{"id":"3d980ff2-b34b-431c-ad40-4a623b76e54e","arxiv_id":"1908.08155","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every finite nearlattice with at least 83 times 2 to the (n minus 8) subnearlattices is planar, and for n at least 9 this bound is sharp.","lead":"This paper proves a sharp limit: any finite algebraic structure called a nearlattice that contains many sub-nearlattices must have a drawing with no crossing lines. The result pins down the exact number of sub-nearlattices that guarantees planarity, and proves the number cannot be improved.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unverified computer σ-values are the pivotal unconfirmed link, and Lemma 4.7's claimed K-independence is false, so the printed outputs need independent rechecking.","rationale":"The central claim is a sharp threshold theorem whose proof is a minimal-counterexample argument that eliminates all possibilities by showing certain qn-lattices have σ≤83. The eliminations rest on a large number of computer-generated σ-values, and the paper does not provide a versioned program or machine-checked proof of those numbers. The reader's weakest assumption correctly identifies this as the main risk. On closer reading, the risk is heightened because Lemma 4.7, which is used to justify the independence of σ(W) from the chosen K, is demonstrably false in general: an element inserted between a and c in a configuration with a∨b=c yields a forced operation d∨b=c, making the subuniverse {b,d} not of the form X∪Y. This does not by itself invalidate the theorem, because the printed output files appear to have K=KW for every case, but it means the reliability of the σ-values hinges entirely on faithful reconstruction of the inputs and a correct closure algorithm. A wrong transcription in any of the roughly one hundred cases would allow a nonplanar counterexample with σ>83, breaking the sharp bound. The paper is otherwise a serious and careful piece of work, with a plausible proof structure and useful geometric lemmas; the concern is mechanical verification, not mathematical fraud or intentional omission. The conditional verdict is therefore appropriate: the theorem is likely true, but independent re-running or formal verification of the σ-values is needed before the result should be accepted unconditionally.","tokens_in":65266,"tokens_out":23617,"duration_ms":208616,"concrete_test":"Implement an independent verifier (e.g., Python) that reads each output file in Appendix 2, reconstructs the element set, order, and jm-constraints, computes the least qn-lattice closure by repeatedly applying axioms (A1)–(A5) as in Lemma 4.5, enumerates all subsets to count subuniverses, and compares the resulting σ with the value printed. Also check that every element appears in at least one constraint; if any file has K≠KW, recompute σ(KW) and flag a mismatch. This would settle whether any reported σ-value is incorrect.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 2.2 (hence Theorem 1.2) depends crucially on dozens of σ-values computed by the program 'sublatts', quoted in Lemmas 5.4–7.17. The program is not bundled with the arXiv submission, the input files must be reconstructed from the printed outputs, and no machine certificate is provided. A single wrong σ-value or a missed parsing case would allow a minimal counterexample to survive and break the theorem. This concern is amplified by an internal error: Lemma 4.7 claims σ(K)=σ(KW) for any K⊇KW, but this is false. If a new element d satisfies a≤d≤c with a∨b=c, axiom (A5) forces d∨b=c, so {b,d} is not closed and σ(K)≠σ(KW). The printed files appear to have K=KW throughout, but the lemma's failure means the program's outputs can only be trusted if every input file indeed has no extraneous elements and the σ-values are recomputed. Without an independent re-run, the sharp threshold 83·2^{n−8} is credible but unconfirmed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a sharp threshold for planarity of finite nearlattices: an n-element nearlattice with at least 83·2^(n−8) subnearlattices must be planar, and for every n ≥ 9 there exists a non-planar n-element nearlattice with exactly 83·2^(n−8)−1 subnearlattices. The proof works with a minimal counterexample and introduces qn-lattices and jm-constraints as technical tools. It then excludes a large finite family of configurations by combining structural lemmas about entries, anchors, and bridges with σ-values computed by the program 'sublatts'. The sharpness part is inherited from the author's earlier theorem for lattices.","tokens_in":65504,"tokens_out":12132,"duration_ms":123695,"significance":"If the result is correct, it is a natural nearlattice analogue of Czédli's theorem on lattices with 83·2^(|L|−8) sublattices and strengthens the author's subsequent semilattice result with threshold 127·2^(n−8). The qn-lattice/jm-constraint framework is a potentially reusable method for reducing subalgebra-counting questions to finite combinatorial checks. The paper is well organized, the minimal-counterexample argument is clearly structured, and the geometric lemmas in Sections 3 and 8 are presented in detail. The main weakness is that the numerical core of the proof is not independently verifiable from the submission: the program is not bundled, the input files are not supplied, and one lemma used to justify the σ-values is false as stated.","major_comments":[{"comment":"The K-independence claim of Lemma 4.7 is false. Let W = {a∨b = c} over KW = {a,b,c}, and let K = KW ∪ {d} with a ≤ d ≤ c in the ambient nearlattice. Axiom (A5) forces d∨b = c, so the subset {b,d} is not a subuniverse of K; consequently |Sub(K)| is not 2^(|K\\KW|)·|Sub(KW)| and σ(K) ≠ σ(KW). Since Lemma 4.8 is stated as a consequence of Lemmas 4.2 and 4.7, the proof of the main exclusion mechanism needs repair. The repair is local: define σ(W) directly as σ(KW) and derive Lemma 4.8 from Lemma 4.2 alone; the false independence statement and the remark about input files containing extraneous elements should then be removed or amended.","section":"Section 4, Lemma 4.7"},{"comment":"The contradiction in the proof of Theorem 2.2 rests on roughly one hundred σ-values computed by the program 'sublatts' (for example, Lemma 5.5 uses ten values and Lemmas 7.13–7.17 use values such as 77, 71.75, 69.5, 75, 74, 78.75, 79, 70.5, 82, and 81.5). The program is not included in the arXiv submission, the input files are not provided (only one sample appears in Section 9), and no machine-checked certificate is supplied. These values are load-bearing: via Lemma 4.8 each value σ ≤ 83 excludes a case, and the final contradiction requires that every listed case really is excluded. Without the program, all input files, and a reproducible way to regenerate the printed outputs, the central claim is credible but unverified by the submitted material. Please supply the program, the complete set of input files, and a script or certificate that reproduces the σ-values in Appendix 2.","section":"Sections 4–7 and Appendix 2"}],"minor_comments":[{"comment":"The sentence 'Since p′(0) > 0 by (3.8)' should refer to p(0) > 0; equation (3.8) establishes the positivity of p at 0, not of p′ at 0.","section":"Section 3, proof of Lemma 3.4"},{"comment":"The input-file syntax is documented only by a running example; a short formal description of the commands and constraints syntax would substantially help independent re-runs of the program.","section":"Section 9, Algorithm 9.1"},{"comment":"The abstract says 'For n > 8, this result is sharp' while Theorem 1.2 says 'For n ≥ 9'; these are equivalent for integer n, but the wording should be unified.","section":"Abstract and Theorem 1.2"},{"comment":"The proof of the meet-reducible case in Remark 1.3 is omitted with 'analogous details are omitted'; since this remark is not used in the main theorem, this is acceptable, but a reference or a brief argument would be preferable.","section":"Remark 1.3"}],"recommendation":"major_revision","confidential_remarks":"I believe the theorem is likely correct and that the main issues are fixable: the false Lemma 4.7 has an obvious local repair, and the computational component can be made verifiable by archiving the program and all input files. I would not reject the paper outright, but acceptance should be conditional on full computational transparency and on correcting Lemma 4.7."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious, sharp extension of Czedli's earlier 83-sublattice theorem to nearlattices, with a reusable qn-lattice machinery. I think it deserves referee time, but the referee needs to recheck the computer part, and one lemma is false as stated.\n\nWhat is genuinely new: Theorem 1.2 is a real generalization—finite nearlattice with at least 83*2^{n-8} subnearlattices is planar, sharp for n>=9. Since Kelly-Rival has no semilattice analogue, the proof instead parses a minimal counterexample using qn-lattices and sigma-values computed by the program. The reduction in Section 8 to two minimal elements, entries, and anchors is intricate and mostly clear. The appendix prints all output files and one sample input, and the program is said to be freely available. That is real, reproducible-in-principle work.\n\nSoft spots, in order of importance. First, Lemma 4.7 is wrong as stated. It claims sigma(K)=sigma(KW) for any K containing KW, but adding an element d with a<=d<=c when a∨b=c forces d∨b=c by axiom (A5), so {b,d} is not closed and the subuniverse count changes. The proof multiplies by 2^{K\\KW} and silently assumes no new axioms fire. The paper's input files appear to use KW only, so the error may not be load-bearing, but as written it is an error and the referee should confirm that no reported sigma-value depends on omitted elements.\n\nSecond, the central contradiction depends on dozens of sigma-values printed in Appendix 2 but produced by an unbundled program. The input files must be reconstructed from output files, and there is no machine-checked certificate. This is not fatal—the computation method is simple and the outputs are there—but it is a genuine mechanical soft spot. One wrong sigma-value or one missed parsing case would break the theorem, and I could not independently audit the values.\n\nThe analytic parts I checked are coherent, and the self-citations to [13] and [14] are appropriate because those are precisely the theorems being generalized. The proof of Remark 1.3 is sketched with one part omitted, which is minor.\n\nBottom line: a lattice theorist will get real value from this paper. Send it to a careful referee who can rerun the program. I would not desk reject.","headline":"A sharp nearlattice analogue of Czedli's 83-sublattice theorem, built on a clever new qn-lattice technique; the proof is credible but rests on unverified computer output and contains one false lemma that needs checking.","tokens_in":66048,"tokens_out":3008,"would_cite":true,"duration_ms":34559,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["06A12","06B75","20M10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A high subnearlattice count forces a planar diagram","keywords":["planar nearlattice","planar semilattice","subnearlattice count","relative number of subuniverses","qn-lattice","computer-assisted proof","sharp threshold","commutative idempotent semigroup"],"falsifier":"Recompute every $\\sigma$-value printed in the output files of Sections 5 to 7 by running 'sublatts' on the corresponding input files; if any value reported as at most 83 is actually larger, or if any printed output does not match the cited case, the exclusion argument breaks.","tokens_in":65023,"feed_emoji":"📐","tokens_out":7742,"duration_ms":72779,"temperature":0.7,"pith_summary":"The paper proves a sharp counting threshold for planarity in finite nearlattices: any $n$-element nearlattice with at least $83\\cdot 2^{n-8}$ subnearlattices must have a planar Hasse diagram. For every $n\\ge 9$ the bound is sharp, because a non-planar $n$-element nearlattice exists with exactly $83\\cdot 2^{n-8}-1$ subnearlattices. Finite nearlattices include finite semilattices, commutative idempotent semigroups, join-enriched meet semilattices, and chopped lattices, so the result transfers to all these forms. The proof works by assuming a smallest counterexample, encoding small configurations as qn-lattices, and using computer-generated subalgebra counts to exclude every possible configuration.","feed_headline":"A high subnearlattice count forces a planar diagram","feed_subtitle":"Any n-element nearlattice with at least 83·2^(n−8) subnearlattices is planar, and one fewer can fail.","key_machinery":"Two devices carry the argument. The first is the relative number of subuniverses, $\\sigma(L)=|\\operatorname{Sub}(L)|\\cdot 2^{8-n}$, which normalizes the count so that the threshold becomes $\\sigma>83$ independent of $n$. The second is the notion of a qn-lattice: a finite poset with partial join and meet operations satisfying five closure axioms, understood as a weak partial subalgebra of a nearlattice. A monotonicity lemma shows that every sub-qn-lattice has $\\sigma$-value at least that of the ambient nearlattice, so a minimal counterexample cannot contain any qn-lattice of $\\sigma\\le 83$. The proof then parses the possible shapes of a minimal counterexample into finitely many cases, computes each case's $\\sigma$-value with the program 'sublatts', and uses geometric lemmas about pointed contours and cells to glue planar pieces whenever a configuration would otherwise be unavoidable.","core_discovery":"Finite upper nearlattices are join-semilattices whose meet is defined exactly when the two elements have a common lower bound; equivalently they are finite semilattices or commutative idempotent semigroups. The main theorem states that if such an $n$-element structure has at least $83\\cdot 2^{n-8}$ subnearlattices, namely nonempty subsets closed under the total join and the partial meet, then its Hasse diagram is planar. Equivalently, writing $\\sigma(L)=|\\operatorname{Sub}(L)|\\cdot 2^{8-n}$, every finite nearlattice with $\\sigma(L)>83$ is planar. For $n\\ge 9$ the statement is sharp: there exists an $n$-element non-planar nearlattice with $\\sigma(L)=83$, that is, with exactly $83\\cdot 2^{n-8}-1$ subnearlattices. Since finite lattices are nearlattices with a smallest element, the theorem generalizes the earlier planar-lattice result, and the proof reduces the lattice case to that prior result.","pith_inferences":["The constant 83 is a computed boundary rather than a value tied to an obvious structural family; a structural description of nearlattices sitting just below the threshold might explain why 83 is the extremal number.","Because the program's input files are not bundled, an independent reimplementation of 'sublatts' or a machine-checked translation of the case trees would turn the computer-assisted portion of the proof into a fully verifiable computation.","The qn-lattice closure construction is general enough that the same normalization-plus-exclusion scheme may yield sharp thresholds for other classes of partial algebras, such as algebras with one partial operation of higher arity."],"forward_implications":["Every finite semilattice, viewed as a nearlattice, has a planar Hasse diagram when its subnearlattice count reaches the $83\\cdot 2^{n-8}$ threshold.","For each $n\\ge 9$, the constant $83\\cdot 2^{n-8}$ cannot be lowered while preserving the implication, since a non-planar example sits exactly one subnearlattice below the threshold.","The theorem extends the planar-lattice threshold to all finite commutative idempotent semigroups and chopped lattices, not only lattices with a least element.","At the smallest sizes the threshold is even stronger in some cases: every nearlattice with at most seven elements is planar, and every eight-element nearlattice with at least 74 subnearlattices is planar."],"supporting_citations":[{"why":"Supplies the prior planar-lattice theorem that handles the lattice case of the main theorem, the sharpness examples for $n\\ge 9$, and several reused $\\sigma$-values.","marker":"[13]"},{"why":"Provides the closely related subsemilattice threshold that motivates the nearlattice formulation and the relative-count normalization.","marker":"[14]"},{"why":"Provides the planar-region lemmas about boundary chains, cells, and the transitivity of 'left of' used in Section 8 to force planarity of a minimal counterexample.","marker":"[46]"},{"why":"Supplies the equivalence between curved and straight planar diagrams used to conclude planarity of glued diagrams.","marker":"[45]"},{"why":"Background lattice theory, including the fact about a boolean sublattice generated by three elements used in one exclusion lemma.","marker":"[21]"}],"fun_headline_variants":["A subnearlattice count threshold forces planar diagrams","83·2^(n−8) subnearlattices force a planar Hasse diagram","Many subnearlattices imply planarity in nearlattices","Sharp bound: subnearlattice count forces planar Hasse diagram"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the correctness of the computer program 'sublatts' and on the faithful transcription of every case tree into its input files; a single wrong reported count or a missed subcase in Sections 5 to 7 could let a minimal counterexample survive.","fun_headline_variants_meta":{"raw":{"variants":["A subnearlattice count threshold forces planar diagrams","83·2^(n−8) subnearlattices force a planar Hasse diagram","Many subnearlattices imply planarity in nearlattices","Sharp bound: subnearlattice count forces planar Hasse diagram"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000901,"raw_usage":{"total_tokens":3834,"prompt_tokens":857,"completion_tokens":2977,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":2898}},"tokens_in":473,"tokens_out":2977,"duration_ms":22086,"temperature":1.0,"reasoning_tokens":2898,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:47:34.767635+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute every $\\sigma$-value printed in the output files of Sections 5 to 7 by running 'sublatts' on the corresponding input files; if any value reported as at most 83 is actually larger, or if any printed output does not match the cited case, the exclusion argument breaks.","supporting_citations":[{"cited_title":"One hundred twenty-seven subsemilattices and planarity","cited_arxiv_id":"1906.12003","evidence_quote":"Provides the closely related subsemilattice threshold that motivates the nearlattice formulation and the relative-count normalization."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the planar-region lemmas about boundary chains, cells, and the transitivity of 'left of' used in Section 8 to force planarity of a minimal counterexample."},{"cited_title":"Discrete Mathematics 63, 197-216 (1987)","cited_arxiv_id":null,"evidence_quote":"Supplies the equivalence between curved and straight planar diagrams used to conclude planarity of glued diagrams."},{"cited_title":"Birkh¨ auser Verlag, Basel (2011)","cited_arxiv_id":null,"evidence_quote":"Background lattice theory, including the fact about a boolean sublattice generated by three elements used in one exclusion lemma."}],"review_version":1}