{ "counts": { "adjacent": 16, "core": 1, "errors": 0, "negative": 72, "total": 89 }, "date": "2026-09-15", "errors": [], "fresh_content_days": 21, "generated_at": "2026-09-15T19:09:04Z", "items": [ { "age_days": 2, "arxiv_id": "2609.14808", "authors": [ "Andre Panossian" ], "content_date": "2026-09-13", "freshness": "fresh", "id": "arxiv:2609.14808", "kind": "paper", "label": "core", "matched_signals": [ "autoformalization", "verifier_guided_reasoning" ], "published": "2026-09-13", "score": 6.5, "source": "arxiv-ai4math-core", "summary": "Scientific autoformalization turns verbal accounts into executable mathematics, but executable code does not settle which model has been constructed. We examine two sources of structural uncertainty: the formalizer that generates a response law, and the recurrence that turns that law into trajectories. In secondary analyses of an openly archived crossed experiment, we studied 320 response maps generated by two pinned language-model formalizers from five engineered cognitive accounts within one sparse quadratic grammar and 16 randomized blocks. With whole blocks held out, source-account identity was recovered at 78.8% accuracy (chance 20.0%) and formalizer identity at 96.3% (chance 50.0%; both p < 0.001). Program size was the stronger single feature family; a pre-specified exploratory comparison found no stable source-account predictive gain from local geometry beyond size. Holding every response map fixed, we then evaluated five recurrence families spanning 33 configurations and 1,013,760 finite-horizon trajectories. Added feedback, projection and leak produced sharply different outcome distributions. The consequential distinction was which comparisons survived: median cross-recurrence rank concordance was 0.73 for endpoint magnitude but 0.05 for settling, among the configuration pairs with defined rankings. Thus a common mathematical language did not erase translation provenance, and robust ordering under one observable did not transfer to another. Scientific autoformalization is usefully studied as model-space construction: the generated ensemble and its dynamical embedding are both part of the specification supporting a scientific claim.", "title": "Transformed in Translation: Two-Stage Structural Uncertainty in LLM-Based Scientific Autoformalization", "updated": "2026-09-13", "url": "https://arxiv.org/abs/2609.14808" }, { "age_days": 2, "arxiv_id": "2609.14351", "authors": [ "Yinjie Li" ], "content_date": "2026-09-13", "freshness": "fresh", "id": "arxiv:2609.14351", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-09-13", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "We prove the Colomo-Pronko conjecture for alternating sign matrices with a prescribed square of zeros at a corner, for all matrix sizes and freezing parameters. A known multiple-integral formula for the frozen-corner count yields determinant representations built from fixed polynomial kernels. We relate these kernels to the conjectured determinant through an inverse identity for the commutator of a signed Pascal matrix with reversal. In odd dimension, the comparison uses the one-dimensional nullspace and projection along it to eliminate the central coordinate. Combined with the asymptotic analysis of Colomo and Pronko, our result removes the conjectural assumption from their GUE Tracy-Widom fluctuation theorem for the intersection of the frozen boundary with the main diagonal in uniformly random alternating sign matrices. The finite-dimensional algebraic core of the proof has been formalized in Lean 4.", "title": "The Colomo-Pronko conjecture for frozen-corner alternating sign matrices", "updated": "2026-09-13", "url": "https://arxiv.org/abs/2609.14351" }, { "age_days": 3, "arxiv_id": "2609.14188", "authors": [ "Romain Popescu" ], "content_date": "2026-09-12", "freshness": "fresh", "id": "arxiv:2609.14188", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-09-12", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "Let $s(m)$ be the Conway subprime function define the binary operation on the natural numbers $x \\circ y= s(x + y)$, and denote by $C_n$, $n \\ge 0$, the sequence of subsets of natural numbers defined by $C_0 = \\{1\\}$, and $C_{n+1} = C_n \\cup (C_n \\circ C_n)$. We prove the conjecture by Caragiu, Vicol and Zaki that $$\\lim_{n\\to \\infty} \\frac{ | C_{n+1}| }{ | C_n |}= \\frac{1+\\sqrt{5}}{2}.$$ The underlying mathematical proof in this paper was constructed with some algorithmic assistance from GPT-6 Astra and its correctness has been formally verified using the Lean 4 proof assistant.", "title": "Golden-ratio growth of Conway's subprime closure", "updated": "2026-09-12", "url": "https://arxiv.org/abs/2609.14188" }, { "age_days": 4, "arxiv_id": "2609.13412", "authors": [ "Yinjie Li" ], "content_date": "2026-09-11", "freshness": "fresh", "id": "arxiv:2609.13412", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning", "lean_formal_proving_agents" ], "published": "2026-09-11", "score": 4.9, "source": "arxiv-ai4math-core", "summary": "Pate proved ordinary irreducible-immanant permanental dominance through order $13$ and identified $(4,4,3,3)$ as the sole remaining order-$14$ case, with $(5,4,3,3)$ and $(3^5)$ forming the order-$15$ frontier. These three cases are settled here; consequently $d_λ(A)/f^λ\\le \\operatorname{per}(A)$ for every partition $λ\\vdash n$ with $n\\le15$ and every complex Hermitian positive-semidefinite matrix $A$. The argument also yields results beyond this finite frontier: an exact four-term bridge for $(4,4,3,3)$, the uniform family $(m,4,3,3)$, a two-parameter family $(a,b,3,3)$ for $a\\ge b\\ge4$ and $5a\\ge8b$, and a long-first-row criterion for arbitrary fixed tails. These results arise from explicit specializations of Pate's $W$-function positivity framework using partial swaps, Young projectors, Pieri--content identities, and branching data. For $(3^5)$, an exact Farkas certificate shows that the central-projector partial-swap cone is insufficient; a branching-refined one-swap construction escapes this obstruction and yields a positive $106+19$-witness certificate. Boundary-compression and node-moving results further describe the reach and limitations of the local-filter method. All finite certificates are checked by exact integer or rational arithmetic and are supplied as ancillary material. The order-$14$ bridge is additionally formalized and kernel-checked in Lean 4 for all complex Hermitian positive-semidefinite matrices, including the exact coefficient normalization and the deduction of $(4,4,3,3)$ permanental dominance from four explicitly stated Pate inequalities.", "title": "Lieb's Permanental Dominance Conjecture for Ordinary Immanants through Order Fifteen", "updated": "2026-09-11", "url": "https://arxiv.org/abs/2609.13412" }, { "age_days": 1, "arxiv_id": "2609.15642", "authors": [ "Henning Ulfarsson" ], "content_date": "2026-09-14", "freshness": "fresh", "id": "arxiv:2609.15642", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-14", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We give an exact algorithm counting the permutations that avoid a fixed pattern from the following family: the direct sum of an increasing pattern and the pattern 231. The first members of the family are 1342 and 12453. For each member, the algorithm computes the number of avoiding permutations of every length up to a given bound using polynomially many arithmetic operations and polynomially many stored integers, with degrees that grow linearly in the length of the pattern. We first obtain an exact recurrence by reading a permutation from left to right and recording, at each step, the constraints that the letters read so far impose on those still unread. Its state space grows exponentially, so evaluating it directly takes exponential time. We then show that part of the state is protected: later steps carry it along unchanged and do not depend on it. Factoring the protected part out turns the recurrence into a dynamic program with polynomially many stored transfer entries, and this gives the polynomial bounds for every member of the family. For the pattern 12453, a translation symmetry sharpens the bounds to degree seven for the operations and degree four for the storage. Separately written implementations and exact Chinese-remainder certification determine the number of 12453-avoiding permutations of every length up to 150. The previously published series reached length 38. The same tables also generate uniformly random avoiders in polynomial time. We illustrate this with a heatmap of one million 12453-avoiding permutations of length 300 sampled with floating-point tables. The counting recurrences for 1342 and 12453 are verified in the Lean 4 proof assistant.", "title": "Protected tails and polynomial-time enumeration of permutations avoiding a direct sum of an increasing pattern and 231", "updated": "2026-09-14", "url": "https://arxiv.org/abs/2609.15642" }, { "age_days": 1, "arxiv_id": "2609.14879", "authors": [ "Aaron Gregory" ], "content_date": "2026-09-14", "freshness": "fresh", "id": "arxiv:2609.14879", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-14", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "A stencil computation repeatedly updates every cell of a grid from its neighbours' values at the previous timestep. Simulating T steps on N cells directly costs Theta(NT), and a line of work beginning with Ahmad et al. reduces this by composing many timesteps into one linear operator and applying it with a Fast Fourier Transform. That technique needs to know which cells will still obey the same operator when the composed step ends, and in a free-boundary problem they do not: the region governed by a given rule is determined by the solution and moves as it evolves. We study one spatial dimension, a three-point stencil with time-varying coefficients, and a computed region that is a single interval whose two endpoints move by arbitrary amounts at every step, revealed online. Let B be the horizon plus the total variation of the boundary trajectory. We give a schedule whose work is O((B+N) log T log(N+B)) and whose span is O(T log T log(N+B)), and we prove that the values it computes are exact. The best existing bound for a region that moves requires its boundary to travel at most one cell per timestep. We drop that requirement and lose nothing by it: a boundary obeying it has B <= 3T, so our bound stays near-linear on every trajectory the earlier result covers. Elsewhere, B grows only by the distance the boundary actually travels -- one jump of width N costs T + 2N. The reason total variation suffices is that everything the two endpoints touch over a time window of any length lies in two intervals, one per endpoint. This cannot be relaxed: with p regions the bound degrades by a factor p, and at p = sqrt(T) there is an instance on which the work is Theta(T^{3/2}) while B + N = Theta(T). All results are machine-checked in Lean 4, apart from the classical convolution bound, which is imported as an interface.", "title": "Fast Stencil Computations on a Single Arbitrarily Moving Interval", "updated": "2026-09-14", "url": "https://arxiv.org/abs/2609.14879" }, { "age_days": 1, "arxiv_id": "2609.14912", "authors": [ "Celio Boulay", "Alexander Chai", "Anthony Chang", "Thomas Moulin" ], "content_date": "2026-09-14", "freshness": "fresh", "id": "arxiv:2609.14912", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-14", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "The mathematics of Origami have been well studied and shown to develop several interesting results. We use Lean 4 tactics and build on Mathlib to redefine the 7 Huzita operations as theorems instead of axioms and prove their existence. We develop proofs for important origami constructions (such as trisecting an angle), implement origami-constructible numbers and prove the associated Cardano's formula, and formalize Haga's theorem. A Crease Pattern Inspector explores physical folding by providing a full pipeline to create and visualize models constrained by the Huzita formalism. The Lean codebase brings 100+ theorems and lemmas.", "title": "A Lean Paper About Paper: A Formal Framework for Origami", "updated": "2026-09-14", "url": "https://arxiv.org/abs/2609.14912" }, { "age_days": 3, "arxiv_id": "2609.14102", "authors": [ "Lars Warren Ericson" ], "content_date": "2026-09-12", "freshness": "fresh", "id": "arxiv:2609.14102", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-12", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "For a finite set $S$ with $\\lvert S\\rvert = N$, the number of families $\\mathcal{B} \\subseteq \\mathcal{P}(S)$ that are topological bases is $\\#(N) = \\sum_{\\mathcal{T} \\in \\operatorname{Top}(S)} 2^{\\lvert\\mathcal{T}\\rvert - \\lvert\\mathcal{M}_{\\mathcal{T}}\\rvert}$, where $\\mathcal{M}_{\\mathcal{T}}$ is the canonical minimal basis of minimal open neighborhoods. The identity is proved in Lean 4 / Mathlib (`CARDB.lean`): bases generating $\\mathcal{T}$ are exactly the sets with $\\mathcal{M}_{\\mathcal{T}} \\subseteq \\mathcal{B} \\subseteq \\mathcal{T}$. The small-$N$ table and the discrete-dominance sandwich are proved in `CARDB/SmallN.lean` and `CARDB/Asymptotics.lean`.", "title": "On the Number of Distinct Topological Bases of a Finite Set of Size $N$", "updated": "2026-09-12", "url": "https://arxiv.org/abs/2609.14102" }, { "age_days": 3, "arxiv_id": "2609.13780", "authors": [ "Shogo Saitou", "Mashu Noguchi" ], "content_date": "2026-09-12", "freshness": "fresh", "id": "arxiv:2609.13780", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-12", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We mechanized proof of Gödel's first and second incompleteness theorems, Solovay's arithmetical completeness theorem of \\mathbf{GL}, and related results in the Lean 4 theorem prover.", "title": "Mechanizing Gödel's incompleteness Theorems and Provability Logic", "updated": "2026-09-12", "url": "https://arxiv.org/abs/2609.13780" }, { "age_days": 3, "arxiv_id": "2609.13703", "authors": [ "Jiarui Yao", "Jiaxi Zhao", "Xiangxin Zhou" ], "content_date": "2026-09-12", "freshness": "fresh", "id": "arxiv:2609.13703", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-12", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "In the best-arm identification problem, we are given $n$ stochastic arms with unknown means and wish to identify the arm with the largest mean with probability at least $1-δ$, using as few samples as possible. We consider independent Gaussian rewards with unit variance and means in $[0,1]$. Chen and Li [2016] conjectured that the instance-wise sample complexity of this problem is characterized by the gap entropy, up to an additive term arising from the two-arm problem. In this paper, we resolve their gap-entropy and almost instance-wise optimality conjectures. For an instance $I$, let $Δ_{[i]}$ be the gap between the largest and the $i$-th largest mean, let $H(I)=\\sum_{i=2}^{n}Δ_{[i]}^{-2}$, and let Ent$(I)$ denote the entropy of the normalized complexities of its dyadic gap groups. For every $0<δ<0.1$, we show that the order-oblivious instance-wise lower bound is $ Θ (H(I)[\\log(1/δ)+Ent(I)]). $ We also give a single $δ$-correct algorithm with expected sample complexity $ O ( H(I)[\\log(1/δ)+Ent(I)] +D\\log(e+\\log(e+D))),D=Δ_{[2]}^{-2}, $ without prior knowledge of the gaps. Our lower bound removes the dyadic-gap and monotonicity restrictions of previous work, and our upper bound removes the additional polylogarithmic factor multiplying the two-arm term. Thus, a single algorithm attains the instance-wise lower bound up to an additive two-arm term. The main theorems have been formalized and proved in Lean 4.", "title": "Gap Entropy and Almost Instance-Wise Optimal Best-Arm Identification", "updated": "2026-09-12", "url": "https://arxiv.org/abs/2609.13703" }, { "age_days": 3, "arxiv_id": "2609.13641", "authors": [ "Angel Ivanov Raychev" ], "content_date": "2026-09-12", "freshness": "fresh", "id": "arxiv:2609.13641", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-12", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "We classify the polyominoes obtained by adjoining four straight arms to a single square, allowing zero arm lengths, according to their ability to tile rectangles, half-strips, bent strips, quadrants, strips, half-planes, and the plane. We also classify their ability to tile an integer enlargement of themselves. Tiles occupy whole square-grid cells; translations, rotations, and reflections are permitted. Exactly five capability profiles occur. For the family $P(n,1,1,0)$, the rectangle profile holds for $n\\le3$ and the bent-strip profile, with no half-strip or rep-tiling, for every $n\\ge4$. A cross with four positive arms tiles the plane precisely when two opposite arms have length one; it never tiles a half-plane. Explicit periodic constructions and geometric obstructions are combined with finite symbolic case certificates. A Lean 4 development verifies the full classification for every natural four-tuple, including the interpretation of the certificates as statements about arbitrary infinite tilings. The account incorporates the author's 2020--2021 L- and T-polyomino work, reconstructs Dahlke's gun argument, and documents the subsequent AI-assisted proof development and formalization.", "title": "Four-arm polyominoes in Golomb's hierarchy: A complete classification with Lean verification", "updated": "2026-09-12", "url": "https://arxiv.org/abs/2609.13641" }, { "age_days": 5, "arxiv_id": "2609.12266", "authors": [ "Behnam Hashemi", "Yuji Nakatsukasa" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "arxiv:2609.12266", "kind": "paper", "label": "adjacent", "matched_signals": [ "lean_formal_proving_agents" ], "published": "2026-09-10", "score": 3.5, "source": "arxiv-ai4math-core", "summary": "It is known that the Sherman--Morrison (SM) formula is not numerically stable. In recent work, we introduced SMIR, an algorithm that incorporates iterative refinement to enhance the SM backward error. In this paper we take a different route: adapting an algorithm of Govaerts, originally designed for general bordered linear systems, we develop a modified Sherman--Morrison (MSM) method whose built-in self-correction makes it surprisingly resilient. Whereas standard SM requires the solution of two $n\\times n$ linear systems, MSM requires three; by contrast, SMIR requires $2+k$ solves, where $k$ is the number of IR steps and can be substantially larger than three, when IR converges slowly. We then derive backward and forward error bounds for both SM and MSM. The SM backward error bound established here is stronger than the one proved in [Hashemi \\& Nakatsukasa 2026]: it accounts for every rounding error and holds with no conditions on the size of the capacitance. From these bounds we extract growth factors that are cheap to compute a posteriori and can be used to certify the backward and forward stability of SM and MSM on a given problem. In our experiments, MSM consistently produces backward stable solutions, and is therefore observed to be forward stable as well. A formal proof of the stability --- or instability --- of MSM remains an open problem.", "title": "Error bounds for the Sherman-Morrison formula and its modification with improved stability", "updated": "2026-09-10", "url": "https://arxiv.org/abs/2609.12266" }, { "age_days": 0, "authors": [ "Jiedong Jiang" ], "content_date": "2026-09-15", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:5121184fc223", "kind": "github_update", "label": "adjacent", "matched_signals": [ "seed_author:Jiedong Jiang" ], "published": "2026-09-15", "repo": "leanprover-community/mathlib4", "score": 2.0, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: feat(CategoryTheory/AB5): AB5 instance of Ab with universe variables (#41737)", "updated": "2026-09-15", "url": "https://github.com/leanprover-community/mathlib4/commit/5121184fc223fa6f3e4239631ec3ccd340faf781" }, { "age_days": 0, "authors": [ "Jiedong Jiang" ], "content_date": "2026-09-15", "freshness": "fresh", "id": "github:leanprover-community/mathlib4:5363421222b9", "kind": "github_update", "label": "adjacent", "matched_signals": [ "seed_author:Jiedong Jiang" ], "published": "2026-09-15", "repo": "leanprover-community/mathlib4", "score": 2.0, "source": "mathlib4-github", "summary": "Recent commit on leanprover-community/mathlib4.", "title": "leanprover-community/mathlib4: chore(Algebra/MonoidHom): rename to `.ofClass` (#43755)", "updated": "2026-09-15", "url": "https://github.com/leanprover-community/mathlib4/commit/5363421222b9775884fce36ee0674a4a4373d530" }, { "age_days": 1, "arxiv_id": "2609.15757", "authors": [ "Julius A. Zeiss" ], "content_date": "2026-09-14", "freshness": "fresh", "id": "arxiv:2609.15757", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning" ], "published": "2026-09-14", "score": 1.9, "source": "arxiv-ai4math-core", "summary": "Entanglement can reduce the communication required for coding tasks, but establishing the minimum achievable cost is essential to understanding its limits. We address this question in a zero-error source-coding task where Alice receives a word and Bob knows an unordered pair of candidates containing it. Alice does not know the pair and must enable Bob to identify her word without error using shared entanglement and one classical message. The candidates satisfy a balanced-difference promise: for words in $\\mathbb{Z}_q^n$ with $n=q\\ell$, each residue modulo $q$ occurs exactly $\\ell$ times in their coordinatewise difference. For all integers $q\\geq2$ and $\\ell\\geq1$, we prove that the task requires exactly $n$ messages when $(q-1)\\ell$ is even and two messages when it is odd. These minima allow arbitrary finite-dimensional shared states independent of the inputs and arbitrary local measurements. In even parity, this establishes optimality of an existing entanglement-assisted protocol. In odd parity, an explicit deterministic protocol achieves the optimum of one bit without entanglement. Our proof combines Fourier analysis with a combinatorial counting argument to determine the smallest eigenvalue of the associated graphs. In even parity, this resolves the spectral assertion of Cao et al.'s Conjecture 6.3 for balanced cyclic generalized Hadamard graphs. Together with an explicit odd-parity bipartition, this determines the quantum chromatic number as $n$ in even parity and $2$ in odd parity, where the classical chromatic number is also $2$. All lemmas, theorems, and corollaries are formalized and verified in Lean.", "title": "Optimal entanglement-assisted source coding under a balanced-difference promise", "updated": "2026-09-14", "url": "https://arxiv.org/abs/2609.15757" }, { "age_days": 1, "arxiv_id": "2609.15554", "authors": [ "Javier Aguilar Martín" ], "content_date": "2026-09-14", "freshness": "fresh", "id": "arxiv:2609.15554", "kind": "paper", "label": "adjacent", "matched_signals": [ "general_ai_math_reasoning" ], "published": "2026-09-14", "score": 1.9, "source": "arxiv-ai4math-core", "summary": "We study packings of annuli (\"rings\") of a common width into a disk, where a ring may nest inside the hole of a strictly larger one, a selection-oriented relative of the Recursive Circle Packing Problem. The two natural objectives, cardinality and contact area, genuinely diverge. For superincreasing radii (each exceeding the sum of all smaller ones) we prove that the descending greedy maximizes every positive, increasing, superadditive objective. Our main structural theorem shows more: the placement rule is irrelevant - any choice among feasible containers yields the lexicographically maximal feasible set, for containers of arbitrary shape and in every dimension. Both hypotheses are sharp: placement irrelevance holds for at most three rings and fails at four, and twin instances rule out every rule that is a function of the observable state. Write $ρ=\\max_i(\\sum_{j>i}r_j)/r_i$ for the violation of superincreasingness. The additive relaxation has universal threshold exactly $ρ=1$. In the geometric model we prove, with no tangency idealization, that the rigid four-ring family has infimum exactly the Tribonacci constant $T\\approx1.83929$. Yet $T$ is not the global threshold: an explicit golden family breaks placement obliviousness at $ρ=\\varphi+3\\varepsilon$ for every small $\\varepsilon>0$, proving $τ\\le\\varphi= 3, and deduce pseudo-unitary existence. The proof has four parts: explicit real coefficients and their quadratic identities; a contour calculation for a range of cubic Fourier coefficients; algebraic completion of all cubics; and categorical reconstruction. Matrix inversion supplies reflection, and an extension of the dimension-field automorphism supplies the positive-dimensional category.", "title": "Cyclic Haagerup-Izumi fusion categories at every odd order", "updated": "2026-09-14", "url": "https://arxiv.org/abs/2609.15986" }, { "age_days": 2, "arxiv_id": "2609.14314", "authors": [ "S P Suresh" ], "content_date": "2026-09-13", "freshness": "fresh", "id": "arxiv:2609.14314", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-13", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We present a new proof of weak normalization for intuitionistic natural deduction. The distinguishing features of this proof are that it works only with cuts rather than cut segments, provides explicit local rules for determining whether to contract a whole proof or reduce one of its subproofs, and in the latter case, which subproof to reduce. We also discuss a formalization of the entire proof in Lean, and present a deterministic algorithm for weak normalization.", "title": "Simplified proofs of Weak Normalization for propositional logic", "updated": "2026-09-13", "url": "https://arxiv.org/abs/2609.14314" }, { "age_days": 2, "arxiv_id": "2609.14525", "authors": [ "Zhipeng Lu", "Sichen Wang" ], "content_date": "2026-09-13", "freshness": "fresh", "id": "arxiv:2609.14525", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-13", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We study how far a fixed Ramsey upper bound can be improved by descending through blue neighborhoods in one vertex set while keeping a second set fixed. A weighted inequality in the two set sizes determines when the descent can stop. For the source bound specified here, the infimum diagonal exponent over all finite derivations lies in $[1.305,\\,1.307]$. A finite derivation gives $R(k,k)\\le3.69507^k$ for all sufficiently large $k$; a concave polygon proves the lower bound for every finite depth. We also characterize the infimum as a greatest fixed point and show that every larger exponent has a finite derivation valid uniformly for nearby clique-size ratios.", "title": "Retained-Set Descent for Diagonal Ramsey Numbers", "updated": "2026-09-13", "url": "https://arxiv.org/abs/2609.14525" }, { "age_days": 3, "arxiv_id": "2609.13946", "authors": [ "Yuning Yang" ], "content_date": "2026-09-12", "freshness": "fresh", "id": "arxiv:2609.13946", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-12", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "This work studies subspace embeddings obtained by two normalized real Walsh transforms, two independent sign diagonals, and uniform coordinate sampling without replacement. The main result shows that the prescribed sample size $k=\\min\\{n,\\lceil Cr/\\varepsilon^2\\rceil\\}$, for a universal constant $C$, suffices to preserve all squared norms on each fixed $r$-dimensional subspace within $1\\pm\\varepsilon$ with probability at least $0.99$. The result holds for every ambient Walsh dimension and all ranks, and answers Problem TR-01 in the Open Problems in Numerical Linear Algebra repository. The proof controls joint entry cumulants of the transformed projection through connected graph contractions and Walsh character identities. These estimates then bound the expected trace of even powers of a product of centered projections. A two-projection decomposition converts this estimate into control of both spectral edges. Bernoulli sampling at arbitrary densities and a deterministic upper bound near full sampling yield the prescribed number of coordinates.", "title": "Subspace embeddings with the rerandomized SRHT", "updated": "2026-09-12", "url": "https://arxiv.org/abs/2609.13946" }, { "age_days": 3, "arxiv_id": "2609.13635", "authors": [ "Thijs Laarhoven" ], "content_date": "2026-09-12", "freshness": "fresh", "id": "arxiv:2609.13635", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-12", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We study the spherical statistics of all the points in a thin shell of a high-dimensional random lattice. The exact relations between lattice points make it unclear when predictions based only on spherical geometry should hold. We answer this question for several statistics, including the number of shell points, the balance of their directions, and the occurrence, repetition, and distribution of differences between them. We identify sharp thresholds as the shell radius grows and show that these statistics undergo several distinct phase transitions. A shell can already agree with one geometric prediction while still differing strongly from another. Our results hold for a single sampled lattice, with probability tending to one as the dimension grows. They include estimates that hold across a complete shell, bounds close to the transition thresholds, and extensions to randomly shifted lattices. A Lean formalization verifies the main results, assuming the classical formulas and probability model stated in the code.", "title": "Spherical statistics and phase transitions in high-dimensional lattices", "updated": "2026-09-12", "url": "https://arxiv.org/abs/2609.13635" }, { "age_days": 3, "arxiv_id": "2609.13879", "authors": [ "Yiling Wu" ], "content_date": "2026-09-12", "freshness": "fresh", "id": "arxiv:2609.13879", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-12", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "An acquired representation can enlarge a system's cognitive repertoire without transferring the capacities exercised in producing that representation. This paper develops a framework for specifying that enlargement and its limits. Its central contribution is a five-part attribution table distinguishing effective tracking, application of acquired structures, acquisition from explicit specifications, acquisition from identifying observations, and retention and reuse. Each entry identifies a positive capacity commitment and a further claim requiring additional support. The argument deliberately grants meaningful content, causal efficacy, and productive inference, so that its conclusion does not depend on treating representations as inert encodings. Map and category examples show why even complete application competence leaves acquisition capacity undetermined, and why acquiring a criterion from its description differs from finding it in examples. Short formal proofs appear in an appendix. The framework is applied to Andrew Ng's world-model interpretation of Othello-GPT and to the specific indicators discussed in contemporary accounts of machine concepts. It preserves demonstrated recognition, classification, inference, and qualified acquisition while specifying what remains unestablished about criterion discovery and accumulation. The result concerns the scope of cognitive attribution rather than the constitutive conditions of concept possession: cognitive achievements deserve credit for the capacities they establish, without silently importing a broader repertoire through the labels attached to them.", "title": "Map Users and Mapmakers: The Scope of Cognitive Attribution from Acquired Representations", "updated": "2026-09-12", "url": "https://arxiv.org/abs/2609.13879" }, { "age_days": 4, "arxiv_id": "2609.13331", "authors": [ "Zhipeng Lu" ], "content_date": "2026-09-11", "freshness": "fresh", "id": "arxiv:2609.13331", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-11", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Let $P_n$ be the matrix of a random permutation of $n$ symbols and let $M_n=\\log\\max_{|z|=1}|\\det(I-zP_n)|$. Cook and Zeitouni proved that $M_n/\\log n$ converges in probability to a constant $x_0$ for a uniform permutation. We show that the $\\sqrt{\\log n}$ fluctuations of $M_n$ are carried entirely by the number of cycles $K_n$. Write $λ(s)=\\log\\{Γ(1+s)/Γ(1+s/2)^2\\}$, let $s_κ$ minimize $(1+κλ(s))/s$ on $(0,\\infty)$, and put $v(κ)=κλ'(s_κ)$ and $a_θ=λ(s_θ)/s_θ$. Under the Ewens measure with any fixed parameter $θ>0$ we prove $M_n=v(θ)\\log n+a_θ(K_n-θ\\log n)+O_P(\\log\\log n)$, so that the standardized pair $(K_n,M_n)$ converges jointly to $(G,G)$ with $G$ standard normal: the maximum and the cycle count are asymptotically perfectly aligned. This is deduced from a statement about the exact conditional law, which does not depend on $θ$: for every compact $[κ_-,κ_+]\\subset(0,\\infty)$ there is a finite $C$ such that $P(|M_n-v(k/\\log n)\\log n|>C\\log\\log n \\mid K_n=k)$ tends to $0$ uniformly over integers $k$ with $κ_-\\log n\\le k\\leκ_+\\log n$, that is, over exact and possibly atypical cycle counts. The proof keeps the size and the cycle count simultaneously in a two-variable coefficient extraction. Cycles longer than $n/(\\log n)^4$ are reserved as an analytic factor whose coefficients are flat under every size shift produced by the shorter cycles; positivity then converts a scalar coefficient asymptotic into a relative comparison of the entire path-constrained measure, with an error that does not degrade with the number of constraints or with the rarity of the event. The constrained lower bound comes from pointwise saddle estimates for killed convolutions along a dyadic chain of endpoint boxes.", "title": "Spectral extremes under exact cycle conditioning", "updated": "2026-09-11", "url": "https://arxiv.org/abs/2609.13331" }, { "age_days": 4, "arxiv_id": "2609.13352", "authors": [ "Sofian Audry", "Stephen Kelly" ], "content_date": "2026-09-11", "freshness": "fresh", "id": "arxiv:2609.13352", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-11", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We present three robotic art installations which explore the aesthetics of adaptive behavior. Through embodied machine leaning and digital evolution, these works draw viewers into an artificial ecosystem in which open-ended novelty, trial-and-error learning, competition, and cooperation emerge in real time. Research-creation practices are examined in relation to these works, focusing on how they redefine the role of artists within a human-machine collective while examining points of convergence and divergence between artistic and engineering approaches to adaptive robotics. The systems in question use learning and evolutionary processes not as a means to optimize a specific solution, but as an aesthetic experience on its own, suggesting new modes of interdisciplinary art-science research. Finally, we discuss strategies and practices to elevate the aesthetic experience for audiences, including contexts of presentation as well as temporal and material considerations for artworks based on embodied adaptive systems.", "title": "Real-time Learning and Evolution in Robotic Art Installations", "updated": "2026-09-11", "url": "https://arxiv.org/abs/2609.13352" }, { "age_days": 4, "arxiv_id": "2609.13359", "authors": [ "Rodrigo Nicolau Almeida", "Søren Brinck Knudstorp" ], "content_date": "2026-09-11", "freshness": "fresh", "id": "arxiv:2609.13359", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-11", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We show that Medvedev's logic of finite problems, a well-known superintuitionistic logic, is undecidable. The key method is a reduction from the periodic tiling problem to non-theoremhood in Medvedev's logic. This settles a longstanding open problem. Using similar techniques, but reducing instead to the ordinary tiling problem, we likewise obtain undecidability of Skvortsov's logic of infinite problems, and the fact that the two logics are distinct -- in fact, they are separated by any aperiodic tiling of the plane. Due to the fact that Medvedev's logic figures in so many different areas, these results have implications for several fields -- for example, the study of schematic fragments of logics such as propositional dependence logic, or the study of internal logics of toposes. The core idea and technical work of the undecidability proof were obtained using ChatGPT Sol 5.6, and formally verified in Lean by Claude Opus 5. A detailed methodology section outlines how such results were obtained.", "title": "Medvedev logic is undecidable", "updated": "2026-09-11", "url": "https://arxiv.org/abs/2609.13359" }, { "age_days": 4, "arxiv_id": "2609.13609", "authors": [ "Michael Hellstern", "Byol Kim", "Ali Shojaie" ], "content_date": "2026-09-11", "freshness": "fresh", "id": "arxiv:2609.13609", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-11", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "Network analysis for multivariate time series is popular in many fields, from neuroscience to seismology. The inverse spectral density is a common choice for time series network analysis due to its representation of the frequency domain correlation between two variables after removing the best linear predictor of all other variables. In many applications, the goal is to study how these networks change across different conditions. For example, in neuroscience, one might be interested in how the brain connectivity network changes before and after stimulation. Towards this goal, we develop an inference framework based on a direct estimate of the difference in two high-dimensional inverse spectral densities. We develop a new Gaussian approximation error bound for any de-biased D-trace estimation procedure which is then leveraged to both inform optimal window sizes of Welch's estimators of the spectral density and establish asymptotic normality of our de-biased D-trace estimator. Moreover, we develop an efficient algorithm based on a generalized D-trace estimation procedure to overcome the computational complexity of high-dimensional inference. The method is illustrated on synthetic data experiments and on experiments with electroencephalography data.", "title": "Assumption-Lean Inference for Spectral Differential Network Analysis of High-Dimensional Time Series", "updated": "2026-09-11", "url": "https://arxiv.org/abs/2609.13609" }, { "age_days": 4, "arxiv_id": "2609.13351", "authors": [ "Özgür Soysal" ], "content_date": "2026-09-11", "freshness": "fresh", "id": "arxiv:2609.13351", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-11", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We prove that the capacity of the binary deletion channel satisfies $C(d)\\le (1-d)/4$ for every $13/20\\le d<1$. The proof describes the output from right to left, using a six-bit context to assign a description length. We bound the increase in expected description length minus output entropy when one input bit is added. A relative-entropy identity reduces this bound to finitely many linear inequalities. A potential on input windows of length 26 makes the inequalities telescope, giving a bound for every input word. Deletion composition extends the result from $d=13/20$ to all larger deletion probabilities. We also obtain finite-block bounds on mutual information and decoding error, with explicit $O(\\log n/n)$ corrections. The finite certificate is checked using exact integer arithmetic, and the proof is formalized end to end in Lean.", "title": "A New Upper Bound on the Binary Deletion Channel Capacity", "updated": "2026-09-11", "url": "https://arxiv.org/abs/2609.13351" }, { "age_days": 5, "arxiv_id": "2609.13317", "authors": [ "Magnus Boman" ], "content_date": "2026-09-10", "freshness": "fresh", "id": "arxiv:2609.13317", "kind": "paper", "label": "negative", "matched_signals": [], "published": "2026-09-10", "score": 0.5, "source": "arxiv-ai4math-core", "summary": "We study fifteen properties of binary relations that have established uses in modal logic, order and preference theory, and relation algebra. The organising question is pragmatic: once some properties of a relation are known, which further properties follow, which combinations force degeneracy, and which properties remain independent? We give a proved Horn basis of elementary and compound entailments, explicit countermodels for non-entailments, and a relation-algebraic translation of all fifteen properties. The modal discussion includes the usual Scott--Lemmon correspondences and the logic of transitive dense frames studied by Ghilardi and Mints. Exhaustive finite enumeration and targeted model search are used for discovery, while the catalogue is certified in Lean. A kernel-checked coverage calculation ranges over all 2^15 = 32,768 antecedent sets and all fifteen possible consequents. Its 51-rule Horn manifest is coupled definitionally to Lean proofs of semantic soundness, and its 49 witness rows are coupled definitionally to Lean-certified finite or infinite relations. Consequently, over non-empty domains, the Horn closure is complete for positive entailment among the fifteen selected properties, and the consistency classification is complete for their positive combinations.", "title": "A Catalogue of Properties of Binary Relations: Entailments, Incompatibilities, and Independence Results", "updated": "2026-09-10", "url": "https://arxiv.org/abs/2609.13317" }, { "age_days": 2, "arxiv_id": "2609.14572", "authors": [ "Sushan Adhikari" ], "content_date": "2026-09-13", "freshness": "fresh", "id": "arxiv:2609.14572", "kind": "paper", "label": "negative", "matched_signals": [ "negative:generic_llm_rag" ], "published": "2026-09-13", "score": -2.5, "source": "arxiv-ai4math-core", "summary": "Teaching abstract theoretical computer science (TCS) concepts such as algorithm analysis and complexity theory is challenging because students must handle formal proofs and asymptotic reasoning that conventional resources rarely explain in an adaptive, on-demand way. We present AlgoRAG, a specialized Retrieval-Augmented Generation (RAG) system that couples a large language model (LLM) with a curated, domain-specific knowledge base to address these challenges. The knowledge base integrates authoritative textbooks, 847 lecture slides, 312 practice problems with solutions, 156 worked proof templates, and 89 complexity worksheets. AlgoRAG incorporates domain-specific optimizations including mathematical entity recognition, notation-aware retrieval, and pedagogical re-ranking. We evaluate AlgoRAG on 179 curated exam-style questions spanning asymptotic analysis, recurrence relations, dynamic programming, graph algorithms, NP-completeness, sorting, and divide-and-conquer. The system achieves a 100% success rate with a mean response time of 38.0 seconds. While BLEU-4 scores are zero -- a known limitation of n-gram matching on mathematical proofs where equivalent reasoning may use entirely different notation -- AlgoRAG attains ROUGE-1 F1 of 0.0963, ROUGE-L F1 of 0.0683, and a pedagogical quality score of 0.7620, indicating that responses are well-structured and didactically sound even when surface wording diverges from reference answers. Performance is especially strong on NP-completeness (ROUGE-1 F1 = 0.1285, pedagogical quality = 0.7643) and graph algorithms (ROUGE-1 F1 = 0.1023, pedagogical quality = 0.8086). These results support the conclusion that RAG is an effective architecture for personalized theoretical-CS instruction, providing correct, context-rich explanations even for highly abstract topics.", "title": "AlgoRAG: Retrieval-Augmented Generation for Theoretical Computer Science Education -- A Comprehensive Evaluation Framework for Algorithm Analysis and Complexity Theory", "updated": "2026-09-13", "url": "https://arxiv.org/abs/2609.14572" } ], "lookback_days": 21, "schema": "ai4math-radar-run-v1", "timezone": "America/Los_Angeles" }