{"id":"068fecc9-9f1d-4792-beaf-55ba23ac4310","arxiv_id":"2607.08473","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Vol3 with its systole or second-shortest geodesic admits no positive spun ideal triangulation, by Choi's criterion applied to explicit shape varieties of census fillings.","lead":"The paper gives the first examples of closed hyperbolic 3-manifolds that admit no positive spun ideal triangulation about a chosen geodesic. The examples are Vol3 with its two shortest geodesics, supporting a conjecture that Vol3 never has such a triangulation.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the only non-purely-algebraic step—the geodesic-length identification performed in a cover that itself admits a positive triangulation—and correctly rates its risk as low and independently re-checkable. The algebraic core of the argument (edge equations, holonomy reduction, vanishing of dH(r), application of Choi) is fully expanded with no free parameters and no circularity. Because that core is self-contained and the residual check is standard SnapPy length-spectrum work already outlined in the proofs of Propositions 3.1 and 3.16, no load-bearing concern arises that would move the verdict. The recommended concrete test simply re-executes the two independent verifications already present in the paper.","tokens_in":16754,"tokens_out":537,"duration_ms":6929,"concrete_test":"Independently recompute the four algebraic solutions of the system (3.3)+(3.4)+(3.6) (or the analogous system for m010) in high-precision arithmetic, evaluate the 3\times3 determinant of Lemma 3.14 (resp. the determinant after (3.22)), and confirm it is identically zero; simultaneously recompute the length spectrum of the cover gLLAQbcedefftoaolonDbCb(1,0) up to three times the core length and verify that no shorter non-covering geodesic projects to Vol3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.1 rests on two fully expanded algebraic certificates (Propositions 3.1 and 3.16). For each, the shape variety is cut out by explicit edge equations (3.3–3.4 and 3.17–3.18), the Dehn-filling holonomy equation is derived and reduced (to y^{2}+y+1=0 and the quartic (3.22)), and a concrete slope r is exhibited whose log-holonomy derivative vanishes on the tangent space (via an explicit 3\times3 determinant that reduces to zero after substitution of the algebraic relations). Smoothness of the varieties is checked by non-vanishing of maximal minors (Macaulay2). Choi’s Corollary 2.9 then applies directly. The residual identification that the filling cores are a systole and second-shortest geodesic is performed in a three-fold cover that does admit a positive spun triangulation (isosig given), so the length-spectrum computation is independent of the non-existence result. No hidden free parameters or circular appeals appear.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper establishes the first examples of pairs (N, γ) consisting of a closed hyperbolic 3-manifold N and a simple closed geodesic γ such that N-γ admits no positive spun ideal triangulation. The main theorem (Theorem 1.1) treats N = Vol3 = m007(3,1) with γ its systole or second-shortest geodesic. The proofs (Propositions 3.1 and 3.16) cut out the shape varieties of the SnapPy triangulations of m007 and m010 by explicit edge equations, adjoin the relevant Dehn-filling holonomy equations, reduce to algebraic relations (y^{2} + y + 1 = 0 and a quartic), exhibit slopes r for which dH(r) vanishes on the tangent space (via 3\times3 determinants that reduce to zero after substitution), verify smoothness by non-vanishing of maximal minors, and invoke Choi’s Corollary 2.9. Length-spectrum checks in a three-fold cover that does admit a positive spun triangulation identify the filling cores as the claimed geodesics. Section 4 reports extensive numerical searches yielding 271 further manifold pairs and 28 orbifold pairs, plus three examples where the angle-structure polytope is non-empty.","tokens_in":16987,"tokens_out":866,"duration_ms":16966,"significance":"If correct, the result supplies the first closed hyperbolic examples without positive spun triangulations and supplies concrete evidence for the long-standing conjecture that Vol3 itself admits none for any geodesic. The algebraic certificates are fully expanded, hand-checkable after Macaulay2 verification of smoothness, and free of free parameters; the length-spectrum arguments are independent of the non-existence claim because they are performed in a cover that does admit a positive spun triangulation. The computational census of further examples and the observation that certain angle polytopes are non-empty yet cannot contain a volume maximizer are valuable secondary contributions that rule out a natural approach to the generalised Casson conjecture.","major_comments":[],"minor_comments":[{"comment":"In the proof of Lemma 3.2 the three edge equations are written out fully and then two are retained; a one-sentence remark that the third is a consequence of the product of all three left-hand sides equalling 1 would make the reduction completely self-contained for a reader who does not open SnapPy.","section":null},{"comment":"Lemma 3.5 and the analogous reduction for m010 both discard real or complete-structure solutions by volume considerations. Adding a short parenthetical that the volumes of the four retained solutions lie in a certified interval about ±Vol(Vol3) (already computed in the cover) would remove any residual doubt.","section":null},{"comment":"The isosig of the positive spun triangulation of the three-fold cover is given only in the proof of Proposition 3.1; repeating it (or giving a SnapPy one-liner) in the proof of Proposition 3.16 would make both length-spectrum arguments independently reproducible.","section":null},{"comment":"Section 4.1 describes the search criterion (4.2) but does not record the floating-point tolerance used to decide that a ratio is “real.” A single sentence stating the threshold (e.g., Im < 10^{-10}) would clarify the numerical evidence for the 271 pairs.","section":null},{"comment":"The ancillary-file citation [10] appears only as “http://arxiv.org/”; a more precise arXiv identifier or DOI once the paper is posted would help future readers locate the lists of 271 + 28 examples.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a clean, self-contained contribution that sits squarely in the journal’s geometric-topology remit. The algebraic core is unusually transparent for a SnapPy-assisted paper; I see no reason to request further verification beyond the already-supplied Macaulay2 checks. Accept as is."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper delivers the first closed hyperbolic 3-manifolds that admit no positive spun ideal triangulation about a chosen geodesic. The main theorems are for Vol3 and its systole and second-shortest geodesic; the proofs are fully expanded algebraic certificates that apply Choi's vanishing criterion.\n\nWhat is new is the successful application to closed manifolds. Choi already had the criterion and a cone-manifold example; the authors produce explicit edge equations, holonomy equations, square-root case analysis, and a concrete slope r whose dH(r) vanishes on the tangent space (via a 3x3 determinant that reduces to zero after substitution). Smoothness is checked by non-vanishing of maximal minors (hand + Macaulay2). The length-spectrum identification that the filling cores are the systole and second systole is done in a three-fold cover that does admit a positive spun triangulation, so it is independent of the non-existence claim. The census search (271 manifold pairs, 28 orbifolds) and the three angle-structure examples that block a naïve approach to the generalised Casson conjecture are cleanly separated from the theorems.\n\nSoft spots are minor. The residual risk is a mis-identification of geodesic lengths in the cover; that computation is re-checkable with the given isosig. The broader Vol3 conjecture remains open, and the numerical evidence for the other 85 fillings of Vol3 cannot invoke Choi. None of this undercuts the two main theorems.\n\nThe math, the data, and the citations look solid. This is for people who work on ideal triangulations, SnapPy census manifolds, or the Casson–Rivin volume functional. It deserves a serious referee and should be engaged with; I would cite the Vol3 certificates and the angle-structure counter-examples.","headline":"First rigorous closed-manifold examples of missing positive spun triangulations, with clean algebraic certificates for Vol3's two shortest geodesics.","tokens_in":17536,"tokens_out":455,"would_cite":true,"duration_ms":4961,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K32","57M50"],"pacs":[],"model":"grok-4.5","headline":"The closed hyperbolic manifold Vol3 has no positive spun ideal triangulation about its shortest or second-shortest geodesic.","keywords":["hyperbolic 3-manifolds","ideal triangulations","spun triangulations","shape variety","Vol3","Dehn filling","Choi criterion","angle structures"],"falsifier":"Exhibit any ideal triangulation of the complement of either geodesic together with positive shape parameters whose holonomies recover the complete structure on Vol3; or show that the length spectrum of Vol3 contains a shorter geodesic than the filling cores used in the paper.","tokens_in":17675,"feed_emoji":"🔺","tokens_out":990,"duration_ms":9443,"temperature":0.7,"pith_summary":"A long-standing conjecture says every finite-volume hyperbolic three-manifold admits a decomposition into positively oriented ideal tetrahedra. For closed manifolds the natural analogue is a positive spun triangulation: tetrahedra that spin around a chosen simple closed geodesic and whose completion recovers the closed manifold. This paper produces the first rigorous examples of closed hyperbolic manifolds where no such positive spun triangulation exists for particular geodesics. The examples are Vol3 (the third-smallest closed hyperbolic three-manifold) spun about its systole and about a second-shortest geodesic. The argument applies Choi's smoothness criterion for shape varieties: at the candidate spun shapes the derivative of the log-holonomy of an auxiliary slope vanishes, so the shapes cannot be positive. The result supplies concrete evidence for the stronger conjecture that Vol3 admits no positive spun triangulation for any geodesic at all, and shows that certain volume-maximization strategies for proving existence must fail.","feed_headline":"Vol3 has no positive spun triangulation about its shortest geodesics","feed_subtitle":"First rigorous closed hyperbolic examples that refuse positive tetrahedra spun about a chosen geodesic","key_machinery":"Choi's criterion (Theorem 2.8 / Corollary 2.9): if a spun shape z lies in the shape variety of an ideal triangulation and the derivative of the log-holonomy of some boundary slope vanishes at z, then no retriangulation can make those shapes positive. The paper exhibits explicit algebraic equations for the shape varieties of m007 and m010, solves the holonomy equations for the relevant fillings, and verifies the vanishing condition by direct linear algebra.","core_discovery":"The pairs (Vol3, shortest geodesic) and (Vol3, second-shortest geodesic) admit no positive spun ideal triangulation. Equivalently, the Dehn fillings m007(3,1) and m010(-1,2) have no positive spun triangulation. This is the first rigorous instance of a closed hyperbolic three-manifold that fails to possess a positive spun triangulation about a chosen geodesic.","pith_inferences":["If the stronger conjecture for Vol3 is true, then positivity of ideal triangulations cannot be expected for every closed hyperbolic three-manifold even after allowing spinning about an arbitrary geodesic.","The same vanishing criterion may obstruct positive spun triangulations for infinite families obtained by hyperbolic Dehn filling of higher-cusped manifolds that share the same algebraic shape equations.","Computational certification of hyperbolic structures that currently depends on positive spun triangulations will need alternative certificates (Dirichlet domains, verified covers, arithmetic methods) for manifolds in the Vol3 class."],"forward_implications":["Vol3 is a concrete candidate for a closed hyperbolic manifold that admits no positive spun triangulation for any geodesic.","Any proof strategy for the generalised Casson conjecture that relies only on maximising volume over angle polytopes and retriangulating must fail for the three census fillings o9_29517(1,1), o9_21590(-1,1) and o11_465181(1,1).","Numerical searches that locate 271 further closed-manifold pairs and 28 orbifold pairs with the same vanishing-holonomy signature become candidates for rigorous non-existence proofs by the same method.","Existence of positive spun triangulations is strictly stronger than existence of angle structures, even when the volume functional has no critical point inside the angle polytope."],"fun_headline_variants":["Vol3 admits no positive spun triangulation about its two shortest geodesics","First closed hyperbolic pairs with no positive spun ideal triangulation on a geodesic","Vol3 systole and second systole block all positive spun triangulations","m007(3,1) and m010(-1,2) have no positive spun triangulations","Closed hyperbolic Vol3 fails positive spun triangulations about chosen geodesics"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The claim that the Dehn-filling cores of m007(3,1) and m010(-1,2) are respectively a systole and a second-shortest geodesic of Vol3 rests on a length-spectrum computation performed in a three-fold cover that does admit a positive spun triangulation.","fun_headline_variants_meta":{"raw":{"variants":["Vol3 admits no positive spun triangulation about its two shortest geodesics","First closed hyperbolic pairs with no positive spun ideal triangulation on a geodesic","Vol3 systole and second systole block all positive spun triangulations","m007(3,1) and m010(-1,2) have no positive spun triangulations","Closed hyperbolic Vol3 fails positive spun triangulations about chosen geodesics"]},"model":"grok-4.5","effort":"low","cost_usd":0.005214,"raw_usage":{"total_tokens":1287,"prompt_tokens":627,"num_sources_used":0,"completion_tokens":104,"cost_in_usd_ticks":52140000,"prompt_tokens_details":{"text_tokens":627,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":556,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":627,"tokens_out":104,"duration_ms":5693,"temperature":1.0,"reasoning_tokens":556,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T06:54:26.640386+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit any ideal triangulation of the complement of either geodesic together with positive shape parameters whose holonomies recover the complete structure on Vol3; or show that the length spectrum of Vol3 contains a shorter geodesic than the filling cores used in the paper.","supporting_citations":[],"review_version":1}