{"id":"6a3207ef-67e7-4b56-8262-a7d1a1c0dbc0","arxiv_id":"1908.04320","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Tropically planar graphs number 152 at genus 6 and 672 at genus 7, grow between roughly 2.47^g and 2^{11g/3+o(g)}, and make up asymptotically 0% of connected trivalent planar graphs.","lead":"Researchers counted the graphs that can appear as skeletons of smooth tropical plane curves, finding 152 of genus 6 and 672 of genus 7, and proved bounds showing such graphs grow much more slowly than all planar cubic graphs. A generalist might read this as a concrete census of a tropical-geometry family, plus evidence that most planar cubic graphs cannot come from tropical curves.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The lower-bound base gamma rests on the unverified assertion that the 75 genus-6, 13 genus-4, and 2 genus-2 pictured tiles are distinct marked regular triangulations of the parallelogram; a duplicate or nonregular tile would shrink the recurrence coefficients and lower the bound.","rationale":"The paper's strongest claims are the zero-density statement, the upper bound T(g) = O(2^{11g/3+O(sqrt g)}), and the lower bound T(g) = Omega(2.47^g). The most load-bearing of these for the paper's novelty is the lower bound, because it improves the previously known Omega(2^g) bound and depends on a concrete combinatorial construction. That construction rests on the tile counts 2, 13, and 75, which enter directly into the recurrence for a_n; any duplicate, nonregular, or otherwise invalid tile changes the recurrence and therefore changes the base gamma. The paper supplies neither the triangulations nor the verification code/data, and the uniqueness assertions in Propositions 5.2 and 5.3 are justified only by reference to the pictures and by the phrase 'by construction.' This is a genuine reproducibility gap in the central lower-bound argument. I do not treat this as a demonstrated error: with the correct code or full triangulation data, the claims may well be true, and the plausible structural arguments in the paper support a conditional acceptance. The Theorem 4.2 gap noted by the reader is also real, but it is not the most load-bearing concern for the main claims, since the zero-density conclusion already follows from the upper bound T(g) = O(2^{11g/3}) together with the known growth P(g) = Omega(15.88^g). The exact counts T(6)=152 and T(7)=672 share the same computational reproducibility issue, but the lower-bound tile inventory is the single point on which the advertised gamma rests. The appropriate verdict remains CONDITIONAL, unchanged from the reader's assessment.","tokens_in":23665,"tokens_out":10354,"duration_ms":112885,"concrete_test":"Independently enumerate all regular unimodular triangulations of the parallelograms P||_2, P||_4, and P||_6 using TOPCOM or another exact enumeration tool with the vertex sets defined in Section 5, compute the marked dual graphs with the L and R attachment points as specified in Figures 23-29, and compare against the 2, 13, and 75 tiles used in the proof. If every listed tile is regular and the marked graphs are pairwise distinct, and the ordered-pair condition in Proposition 5.3 holds, recompute the recurrence and verify alpha ≈ 6.1233; otherwise re-derive Corollary 5.5 with the corrected tile counts and recurrence coefficients.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main lower-bound construction in Section 5 depends entirely on the tile inventory: 2 genus-2 tiles, 13 genus-4 tiles, and 75 genus-6 tiles, which yield the recurrence a_n = 2a_{n-1} + 13a_{n-2} + 75a_{n-3} in Proposition 5.4 and the claimed base gamma = sqrt(alpha) ≈ 2.47 in Corollary 5.5. The proof that each tile is realizable as a regular unimodular triangulation of the relevant parallelogram, and that the tiles give pairwise distinct marked graphs satisfying the ordered-pair uniqueness condition in Proposition 5.3, is asserted but not demonstrated: the TOPCOM search is described without supplying code, input files, or output logs, and Appendix A pictures only the dual graphs with L/R markings, not the actual triangulations of P||_2, P||_4, and P||_6. A duplicate marked graph would inflate the recurrence coefficient, and a nonregular tile would be invalid for the gluing construction; either defect changes alpha and could lower gamma below the claimed value or even below the previously known Omega(2^g) bound. Since the advertised improvement over the previous lower bound is precisely this larger base, the lower-bound theorem is conditional on the unverified tile inventory.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies trophically planar graphs, defined as skeletons of smooth tropical plane curves. It develops new necessary conditions for a graph to be trophically planar (notably the TIE-fighter obstruction), computes the exact numbers of trophically planar graphs of genus 6 and 7 (152 and 672), and proves asymptotic estimates: an upper bound T(g) = O(2^{11g/3+O(√g)}) and a lower bound T(g) = Ω(γ^g) with γ ≈ 2.47. From these results the authors conclude that, asymptotically, 0% of connected trivalent planar graphs are trophically planar.","tokens_in":23904,"tokens_out":18332,"duration_ms":181399,"significance":"If the claims hold, the paper makes a substantial contribution to the quantitative study of trophically planar graphs. The upper bound is a nontrivial exponential estimate, and the lower bound improves the previously known Ω(2^g) to a base γ ≈ 2.47, a genuine advance. The zero-density statement is a strong negative result. The TIE-fighter obstruction is an elegant new structural tool, and the exact counts for genus 6 and 7 provide valuable data. The paper also includes a large set of computational tiles, but the reproducibility of that finite verification is a major weakness, as detailed below.","major_comments":[{"comment":"The lower bound rests on an unverified finite tile inventory. The paper asserts that a TOPCOM search shows that the parallelograms P^||_2, P^||_4, and P^||_6 admit no non-regular triangulations, and that the 2+13+75 pictured tiles are distinct marked graphs, but it supplies no code, input files, output logs, or the actual triangulations. Appendix A presents only the marked skeletons, not the triangulations of the parallelograms. A duplicate or non-regular tile would change the coefficients of the recurrence a_n = 2a_{n-1} + 13a_{n-2} + 75a_{n-3} in Proposition 5.4 and could lower γ below the claimed value of √α ≈ 2.47, potentially even below the previously known Ω(2^g) bound. The lower-bound theorem is therefore conditional on a finite verification that is not made reproducible.","section":"Section 5, Corollary 5.5"},{"comment":"The proof of Proposition 5.3 relies on two unproved assertions about the tile set: that no two tiles give the same ordered pair of marked graphs, and that no tile contributing two 2-edge-connected components contributes a component that is also available from a single-tile component. These are nontrivial combinatorial properties over a set of 90 tiles. The paper neither proves them nor provides a machine-checkable certificate. Since the injectivity of the construction (and hence the lower bound on T(g)) depends on these properties, they should be verified explicitly, for example by a table of the ordered pairs or a reproducible script.","section":"Section 5, Proposition 5.3"}],"minor_comments":[{"comment":"There is a typo in the sentence 'a tropical curves has one vertex for each subpolygon in the subdivision'; it should read 'a tropical curve has'.","section":"Section 2.3"},{"comment":"The sentence 'It is not always immediately obvious is a graph if crowded' is grammatically garbled; it should likely read 'It is not always immediately obvious whether a graph is crowded.'","section":"Section 3"},{"comment":"In the last paragraph, the text says 'H1,··· ,Hm are the graphs arising from the tiles T′_1,··· ,T′_k' but the index should be m, not k.","section":"Section 5, proof of Proposition 5.2"},{"comment":"There is a duplicated word: 'we have have T(g) ≥ ...' should read 'we have T(g) ≥ ...'.","section":"Section 5, proof of Corollary 5.5"},{"comment":"The derivation of the asymptotic lower bound uses numerical approximations A ≈ 0.49999, B ≈ 0.25001 + 0.00543i, α ≈ 6.1233, and r ≈ 3.4998. Since the conclusion a_n = Ω(α^n) requires exact inequalities such as A > 0 and α > r, the authors should replace these floating-point approximations with rigorous bounds, e.g. via interval arithmetic or explicit algebraic estimates.","section":"Section 5, Proposition 5.4"},{"comment":"The claim that any trivalent graph containing a copy of H is sprawling is not immediate because H contains a degree-1 vertex and the copy need not be induced. The statement is true, since all other vertices of H have degree 3 in H and hence are saturated in any ambient trivalent graph, but a brief justification would make the proof self-contained.","section":"Section 4, Theorem 4.2"},{"comment":"The markings L and R are used throughout the construction, but the paper does not formally define a 'marked graph' or what it means for two marked graphs to be isomorphic. A precise definition would improve rigor and readability.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The lower-bound construction depends on a finite tile database that is not included in the paper or ancillary files. The authors should be asked to provide the TOPCOM scripts, the triangulations for all tiles, and a machine-checkable verification of the distinctness and ordered-pair uniqueness properties. I also note that one of the authors is a co-author of reference [6], on whose computational framework the paper relies; this overlap is not improper, but it makes independent verifiability of the computational assertions especially important. The upper-bound and zero-density arguments appear sound and do not depend on these computational issues."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper is worth serious attention. The new upper bound on T(g) is the strongest piece—stratifying by lattice width and bounding triangulations per polygon is a clean, checkable argument, and the base 2^{11/3} with the O(sqrt(g)) correction is genuinely new. The TIE-fighter obstruction is a nice geometric addition, and bridge reduction is a useful operation that connects bridged and 2-edge-connected troplanar graphs. The exact counts through genus 7, built on the Brodsky–Joswig–Morrison–Sturmfels pipeline, are new, and the counts themselves look plausible.\n\nThe main soft spot is the lower-bound construction in Section 5. The tiling argument is elegant, but the entire base gamma rests on the tile inventory: 2 genus-2, 13 genus-4, and 75 genus-6 tiles. The paper asserts that TOPCOM found no non-regular triangulations of these parallelograms and that the tiles are distinct as marked graphs, but it does not supply the actual triangulations (the appendix shows only dual graphs with markings), the TOPCOM input/output, or a reproducibility script. A duplicate or non-regular tile would inflate the recurrence coefficients and lower gamma. I do not think this is a fatal flaw—the method is sound and the claim is probably correct—but a careful referee should demand the tile data before signing off on the lower bound.\n\nThere is also a genuine gap in the proof of Theorem 4.2. The assertion that every trivalent graph containing a copy of the sprawling graph H is itself sprawling is false for non-induced subgraph copies: a 2-edge-connected cubic graph can easily contain such an H with the branches reconnected elsewhere. This matters for that proof, but not for the paper's main zero-density claim, since Theorem 4.14 plus the known growth rate of P(g) already gives T(g)/P(g) -> 0. The authors should either fix the proof or acknowledge the independent route.\n\nMinor note: the statement that hyperelliptic polygons produce no new graphs beyond those from maximal nonhyperelliptic polygons for genus 6 and 7 is asserted as a computational fact without details; again, this is checkable but not shown.\n\nWho is this for? Tropical geometers and graph enumerators. It deserves a serious referee and, with the computational inputs released, could become a solid reference for the asymptotic census of tropically planar graphs.","headline":"Solid new upper bound and a plausible lower bound that needs computational verification; the zero-density result is correct despite a gap in one of its proofs.","tokens_in":24472,"tokens_out":5752,"would_cite":true,"duration_ms":60064,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14T05","05C30","05C10","52B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that tropically planar graphs are asymptotically 0% of connected trivalent planar graphs, with explicit upper and lower bounds on their number.","keywords":["tropically planar graphs","tropical plane curves","skeleton of a tropical curve","regular unimodular triangulations","lattice polygons","trivalent planar graphs","asymptotic enumeration","lattice width"],"falsifier":"Independently enumerate all regular unimodular triangulations of the parallelograms $P^{\\parallel}_2$, $P^{\\parallel}_4$, and $P^{\\parallel}_6$ and compare the resulting marked skeletons against the paper's Appendix A tiles. A duplicate marked graph, a tile that is not a regular triangulation, or a genus-6 or genus-7 total different from 152 or 672 would force the corresponding bound or count to be revised downward.","tokens_in":23447,"feed_emoji":"🌴","tokens_out":11404,"duration_ms":98865,"temperature":0.7,"pith_summary":"Tropically planar graphs are the graphs that appear as skeletons of smooth tropical plane curves. This paper proves that, as the genus grows, these graphs become vanishingly rare among connected trivalent planar graphs: the ratio $T(g)/P(g)$ tends to 0. It establishes quantitative bounds, $T(g)=O(2^{11g/3+O(\\sqrt g)})$ and $T(g)=\\Omega(\\gamma^g)$ with $\\gamma\\approx 2.47$, and it extends the exact enumeration from genus 5 to genus 6 and 7, finding 152 and 672 tropically planar graphs respectively. Along the way it develops new necessary conditions, including a forbidden TIE-fighter configuration and a bridge-reduction surgery, that rule out many non-tropically planar graphs.","feed_headline":"Tropically planar graphs vanish to 0% among planar graphs","feed_subtitle":"Exact counts reach 672 at genus 7; bounds settle growth between 2.47^g and 12.7^g.","key_machinery":"The argument runs on three engines. First, the duality between a smooth tropical plane curve and a regular unimodular triangulation of its Newton polygon: each troplanar graph is the skeleton dual to such a triangulation, so counting troplanar graphs becomes counting regular unimodular triangulations of lattice polygons of genus $g$ up to the graphs they yield. Second, an upper-bound engine that stratifies polygons by lattice width $\\ell$: width-2 (hyperelliptic) polygons contribute $O(2^g)$ graphs, width-3 polygons contribute $O(8^g\\sqrt g)$ each via the binomial coefficient $\\binom{g-2}{a}$ of unimodular triangulations of the interior trapezoid, and width-at-least-4 polygons are bounded using $r\\le 2g/\\ell+4\\sqrt{g+8/3}+2$ boundary points together with the general triangulation bound $2^{3g+r-3}$. Third, a lower-bound engine that tiles the parallelogram $P^{\\parallel}_{2n}$ with 2 tiles of genus 2, 13 of genus 4, and 75 of genus 6; regularity is preserved when gluing along lattice-length-1 edges, and Proposition 5.1 ensures different tile sequences yield non-isomorphic graphs, producing the recurrence $a_n=2a_{n-1}+13a_{n-2}+75a_{n-3}$ whose dominant root $\\alpha\\approx 6.1233$ gives $\\gamma=\\sqrt{\\alpha}\\approx 2.47$. A supporting surgery, bridge reduction, uses a bistellar flip to show that reducing all bridges of a troplanar graph yields a 2-edge-connected troplanar graph, giving $T(g)\\le 2^{g-1}T^{(2)}(g)$.","core_discovery":"On the paper's own terms, the central discovery is that troplanar graphs are exponentially rare among all connected trivalent planar graphs: $\\lim_{g\\to\\infty} T(g)/P(g)=0$, with the quantitative bounds $T(g)=O(2^{11g/3+O(\\sqrt g)})$ and $T(g)=\\Omega(\\gamma^g)$, where $\\gamma=\\sqrt{\\alpha}\\approx 2.47$ and $\\alpha$ is the unique real root of $x^3-2x^2-13x-75$. The upper bound comes from stratifying lattice polygons by lattice width and bounding the number of their unimodular triangulations; the lower bound comes from tiling a genus-$2n$ parallelogram with fixed tiles of genus 2, 4, and 6, giving a recurrence whose growth rate is governed by $\\alpha$. The paper also settles the exact enumeration through genus 7: $T(6)=152$ and $T(7)=672$, continuing the sequence 2, 4, 13, 38, 152, which does not appear in the OEIS. These results supersede the previously known lower bound $T(g)=\\Omega(2^g)$ from hyperelliptic chains.","pith_inferences":["If regular triangulations are as rare among all unimodular triangulations as numerical experiments suggest, the true exponential base of $T(g)$ could be much closer to the lower bound 2.47 than to the upper bound 12.7; this is an extrapolation, not a paper claim.","Extending the tiling construction to tiles of genus 8 or higher would change the recurrence's characteristic polynomial, and the growth base of the lower bound is the dominant root of that polynomial, so each new tile family is a lever for raising $\\gamma$; the authors mention this direction only in passing.","The paper notes that proving a 'two loops in a row' obstruction analogous to Corollary 3.5 would remove 18 of the 28 genus-6 graphs not ruled out by any known criterion; supplying that proof is a direct way to test whether $T(6)=152$ can be improved.","Because the upper bound counts all unimodular triangulations rather than only regular ones, a sharper census of regular triangulations of lattice polygons could both validate the lower-bound tile list and narrow the gap between 2.47 and 12.7."],"forward_implications":["If the bounds are correct, troplanar graphs are not just a minority but an exponentially negligible proportion of connected trivalent planar graphs: the ratio $T(g)/P(g)$ tends to 0 at an exponential rate.","The exact counts $T(6)=152$ and $T(7)=672$ give the first data beyond genus 5, and the sequence 2, 4, 13, 38, 152 matches no OEIS sequence, so any proposed formula for $T(g)$ must reproduce these values.","The upper bound $O(2^{11g/3+O(\\sqrt g)})$ improves on the generic planar-graph bound (base roughly 15.88) and shows the exponential base of $T(g)$ is at most $2^{11/3}\\approx 12.7$.","The lower bound $\\Omega(\\gamma^g)$ with $\\gamma\\approx 2.47$ supersedes the hyperelliptic-chain lower bound $\\Omega(2^g)$, so the true exponential base lies somewhere in $[2.47,12.7]$.","New necessary conditions, namely that TIE-fighter graphs are never troplanar and that bridge reduction preserves troplanarity, give practical tests for ruling graphs in or out at any fixed genus."],"supporting_citations":[{"why":"Provides the enumeration algorithm and the maximal-polygon reduction that the genus-6 and genus-7 computations build on.","marker":"[6]"},{"why":"Supplies the random cubic planar graph theorem used to show most planar trivalent graphs are sprawling, hence not troplanar.","marker":"[5]"},{"why":"Gives the asymptotic growth of $P(g)$ and $P^s(g)$, the comparison bases for the 0% limit and the upper bound.","marker":"[22]"},{"why":"Provides the general bound on unimodular triangulations ($2^{3g+r-3}$) and the regularity facts used in the upper bound and bridge-reduction proof.","marker":"[13]"},{"why":"Counts the unimodular triangulations of the interior trapezoid and states that regular triangulations glue along lattice-length-1 edges, used in both bounds.","marker":"[18]"},{"why":"The computational tool used to enumerate regular unimodular triangulations of the maximal polygons and to check the tiles for regularity.","marker":"[23]"}],"fun_headline_variants":["Tropically planar graphs vanish to 0% in the limit","0% of planar graphs are tropically planar asymptotically","Tropically planar graph counts: 152 at genus 6, 672 at genus 7","Exponentially few tropically planar graphs as genus grows","Tropically planar graphs: growth rate between 2.47^g and 12.7^g"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The lower bound assumes that the 75 genus-6 tiles and 13 genus-4 tiles listed in the appendix are all distinct as marked graphs and all realizable as regular triangulations of the parallelogram; if any two are actually the same, or one fails regularity, the tiling count and the base $\\gamma\\approx 2.47$ would shrink.","fun_headline_variants_meta":{"raw":{"variants":["Tropically planar graphs vanish to 0% in the limit","0% of planar graphs are tropically planar asymptotically","Tropically planar graph counts: 152 at genus 6, 672 at genus 7","Exponentially few tropically planar graphs as genus grows","Tropically planar graphs: growth rate between 2.47^g and 12.7^g"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001157,"raw_usage":{"total_tokens":4749,"prompt_tokens":854,"completion_tokens":3895,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":3792}},"tokens_in":470,"tokens_out":3895,"duration_ms":27290,"temperature":1.0,"reasoning_tokens":3792,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:46:07.351146+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently enumerate all regular unimodular triangulations of the parallelograms $P^{\\parallel}_2$, $P^{\\parallel}_4$, and $P^{\\parallel}_6$ and compare the resulting marked skeletons against the paper's Appendix A tiles. A duplicate marked graph, a tile that is not a regular triangulation, or a genus-6 or genus-7 total different from 152 or 672 would force the corresponding bound or count to be revised downward.","supporting_citations":[{"cited_title":"Brodsky, M","cited_arxiv_id":null,"evidence_quote":"Provides the enumeration algorithm and the maximal-polygon reduction that the genus-6 and genus-7 computations build on."},{"cited_title":"Bodirsky, M","cited_arxiv_id":null,"evidence_quote":"Supplies the random cubic planar graph theorem used to show most planar trivalent graphs are sprawling, hence not troplanar."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic growth of $P(g)$ and $P^s(g)$, the comparison bases for the 0% limit and the upper bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the general bound on unimodular triangulations ($2^{3g+r-3}$) and the regularity facts used in the upper bound and bridge-reduction proof."},{"cited_title":"Kaibel and G","cited_arxiv_id":null,"evidence_quote":"Counts the unimodular triangulations of the interior trapezoid and states that regular triangulations glue along lattice-length-1 edges, used in both bounds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The computational tool used to enumerate regular unimodular triangulations of the maximal polygons and to check the tiles for regularity."}],"review_version":1}