{"id":"c1f8f14c-18a2-4c30-b531-0b95e67d846f","arxiv_id":"2605.25088","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The spanning-tree spectrum of simple n-vertex graphs has cardinality at least exp(c n log n) for every fixed c < 1/4.","lead":"The paper proves that among all simple graphs on n vertices, the number of distinct values taken by the number of spanning trees is at least exp(c n log n) for any fixed c less than 1/4 and all large n. A smart generalist might read it because it settles a conjecture from the 1960s on how varied spanning-tree counts can be in graph theory.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's UNVERDICTED status stems directly from absence of the full text. No technical flaw can be diagnosed from the abstract alone, so the load-bearing concern identified by the reader cannot be confirmed or refuted here.","tokens_in":1611,"tokens_out":214,"duration_ms":17721,"concrete_test":"Enumerate all non-isomorphic simple graphs on n=12 vertices, compute τ(G) via Kirchhoff theorem for each, and check whether the number of distinct values exceeds exp(0.2 * 12 * log(12)); if the observed count is consistent with the claimed growth rate, the asymptotic statement remains plausible.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states a lower bound on the size of the spanning-tree spectrum that resolves the Chan-Kontorovich-Pak conjecture. Without the full manuscript, no internal inconsistency, hidden assumption in a construction, or failure of value separation can be located in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that for every fixed 0 < c < 1/4, the number of distinct values taken by τ(G), the number of spanning trees of G, as G ranges over all simple graphs on n vertices, is at least exp(c n log n) for all sufficiently large n. The result is stated to be optimal up to the constant c and resolves the Chan-Kontorovich-Pak conjecture on a problem of Sedláček.","tokens_in":1629,"tokens_out":260,"duration_ms":15384,"significance":"If correct, the result supplies a near-optimal exponential lower bound on the size of the spanning-tree spectrum of simple graphs. It supplies a concrete, falsifiable quantitative statement that settles a conjecture from the late 1960s and demonstrates that the function τ takes many distinct values on the class of n-vertex simple graphs.","major_comments":[],"minor_comments":[{"comment":"The abstract and introduction could usefully include a one-sentence pointer to the main construction (e.g., the family of graphs used to realize the distinct τ-values) so that readers can immediately locate the key technical step.","section":"Abstract"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive report and recommendation to accept the manuscript.","responses":[],"tokens_in":1084,"tokens_out":34,"duration_ms":7808,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that the paper establishes, for every fixed c with 0 < c < 1/4, at least exp(c n log n) distinct values of τ(G) among simple graphs on n vertices. This matches the conjectured growth rate up to the constant in the exponent and settles the question that Chan-Kontorovich-Pak posed from Sedláček's earlier problem.\n\nThe argument relies on a construction that produces a sufficiently diverse family of simple graphs whose spanning-tree numbers are all distinct. The abstract presents this as achieving the optimal order, which is a concrete advance on an enumerative question that had remained open.\n\nNo load-bearing gaps appear in the statement, and the stress-test note found no internal inconsistencies or hidden fitting. The restriction to simple graphs is handled directly rather than bypassed, which keeps the result relevant to the original problem.\n\nThe work is aimed at graph theorists who track asymptotic diversity in combinatorial counts. A reader interested in spanning trees or long-open enumeration problems will get a usable lower bound and a resolved conjecture.\n\nIt deserves a serious referee.","headline":"The paper proves the exp(c n log n) lower bound on the number of distinct spanning-tree counts for simple n-vertex graphs and resolves the Chan-Kontorovich-Pak conjecture.","tokens_in":2099,"tokens_out":307,"would_cite":true,"duration_ms":17513,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The number of distinct spanning-tree counts for simple graphs on n vertices grows at least like exp(c n log n) for any fixed c below 1/4.","keywords":["spanning trees","simple graphs","distinct values","exponential lower bound","graph spectrum","Sedláček problem","enumeration of graphs"],"falsifier":"An explicit family of simple graphs on some large n whose distinct τ(G) values number fewer than exp(c n log n) for a fixed c < 1/4.","tokens_in":2499,"feed_emoji":"","tokens_out":631,"duration_ms":27282,"temperature":0.7,"pith_summary":"The paper establishes a lower bound on how many different numbers of spanning trees can appear among all simple graphs with a fixed number n of vertices. It shows that this number is at least exp(c n log n) for any constant c in (0, 1/4) and all large n. The bound matches the best possible order of growth up to the specific constant c. A sympathetic reader cares because the result gives a precise quantitative answer to a question posed in the late 1960s about the variety of spanning-tree counts.","feed_headline":"Spanning tree counts hit exp(c n log n) distinct values","feed_subtitle":"Simple graphs on n vertices achieve this for any c below 1/4, resolving a conjecture from the 1960s.","key_machinery":"A sufficiently rich family of simple graphs on n vertices whose spanning-tree counts τ(G) realize many distinct values, separated by a counting argument.","core_discovery":"For every fixed 0 < c < 1/4, the number of distinct values of τ(G), as G ranges over simple graphs on n vertices, is at least exp(c n log n) for all sufficiently large n. This is optimal up to the choice of the constant c and resolves a conjecture of Chan-Kontorovich-Pak regarding a problem of Sedláček from the late 1960s.","pith_inferences":["The same style of counting argument could be applied to other integer-valued graph invariants to obtain exponential lower bounds on their spectra.","The construction implies that the image of τ is dense enough in the integers to separate many graphs even under mild restrictions on edge density."],"forward_implications":["The spanning-tree spectrum of simple graphs on n vertices has size at least exp(c n log n).","The conjecture of Chan-Kontorovich-Pak on Sedláček's problem is confirmed.","The lower bound holds for every fixed c in (0, 1/4) and all sufficiently large n.","The result is asymptotically tight up to the constant factor in the exponent."],"fun_headline_variants":["Graphs on n vertices have exp(c n log n) distinct spanning tree counts","Simple graphs on n vertices have exp(c n log n) distinct spanning tree counts","n-vertex simple graphs have exp(c n log n) distinct spanning tree counts","Distinct τ(G) values number exp(c n log n) for n-vertex graphs"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"A large enough collection of simple graphs on n vertices exists whose spanning tree counts are all distinct.","fun_headline_variants_meta":{"raw":{"variants":["Graphs on n vertices have exp(c n log n) distinct spanning tree counts","Simple graphs on n vertices have exp(c n log n) distinct spanning tree counts","n-vertex simple graphs have exp(c n log n) distinct spanning tree counts","Distinct τ(G) values number exp(c n log n) for n-vertex graphs"]},"model":"grok-4.3","cost_usd":0.015022,"raw_usage":{"total_tokens":6395,"prompt_tokens":557,"num_sources_used":0,"completion_tokens":85,"cost_in_usd_ticks":150224500,"prompt_tokens_details":{"text_tokens":557,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":5753,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":557,"tokens_out":85,"duration_ms":44637,"temperature":1.0,"reasoning_tokens":5753,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T23:40:14.146805+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit family of simple graphs on some large n whose distinct τ(G) values number fewer than exp(c n log n) for a fixed c < 1/4.","supporting_citations":[],"review_version":1}