{ "counts": { "adjacent": 14, "core": 3, "errors": 0, "negative": 73, "total": 90 }, "date": "2026-09-06", "errors": [], "fresh_content_days": 21, "generated_at": "2026-09-06T17:39:17Z", "items": [ { "age_days": 6, "arxiv_id": "2608.30238", "authors": [ "Gregory Morse" ], "content_date": "2026-08-31", "freshness": "fresh", "id": "arxiv:2608.30238", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "verifier_guided_reasoning" ], "published": "2026-08-31", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Classical lower bounds show that multiplying two degree-three polynomials over $\\mathbb F_2$ requires nine scalar products in bilinear or quadratic models. They do not settle unrestricted Boolean multiplicative complexity: an XOR--AND circuit may reuse nonlinear intermediate wires, and Boolean equality is taken modulo $x_i^2=x_i$, so a multiplication can lower algebraic degree. Let $\\operatorname{Mul}_4:\\mathbb F_2^8\\to\\mathbb F_2^7$ output the seven coefficients of the product of two four-term binary polynomials. We prove that its unrestricted XOR--AND multiplicative complexity is exactly nine. This resolves, for a natural vector-valued quadratic function, the Boyar--Find question of whether a quadratic-circuit lower bound can persist against unrestricted nonlinear reuse. The proof is structural rather than exhaustive. A useful purely quadratic prefix is forced onto the three rational places of $\\mathbb P^1(\\mathbb F_2)$. In a hypothetical eight-AND circuit, the unique non-useful gate must carry a cubic high part. Any useful continuation then forces a rational tangent and exposes a first Hasse jet, while exterior jet separation together with Boolean idempotence prevents the same defect from exposing the second Hasse jet. The required useful suffix therefore cannot exist. A complete Lean 4 formalization verifies the Boolean-ANF semantics, the unrestricted circuit model, and the exact theorem; it uses no project-specific axiom or native decision procedure. The same zero-defect flag argument gives multiplicative complexity six for three-term multiplication, and the method isolates the multi-defect obstruction for five terms.", "title": "Unrestricted Boolean Multiplicative Complexity of Four-Term Binary Polynomial Multiplication: Rational Places, Hasse Jets, and the Failure of Nonlinear Feedback", "updated": "2026-08-31", "url": "https://arxiv.org/abs/2608.30238" }, { "age_days": 8, "arxiv_id": "2608.29270", "authors": [ "Hojae Han", "Jongyoon Kim", "Sanghyeok Park", "Dongwook Cheon", "Yeachan Park", "Myung Jae Jeon", "Sunjong Choe", "Soonho Kong", "Wonseok Hur", "Seung-won Hwang", "Donghoon Hyeon" ], "content_date": "2026-08-29", "freshness": "fresh", "id": "arxiv:2608.29270", "kind": "paper", "label": "core", "matched_signals": [ "autoformalization", "lean_formal_proving_agents" ], "published": "2026-08-29", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Autoformalization translates informal mathematical theorems into code for proof assistants such as Lean. A central challenge is that current evaluation metrics can accept type-correct but misaligned statements or reject correct statements written in a different formulation. Inspired by Pass@$k$, we propose SA-Pass (*Semantic Alignment Pass*), which tests formal statements using auxiliary statements called *shadows* that characterize the intended statement. A generated statement receives full credit only when it compiles, implies each shadow (forward check), and is implied by their conjunction (backward check). We instantiate SA-Pass in ShadowBench, a Lean 4 full autoformalization benchmark of 178 postgraduate- to research-level problems spanning eight mathematical areas. Claude Code (Opus 4.8) with Numina-Lean-Agent reaches $61.8\\%$ compile rate and $11.2\\%$ SA-Pass. Across outputs generated by six agentic configurations, SA-Pass achieves $98.8\\%$ binary agreement with expert judgments. An early version of ShadowBench served as the benchmark for Track 4 of the ICML 2026 AI4Math Challenge.", "title": "SHADOWBENCH: Toward Reliable Automatic Evaluation of Semantic Alignment in Autoformalization", "updated": "2026-09-02", "url": "https://arxiv.org/abs/2608.29270" }, { "age_days": 9, "arxiv_id": "2608.28433", "authors": [ "Shuze Chen", "Kunal Marwaha", "Xiaoyang Lu", "Henry Yuen", "Tianyi Peng" ], "content_date": "2026-08-28", "freshness": "fresh", "id": "arxiv:2608.28433", "kind": "paper", "label": "core", "matched_signals": [ "lean_formal_proving_agents", "verifier_guided_reasoning" ], "published": "2026-08-28", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Proof assistants such as Lean 4 promise the paradigm of formally verified mathematics, but large-scale formalization projects have faced major barriers to entry, including the need for expertise in formal verification (as well as the underlying mathematics) and the significant time required for writing formal proofs. AI coding agents have dramatically reduced these barriers; human users can now use natural language to prompt agents to write complex proofs in Lean. This opens up the intriguing possibility of internet-scale mathematical collaboration involving both humans and AI agents, where correctness is machine-checked. To realize this possibility, we introduce Prove2Me (https://prove2.me), an open collaborative platform for formalizing mathematics. Users launch formalization \"missions\", to which AI agents contribute formal proofs toward completion. We designed mechanisms and a specialized harness in Prove2Me that enable large-scale collaboration so that agents can build on one another's work and freely reuse existing results. In doing so, Prove2Me aims to turn math formalization into a scalable, crowd-sourced effort open to anyone with an agent.", "title": "Prove2Me: An Open Collaborative Platform for Scaling Math Formalization", "updated": "2026-08-31", "url": "https://arxiv.org/abs/2608.28433" }, { "age_days": 3, "arxiv_id": "2609.03478", "authors": [ "Ján Pastorek" ], "content_date": "2026-09-03", "freshness": "fresh", "id": "arxiv:2609.03478", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-09-03", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "We report on our ongoing project to develop a computational pipeline, AutoGraphForge, for an automated graph-theoretic conjecturing-refuting-formalizing-proving system. Conjecture generation is counterexample-guided and runs in rounds: a Graffiti3 generator proposes conjectures over a small, evolving snapshot table $T$ (initially a few hundred graphs with their computed invariants) that grows only by counterexamples to its own conjectures. A novelty filter of $559$ classical and folklore relations, closed under transitive composition and linear identity substitution, decides via a linear program whether a candidate is already implied by known results. Surviving candidates are tested against a dataset of about $348,000$ graphs, unioning the complete House of Graphs invariant export, the exhaustive census of all connected graphs on at most nine vertices, several extremal families (strongly regular, minimal Ramsey, Cayley, cages, barbells, lollipops, spiders), and random models. Counterexample-search algorithms then attack the remainder. Run for several rounds on an HPC cluster, the loop yields $6,522$ conjectures that survived the refutation dataset, the novelty filter and every active-search run -- among them nontrivial relations between the annihilation number and the edge-cover number for bipartite and regular graphs, which we prove by hand. A subsequent formalization and proving stage deterministically translates each surviving conjecture into a Lean 4 statement skeleton; every candidate proof is kernel-verified against a pinned mathlib4 and our custom invariant preamble. This stage integrates two neural provers -- DeepSeek-Prover-V2-671B (served with vLLM) and the Lean-specialised OProver-32B -- behind the independent kernel check. It is implemented end-to-end and passes initial sanity checks, with the full pipeline currently running on the cluster.", "title": "AutoGraphForge: Towards Automated Graph Theory Discovery", "updated": "2026-09-03", "url": "https://arxiv.org/abs/2609.03478" }, { "age_days": 7, "arxiv_id": "2608.29750", "authors": [ "Yinjie Li" ], "content_date": "2026-08-30", "freshness": "fresh", "id": "arxiv:2608.29750", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-08-30", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\\in\\mathbb R^{n\\times n}$ satisfies $\\|A^{-1}\\|_F\\ge 2n(n+1)^{-1}\\|A\\|_{\\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Cheng proved the conjecture in odd dimensions, while Frankel and Urschel proved the even-dimensional case for $n\\ge1000$. We complete the remaining even-dimensional cases. Starting from the structural identities in Frankel--Urschel Lemma 2.1, we derive an exact global defect budget and combine binary rounding with Gram projection. A refined ten-row obstruction handles every even $n\\ge66$; a finite exact calculation handles $4\\le n\\le64$, $n\\ne6$; and a separate multi-column energy argument treats $n=6$. The order-two case follows from a direct calculation. The new even-dimensional proof has been formalized in Lean 4, with Frankel--Urschel Lemma 2.1 as its sole external mathematical input. Together with Cheng's odd-dimensional theorem, this proves the S-matrix conjecture in every dimension.", "title": "The S-matrix conjecture", "updated": "2026-08-30", "url": "https://arxiv.org/abs/2608.29750" }, { "age_days": 7, "arxiv_id": "2608.29592", "authors": [ "Jungyeom Kim", "Jihyeok Park" ], "content_date": "2026-08-30", "freshness": "fresh", "id": "arxiv:2608.29592", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-08-30", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "We resolve three open problems concerning parsing expression grammars (PEGs). We construct a single language $C$ satisfying $C\\in\\mathsf{LIN}\\cap\\mathsf{PEG}$ and $C^R\\in\\mathsf{LIN}\\setminus\\mathsf{PEG}$. This proves that some linear context-free language is not a PEG language and that PEG languages are not closed under reversal, confirming a conjecture of Loff, Moreira, and Reis. Factoring the same witness resolves the concatenation-closure problem of Rubtsov and Chudinov negatively, in the strong form $\\mathsf{PEG}\\cdot\\mathsf{REG}\\not\\subseteq\\mathsf{PEG}$ despite $\\mathsf{REG}\\cdot\\mathsf{PEG}\\subseteq\\mathsf{PEG}$. It also refutes closure under Kleene star, homomorphisms, and substitutions. Our main technique converts scaffolding automata (SCAs), which characterize reversals of PEG languages, into dynamic data structures in the cell-probe model. For any suitably local serialization of a problem with preprocessing, updates, and a final Boolean query, an SCA recognizer yields an exact deterministic cell-probe data structure whose operation costs are proportional to the corresponding encoding lengths. Cell-probe lower bounds can therefore prove SCA non-membership and, by reversal, PEG non-membership. We apply this transfer to Multiphase Inner Product using one-symbol update blocks and a query suffix of length $O(\\log n)$, while keeping both the language and its reversal linear context-free. Ko's cell-probe lower bound then yields the witness above. The arguments are additionally formalized in Lean 4.", "title": "Separating Parsing Expression Grammars using Cell-Probe Lower Bounds", "updated": "2026-08-30", "url": "https://arxiv.org/abs/2608.29592" }, { "age_days": 8, "arxiv_id": "2608.28997", "authors": [ "Maher Kallel", "Mohamed El Louadi" ], "content_date": "2026-08-29", "freshness": "fresh", "id": "arxiv:2608.28997", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-08-29", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "In May 2026 an OpenAI model produced a counterexample to the Erdős unit distance conjecture. Five mathematicians published a human-verified version the same day, and the result entered the literature within weeks. In August 2026 the same laboratory published ten mathematical and theoretical computer science results, each accompanied by a machine-checkable Lean 4 certificate with no unproved steps. Four weeks later, one remained the subject of an unresolved dispute over whether its formalization meant what it claimed. We argue that this difference is structural. We distinguish three layers of verification: derivational validity, which a kernel checks; representational fidelity, whether the formal statement means the intended question; and epistemic significance. Only the first is mechanizable. Making it effectively free therefore does not eliminate verification work but shifts the burden to layers dependent on scarce expert attention. Measurements of the August corpus illustrate the shift. The kernel-checked proofs total 20.6 MB, while the statements requiring human audit total 55.6 KB, a ratio of 379 to 1. Yet those statements contain 218 bespoke definitions rather than relying on community-vetted ones. The audit surface is therefore small in volume but irreducibly expert. We argue that machine checking produces verification abundance while leaving adjudication scarce. We propose a six-category taxonomy of representational mismatch, a disclosure schema for machine-generated mathematical claims, and implications for software, cryptography, and regulated decision systems.", "title": "Verification abundance, adjudication scarcity: what happens to mathematical knowledge when proof checking becomes free", "updated": "2026-08-29", "url": "https://arxiv.org/abs/2608.28997" }, { "age_days": 4, "arxiv_id": "2609.02978", "authors": [ "Diar Heidary" ], "content_date": "2026-09-02", "freshness": "fresh", "id": "arxiv:2609.02978", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-02", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "The fully independent SparseStack sketch is a vertical stack of $s$ independent CountSketch matrices, scaled by $s^{-1/2}$, so that every column has exactly $s$ nonzero entries. We prove that it is an oblivious subspace embedding for $d$-dimensional subspaces with distortion $ε$ and failure probability $δ$ when $m = O((d+\\log(1/δ))/ε^2)$ and $s = O(\\log(d/δ)/ε)$, with explicit constants. These are the parameters conjectured by Nelson and Nguyen (FOCS 2013) for this construction; the row count is optimal by their lower bound. The proof bounds the even moments of the Gram error. A conditional-expectation coupling replaces each signed one-hot column selector by a vector with independent three-point entries, at the cost of a constant factor per moment order. The three-point law has a three-dimensional $L_2$ space, so multiplication by an entry is a $3 \\times 3$ Jacobi matrix, and the $2q$-th moment becomes a vacuum matrix element of a deterministic operator on a finite tensor product, graded by total occupation. The operator has three grade bands, and we bound each band on the grade-$ν$ sector by $C(\\sqrt{(d+ν+1)/m} + (d+ν+1)/m + (ν+1)/s)$ with $C = 3+\\sqrt{2}$. The key step is a shared-factor inequality: each row block of the positive operator attached to one tensor slot is rank one with trace $d$, and the sum over $\\ell$ slots sharing the same external factor has norm at most $d+\\ell-1$. A $2q$-step expansion and Markov's inequality complete the argument, which is finite-dimensional and does not use Gaussian comparison. The theorem has been formally verified in Lean 4. The proof was developed with AI systems under the author's direction, as disclosed in the paper.", "title": "SparseStack Is an Optimal Oblivious Subspace Embedding", "updated": "2026-09-02", "url": "https://arxiv.org/abs/2609.02978" }, { "age_days": 6, "arxiv_id": "2609.00101", "authors": [ "Qianli Ma" ], "content_date": "2026-08-31", "freshness": "fresh", "id": "arxiv:2609.00101", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-08-31", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We completely solve Colombo's 1928 determinant problem. For distinct real $x_1,\\ldots,x_N$, $N\\geq 2$, and an integer $D\\geq 1$, we prove that $\\det[(x_j-x_i)^D]\\neq 0$ if and only if $D\\geq N-1$ and either $N$ is even or $D$ is even. The even-exponent case follows from Dyn--Goodman--Micchelli (1986); the remaining odd case is proved by a strict Pfaffian sign theorem. The new odd-exponent theorem and its complete proof chain have also been formalized in Lean 4.", "title": "Colombo's Determinant Problem", "updated": "2026-08-31", "url": "https://arxiv.org/abs/2609.00101" }, { "age_days": 9, "arxiv_id": "2608.28133", "authors": [ "Zhaorui Wu" ], "content_date": "2026-08-28", "freshness": "fresh", "id": "arxiv:2608.28133", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-08-28", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "A $p$-ary bent partition of $\\mathbb{F}_p^n$ is a partition into $K$ nonempty cells such that every balanced assignment of its cells to $\\mathbb{F}_p$ produces a bent function. It was asked whether every possible depth $K$ is a power of $p$; for general $p$, previous affirmative results required regularity or cell-symmetry hypotheses. We prove the stronger unconditional statement that, for every nonzero $h$, exactly $p^n/K$ points remain in the same fine cell under translation by $h$. Thus the fine cells form a partitioned difference family and the fine label map is zero-difference balanced. Consequently $K\\mid p^n$, so $K=p^t$; nonempty cells further give $1\\le t