{"id":"fcb032b1-317a-4448-a58e-b0381e7f4ea4","arxiv_id":"2506.12465","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A filling multi-geodesic on a genus g hyperbolic surface has length at least half the perimeter of a regular right-angled (8g-4)-gon, and this bound is sharp.","lead":"The authors prove that the shortest possible filling curve on a closed hyperbolic surface of genus g has length exactly half the perimeter of a regular right-angled polygon with 8g-4 sides, and they exhibit a single geodesic achieving this minimum. This answers a natural extremal question in hyperbolic geometry that was previously known only for pairs of simple filling curves.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 is only proved for regular polygons; the reduction from arbitrary hyperbolic polygons to regular ones is missing, so the lower-bound proof of Theorem 1.1 is incomplete as written.","rationale":"The central claim has two parts: a lower bound over all filling multi-geodesics and a sharpness construction. The lower bound depends on Theorem 2.1 (graph reduction) and Theorem 3.1 (isoperimetric inequality). The reader focused on Theorem 2.1 as the weakest assumption; I agree that the intricate Section 2.3 proof is hard to verify, but the more immediate logical gap is in Theorem 3.1. The theorem is stated for arbitrary polygons, yet every lemma used in its proof assumes the polygons are regular, and no reduction is supplied. This is not a disagreement with known mathematics: the hyperbolic polygon isoperimetric inequality is classical, and the inequality may well be true; the problem is that the manuscript does not prove or even cite it at the point of use. Because this missing step is repairable and no counterexample is identified, the verdict should remain CONDITIONAL rather than ACCEPT or REJECT. The Section 4 double-point count (2g-3 versus the Euler-characteristic-required 2g-1 for an (8g-4)-gon complement) is a separate concrete error that also needs correction, but it affects the sharpness/existence direction and is easier to fix than the missing isoperimetric argument. Hence I recommend no change to the reader's CONDITIONAL verdict, with the understanding that both repairs are prerequisites for acceptance.","tokens_in":16467,"tokens_out":11984,"duration_ms":145587,"concrete_test":"Supply and verify a regularization lemma: for every convex hyperbolic m-gon P with area A, Perim(P) >= P_m(A), where P_m(A) is the perimeter of the regular hyperbolic m-gon of area A. Then re-run the induction in Theorem 3.1 after replacing each Di by the regular mi-gon of the same area. If such a lemma is absent, Theorem 3.1 remains unproven; if a concrete non-regular m-gon with area A and perimeter below P_m(A) can be found, the theorem is false. A numerical check with a non-regular right-angled octagon (area 2 pi) against the regular right-angled octagon would test the k = 1 case, but a proof from [10] is required for the general case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the main lower bound passes through Theorem 3.1, which is stated for arbitrary hyperbolic mi-gons with mi >= 4. However, the proof in Section 3 treats the Di as regular from Proposition 3.8 onward: Proposition 3.8 assumes 'regular hyperbolic m1- and m2-gons', Lemma 3.10 assumes 'Di regular hyperbolic mi-gons', and the induction in the proof of Theorem 3.1 compares regular polygons eD_j. No statement or proof is given that for a fixed number of sides and fixed area, the regular polygon minimizes perimeter among all hyperbolic polygons. The reference [10] to Bezdek is never used in the proof. Without this regularization step, the key inequality ell(G) = (1/2) sum Perim(D_i) >= (1/2) P_{8g-4} does not follow, because the components produced by Theorem 2.1 are not regular. This is the load-bearing step connecting the graph reduction to the numerical lower bound, and it is not established as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to determine the minimal length of a filling multi-geodesic on a closed orientable hyperbolic surface of genus g, minimized over moduli space. Theorem 1.1 states this minimum equals half the perimeter of a regular right-angled (8g−4)-gon, and that it is realized by a single filling geodesic. The proof has two main ingredients: Theorem 2.1, which replaces a filling multi-geodesic by a filling geodesic graph of no larger length whose complementary components are polygons with at least five sides and satisfy a side-count identity, and Theorem 3.1, an isoperimetric inequality comparing the perimeter sum of such polygons with the perimeter of a regular polygon of the same area. Section 4 constructs a hyperbolic surface and a single filling geodesic whose complement is the desired regular polygon.","tokens_in":16609,"tokens_out":7064,"duration_ms":85657,"significance":"If correct, the result settles a natural question about filling geodesics and extends the Aougab–Huang conjecture from filling pairs to arbitrary filling multi-geodesics. The construction of a single filling geodesic attaining the bound is explicit, and the corollaries on the kissing number of surfaces with filling systole are of independent interest. The paper also provides a clean statement of the isoperimetric inequality for polygons with at least four sides, which is a useful contribution in itself. However, the proof of the isoperimetric theorem is currently incomplete, so the main result is not yet established as written.","major_comments":[{"comment":"The theorem is stated for arbitrary hyperbolic mi-gons, but the proof from Proposition 3.8 onward only treats regular polygons. Proposition 3.8 assumes D1 and D2 are regular hyperbolic polygons, Lemma 3.10 assumes each Di is regular, and the induction in the proof of Theorem 3.1 compares the regular replacements eD_j. There is no argument showing that, among hyperbolic polygons with a fixed number of sides and fixed area, the regular polygon minimizes perimeter. Reference [10] (Bezdek) is listed but never cited in the proof. Consequently the key step in the proof of Theorem 1.1, namely ℓ(G) = 1/2 Σ Perim(D_i) ≥ 1/2 P_{8g−4}, does not follow for the arbitrary polygons produced by Theorem 2.1. This is the load-bearing connection between the graph reduction and the numerical lower bound, so the proof of Theorem 3.1 is incomplete as written.","section":"Section 2.3, proof of Theorem 2.1"},{"comment":"The iterative construction ends with the assertion: 'Since Γ is a finite filling graph, the repeating process will eventually stop.' This is not justified in the general case. The replacement step replaces an arc β by a shortest proper geodesic arc β′ that may not lie in Γ, so the intermediate graph Θ_i is no longer a subgraph of Γ. Finiteness of Γ therefore does not, by itself, bound the number of iterations. A rigorous termination argument—for instance, a monotone complexity measure for the pair (Θ_i, G_i) or a bound on the number of essential cutting curves needed to fill the surface—is required. The termination of the process is essential to the existence of the graph G, so this gap also affects the lower bound.","section":"Section 2.3, proof of Theorem 2.1"}],"minor_comments":[{"comment":"Reference [10] (Bezdek) is never cited in the body of the paper. Either it should be used to supply the missing regularization step in Theorem 3.1, or it should be removed.","section":"References"},{"comment":"The claim that the curve α in Figure 12 is filling and that its complement is an (8g−4)-gon is stated without proof. Since the construction of the extremal surface depends on this assertion, a brief justification or a more detailed reading of the figure would be helpful.","section":"Section 4"},{"comment":"There are several typographical errors, including 'SHOR TEST' in the title, 'olygons' in Section 2.2, 'essencity' in Section 2.3, and some malformed summation indices such as 'kX k=1'. The manuscript would benefit from a careful proofreading pass.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main result is plausible and potentially important, and the missing step in Theorem 3.1 appears to be repairable: the needed statement—that regular polygons minimize perimeter among polygons with the same number of sides and same area—is a known isoperimetric-type result, and reference [10] is likely intended for this purpose. The termination issue in Theorem 2.1 also needs a clearer argument. Because both gaps are in load-bearing positions, I cannot recommend acceptance in the current form, but the manuscript is a good candidate after a substantial revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read the paper with interest. The main result — the filling length of a hyperbolic surface of genus g is exactly 1/2 P_{8g-4}, realized by a single filling geodesic — is a clean statement and likely true. The generalization from filling pairs to arbitrary filling multi-geodesics is a real step, and the construction of the explicit geodesic with right-angled regular polygon complement is nice. The kissing-number corollary (3.525g/log g) is also a worthwhile bonus.\n\nThe proof strategy is sensible: reduce to a triangle-free filling graph, then use an isoperimetric inequality. The graph reduction in Section 2 is intricate and genuinely new as far as I can tell.\n\nHowever, there are two soft spots that need attention.\n\nFirst, the proof of Theorem 3.1 as written only proves the inequality for regular polygons. Theorem 3.1 is stated for arbitrary hyperbolic m_i-gons, and the proof constructs regular polygons \\tilde D_j with the same total area and side counts, but then invokes Proposition 3.8 to compare Perim(\\tilde D_2) with Perim(D_1)+Perim(D_2). Proposition 3.8 is for regular D_1, D_2. No step is given showing that replacing the arbitrary D_i by the regular \\tilde D_i with the same area decreases (or does not increase) the total perimeter. The reference to Bezdek [10] is never used. This is load-bearing: without it, the inequality \\ell(G) >= 1/2 P_{8g-4} does not follow from Theorem 2.1 because the complementary polygons are not regular. I'm fairly sure it's fixable via the hyperbolic isoperimetric inequality, but it needs to be stated and proved.\n\nSecond, the construction in Section 4 has a concrete arithmetic error. The text says the filling curve \\alpha has 2g-3 double self-intersection points for g>2, but a single filling curve with only double points must have 2g-1 self-intersections to yield one complementary (8g-4)-gon. Check Euler characteristic: for d double points, one complementary face gives d = 2g-1. The g=2 case is given as 3, which matches 2g-1, so the formula 2g-3 is likely a typo for g>2, but as written the construction doesn't have the claimed property.\n\nThere are also minor issues: the proof of Theorem 2.1 is hard to check, especially Section 2.3, and the average-angle argument leading to (4) is terse. But these are less concerning.\n\nOverall: the central claim is probably correct, the approach is new, and the paper deserves careful refereeing. But it is not ready in its current form. The authors need to fix the regularization gap in Theorem 3.1 and correct the self-intersection count. After that, I'd be happy to see it published.","headline":"Likely-true main theorem, but the proof has a load-bearing gap in the isoperimetric inequality and a concrete miscount in the construction; worth refereeing after revision.","tokens_in":17193,"tokens_out":5610,"would_cite":false,"duration_ms":60644,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K20","53C22","51M10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every closed orientable hyperbolic surface of genus g, the shortest total length of a filling multi-geodesic is exactly half the perimeter of the regular right-angled hyperbolic (8g−4)-gon, and a single filling geodesic attains it.","keywords":["filling geodesics","hyperbolic surfaces","isoperimetric inequality","right-angled polygon","perimeter","geodesic graph","moduli space","kissing number"],"falsifier":"Construct a genus-g hyperbolic surface and a filling multi-geodesic of total length strictly less than 1/2 P_{8g−4}, or exhibit a filling multi-geodesic of length exactly 1/2 P_{8g−4} whose complement is not a regular right-angled (8g−4)-gon.","tokens_in":16188,"feed_emoji":"📐","tokens_out":6756,"duration_ms":65756,"temperature":0.7,"pith_summary":"This paper answers a natural question in hyperbolic geometry: among all genus-g hyperbolic surfaces, what is the minimal total length of a set of closed geodesics that cuts the surface into disks? The answer is exactly half the perimeter of the regular right-angled hyperbolic (8g−4)-gon, and the minimum is realized by a single filling geodesic. The proof combines a reduction that eliminates triangles from the complementary regions with a new isoperimetric inequality for hyperbolic polygons.","feed_headline":"The shortest filling geodesic is half a (8g−4)-gon's perimeter","feed_subtitle":"A sharp lower bound on geodesics that cut a genus-g hyperbolic surface into disks, attained by a single curve.","key_machinery":"The argument rests on two mechanisms: a reduction (Theorem 2.1) that converts any filling multi-geodesic into a filling geodesic graph of no greater length whose complementary polygons each have at least five sides, with the side-count identity sum(m_i − 4) = 8g − 8; and an isoperimetric inequality (Theorem 3.1) that bounds the total perimeter of such a polygon family below by P_{8g−4}. The inequality is proved by studying perimeter functions of regular hyperbolic polygons and relies on a generalization of Sanki-Vadnere's even-polygon isoperimetric result.","core_discovery":"The central discovery is that the filling length infimum over the moduli space of genus-g hyperbolic surfaces equals 1/2 P_{8g−4}, where P_k is the perimeter of the regular right-angled hyperbolic k-gon. The proof proceeds in two steps: first, any filling multi-geodesic is shortened to a filling geodesic graph whose complement has only polygons with at least five sides and satisfies sum(m_i − 4) = 8g − 8; second, an isoperimetric inequality shows the total perimeter of such a collection is at least P_{8g−4}, with equality only for a single regular right-angled (8g−4)-gon. An explicit construction exhibits a single filling geodesic of this length, so the bound is sharp.","pith_inferences":["The triangle-free condition in the reduction is essential, since the isoperimetric inequality fails for triangles; this suggests that any relaxation of the filling-graph hypothesis would need an explicit correction term that accounts for triangular regions.","The same two-step strategy (reduce the graph, then apply a polygon isoperimetric inequality) could be adapted to other settings, such as filling curves on punctured surfaces or on non-orientable surfaces, where the relevant regular polygon side count would change.","The equality case forces the surface to be a regular polygonal gluing, hinting that global minimizers of filling length in moduli space are highly symmetric; it would be natural to ask whether these are the only critical points of the filling-length function."],"forward_implications":["The bound is sharp for every genus g ≥ 2: a single filling geodesic whose complement is a regular right-angled (8g−4)-gon has length exactly 1/2 P_{8g−4}.","Equality is rigid: a filling multi-geodesic of length 1/2 P_{8g−4} must be a single geodesic whose complement is the regular right-angled (8g−4)-gon.","For surfaces whose systole set fills, the kissing number satisfies sys(X) · kiss(X) ≥ 1/2 P_{8g−4}, yielding kiss(X) ≥ (3.525) g / log g for large g, improving the previous constant.","The reduction and isoperimetric inequality apply to arbitrary filling multi-geodesics, including those with multiple intersection points and self-intersections, settling a conjecture previously known only for filling pairs."],"supporting_citations":[{"why":"Supplies the even-polygon isoperimetric inequality and the perimeter-function lemmas (Proposition 3.5) that the new inequality generalizes.","marker":"[29]"},{"why":"States the filling-pair conjecture that Theorem 1.1 resolves in the general multi-geodesic case.","marker":"[1]"},{"why":"Provides an independent proof of the filling-pair case, establishing the baseline that this paper extends.","marker":"[17]"},{"why":"Constructs a filling curve with 2g−1 self-intersections, a similar explicit realizing example that the paper's gluing construction parallels.","marker":"[3]"}],"fun_headline_variants":["Minimum filling geodesic length: half a (8g-4)-gon's perimeter","Sharp bound: shortest filling geodesic is half a (8g-4)-gon's perimeter","Exact minimum: filling geodesic = 1/2 perimeter of regular (8g-4)-gon","Minimal filling curve: half perimeter of a regular (8g-4)-gon","Filling geodesic minimum equals half the regular (8g-4)-gon's perimeter"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The lower bound rests on the reduction in Theorem 2.1, which asserts that every filling multi-geodesic can be replaced by a filling geodesic graph of no greater length whose complementary polygons all have at least five sides; if that reduction produced triangles or increased length, the isoperimetric inequality of Theorem 3.1 could not be applied and the bound would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Minimum filling geodesic length: half a (8g-4)-gon's perimeter","Sharp bound: shortest filling geodesic is half a (8g-4)-gon's perimeter","Exact minimum: filling geodesic = 1/2 perimeter of regular (8g-4)-gon","Minimal filling curve: half perimeter of a regular (8g-4)-gon","Filling geodesic minimum equals half the regular (8g-4)-gon's perimeter"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001495,"raw_usage":{"total_tokens":5899,"prompt_tokens":744,"completion_tokens":5155,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":360,"completion_tokens_details":{"reasoning_tokens":5036}},"tokens_in":360,"tokens_out":5155,"duration_ms":44421,"temperature":1.0,"reasoning_tokens":5036,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:50:34.665001+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a genus-g hyperbolic surface and a filling multi-geodesic of total length strictly less than 1/2 P_{8g−4}, or exhibit a filling multi-geodesic of length exactly 1/2 P_{8g−4} whose complement is not a regular right-angled (8g−4)-gon.","supporting_citations":[{"cited_title":"A conjecture on the lengths of filling pairs","cited_arxiv_id":null,"evidence_quote":"Supplies the even-polygon isoperimetric inequality and the perimeter-function lemmas (Proposition 3.5) that the new inequality generalizes."},{"cited_title":"Minimally intersecting filling pairs on surfaces","cited_arxiv_id":null,"evidence_quote":"States the filling-pair conjecture that Theorem 1.1 resolves in the general multi-geodesic case."},{"cited_title":"A short proof of a conjecture of Aougab-Huang","cited_arxiv_id":null,"evidence_quote":"Provides an independent proof of the filling-pair case, establishing the baseline that this paper extends."},{"cited_title":"The geometry and combinatorics of closed geodesics on hyperbolic surfaces","cited_arxiv_id":null,"evidence_quote":"Constructs a filling curve with 2g−1 self-intersections, a similar explicit realizing example that the paper's gluing construction parallels."}],"review_version":1}