arXiv subjects

Tong Niu

Publications and source records attributed to Tong Niu.

At least 19 recordsLinked to original sources

arXiv2026

EgoBench: An Interactive Egocentric Multimodal Benchmark for Tool-Using Agents

As AI agents increasingly operate in open, real-world environments, they require a deep synergy of multimodal perception, tool invocation with multi-hop reasoning, and dynamic interaction with users. However, existing benchmarks fail to jointly evaluate these capabilities due to challenges in designing strictly coupled multi-capability tasks, simulating natural and task-constrained user feedback, and ensuring objective evaluation of dynamic interaction. To bridge this gap, we introduce EgoBench, the first interactive multimodal benchmark for tool-using agents. EgoBench comprises 1,045 egocentric-video-grounded tasks covering four daily scenarios, along with a user-agent-tool interactive environment for evaluation. We implement a three-stage synergistic pipeline through which each task is designed to enforce the joint application of visual perception and tool-augmented multi-hop reasoning. We additionally develop a multi-agent simulated user within EgoBench to evaluate agents' interaction capabilities, which generates high-fidelity, task-aligned responses to agents. Furthermore, we establish a deterministic joint validation framework that guarantees objective assessment through process-based and result-based equivalence. Benchmarking eight SOTA video-MLLM agents on EgoBench reveals a severe performance ceiling: the best model achieves only 30.62% accuracy in the best-performing scenario, averaging 19.43% across all four scenarios. Finally, we conduct a multi-dimensional error analysis to disentangle failure modes, exposing capability bottlenecks for advancing future AI agents.

cs.AI↗

arXiv2026

Toward a Formally Verified Optimality Certificate for OGR(29): A SAT-Encoding Methods Note with Small-Case Demos

A Golomb ruler of order~$n$ is an integer set $\{a_0<a_1<\cdots<a_{n-1}\}$ whose $\binom{n}{2}$ pairwise differences $a_j-a_i$ ($i<j$) are all distinct. The optimal Golomb ruler problem asks for $\mathrm{OGR}(n)=\min\{a_{n-1}-a_0\}$ and is a classical combinatorial benchmark. The values $\mathrm{OGR}(2),\dots,\mathrm{OGR}(28)$ are settled through distributed volunteer search (the Distributed.net OGR project); $\mathrm{OGR}(29)$ is in active computation, with verification expected in late 2026 or early 2027. The recent upper bound $\mathrm{OGR}(29)\le 757$ of Lee, Park, and Kim (arXiv:2510.0122, October~2025) tightens the search window. This note describes a compact CNF encoding of the decision problem $\mathrm{GR}(n,L)$ (``is there a Golomb ruler of order~$n$ with length exactly~$L$?'') with $O(n^2 L)$ clauses, together with a small-case verification sweep that emits machine-checkable LRAT certificates of optimality for $\mathrm{OGR}(n)$ at $n\le 12$. We then give closed-form encoding-size estimates for $\mathrm{OGR}(29, L=757)$ and propose a cube-and-conquer decomposition aimed at a Mallob-style parallel run on commodity multi-core hardware. The certificate pipeline (CaDiCaL with --lrat=true, a structural sanity-check, then formal validation by drat-trim or cake\_lpr) is end-to-end. A formally verified optimality proof for any single $\mathrm{OGR}(n)$ value beyond the trivial $n\le 5$ would be a first in the field.

cs.DC↗

arXiv2026

Three short proofs of Mathar's 2014 conjecture for OEIS A002627

For the OEIS sequence A002627, defined by the inhomogeneous first-order recurrence $a(n) = n\,a(n-1) + 1$ with $a(0) = 0$, R.~J.~Mathar recorded in February 2014 the conjectured second-order homogeneous recurrence \[ a(n) - (n+1)\,a(n-1) + (n-1)\,a(n-2) = 0, \qquad n \ge 2, \] which has remained marked as a conjecture on the OEIS for over a decade. We give three short proofs. The first is two lines: subtract the defining recurrence at adjacent indices and the constant cancels (we call this homogenisation). The second reads off the same relation from the exponential generating function $F(x) = (e^x-1)/(1-x)$. The third is a Pascal-rule telescoping on the binomial-sum form $a(m) = \sum_{k=0}^{m-1} k!\binom{m}{k}$. All three derivations are elementary, requiring nothing beyond undergraduate techniques. We remark that the same homogenisation trick clears an entire class of ``Conjecture: \dots'' entries on the OEIS, namely sequences satisfying $a(n) = p(n)\,a(n-1) + q(n)$ with simple $q$.

math.CO↗

arXiv2026

An explicit algebraic generating function for OEIS A348410

For the OEIS sequence A348410, P. Bala recorded in February 2022 two equivalent closed forms, $a(n) = [x^{n}] ((1-x)(1-x^2))^{-n}$ and a single-index binomial sum. R. J. Mathar (October 2021) and V. Kotesovec (November 2021) each contributed a conjectured P-recursive recurrence -- Mathar's of order $4$, Kotesovec's of order $2$. We apply Lagrange-B\"urmann inversion to Bala's $[x^n]$ form to derive the parametric expression $A(t) = (1 - y^2)/(1 - y - 4 y^2)$, where $y = y(t)$ is implicit by $y(1-y)^2(1+y) = t$. Eliminating $y$ via resultant gives the explicit algebraic equation $P(t, A) = 0$ of degree $4$ in $A$ and degree $2$ in $t$. As an immediate corollary (Stanley's classical algebraic-implies-D-finite theorem), $A(t)$ is D-finite. Mathar's and Kotesovec's specific recurrences are not directly proven here; we only verify Kotesovec's order-$2$ recurrence numerically for $n = 3, \ldots, 1000$ and observe that an explicit ODE-and-recurrence extraction from $P(t, A) = 0$ via the standard Bostan-Chyzak-Salvy algebraic-to-holonomic procedure would close both conjectures. The supplementary archive contains a SymPy script which derives $P(t, A)$ and checks the numerical evidence.

math.CO↗

arXiv2026

A short proof of Mathar's 2013 recurrence conjecture for the reversible-binary-string sequence A032123

For the OEIS sequence A032123, the number of length-$2n$ black-and-white strings with $n$ black beads, considered up to reversal, R. J. Mathar contributed in November 2013 the conjectured order-5 P-recursive recurrence \[ \begin{aligned} &n(n-1)\,a(n) - 2(n-1)(3n-4)\,a(n-1) + 4(2n^{2}-14n+19)\,a(n-2) &\qquad + 8(n^{2}+5n-19)\,a(n-3) - 16(n-3)(3n-10)\,a(n-4) &\qquad + 32(n-4)(2n-9)\,a(n-5) \;=\; 0, \qquad n \ge 6. \end{aligned} \] We give a short proof. Burnside's lemma applied to the reversal action gives the closed form $a(n) = \tfrac{1}{2}\bigl(\binom{2n}{n} + [n \text{ even}]\binom{n}{n/2}\bigr)$; the two summands satisfy elementary recurrences of order $1$ and $2$ respectively; and Mathar's order-5 operator, applied to each summand separately, reduces to a polynomial identity that simplifies to zero after a brief calculation. The supplementary archive includes a SymPy script which verifies the polynomial identities symbolically and checks Mathar's recurrence numerically for $n = 6, \ldots, 5000$.

math.CO↗

arXiv2026

A short proof of Mathar's 2021 recurrence conjecture for the Lehmer-Comtet diagonal A045406

For OEIS sequence A045406, the column-2 diagonal of the Lehmer-Comtet triangle A008296, R. J. Mathar contributed in September 2021 the conjectured order-2 P-recursive recurrence \[ a(n) + (2n-7)\,a(n-1) + (n-4)^{2}\,a(n-2) \;=\; 0,\qquad n \ge 5. \] We give a short proof. Detlefs's harmonic-number closed form $a(n) = (-1)^n (2 H_{n-3} - 3)(n-3)!$ for $n \ge 3$ collapses the left-hand side, after factoring out $(-1)^n (n-5)! (n-4)$, to a polynomial identity in $n$ with coefficient $H_{n-4}$. The $H_{n-4}$-coefficient simplifies to $(n-3) - (2n-7) + (n-4) = 0$ (using $H_{n-3} = H_{n-4} + 1/(n-3)$ and $H_{n-5} = H_{n-4} - 1/(n-4)$); the constant remainder is $0$ for the same reason. The supplementary archive contains a SymPy script verifying both pieces symbolically, the e.g.f.\ expansion against the harmonic closed form, and Mathar's recurrence numerically for $n = 5, \ldots, 5000$.

math.CO↗

arXiv2026

A short proof of Mathar's 2020 recurrence conjecture for the generalized-Stirling sequence A001711

For the OEIS sequence A001711, contributed by N. J. A. Sloane long before the on-line era and identified there as the diagonal $T(n+4, 4)$ of a generalized-Stirling triangle, R. J. Mathar contributed in February 2020 the conjectured order-2 P-recursive recurrence \[ a(n) - (2n+5)\,a(n-1) + (n+2)^{2}\,a(n-2) \;=\; 0,\qquad n \ge 2. \] We give a one-page proof. Detlefs's harmonic-number closed form $a(n) = \tfrac{1}{4}(n+3)!\,(2 H_{n+3} - 3)$ collapses the left-hand side, after dividing through by $(n+1)!/4$, to a polynomial identity of $n$ with coefficient $H_{n+2}$. The harmonic-number coefficient simplifies to $(n+3) - (2n+5) + (n+2) = 0$ (using $H_{n+3} = H_{n+2} + \tfrac{1}{n+3}$ and $H_{n+1} = H_{n+2} - \tfrac{1}{n+2}$); the constant remainder is $-3 \cdot 0 = 0$ for the same reason. The supplementary archive contains a SymPy script verifying both pieces symbolically, the e.g.f.\ expansion against the harmonic closed form, and Mathar's recurrence numerically for $n = 2, \ldots, 5000$.

math.CO↗

arXiv2026

Parity vectors and paradoxical sequences in the accelerated Collatz map

This note studies parity vectors and paradoxical sequences in the accelerated Collatz iteration $T(n) = (3n+1)/2$ for $n$ odd, $T(n) = n/2$ for $n$ even. Building on Rozier and Terracol (arXiv:2502.00948, 2025), Terras (1976), Lagarias (1985), and Tao (2019), we prove three theorems and add one numerical observation. The first is a sharp finitary form of Terras's parity-vector density; the second is a closed-form analytic count of paradoxical $\Omega_k(n)$ for each fixed length $k$. The third is a density-zero theorem for bounded-length paradoxical sequences with explicit constant. As for the numerical piece, among the seven $(j, q)$ pairs that show up in the Rozier-Terracol enumeration with first term $n \le 10^9$, every paradoxical reduced ratio $q/j$ turns out to be a left convergent, a left semiconvergent, or a Stern-Brocot mediant of adjacent convergents/semiconvergents of $\log_3 2$. The three theorems are unconditional. The fourth observation is verified for $n \le 10^7$ and conjectured for all $n$. We make no claim toward the Collatz conjecture or Terras's coefficient-stopping-time conjecture.

math.NT↗

arXiv2026

A short proof of Mathar's 2013 recurrence conjecture for the Laguerre sequence~A025166

For the OEIS sequence A025166, defined by $a(n) = -n!\,2^{n}\,L_{n}(1/2)$ where $L_{n}$ is the Laguerre polynomial of degree $n$, R.~J.~Mathar contributed in February 2013 the conjectured order-2 P-recursive recurrence \[ a(n) + (-4n+3)\, a(n-1) + 4(n-1)^{2}\, a(n-2) \;=\; 0, \qquad n \ge 2. \] We give a one-page proof. The exponential generating function $F(x) = -\exp\!\big(-x/(1-2x)\big)/(1-2x)$ satisfies the first-order linear ODE $(1-2x)^{2} F'(x) = (1-4x)\, F(x)$, and Mathar's recurrence then falls out by reading off the coefficient of $x^{n}/n!$. Both steps are short. The supplementary archive includes a SymPy script which checks the ODE identically and the recurrence numerically up to $n = 5000$.

math.CO↗

arXiv2026

The vertex Folkman number $F_v(3,3;5)$ equals~$8$

The vertex Folkman number $F_v(s,t;k)$ is the smallest $n$ for which there exists a $K_k$-free graph on $n$ vertices whose vertices cannot be $2$-colored without producing a monochromatic copy of $K_s$ or $K_t$. We show $F_v(3,3;5)=8$. The witness is the cone $K_1 \vee \overline{C_7}$, a single universal vertex joined to the complement of a $7$-cycle. That this graph is $K_5$-free and arrows $(3,3)^v$ follows from a short independence-number argument. The matching lower bound -- no $K_5$-free graph on $7$ or fewer vertices works -- comes from exhaustive enumeration via nauty and a SAT check using Glucose\,4. The appendix has a self-contained Python script for verification.

math.CO↗

arXiv2026

Improved Upper Bounds on the Pebbling Numbers of the Blanu\v{s}a Snarks

The two Blanu\v{s}a snarks $B_1$ and $B_2$ are 3-regular graphs on 18 vertices. Dantas, Lordelo, Niedermaier and Nogueira (Discrete Appl. Math. 361, 2025, pp. 336-346) established the first systematic bounds $23 \le \pi(B_i) \le 34$ for $i=1,2$. Bridi, Marquezino and Figueiredo (arXiv:2505.16050, 2025) then sharpened the upper side to $\pi(B_1) \le 31$ and $\pi(B_2) \le 30$ via a Weight Function Lemma heuristic. We push the upper bounds further to $\pi(B_1) \le 28$ and $\pi(B_2) \le 29$. The route is again Hurlbert's Weight Function Lemma, but applied one automorphism orbit at a time, with optimal weight functions coming from a linear program over a corpus of roughly $30{,}000$ rooted-subtree strategies per target. For the lower bound $\pi(B_i) \ge 23$ we re-derive the witnesses of Dantas et al. and re-verify them with two independent oracles: an exhaustive forward state-space search, and a sound-and-complete MILP encoding whose acyclicity constraint is motivated by the Milans-Clark No-Cycle Lemma. The interval for $B_1$ shrinks from $[23, 31]$ to $[23, 28]$, and for $B_2$ from $[23, 30]$ to $[23, 29]$.

math.CO↗

arXiv2026

A short proof of Mathar's 2016 recurrence conjecture for OEIS A176677

For the OEIS sequence A176677, defined by the quadratic convolution recurrence $a(0) = a(1) = 1$ and $a(n+1) = \sum_{p=0}^n a(p) a(n-p) - 1$ for $n \ge 1$, R.~J.~Mathar contributed in March 2016 the conjectured order-4 P-recursive recurrence \[ (n+1)\,a(n) + 2(-3n+1)\,a(n-1) + (9n-13)\,a(n-2) - 4\,a(n-3) + 4(-n+4)\,a(n-4) = 0. \] We give a short proof. The convolution recurrence translates directly into the algebraic equation $z(1-z) G(z)^2 - (1-z) G(z) + (1 - z - z^2) = 0$ for the ordinary generating function $G(z)$, and Mathar's recurrence then drops out as the coefficient form of a 1st-order linear inhomogeneous ODE $q_0(z) G(z) + q_1(z) G'(z) = R(z)$ that we verify by polynomial division modulo the algebraic equation. The polynomial $q_1(z)$ admits the factorization $q_1(z) = -z(z-1)(2z-1)(2z^2 + 3z - 1)$, whose roots are exactly the singularities of $G$. Deutsch's combinatorial interpretation (Motzkin paths of length $n-1$ with two-coloured level-zero horizontal steps) is preserved.

math.CO↗

arXiv2026

The pebbling number of Fibonacci cubes

The $n$-th Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by binary strings with no two consecutive ones. We determine $\pi(\Gamma_n) = 2^n$ for $n \le 6$, so the pebbling number of $\Gamma_n$ equals that of the ambient hypercube $Q_n$ despite $\Gamma_n$ having far fewer vertices. The lower bound is a standard potential argument. For the upper bound, the Weight Function Lemma yields $2^n+1$ -- one too many -- so we close the gap by exhaustive MILP verification. We conjecture $\pi(\Gamma_n) = 2^n$ for all $n$.

math.CO↗

arXiv2026

Graceful labelings of spiders with three-edge legs and pendant leaves at the center

A graph $G$ on $m$ edges is graceful if there is an injection $f : V(G) \to \{0, 1, \ldots, m\}$ whose induced edge labels $\{|f(u) - f(v)| : uv \in E(G)\}$ are exactly $\{1, 2, \ldots, m\}$. Ringel and Kotzig conjectured in 1964 that every tree is graceful. A computer check has confirmed this for all trees on at most 35 vertices (Fang 2010), but no general proof is known. Here we exhibit an infinite family of trees that escapes the named spider results of Bahls--Lake--Wertheim, Panpa--Poomsa-ard, and Panpa--Imnang--Wasuanankul: the family $T(k, m)$ of spiders with $k$ legs of length $3$ together with $m$ pendant leaves at the centre. We prove every such tree is graceful for all $k \ge 1$ and $m \ge 0$. The argument splits into two short lemmas. The first is a pendant-extension lemma that applies whenever the underlying graceful labeling sends the centre to $0$; the second is the base case, namely that $S_{k, 3}$ admits exactly such an apex-zero labeling, a fact already implicit in Bahls--Lake--Wertheim (2010). What is new is the explicit identification of $T(k, m)$ and the observation that the family is closed under both pendant addition and the apex-zero condition, so it includes infinitely many trees not handled by the named theorems above.

math.CO↗

arXiv2026

A short proof of Mathar's 2013 recurrence conjecture for the Meixner sequence A214615

For the OEIS sequence A214615, defined by $a(n) = M_{n}(1)$ where $M_{n}$ is the $n$-th Meixner polynomial satisfying $M_{n+1}(x) = x\,M_{n}(x) - n^{2}\,M_{n-1}(x)$, R.~J.~Mathar contributed on 6~March 2013 the conjectured order-2 P-recursive recurrence $a(n) - a(n-1) + (n-1)^{2}\,a(n-2) = 0$ for $n \ge 2$. We give a one-page proof. The exponential generating function $F(t) = \exp\!\bigl(\arctan t\bigr)/\sqrt{1+t^{2}}$ satisfies the first-order linear ODE $(1+t^{2})\,F'(t) = (1-t)\,F(t)$, and Mathar's recurrence then falls out by reading off the coefficient of $t^{n}/n!$. Both steps are short. The supplementary archive includes a SymPy script that checks the ODE identically and the recurrence numerically up to $n = 500$.

math.CO↗

arXiv2026

An exact small-$n$ computation of the minimum 2-coloring discrepancy of $K_n^{(3)}$

For an integer $r \ge 2$ and an order $n \equiv 1, 3 \pmod{6}$, write $\delta_r(n)$ for the minimum, over all $r$-colourings $\chi : \binom{[n]}{3} \to [r]$, of $\max_{\mathcal{S}} \mathrm{disc}(\mathcal{S}, \chi)$, where the maximum is over labelled Steiner triple systems $\mathcal{S}$ of order $n$ and $\mathrm{disc}(\mathcal{S}, \chi) = \max_c |\#\{T \in \mathcal{S} : \chi(T) = c\} - |\mathcal{S}|/r|$. Following Gishboliner, Glock, and Sgueglia \cite{GishbolinerGlockSgueglia2025}, the bulk of the recent work on this quantity has been on lower bounds for $r \ge 3$ (proving $\delta_r(n) = \Omega(n^2)$) and on structural characterisation of the low-discrepancy 2-colourings. We give three small computational contributions in the small-$n$ regime $n \in \{7, 9, 13, 15, 19, 21\}$: An exact value of $\delta_2(n)$ for each such $n$, matching the formula $\delta_2(n) = \min_{x \in [0, n] \cap \mathbb{Z}} |x(n-x)/2 - n(n-1)/12|$ obtained by optimising the GGS Example 1.1 family. Rigorous for $n \in \{7, 9\}$ via exhaustive search over labelled STSs ($30$ resp. $840$ systems) and over all $2$-colourings; computational for $n \in \{13, 15, 19, 21\}$ by simulated-annealing search; A wide near-optimal basin: at $n = 9$, every two-colour-flip neighbour of the optimal Example~1.1 colouring that maintains discrepancy $1.0$ exists; about $34\%$ of two-flip perturbations preserve optimality; Random-colouring statistics for $r \in \{2, 3, 4\}$: $\langle\max_{\mathcal{S}}\mathrm{disc}\rangle$ grows linearly in $n$, in agreement with a heuristic Gaussian estimate $n / \sqrt{6r} \cdot \sqrt{2 \log K}$ over $K$ sampled labellings; the typical-case discrepancy is far below the GGS worst-case $\Omega(n^2)$. We additionally state a conjectural exact formula for $\delta_2(n)$ that holds for every $n \equiv 1, 3 \pmod{6}$.

math.CO↗

arXiv2026

Size-4 Counterexamples to the Sidon-Extension Conjecture

A finite set $S \subset \mathbb{Z}$ is a Sidon set if its pairwise differences are distinct. Recall that a perfect difference set (PDS) of order $n$ is a set $B \subset \mathbb{Z}_v$ ($v = n^2 - n + 1$) of size $n$ such that every nonzero residue arises exactly once as a difference of two elements of $B$. Erd\H{o}s's \$1000 conjecture -- that every finite Sidon set extends to a finite PDS -- was disproved by Alexeev and Mixon (arXiv:2510.19804, October 2025), via the size-5 counterexamples $\{1,2,4,8,13\}$ and Hall's earlier $\{1,3,9,10,13\}$; they then asked: what is the smallest size $s$ of a non-extending Sidon set? The trivial bounds give $3 \le s \le 5$. Our evidence points to $s = 4$. We exhibit two integer Sidon sets, \[ A = \{0, 1, 3, 11\}, \qquad B = \{0, 1, 4, 11\}, \] together with the apparent infinite family of dilations $kA$, $kB$ and their reflections, all of which fail to extend for every prime power $q \le 317$ via the Singer affine-orbit check (rigorous under Hall's 1947 uniqueness for Desarguesian cyclic planes through $q \le 40$ and under the prime-power conjecture beyond that), and unconditionally for every modulus $v \le 133$ via brute-force depth-first search. We also report the exact density $N_{\text{ne}}(N) = 4 \lfloor N / 11 \rfloor$ of non-extending size-4 Sidon sets in $[0, N]$ for $N \le 50$ -- the match is exact, which suggests the $kA, kB$ family is complete in this range. A complete proof, perhaps in the spirit of Alexeev--Mixon's polarity argument or via a multiplier descent, remains open.

math.CO↗

arXiv2026

A Proof of Bala's General-$m$ Representation of the Harmonic Numbers

For every nonzero integer $m$ and every integer $n \ge 1$, the $n$\textsuperscript{th} harmonic number $H_n = 1 + \tfrac12 + \dots + \tfrac1n$ satisfies the identity \[ H_n \;=\; \frac{1}{m}\,\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k}\, \binom{m k}{k}\binom{n + (m-1)k}{n - k}. \] The cases $m = 1$ and $m = 2$ are classical; for general nonzero integer $m$ the identity was conjectured by P.~Bala in the OEIS entry A001008 in 2022 and remained open. We prove it here, working throughout in $\QQ[[x]]$. The proof reduces, via a substitution $u = x/(1-x)^m$, to two formal-power-series identities: a Lagrange--B\"urmann evaluation of $\sum_{k\ge1} \binom{mk}{k} u^k / k$, and the fixed-point fact that under that substitution the unique solution $v(u)$ of $v = u(1-v)^{m}$ is $v = x$. The argument extends verbatim to arbitrary complex $m \ne 0$.

math.NT↗