{"id":"029bdddc-a24f-42cc-8463-7c7fe6c4c7cd","arxiv_id":"2607.23560","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Near-maximal tensor spectral radius forces a k-graph with matching number ≤β to be structurally close to S_{n,k,β} for large n.","lead":"A stability theorem shows that k-uniform hypergraphs with bounded matching number and near-maximal tensor spectral radius must be edge-close to the classical extremal construction S_{n,k,β}. This gives a new eigenvector-based proof of the spectral Erdős matching conjecture for large n.","discovery_kind":"extension","skeptic_critique":null,"referee_report":null,"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece is Theorem 1.1: a genuine stability statement. For fixed k≥3, β≥2 and large n, any k-graph with matching number ≤β whose tensor spectral radius is within (1-δ) of the extremal value must have a β-set that hits every edge, and after labeling it differs from S_{n,k,β} in at most Cδ n^{k-1} edges. The exact spectral Erdős-matching corollary (already proved by Kang–Lu–Yuan–Zhou via shifting) drops out immediately at δ=0; they cite that work cleanly and position their contribution as the stability upgrade plus an alternative method.\n\nWhat they do well is keep the argument elementary and self-contained. They take a normalized nonnegative principal eigenvector, cut at a small threshold η to form the candidate set W, use the eigenvalue equation plus a greedy matching argument to force |W|=β and that every edge meets W, then convert the spectral gap into an edit-distance bound via Hölder/Maclaurin and the elementary inequality (1-s)^{(k-1)/k}≤1-((k-1)/k)s. No black boxes, no circular reductions, no free parameters. The citation pattern is honest about the concurrent exact result and about the classical combinatorial literature.\n\nThe only softness is the usual “n large enough” absorption of lower-order terms (the multi-core edges O(n^{(k-1)(k-2)/k}) and the η-contribution). That is standard for this style of asymptotic spectral extremal work and is not a derivation gap; the coefficient comparison after (9) and the uniform o(1) in (11) are written so that the main-term gap wins once n is large. Constants are existential, not effective, which is fine for the claim as stated.\n\nThis is for people who already work in spectral extremal hypergraph theory or who want a clean eigenvector-mass template they can try on other forbidden configurations. It does not settle the full combinatorial Erdős matching conjecture for all n, nor does it claim to. I would send it to a serious referee without hesitation; the math is solid and the novelty is real relative to the shifting paper.","headline":"New spectral stability theorem for bounded-matching hypergraphs; clean eigenvector proof that also recovers the known exact extremal result for large n.","tokens_in":13187,"tokens_out":551,"would_cite":true,"duration_ms":11424,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C35","15A69"],"pacs":[],"model":"grok-4.5","headline":"Near-maximal tensor spectral radius forces a k-uniform hypergraph with bounded matching number to sit inside the star-like extremal example S_{n,k,β} and miss only O(δ n^{k-1}) of its edges.","keywords":["uniform hypergraph","adjacency tensor","spectral radius","stability","matching number","Erdős matching conjecture"],"falsifier":"Exhibit a single infinite family of k-graphs with matching number ≤β whose spectral radius is (1-o(1)) times that of S_{n,k,β} yet whose edge sets remain a positive-density distance away from every copy of S_{n,k,β}.","tokens_in":13477,"feed_emoji":"⬡","tokens_out":1067,"duration_ms":21132,"temperature":0.7,"pith_summary":"The paper proves a stability theorem for the tensor spectral radius of k-uniform hypergraphs whose matching number is at most a fixed β. If an n-vertex example has spectral radius within a small factor of the known maximum, then there is a set of exactly β vertices that meets every edge, and the hypergraph differs from the classical extremal construction S_{n,k,β} by at most a constant times δ n^{k-1} edges. The argument works by reading the principal eigenvector: vertices with large coordinates must form a vertex cover of size β, after which a quantitative comparison of Rayleigh quotients bounds the missing edges. As a direct corollary one recovers the exact spectral Erdős matching theorem for all large n, without shifting. A sympathetic reader cares because spectral radius is often easier to compute or bound than edge counts, yet here it still forces the same rigid structure that the classical extremal problem predicts.","feed_headline":"Spectral near-max forces hypergraphs into the star extremal","feed_subtitle":"Bounded matching number plus near-optimal tensor radius pins down the edge set up to o(n^{k-1}) errors","key_machinery":"The normalized nonnegative principal eigenvector of the adjacency tensor. Vertices whose coordinates exceed a small threshold η form a set W of size exactly β that covers every edge; the same vector then supplies the coefficient gap that converts a spectral deficit into an edge-deficit bound.","core_discovery":"For fixed k≥3 and β≥2 and all sufficiently large n, every n-vertex k-graph H with matching number at most β and tensor spectral radius at least (1-δ)ρ(S_{n,k,β}) admits a β-set W that intersects every edge of H; after relabeling, the symmetric difference of the edge sets of H and S_{n,k,β} has size at most Cδ n^{k-1}.","pith_inferences":["The same eigenvector-threshold argument should adapt to other hereditary properties whose extremal examples are complete multipartite or starring constructions.","Once n is large, the stability constant C can be tracked explicitly in terms of k and β, opening the door to effective (computer-checkable) bounds for moderate n.","Combining the stability theorem with existing edge-stability results for the Erdős matching conjecture would yield a two-way dictionary between spectral and combinatorial closeness."],"forward_implications":["The exact spectral Erdős matching conjecture holds for all sufficiently large n: ρ(H)≤ρ(S_{n,k,β}) with equality only for H isomorphic to S_{n,k,β}.","Any hypergraph attaining spectral radius within o(n^{(k-1)^2/k}) of the extremal value must already be a spanning subgraph of some S_{n,k,β}.","The missing-edge count is linearly controlled by the spectral deficit, giving a quantitative stability form rather than a mere qualitative one.","The eigenvector method supplies an alternative to shifting that may extend to other bounded-matching spectral problems."],"fun_headline_variants":["Near-max tensor radius locks k-graphs to star extremal","Bounded matching plus spectral near-max forces star form","Tensor stability pins small-matching hypergraphs to S_{n,k,β}","Spectral near-optimum implies edge set near the star extremal","Near-max radius forces every edge through a fixed β-set"],"cache_read_input_tokens":128,"weakest_assumption_plain":"n must be large enough that all lower-order error terms stay strictly smaller than the main-term gap between β and β-1 in the spectral-radius asymptotics.","fun_headline_variants_meta":{"raw":{"variants":["Near-max tensor radius locks k-graphs to star extremal","Bounded matching plus spectral near-max forces star form","Tensor stability pins small-matching hypergraphs to S_{n,k,β}","Spectral near-optimum implies edge set near the star extremal","Near-max radius forces every edge through a fixed β-set"]},"model":"grok-4.5","effort":"low","cost_usd":0.003221,"raw_usage":{"total_tokens":1102,"prompt_tokens":739,"num_sources_used":0,"completion_tokens":92,"cost_in_usd_ticks":32208000,"prompt_tokens_details":{"text_tokens":739,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":271,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":739,"tokens_out":92,"duration_ms":5349,"temperature":1.0,"reasoning_tokens":271,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T19:00:39.616024+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a single infinite family of k-graphs with matching number ≤β whose spectral radius is (1-o(1)) times that of S_{n,k,β} yet whose edge sets remain a positive-density distance away from every copy of S_{n,k,β}.","supporting_citations":[],"review_version":1}