{"id":"ae53d3ce-f89e-4bba-bcba-13c175b62230","arxiv_id":"2607.16103","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Every n≥8 vertex planar graph with no 8-cycle has at most 69/25 (n−2) edges, improving the previous best coefficient ≈2.99 to 2.76.","lead":"A new proof shows every planar graph on n vertices with no 8-cycle has at most (69/25)(n−2) edges, improving the previous leading coefficient from ≈2.99 to 2.76. The bound narrows the gap in a long-open extremal problem and is certified by audited computer searches.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 rests on unverified finite certificates; a missing configuration in (C5)'s 9/2 load bound would invalidate the discharging.","rationale":"We read the proof in good faith. The discharged arithmetic and induction are correct; the only non-transparent components are the six certificates. We examined the lifting arguments in the appendix, in particular Lemma 16 (merged-H distances) and Lemma 17 (layer-cutoff audit), and found no explicit mathematical error. However, the theorem's validity depends entirely on the exhaustiveness of these finite searches. This is not a disagreement with consensus; it is a correctness risk specific to computer-assisted proofs. The paper's documentation is exemplary by the genre's standard, but the code is not archived with a hash or DOI, and none of the lifting is machine-checked. Therefore the certificate maxima—especially the 9/2 bound in (C5)—are the single load-bearing assumption. The proposed independent re-execution would settle this: if it reproduces the certificates, our concern is resolved; if it finds a discrepancy, the discharging fails. We therefore maintain the reader's CONDITIONAL verdict.","tokens_in":19377,"tokens_out":23554,"duration_ms":208400,"concrete_test":"Independently implement Algorithm 5 (sealed three-root-edge search) from the pseudocode in §A6 in a separate codebase, without using the authors' source or certificate files, and enumerate the full transition tree. Compare the reported totals (223,766 states, 136,987 leaves) and the terminal load maximum (≤9/2). Also re-run the full verification suite (C1)-(C6) using the archived code from an independent machine/compiler. If the independent run reproduces the exact certificate (max 9/2, zero omitted layer branches, zero overbound terminals), the finite bound is confirmed. If it finds any terminal state with load >9/2 or any valid branch above layer 6, Theorem 1 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 is proven by discharging, and every discharge inequality depends on numeric maxima from certificates (C1)-(C6). The finite searches are exhaustive only under the lifting arguments of Propositions 1-2 and Lemmas 6-17. The most delicate is (C5): the sealed three-root-edge search (Algorithm 5) bounds L_f(e) ≤ 9/2 for every edge of a face f with d(f)≥9. This search uses a compressed root-boundary model, H/R/O-seals, merged-H distances, and a construction-layer cutoff at 6. If any genuine local configuration is not represented in the terminal states—e.g., because the construction-layer audit of Lemma 17 misses a legal relevant branch above layer 6, or because the minimum-layer parent rule in the ordinary searches prunes a valid BFS exposure—the certified bound could be too low, and Lemma 4 would fail. Then the final charge of large faces could become negative and the edge bound 69/25(n−2) would not follow. The same structural risk applies to (C2)-(C4), whose maxima (7,7,5,4,6) are computed under the closure conditions of Lemma 12. No part of the lifting is machine-checked, and the code/certificates are only available on a personal homepage without a hash or archive DOI (Data and code availability, p.7). Thus the theorem's correctness depends entirely on the correctness and exhaustiveness of these unverified finite computations.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 1: every n-vertex simple planar graph with no (not necessarily induced) copy of C_8 has at most (69/25)(n−2) edges for every n ≥ 8. This improves the previous best bound (323/108)n − 6. The proof reduces to the 2-connected, minimum-degree-at-least-3 case by an induction on n, and then applies discharging. Faces receive initial charge 2d−4; each face sends α = 50/69 to each incident vertex; triangular faces are compensated from their nearest 4+-faces according to an equal-split load. The load bounds used in the discharging are ℓ(f) ≤ 19/3 for 4-faces, ℓ(f) ≤ 7,7,5,4 for faces of degree 5,6,7, and ℓ(f) ≤ 9d/2 for d ≥ 9. These bounds are derived from six finite certificates (C1)–(C6), which are asserted in Theorem 2 and verified by exhaustive plane-patch searches described at length in the appendix. The discharging then gives e(G) ≤ (2/α)(n−2) = (69/25)(n−2).","tokens_in":19673,"tokens_out":15447,"duration_ms":139230,"significance":"If the result holds, it is a meaningful improvement in the planar Turán problem for C_8, lowering the upper-bound coefficient from 323/108 ≈ 2.991 to 69/25 = 2.76 and narrowing the gap to the construction with coefficient 21/8 = 2.625. The paper's hand-checkable mathematics is sound: I checked the induction steps, the cut-vertex and low-degree reductions, the base case for n = 8, and the discharging algebra. The coefficients α = 50/69, 4/23, 19/3, and 9d/2 all work out as stated. The finite computations are described in unusual detail for a math paper, with pseudocode, state counts, and certificate-checking claims. The main weakness is not the mathematical framework but the verifiability of the finite certificates: the actual certificate files and code are not included in the manuscript and are only available on a personal homepage, and several load-bearing lifting steps are argued informally rather than machine-checked.","major_comments":[{"comment":"The theorem's central claim depends on the six finite certificates (C1)–(C6). The manuscript reports aggregate maxima and state counts, but the actual certificate files and source code are only available on a personal homepage, without a hash, archive DOI, or versioned repository. For a computer-assisted proof in a journal, this is not reproducible. Please deposit the code, recorded outputs, and machine-readable certificates in a permanent archive (e.g., arXiv ancillary files or Zenodo) with a checksum/DOI, or include the certificate data as supplementary material. Without this, an independent reader cannot verify that the reported maxima (7, 7, 5, 4, 6, 19/3, 9/2) are truly outputs of the described exhaustive searches.","section":"§2, Theorem 2; Data and code availability, p.7"},{"comment":"The quantities p_S(g), q_S(g), s_S(g), and λ_S(g) used to certify (C5) are not formally defined. Section 2 defines S_f(g), c_{f,e}(g), and L_f(e), but the appendix refers only to 'visible distinguished-entry count', 'visible shortest-entry count', and 'merged-nearest-face count' as if they had been defined in Section 2. Lemma 16's inequality c_{f,e}(g) ≤ λ_S(g) is load-bearing for the large-face bound, so the sealed dual graph D_S, the merging of H-seals, and the exact definitions of p_S, q_S, s_S must be stated precisely. As written, a reader cannot recompute or audit the terminal load values from the certificate.","section":"A6, Algorithm 5 and Lemma 16"},{"comment":"The layer-cutoff audit is described only in prose: at every reachable state the verifier 'retries every admissible third-vertex choice there with the construction-layer bound relaxed' and 'finds no legal relevant triangular branch above layer 6.' This audit is exactly what ensures that the construction-layer cutoff at 6 omits no genuine relevant configuration. The statement is not accompanied by a pseudocode block or a machine-checked certificate of its own, and the notion of 'legal relevant branch' is not formalized. Since Lemma 18 uses this audit to conclude that the sealed search is exhaustive, this gap is load-bearing and needs to be closed either by a precise algorithmic specification or by an independently checkable certificate.","section":"A6, Lemma 17(2)"}],"minor_comments":[{"comment":"The notation p(m) is used before it is defined; it should be stated explicitly that p(m) denotes the maximum number of edges in an m-vertex planar graph (with p(1)=0, p(2)=1, p(m)=3m−6 for 3≤m≤7).","section":"§3, proof of Theorem 1"},{"comment":"In the case u ≤ 5, the inequality is correct but could be written more clearly as ℓ(f) ≤ u + (7−u)/2 ≤ 6, which makes the use of the bound b(f) ≤ 7 transparent.","section":"Lemma 3"},{"comment":"The statement 'A face of degree 8 cannot occur' relies on the fact that in a 2-connected plane graph every face boundary is a cycle; this should be said explicitly at that point for the reader's convenience.","section":"§3, discharging"},{"comment":"The URL is given with a line break and no version information. If the repository is updated, the certificate outputs in the paper could become impossible to match. A stable identifier and a note on the exact software versions used would help.","section":"Data and code availability"}],"recommendation":"major_revision","confidential_remarks":"The mathematical skeleton is sound and the paper is likely publishable if the computational artifacts are made verifiable and the formal definitions in A6 are tightened. I do not recommend rejection: the discharging proof itself is correct, and the computational claims are plausible and unusually well documented. However, the current reliance on unarchived personal-homepage code and on hand-written lifting arguments that are not machine-checked is too fragile for the central theorem to be accepted as it stands."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"One thing to know: this is a genuine advance, not a fitted coefficient. The new bound 69/25(n−2) improves the previous 323/108 n − 6 substantially, and the proof's mathematical skeleton is sound. The thing to watch is the finite-search layer, exactly as the stress-test note says.\n\nWhat's new: Theorem 1 is a true C8-specific bound, derived from six finite certificates. I checked the discharging arithmetic: α=50/69, triangles end at 0, 4-faces reach 0 using ℓ≤19/3, 5-7 faces are positive, and large faces use L_f(e)≤9/2. The reduction to 2-connected δ≥3 via induction is clean, and the base case n=8 is independently verified. The certificates are not circular: the searches reject states containing an 8-cycle and then output maxima; 69/25 is forced by those maxima.\n\nStrength: the appendix documents the certificate searches far better than most computer-assisted discharging papers: exact transition tree, a re-parser certificate checker, malformed-certificate tests, and a no-layer-limit retry audit. That is real evidence of care.\n\nSoft spots: the central claim rests on the exhaustiveness of the sealed search for C5 and on an informal lifting argument (Propositions 1–2 and Lemmas 6–17). If a legal triangular continuation is missing, the load bound 9/2 could fail and the whole discharging collapses. That is a genuine risk, but it is not a red flag. It is the standard burden of computer-assisted proof, and this paper goes further than most to meet it. The code and certificates, however, are only on the corresponding author's homepage with no hash or archive DOI, and the lifting is not machine-checked. I would want an independent rerun of the 854.7-second suite, or a formalization of the lifting, before treating the bound as settled.\n\nThe citation pattern is fine, and the framing is honest about this being an upper bound, not a determination. Audience: planar Turán researchers and people who care about credible computer-assisted discharging. I would send it to peer review and ask for archive-stable artifacts and independent verification. Desk rejection would be a mistake.","headline":"A real improvement on C8 planar Turán: the human-checkable discharging is correct, and the finite-certificate layer is transparent enough that the only open question is independent verification.","tokens_in":20318,"tokens_out":4112,"would_cite":true,"duration_ms":40414,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C10","05C38"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every planar graph with no 8-cycle has at most 2.76(n−2) edges.","keywords":["planar Turán number","C8-free planar graphs","discharging method","computer-assisted proof","extremal graph theory","cycle subgraphs","edge bounds"],"falsifier":"Find a simple planar graph on n vertices with no 8-cycle and more than 69/25(n−2) edges, or exhibit a local rooted configuration that violates one of the certificates: a 4-face with seven triangles for which it is the unique nearest 4+-face, a boundary edge of a face of degree at least 9 carrying load above 9/2, or a planar non-Hamiltonian graph on eight vertices with 17 or 18 edges.","tokens_in":19176,"feed_emoji":"📐","tokens_out":4340,"duration_ms":34219,"temperature":0.7,"pith_summary":"This paper proves an upper bound on how many edges a planar graph can have without containing an 8-cycle: every n-vertex such graph has at most 69/25 (n−2), about 2.76(n−2), edges. That improves the previous best bound, whose leading coefficient was about 2.99, for every n at least 8. The proof combines a discharging argument on faces with six finite computer-checked certificates about local configurations. If correct, it brings the C8 planar Turán number closer to the known construction with coefficient 2.625, though it does not settle the exact value.","feed_headline":"Planar graphs with no 8-cycle top out at 2.76(n−2) edges","feed_subtitle":"New bound improves the best known upper limit for the C8 planar Turán problem and closes part of the gap to the construction.","key_machinery":"The proof is a discharging argument on the faces of a 2-connected plane graph. Each face starts with charge 2d(f)−4; every face sends α = 50/69 to each incident vertex, and every triangular face receives compensation (4/23)/|N(g)| split equally among its nearest 4+-faces, defining the equal-split load ℓ(f). The load is controlled by six finite certificates: bad-count maxima for small faces, at most six unique-nearest contributors to a 4-face, the dangerous 4-face case, a per-edge load bound of 9/2 for faces of degree at least 9, and an eight-vertex base case. These local bounds force every face to end with nonnegative charge, so Euler's formula gives 4n−8 ≥ 2α e(G), hence e(G) ≤ 69/25 (n−2).","core_discovery":"The central claim is Theorem 1: for every n≥8, the planar Turán number ex_P(n, C8) is at most 69/25 (n−2). In plain terms, any simple planar graph on n vertices that contains no subgraph isomorphic to an 8-cycle has at most 2.76(n−2) edges. This supersedes the earlier bound (323/108)n − 6, which held for n≥27, and extends the bound to all n≥8. The authors are explicit that this is an improvement, not an exact determination: the best known construction still has leading coefficient 21/8 = 2.625, and equality cases are not characterized.","pith_inferences":["If the local certificates could be sharpened, the same discharging scheme might push the coefficient below 2.76, and the natural limit to test is the conjectured 21/8 from the construction.","The heavy reliance on exhaustive local searches suggests the method could transfer to C9 or longer cycles, but the search space and certificates would grow substantially.","A possible testable extension: run the same finite search on the dangerous 4-face case with a larger radius to see whether the extra nearest 4+-face can be forced at distance 2 rather than 3, which would improve the four-face load bound from 19/3."],"forward_implications":["For every n≥8, no planar C8-free graph can exceed about 2.76(n−2) edges, improving the previous leading coefficient 323/108 ≈ 2.99 for n≥27.","The gap between the upper bound 2.76 and the construction lower bound 2.625 remains open; the exact planar Turán number of C8 is not determined by this paper.","The induction base handles n=8 via the finite fact that every planar graph on eight vertices with 17 or 18 edges is Hamiltonian.","Degree-8 faces cannot appear in the extremal discharging argument, since an 8-face would itself be an 8-cycle; the bound survives through that exclusion."],"fun_headline_variants":["C8-free planar graphs edge bound tightened to 2.76(n−2)","New best upper bound for planar Turán number of 8-cycles","No 8-cycle planar graphs: edge count cut to 2.76(n−2)","Improved planar Turán bound for C8: at most 69/25 edges per vertex","Tighter cap on edges in planar graphs without 8-cycles"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof depends on the finite computer searches being exhaustive for every local configuration that can actually occur in a 2-connected, C8-free planar graph with minimum degree 3; the hand-written lifting argument that transfers those enumerated bounds to all such graphs is not machine-checked.","fun_headline_variants_meta":{"raw":{"variants":["C8-free planar graphs edge bound tightened to 2.76(n−2)","New best upper bound for planar Turán number of 8-cycles","No 8-cycle planar graphs: edge count cut to 2.76(n−2)","Improved planar Turán bound for C8: at most 69/25 edges per vertex","Tighter cap on edges in planar graphs without 8-cycles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001388,"raw_usage":{"total_tokens":5385,"prompt_tokens":603,"completion_tokens":4782,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":347,"completion_tokens_details":{"reasoning_tokens":4677}},"tokens_in":347,"tokens_out":4782,"duration_ms":29286,"temperature":1.0,"reasoning_tokens":4677,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T21:19:37.362107+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a simple planar graph on n vertices with no 8-cycle and more than 69/25(n−2) edges, or exhibit a local rooted configuration that violates one of the certificates: a 4-face with seven triangles for which it is the unique nearest 4+-face, a boundary edge of a face of degree at least 9 carrying load above 9/2, or a planar non-Hamiltonian graph on eight vertices with 17 or 18 edges.","supporting_citations":[],"review_version":1}