arXiv subjects

Ji Ho Bae

Publications and source records attributed to Ji Ho Bae.

6 recordsLinked to original sources

arXiv2026

Unbounded logarithmic limsup in Erdős Problem 684 via shifted carry scheduling

For $1\leq k\leq n$, let $u(n,k)=\prod_{p\leq k}p^{ν_p\binom nk}$ and $f(n)=\min\{1\leq k\leq n:u(n,k)>n^2\}$. The minimum is interpreted as $+\infty$ if the set is empty. Here $ν_p(m)$ denotes the exponent of the prime $p$ in $m$. Erdős Problem 684 asks for bounds on $f(n)$. We prove $\limsup_{n\to\infty} \frac{f(n)}{\log n}\frac{\log\log\log n}{\log\log n}\geq\frac12$. In particular, $\limsup_{n\to\infty}f(n)/\log n=\infty$, so no uniform estimate $f(n)=O(\log n)$ is possible. The proof constructs integers $n=tL_M-h-1$, where $L_M=\operatorname{lcm}(1,\ldots,M)$. Product-cell coding yields simultaneous two-sided approximations to $tL_M$ modulo every prime in $(M,K]$. Writing $n+1=tL_M-h$, the shift by $h$ folds both signs into the same one-sided carry region. The carries at levels at most $M$ are bounded by $\log\binom{h+k}{h}$. For the prime powers above $K$, a truncated CRT witness has modulus below the search range; exponential weighting then gives an exponentially small exceptional set. An elementary anchored-fibre lemma selects a multiplier satisfying both requirements. Together, these ingredients prove the stated bound unconditionally. The theorem has been formally verified in Lean 4 with Mathlib.

math.NT↗

arXiv2026

Anti-Periodic Positional Encoding: Möbius Boundary Conditions Make In-Context Retrieval Reliable

Möbius RoPE is a rotary positional encoding built on the anti-periodic frequency ladder $θ_i=π(2i+1)/N$: every rotation plane advances by an odd multiple of $π$ across the training context, so the positional holonomy is $-1$ and the two ends of the sequence are deterministically coupled through a closed-form Dirichlet "dipole"; to our knowledge this is the first anti-periodic boundary condition in positional encoding. We verify the theory numerically to $\sim 10^{-6}$ and pretrain 48 models spanning six 160M-class and three 410M-class arms (2B FineWeb-Edu tokens each; the hybrid arm puts Möbius frequencies on 25% of heads). Hybrid perplexity is unchanged (29.66 vs. 29.72), but needle-in-a-haystack retrieval becomes reliable: $90.3\pm5.7\%$ versus $63.3\pm31.4\%$ at context 512 ($n=6$ seeds), observed worst seed 86% versus 14%, robust variance tests $p=0.013$-$0.029$ (unadjusted), recurring at 410M (Levene $p=0.040$). Matched controls isolate the mechanism: an aperiodic ladder in the same frequency band reproduces none of the effect, and a periodic (holonomy $+1$) ladder only a fraction. Swapping trained models' frequency table back to standard RoPE (weights frozen) collapses retrieval, with damage concentrated on far needles: trained models depend on this long-range geometry. A NoPE arm is even more reliable at short context but pays a 13% perplexity tax and extrapolates worst; only the anti-periodic hybrid pairs baseline perplexity with a high reliability floor. The effect is scoped to single-needle retrieval within the training window; a one-line frequency swap thus provides zero-cost insurance against the retrieval seed lottery.

cs.CL↗

arXiv2026

A resolution of Erdős Problem #190 via Erdős-Lovász, BCT, and Baker-Harman-Pintz

Let H(k) be the smallest N such that every finite coloring of [N] contains a monochromatic or rainbow k-term arithmetic progression. Erdős and Graham asked whether $H(k)^{1/k}/k \to \infty$ (Problem #190 of the Erdős Problems database). We prove that there is an absolute constant $k_0 \ge 2$ such that for all $k \ge k_0$, \[ H(k)^{1/k}/k \ge (1/e - \varepsilon(k)) \cdot k/\log k, \qquad \varepsilon(k) = O(k^{-0.475} \log k) \to 0 \text{ as } k \to \infty; \] in particular $H(k)^{1/k}/k = Ω(k/\log k)$ and $\lim_{k\to\infty} H(k)^{1/k}/k = \infty$, resolving the positive direction of the Erdős-Graham question. The argument combines three standard ingredients -- the symmetric Lovász Local Lemma applied to the k-AP hypergraph on $[N]$, the restricted form of the Blankenship-Cummings-Taranchuk recurrence, and the Baker-Harman-Pintz prime-gap theorem -- together with the pigeonhole reduction $H(k) \ge W(k-1,k)$, and uses BHP as the only analytic black box. Previous applications of Erdős-Lovász had fixed $r$; the improvement here is that the $r^{k-1}$ base dominates once one allows the color count $r_0 = \lfloor k / \log k \rfloor$ to grow with $k$. No matching upper bound on $H(k)^{1/k}/k$ is known.

math.CO↗

arXiv2026

Vertex-minor Ramsey numbers: exact values and extremal structure

We determine the vertex-minor Ramsey number $\Rvm(4)=11$, where $\Rvm(k)$ is the smallest~$n$ such that every $n$-vertex graph contains the edgeless graph~$E_k$ as a vertex-minor. We prove this by an exhaustive classification of the graphs on~$10$ and~$11$ vertices under local complementation. At the extremal order $n=10$, exactly six non-isomorphic graphs avoid~$E_4$ as a vertex-minor; up to isomorphism, they represent five LC-equivalence classes, and each labeled LC orbit has cardinality~$8{,}712$. Thus $k=4$ is the first case in which the general upper bound $2^k-1$ is not attained. Using the extremal graphs as building blocks, we derive explicit lower bounds on~$\Rvm(k)$ that surpass the leading term of the asymptotic bound for all $k\leq 9$; in particular, $\Rvm(5)\geq 13$. We also describe structural properties of the six extremal graphs and formulate the next open problem, whether $\Rvm(5)=15$.

math.CO↗

arXiv2026

Quantum Query Complexity of the Hyperoctahedral Group

We determine the quantum query complexity of oracle identification on the hyperoctahedral group $B_N = \{\pm 1\}^N \rtimes S_N$ with respect to the natural representation: $Q_{LV}(B_N) = 2(N-1)$ for all $N \ge 2$. This is twice the symmetric-group value $Q_{LV}(S_N) = N-1$; the doubling arises from an $\varepsilon$-parity obstruction that restricts the bottleneck representation $\operatorname{sgn}(σ)$ to even tensor powers. The proof combines a reduction to $S_N$ Kronecker products via Rademacher moment polynomials with the bipartition distance formula $d_T(((N),\varnothing),(α,β)) = 2(N-α_1)-|β|$ in the tensor product graph. A closed-form generating function yields the first-appearance multiplicity $(2N-3)!!$. We also show $Q_{\mathrm{decomp}}(φ) \le 2\,Q_{\mathrm{signed}}(φ)$, with equality on $B_2$, and conjecture a link between the adversary bound and the graph eccentricity.

math.CO↗

arXiv2026

When Self-Reference Fails to Close: Matrix-Level Dynamics in Large Language Models

We investigate how self-referential inputs alter the internal matrix dynamics of large language models. Measuring 106 scalar metrics across up to 7 analysis passes on four models from three architecture families -- Qwen3-VL-8B, Llama-3.2-11B, Llama-3.3-70B, and Gemma-2-9B -- over 300 prompts in a 14-level hierarchy at three temperatures ($T \in \{0.0, 0.3, 0.7\}$), we find that self-reference alone is not destabilizing: grounded self-referential statements and meta-cognitive prompts are markedly more stable than paradoxical self-reference on key collapse-related metrics, and on several such metrics can be as stable as factual controls. Instability concentrates in prompts inducing non-closing truth recursion (NCTR) -- truth-value computations with no finite-depth resolution. NCTR prompts produce anomalously elevated attention effective rank -- indicating attention reorganization with global dispersion rather than simple concentration collapse -- and key metrics reach Cohen's $d = 3.14$ (attention effective rank) to $3.52$ (variance kurtosis) vs. stable self-reference in the 70B model; 281/397 metric-model combinations differentiate NCTR from stable self-reference after FDR correction ($q < 0.05$), 198 with $|d| > 0.8$. Per-layer SVD confirms disruption at every sampled layer ($d > +1.0$ in all three models analyzed), ruling out aggregation artifacts. A classifier achieves AUC $0.81$-$0.90$; 30 minimal pairs yield 42/387 significant combinations; 43/106 metrics replicate across all four models. We connect these observations to three classical matrix-semigroup problems and propose, as a conjecture, that NCTR forces finite-depth transformers toward dynamical regimes where these problems concentrate. NCTR prompts also produce elevated contradictory output ($+34$-$56$ percentage points vs. controls), suggesting practical relevance for understanding self-referential failure modes.

cs.CL↗