lies within trace distance $a$ of them, giving a quantitative counterpart to Fawzi's theorem that the separable set has no exact semidefinite representation. Our proof combines the quantitative pseudo-density theorem of Lee, Raghavendra, and Steurer with explicit block-positive operators and Chebyshev amplification. Our main results are supported by Lean proofs.",
      "title": "Semidefinite extension complexity of the separable set, with applications to approximate disentanglers",
      "updated": "2026-09-08",
      "url": "https://arxiv.org/abs/2609.09033"
    },
    {
      "age_days": 1,
      "arxiv_id": "2609.08637",
      "authors": [
        "Luka Debevc",
        "Nishan Chatterjee",
        "Antoine Doucet",
        "Senja Pollak",
        "Matej Martinc"
      ],
      "content_date": "2026-09-08",
      "freshness": "fresh",
      "id": "arxiv:2609.08637",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-08",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Large Language Models are increasingly deployed as information intermediaries, yet measuring their political behavior remains fragile because questionnaire results mix model dispositions with measurement artifacts and response-elicitation biases. We introduce a robust Political Compass Test evaluation framework that samples 300 configurations across an eight-dimensional perturbation space varying language, framing, instructions, answer format, option order, and persona wording. We evaluate eight Gemma 3 and Qwen 3 models across 14 languages and three quantization levels, obtaining design-averaged political coordinates with quantified uncertainty. Most models lean Libertarian-Left on average, but instruction phrasing, language, and answer format significantly affect recovered coordinates. Cross-lingual differences primarily reflect coordinate drift rather than distinct cultural reasoning. Reverse-engineering the test also exposes axis-weighting imbalances and the collapse of degenerate responses toward the center, so near-origin estimates for the smallest models can reflect weak signal rather than centrism. Free-text reasoning and chat-then-classify elicitation alter recovered coordinates, and larger models show clearer persona separation, with a specific failure of the Authoritarian-Left persona to move most models in the intended social direction. In downstream tasks, persona effects are modest relative to model size and target group for hate-speech detection, while base and centrist prompts give the highest agreement for topic-level sentiment. Political role prompting therefore has measurable but task- and dataset-specific downstream effects.",
      "title": "Navigating the digital spectrum: Assessing political bias, stability, and downstream fairness in Large Language Models",
      "updated": "2026-09-08",
      "url": "https://arxiv.org/abs/2609.08637"
    },
    {
      "age_days": 1,
      "arxiv_id": "2609.08885",
      "authors": [
        "Eren Ercan"
      ],
      "content_date": "2026-09-08",
      "freshness": "fresh",
      "id": "arxiv:2609.08885",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-08",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Let $A\\in\\mathbb{R}^{m\\times n}$ have columns of Euclidean norm at most one. We prove that $\\operatorname{disc}(A)\\le2395\\left(1+\\log_+\\frac n9\\right)^{1/4}+2\\sqrt2$. Here $\\log_+t=\\max\\{0,\\log t\\}$. Building on Bansal and Jiang's affine spectral independence framework, we remove the $(\\log\\log n)^{7/4}$ factor from their bound. The fourth root comes from balancing the logarithmic decrease in the alive dimension against the fourth power of the row thresholds. Historical exponential sums control the covariance budget across size classes with summable thresholds. An exact threshold-sum certificate gives the coefficient $2395$, and rounding at most eight remaining fractional coordinates costs $2\\sqrt2$. The finite construction also gives partial colourings from any prescribed starting point and at any prescribed depth, preserving existing signs. We formalize the partial- and full-colouring theorems in Lean, including the finite trajectory, exact threshold sum and final rounding, with Bansal--Jiang Theorem A.4 as the sole external research theorem assumption.",
      "title": "A $(\\log n)^{1/4}$ Bound for the Komlós Problem",
      "updated": "2026-09-08",
      "url": "https://arxiv.org/abs/2609.08885"
    },
    {
      "age_days": 2,
      "arxiv_id": "2609.07621",
      "authors": [
        "Ahmed Bou-Rabee",
        "Yuval Peres"
      ],
      "content_date": "2026-09-07",
      "freshness": "fresh",
      "id": "arxiv:2609.07621",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-07",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Independently orient each horizontal and vertical line of $\\mathbb{Z}^2$ by a fair coin. A walker chooses one of the two lines through its current position with equal probability and takes one step in the direction of that line. Redner (1989) introduced this walk as a model of transport in an isotropic random velocity field and predicted that its root-mean-square displacement grows like $n^{2/3}$. We prove that the walk is transient almost surely. The proof is a short variational argument.",
      "title": "The randomly oriented Manhattan lattice in 2D is transient",
      "updated": "2026-09-07",
      "url": "https://arxiv.org/abs/2609.07621"
    },
    {
      "age_days": 2,
      "arxiv_id": "2609.07934",
      "authors": [
        "Bruno P. Chaves",
        "Jérémy Saucourt",
        "Van Thuy Hoang",
        "Alexis Bougaud",
        "Yassin Boussafa",
        "Erwan Ferrandon",
        "Sai T. Chu",
        "Brent E. Little",
        "David J. Moss",
        "Roberto Morandotti",
        "Vincent Couderc",
        "Benjamin Wetzel"
      ],
      "content_date": "2026-09-07",
      "freshness": "fresh",
      "id": "arxiv:2609.07934",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-07",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "The ability to precisely shape the properties of light over multiple degrees of freedom constitutes the foundation of modern photonic architectures. Broadband sources with addressable spectro-temporal content are for instance critical for applications spanning biomedical imaging, material processing, lidar, and ultrafast spectroscopy, but also key enabling technologies in quantum information processing, high-capacity communications, and optical computing. Yet, flexible and adjustable spectro-temporal shaping so far remains a significant challenge, as it requires the simultaneous manipulation of both frequency and temporal domains across broad spectral bandwidths and extended timescales, from femtoseconds to nanoseconds. Importantly, conventional approaches to spectro-temporal processing typically rely on bulky and complex optical systems that lack scalability and flexibility, ultimately hampering their utility for applications requiring addressable multiphotonic processes. Here, we present a framework for spectro-temporal wavepacket control over a broad bandwidth, merging programmable integrated photonics with high-speed optical characterization. Leveraging machine leaning for tailoring nonlinear pulse propagation, we experimentally report scalable and on-demand shaping of broadband ultrafast pulse patterns. We demonstrate the efficacy of our method through reconfigurable power across 400 nm bandwidth with picosecond resolution, with promises for applications where precise control over light is paramount, in particular those needing versatile and controllable multiphoton excitations.",
      "title": "Spectro-temporal shaping of broadband optical wavepackets via programmable on-chip photonics",
      "updated": "2026-09-07",
      "url": "https://arxiv.org/abs/2609.07934"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.06739",
      "authors": [
        "Jacek P. Dmochowski"
      ],
      "content_date": "2026-09-06",
      "freshness": "fresh",
      "id": "arxiv:2609.06739",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-06",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Sports betting moves money continuously from a large public to a small number of firms. The most influential account of that flow, due to Levitt (2004), holds that books price away from the market-clearing point to exploit predictable public biases. It computes profit from two numbers (the probability that a side wins the proposition and the fraction of handle it attracts), treating the share on a side as independent of the outcome. Here we relax that assumption and derive a profit-bias identity: profit is affine and increasing in the expected share of handle on the losing side, with Levitt's expression as the special case of independence. The identity resolves the book's margin into exactly three channels: its hold, the product of its price shading and the public's lean, and the covariance between bet share and outcome. A lean is thus worthless without shading, and shading is worthless without a lean. Under a public-belief model, the profit driver is the public's Bayes error: the probability of a representative bettor selecting the losing side. We provide necessary and sufficient conditions for a \"Goldilocks Zone\": prices at which book and bettor both profit. Testing these predictions on 1,139 Major League Baseball games, we find that the apparent dependence between bet share and outcome is a Simpson's paradox: present when games are pooled, but absent once they are separated by which side the book favored. The public leans heavily toward favorites, but we detect no matching shading, and the realized margin is indistinguishable from the hold.",
      "title": "The profit-bias identity in sports betting: bookmaker profit as the public's prediction error",
      "updated": "2026-09-06",
      "url": "https://arxiv.org/abs/2609.06739"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.06770",
      "authors": [
        "Michael S. Bernstein"
      ],
      "content_date": "2026-09-06",
      "freshness": "fresh",
      "id": "arxiv:2609.06770",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-06",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Our community creates interfaces---pre-constructed, static artifacts. These interfaces are effortful to create, so we invest great effort and care into each of them. Yet the equilibrium is shifting: we can now construct nearly any software, bespoke, within moments. When construction is cheap, we may soon live in a future in which interfaces are constructed in real time. This is a world of just-in-time interfaces: rapid, discardable, on-demand interfaces tailored to our needs in the moment. Will our community still create traditional interactive systems in this world? Or will we create interface generators, ``interface.md'' guides for AI consumption? I argue that we ought to lean into a future of rapid, bespoke, on-demand interfaces.",
      "title": "The Interface of Theseus: The Rise of Just-In-Time Interfaces",
      "updated": "2026-09-06",
      "url": "https://arxiv.org/abs/2609.06770"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.06747",
      "authors": [
        "Zhipeng Lu"
      ],
      "content_date": "2026-09-06",
      "freshness": "fresh",
      "id": "arxiv:2609.06747",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-06",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "For a connected regular graph G and a vertex a, we study the algebra generated by the adjacency and degree matrices of G-a and its cyclic module P_a generated by the all-ones vector. Our main theorem determines dim P_a for Cartesian products whose factors have equitable distance partitions at the chosen roots. A normalized logarithmic derivative of the local spectral generating function partitions the factors into boundary classes. We identify the boundary-return space exactly and express dim P_a as a sum of affine ranks on additive spectral fibres. For a distance-regular factor with distinct spectrum Θ, this gives dim P_a(F^{\\square m}) = |mΘ| - 1. For products of powers of two distinct complete graphs, we evaluate the fibre formula in closed form. We also determine the full punctured algebras of all Hamming graphs: equality with the compressed Terwilliger algebra holds precisely in dimensions at most four for the hypercube and at most two for larger alphabets. For distance-regular graphs, adjacency moments alone determine the intersection array, with an explicit finite reconstruction. Finally, Cartesian stabilizer formulas separate metric loss from orbit splitting; on Doob graphs their distance-graded defect recovers the number of Shrikhande factors.",
      "title": "Punctured adjacency-degree algebras of Cartesian products",
      "updated": "2026-09-06",
      "url": "https://arxiv.org/abs/2609.06747"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.06461",
      "authors": [
        "YuJian Geng",
        "Dong Qiu"
      ],
      "content_date": "2026-09-06",
      "freshness": "fresh",
      "id": "arxiv:2609.06461",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-06",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Let $R_n$ be the infimum of the inradii of $\\{z:|p(z)|<1\\}$ over monic degree-$n$ polynomials whose zeros lie in the closed unit disk. We prove $nR_n\\toπ/2$, matching the asymptotic obstruction supplied by $z^n-1$. We first establish the exact universal radius $2^{1/n}-1$ for disks centered at zeros, which recovers the $(\\log2)/n$ bound. Small inradius then forces radial concentration of the zeros and decay of their low reciprocal moments. These estimates give an entire limit with a modulus reflection identity; a second rescaling produces an exponential tangent and strict sublevel disks of every radius below $π/2$. The proof is accompanied by a Lean~4 formalization and a step-by-step source index.",
      "title": "From $\\log 2$ to $π/2$: the sharp asymptotic inradius of polynomial lemniscates",
      "updated": "2026-09-06",
      "url": "https://arxiv.org/abs/2609.06461"
    },
    {
      "age_days": 3,
      "arxiv_id": "2609.06412",
      "authors": [
        "Masato Kamba"
      ],
      "content_date": "2026-09-06",
      "freshness": "fresh",
      "id": "arxiv:2609.06412",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-06",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Hardware for lattice-based post-quantum cryptography spends a large share of its area on the number-theoretic transform (NTT), dominated by modular multipliers and twiddle storage. FoldNTT is a redesign of the released radix-2 CFNTT accelerator (TCHES 2022) for the Falcon / FN-DSA prime q = 12289 with one hardware multiplier per butterfly instead of three and about half the stored twiddle constants. The Proth shape q = 3*2^12 + 1 turns modular reduction into shift-and-add K-RED folds, and the bit-reversed twiddle table obeys w[N/2+j] = psi*w[j], so half the table is derived without a multiplier. Checking the released RTL against the mathematics also exposed a bug: its inverse transform omits a per-stage halving and returns 2^10*x, which the retrofit corrects. Every arithmetic block is proven by exact-width SMT and compositional SymbiYosys proofs, control-safety invariants by k-induction; the proofs are mutation-tested and the composed transform is validated by simulation, all rerun by CI. On Artix-7 in a fully open flow, the retrofit costs 3->1 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": 3,
      "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"
    },
    {
      "age_days": 5,
      "arxiv_id": "2609.05386",
      "authors": [
        "Jan Snellman"
      ],
      "content_date": "2026-09-04",
      "freshness": "fresh",
      "id": "arxiv:2609.05386",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-04",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "For $2 \\times 2$ matrices $A_1,\\dots,A_k$, write $[A_1,\\dots,A_k]$ for the left-normed iterated commutator $[\\dots[[A_1,A_2],A_3],\\dots,A_k]$, and $I_k$ for the ideal, in the $3k$-variable reduced-coordinate polynomial ring $R_k$, cutting out its vanishing locus. We prove, for every $k \\geq 2$ over any field of characteristic $\\neq 2$, and as four independently-established results rather than one bundled claim: $I_k$ has codimension 2; $I_k$ has exactly 3 minimal generators; $R_k / I_k$ is Cohen-Macaulay; and $I_k$ is radical. The last of these, together with an explicit component count resting on a non-containment argument, assembles into the Primary Decomposition Theorem: $I_k = P_2 \\cap \\cdots \\cap P_k$ is an irredundant primary decomposition into exactly $k-1$ primes, following an explicit recursive block-involvement pattern. The proof identifies $I_k$ as the ideal of $2 \\times 2$ minors of an explicit $2 \\times 3$ matrix (a determinantal ideal, not merely one that looks determinantal), and invokes classical determinantal-ideal theory (Bruns-Vetter) and an explicit rank-2 Jacobian witness on every component (Serre's criterion) for the algebraic and radicality halves respectively.",
      "title": "The coordinate ring of the k-fold iterated commutator locus for 2x2 matrices",
      "updated": "2026-09-04",
      "url": "https://arxiv.org/abs/2609.05386"
    },
    {
      "age_days": 5,
      "arxiv_id": "2609.05652",
      "authors": [
        "Xiyu Hu"
      ],
      "content_date": "2026-09-04",
      "freshness": "fresh",
      "id": "arxiv:2609.05652",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-04",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "We develop two transition principles for lower-bounding value sets generated by structured sequences over prime fields. A reciprocal-affine family with $M$ internal transitions and bounded quotient multiplicity has image size $\\gg \\min{M,p}^{8/15}$. This recovers the factorial-residue bound and yields the same exponent for arithmetic Pochhammer products, Gaussian $q$-factorials, derangement numbers, and the numbers of ordered subsets. A second theorem treats nonzero sequences whose consecutive ratios evolve under a nondegenerate M\"obius transformation: their value sets have size $\\gg \\min{M,p}^{1/2+η}$ for an absolute constant $η>0$. As consequences, fixed rows of Pascal's triangle and the initial half-blocks of the Catalan and central binomial sequences exceed the square-root scale. The proofs combine transition quotients with, respectively, Cartesian-product point-line incidence geometry and Bourgain's expansion-based incidence theorem in $\\mathrm{SL}_2(\\mathbb{F}_p)$.",
      "title": "Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem",
      "updated": "2026-09-04",
      "url": "https://arxiv.org/abs/2609.05652"
    },
    {
      "age_days": 5,
      "arxiv_id": "2609.05005",
      "authors": [
        "Jineon Baek",
        "Byung-Hak Hwang",
        "Joonhyun La",
        "Hongseok Yang"
      ],
      "content_date": "2026-09-04",
      "freshness": "fresh",
      "id": "arxiv:2609.05005",
      "kind": "paper",
      "label": "negative",
      "matched_signals": [],
      "published": "2026-09-04",
      "score": 0.5,
      "source": "arxiv-ai4math-core",
      "summary": "Algebraic combinatorics often seeks bijections that explain identities between distributions object by object. Encoding combinatorial objects as words lets automata theory study such a bijection as a word-to-word computation and measure its memory, input access, and control of output order. This refines existence questions by asking which computational mechanisms a bijection requires. We develop this viewpoint for Dyck paths. Our motivating example is the $q,t$-Catalan polynomial. Let $D_n$ be the set of Dyck paths of semilength $n$, let $D=\\bigcup_{n\\ge 0}D_n$, and let $area, dinv, bounce \\colon D\\to\\mathbb{N}$ be the standard statistics. Then, \\[ C_n(q,t)=\\sum_{P\\in D_n}q^{area(P)}t^{bounce(P)} =\\sum_{P\\in D_n}q^{dinv(P)}t^{area(P)}. \\] Haglund's zeta map $ζ\\colon D\\to D$ gives a bijective proof: it preserves semilength and sends $(dinv,area)$ to $(area,bounce)$. By contrast, the full symmetry $C_n(q,t)=C_n(t,q)$ still lacks a direct explanation: no explicit, uniform, semilength-preserving bijection is known that swaps area and dinv on every Dyck path. Polyregular maps from automata theory provide a natural computational starting point, but we prove that neither $ζ$ nor the classical height-sweep bijection witnessing Narayana symmetry is polyregular. The missing mechanism is global ordering by numerical levels whose range grows with the input. We call this a \\emph{rank sort} and introduce \\emph{weighted-rank polyregular maps} (WRP), extending polyregular maps by one such sort and containing both bijections. Nevertheless, WRP is a proper subclass of deterministic logspace. We prove that $ζ^{-1}$ lies outside WRP and that no WRP map can realise a semilength-preserving area-dinv swap. Thus the rank-sorting strategy behind $ζ$ cannot be extended within WRP to exchange the two statistics.",
      "title": "A Computational Obstruction to Swapping Area and Dinv: An Automata-Theoretic View of the $q,t$-Catalan Symmetry",
      "updated": "2026-09-04",
      "url": "https://arxiv.org/abs/2609.05005"
    }
  ],
  "lookback_days": 21,
  "schema": "ai4math-radar-run-v1",
  "timezone": "America/Los_Angeles"
}