bibkey: “narang2026stars” authors: “Ijay Narang; Will Perkins; Timothy L. H. Wee” year: 2026 title: “Optimal detection of planted stars via a random energy model” doi: “10.48550/arXiv.2602.15585” url: “https://arxiv.org/abs/2602.15585v1” claim: “Theorem 1.1 states a Gaussian total-variation window for detecting a planted star with one uniform hidden hub, and Corollary 1.8 states the corresponding hub-recovery probability.” strata_touched: [] license: “citation-only” triage: “anchor”

Optimal detection of planted stars via a random energy model

The inspected source is arXiv:2602.15585v1, submitted 17 February 2026, 34 pages. Theorem 1.1, page 3, considers a star of size k planted around a single uniform hidden hub in a graph with n vertices and m edges. Under (log n)^2 << k << sqrt(n) and m = k^2 n / (4 log n) * (1 + gamma / sqrt(log n)), it states TV(P_1,P_0) = 1 - Phi(gamma/sqrt(2)) + o(1). Corollary 1.8, page 5, states the same limiting hub-recovery probability. These are close prior statements about a Gaussian window and one hidden location, so those general mechanisms are not claimed as new here.

This version contains a statement inconsistency: Remark 1.6 says that the null likelihood concentrates at one half throughout that entire window, using Proposition 1.5(ii). Since TV(P_1,P_0) = 1 - E_0 min(1,L), convergence L -> 1/2 in null probability would instead force the total variation to tend to one half throughout the window. This conflicts with the varying limit in Theorem 1.1 away from gamma = 0. The note records the statements as published; it does not resolve which assertion requires correction.

The paper is cited as related literature, not used as a proof premise. The parity-kernel theorem concerns known amplitude and one fixed unknown spike, two orientations, independent pairs or a full Markov path, and uniform control as amplitude approaches either endpoint. Its double-tilt moment bounds and overlap calculation are proved separately as repo-derived deductions. No equivalence with the planted-star experiment or global originality is asserted.