{"id":"3d8e0eee-4ec5-4cd4-9fb0-fecd3ab7544b","arxiv_id":"2608.03575","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a unimodular random trivalent hyperbolic polyhedron, the expected local angle defect equals 2 pi minus pi/3 times the expected face degree, and its sign (zero or negative) exactly determines parabolic versus hyperbolic conformal type.","lead":"A local angle defect computed from the corners of a random infinite polyhedron determines, on average, whether the polyhedron is parabolic or hyperbolic. The paper also shows hyperbolic examples cannot be limits of finite polyhedra, and that the face random walk has positive speed.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4's dichotomy rests on the unstated theorem [26, Thm 1.4] for VEL/RCP type; if that theorem has extra hypotheses (bounded degree, positive angles), the central claim is unsupported.","rationale":"The mass-transport proof of Theorem 3.1 is solid: checking the sent and received masses gives 3S and π·deg(u), respectively, so the unimodular Gauss-Bonnet formula is not the weak point. The finite Gauss-Bonnet formula in Proposition 4.2 also follows from Euler's formula and the local identity θ1+θ2+θ3=π. The load-bearing issue is the next step: converting the identity E[L]=2π−(π/3)E[deg] into a statement about conformal type. That conversion is entirely delegated to [8] and the authors' own unpublished [26]. The paper defines RCP type by existence of a realization, so well-definedness and agreement with VEL type are nontrivial; without [26, Thm 1.4], the central dichotomy is an unsupported hypothesis. This is not an allegation of error in the cited papers; it is a completeness problem that a single theorem statement in the present paper could fix. The reader's (Z2') concern is legitimate but secondary: it governs only the boundary/speed results, not Theorem 1.4. The proposed check settles the issue by direct comparison of hypotheses or by a counterexample. Since the present version is CONDITIONAL and this concern explains that condition, I recommend no verdict change.","tokens_in":26481,"tokens_out":22772,"duration_ms":217462,"concrete_test":"Obtain [26] and [8]; check the exact hypotheses of [26, Thm 1.4] and the [8] dichotomy against the class in Theorem 1.3: infinite, locally finite, simple, one-ended, tame ADTs with Θ∈[0,π−ε], Z1–Z4, unbounded degrees, and only E[deg]<∞. Verify in particular that VEL-parabolic ⇔ RCP-parabolic is proved for weak realizations and allows Θ=0. If any hypothesis is missing, construct or find a tame ADT with a zero angle and E[deg]=6 whose RCP carrier is C but whose VEL type is hyperbolic; such an example would refute Theorem 1.3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main dichotomy is not self-contained at its decisive equivalence. Theorem 3.2 proves (i)↔(ii) by mass transport (clean), but then asserts (ii)↔(iii) and (ii)↔(v) by citing [8], and asserts (iii)↔(iv) by citing [26, Thm 1.4]. None of these cited statements is reproduced in the paper. In particular, RCP-parabolic/RCP-hyperbolic is defined in §2.2 by existence of a realization with carrier C or D, and the paper does not prove that this type is well-defined or agrees with VEL type under the paper's standing assumptions (tame, Θ∈[0,π−ε], unbounded degrees, possibly weak rather than exact RCP realization, only E[deg]<∞). The reader cannot check whether [26, Thm 1.4] applies to this class; if it assumes bounded degree, strict angle positivity, or exact realizations, the equivalences in Theorem 1.3 fail precisely in the advertised regime. This is more load-bearing than (Z2'): Z2' only affects Section 6, while this dependency is what turns the local Gauss-Bonnet formula into the global conformal dichotomy. The dichotomy also silently leaves the case E[L]>0 unaddressed, since the paper never proves E[deg]≥6 except by the same citations.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a probabilistic geometric theory for unimodular random infinite trivalent hyperbolic polyhedra. It defines a local face characteristic L_f(P) from dihedral angles and proves the unimodular Gauss-Bonnet identity E[L_f(P)] = 2π - (π/3) E[deg(f)] by a mass transport argument. The sign of this expectation is then claimed to determine the conformal type: zero expectation corresponds to parabolicity and negative expectation to hyperbolicity, with equivalent characterizations via vertex extremal length, regular circle pattern type, and invariant amenability. The paper also proves that Benjamini-Schramm limits of uniformly face-rooted finite trivalent hyperbolic polyhedra are parabolic, establishes a refined ring lemma for regular circle patterns under a strengthened angle condition (Z2'), and uses it to identify the Poisson boundary with the circle at infinity and to prove positive speed for the face random walk.","tokens_in":26757,"tokens_out":3771,"duration_ms":37017,"significance":"The claimed dichotomy is attractive and potentially important: it would reduce the global conformal type of a unimodular random hyperbolic polyhedron to the expectation of a purely local angle defect. The proof of Theorem 3.1 is a clean, fully written mass transport argument, and the finite Gauss-Bonnet formula in Proposition 4.2 together with the Benjamini-Schramm parabolicity theorem is a genuinely interesting and non-obvious obstruction. The refined ring lemma is a genuinely useful quantitative tool for unbounded-degree circle patterns. The main caveat is that the global dichotomy is not self-contained: its decisive equivalences are delegated to external preprints and papers without reproducing the hypotheses or conclusions, and several Section 6 arguments are too terse to verify as written.","major_comments":[{"comment":"The central equivalence (iii)↔(iv) in Theorem 3.2 is asserted by citing [26, Theorem 1.4], but that theorem is neither stated nor checked against the present hypotheses. The paper works with tame ADTs satisfying (Z1)–(Z4), allows unbounded degrees, only assumes E[deg(ρ)]<∞, admits weak RCP realizations, and allows angle values in [0,π−ε]. If [26, Theorem 1.4] requires bounded degree, strictly positive angles, or exact realizations, then the equivalence fails precisely in the advertised regime. This dependency is load-bearing because it is what turns the local Gauss-Bonnet identity into the global conformal dichotomy. The authors should either state the exact theorem and verify its hypotheses, or prove the needed equivalence directly.","section":"§3, Theorem 3.2"},{"comment":"The paper defines RCP-parabolic/RCP-hyperbolic by existence of a realization with carrier C or D, but does not prove that this type is well-defined under the standing assumptions, nor that it agrees with VEL type for tame ADTs with unbounded degrees and weak realizations. The assertion 'this analytic type agrees with the VEL type' is made without proof. Since this agreement is used in the proof of Theorem 1.4 and in Section 6, it must either be proved or precisely attributed with a verifiable statement. As written, the reader cannot determine what assumptions the cited uniformization theorem needs.","section":"§2.2, Definition 2.11 and following paragraph"},{"comment":"The stochastic half of the paper, including the refined ring lemma and the entire boundary theory in Section 6, depends on the strengthened cycle condition (Z2'), but the paper gives no argument that a typical unimodular random tame THP or ADT satisfies a uniform margin ε0>0. Without such an argument, Lemma 5.1, Corollary 1.6, and Corollary 1.7 are conditional on a hypothesis that is not established for the main class. In addition, Section 6 assumes E[deg(ρ)^3]<∞, which is stronger than the E[deg(ρ)]<∞ used in the dichotomy; the paper should clarify how these assumptions are to be satisfied in intended applications.","section":"§5, Lemma 5.1 and Definition 2.2"},{"comment":"The proof of non-atomicity in Lemma 6.14 relies on the assertion that the straight-line realization obtained from the RCP is a planar embedding compatible with the underlying triangulation; this is not proved and is not immediate for weak realizations or unbounded-degree patterns. The proof of full support in Lemma 6.15 is also extremely compressed: the 'standard planar mass-transport argument' is not described, and it is not clear why a boundary component omitted from the support forces a separator to receive infinite mass. These two lemmas are essential for the Poisson boundary identification, so the arguments need to be written out in full.","section":"§6.3, Lemmas 6.14 and 6.15"},{"comment":"The proof of Theorem 1.4 says that 'under the THP/ADT correspondence, the parabolic/hyperbolic type of P agrees with the RCP type of (G,Θ)', but this correspondence is not established in the manuscript. Theorem 2.10 constructs a THP from an RCP and identifies dihedral angles, but it does not prove that the parabolic/hyperbolic type of the resulting polyhedron matches the RCP carrier type under the present hypotheses (which include ordinary, ideal, and hyperideal vertices). This is a second, independent gap in the chain from the local Gauss-Bonnet formula to the THP dichotomy.","section":"§3, proof of Theorem 1.4"}],"minor_comments":[{"comment":"The wording of (Z3) is confusing: 'homologically non-adjacent edges' is defined through an arc formed by two adjacent edges, but the phrase 'homologically adjacent' suggests a homology condition rather than a combinatorial one. Please rephrase or add a figure.","section":"§2.1, Definition 2.2"},{"comment":"In Lemma 5.3, the set E_u(x) is defined as a union of D_v\\D_u, but the proof later uses points P_xy and P_xz that lie on boundaries of neighboring disks; the inclusion of these points in E_u(x) should be stated explicitly.","section":"§5, Lemma 5.3"},{"comment":"The phrase 'if it lies inside the disk D_a, the intersection graph of centers a,b,y,u would form a planar K4 with b in the center' is not clear; this step needs a more detailed geometric explanation.","section":"§5.4 (Lemma 5.4)"},{"comment":"The compatibility of the family of metrics {d_G^{(u)}} is asserted but the 'remaining rotational ambiguity' is not fully discussed. Since the boundary identification depends on this compatibility, a few sentences of justification would help.","section":"§6.3, Lemma 6.12"},{"comment":"There are several small formatting issues, including 'P^{trun}' in Definition 2.11 and the unlabeled equation (2) referenced in the proof of Theorem 3.1; please ensure all displayed equations are numbered consistently.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on two unpublished preprints by the same group, [24] and [26], for existence, rigidity, and the crucial VEL/RCP equivalence. The reader cannot check whether the cited theorems apply to the generality claimed here. I would recommend the editor insist that the authors either include the precise statements and hypotheses of the cited theorems, or prove the missing equivalences within the paper. The dependence on [26] affects not only Theorem 1.3 but also Theorem 4.3, since the proof of the ergodic-component inequality m(ξ)≥6 uses Theorem 1.3. The mass transport argument and the finite polyhedral Gauss-Bonnet theorem are solid and publishable on their own; the revision should preserve those strengths while making the external dependencies transparent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The cleanest thing in this paper is Theorem 3.1: the mass-transport proof of E[L_f] = 2π − (π/3)E[deg(f)] is short, transparent, and checkable. That is a genuine local result, and it extends the ideal-angle defect from the same group's earlier work in a natural way. The soficity section is also genuinely nice: the finite Gauss–Bonnet identity sum_f L_f = 4π is independent of the angles, so any Benjamini–Schramm limit of uniformly face-rooted finite trivalent hyperbolic polyhedra must have E[deg] = 6 and be parabolic. That gives a clean, self-contained obstruction to hyperbolic polyhedral limits. The refined ring lemma is a plausible technical contribution, though it is conditional on the strengthened cycle condition (Z2') and the proof is terse in places.\n\nThe load-bearing soft spot is exactly the one the stress-test flags. Theorem 3.2 asserts the equivalence between the expected-degree condition and VEL/RCP type by citing [8] and [26, Thm 1.4], but the cited theorem is not stated and its hypotheses are not checked against the paper's standing assumptions: tame angles, unbounded degrees, possibly weak circle-pattern realizations, only E[deg] < ∞. The paper's own definition of RCP-parabolic/hyperbolic is by existence of a realization with carrier C or D, and nothing in the text proves this is well-defined or agrees with VEL type in this generality. If [26, Thm 1.4] needs bounded degree or strict angle positivity, the dichotomy in Theorem 1.4 fails precisely in the advertised regime. This is more serious than the dependence on (Z2'), because (Z2') only affects Section 6, whereas the [26] dependency is what converts the local Gauss–Bonnet formula into the global conformal dichotomy. The case E[L] > 0 is also left hanging, since the paper never proves E[deg] ≥ 6 except by the same external citations.\n\nSection 6 is honestly marked as conditional and should be read that way. The boundary identification and positive speed depend on (Z2'), the third moment assumption, and a handful of lemmas—especially the planarity assertion in Lemma 6.14 and the full-support argument in Lemma 6.15—that are sketched rather than fully demonstrated. These are referee-able gaps, not obvious errors.\n\nThe authors lean heavily on their own preprints [24] and [26]. That is not by itself a flaw, but it means the central dichotomy cannot be independently checked from this paper. If the cited results are correct, this is a real advance. The referee should be asked to verify the exact applicability of [26, Thm 1.4] and to expand the terse parts of Section 6.","headline":"The local Gauss–Bonnet formula and the soficity obstruction are solid and worth publishing; the advertised global dichotomy is a bridge over two unpublished preprints and needs external verification before it can be taken at face value.","tokens_in":27296,"tokens_out":1872,"would_cite":true,"duration_ms":19546,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60D05","52C26"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that for tame, ergodic, unimodular random trivalent hyperbolic polyhedra with finite expected face degree, the global conformal type is parabolic exactly when the expected facewise angle defect $\\mathbb{E}[L_f(P)]$ is…","keywords":["unimodular random graphs","trivalent hyperbolic polyhedra","circle patterns","Gauss-Bonnet formula","conformal type","Poisson boundary","ring lemma","Benjamini-Schramm limits"],"falsifier":"Sample an explicit ergodic unimodular random angled disk triangulation (for instance, a random disk triangulation with admissible i.i.d. angle marks), solve its regular circle pattern numerically, and compare root expected degree with the computed VEL or circle-packing type: a realization with $\\mathbb{E}[\\deg(\\rho)]=6$ that is RCP-hyperbolic, or with $\\mathbb{E}[\\deg(\\rho)]>6$ that is RCP-parabolic, would falsify Theorem 3.2 and the dichotomy it drives.","tokens_in":26275,"feed_emoji":"🔺","tokens_out":11997,"duration_ms":98005,"temperature":0.7,"pith_summary":"The paper tries to establish a local-to-global principle: for an infinite random hyperbolic polyhedron built as the intersection of half-spaces in $\\mathbb{H}^3$, the global conformal type—parabolic, where the truncated polyhedron escapes to a single point on the sphere at infinity, or hyperbolic, where it escapes to a circle—is determined by the expectation of a single local quantity attached to the root face. That quantity, the geometric characteristic number $L_f(P) = 2\\pi - \\sum_{v\\in f}\\theta_v^f$, is a facewise angle defect computed from the dihedral angles along the edges meeting each vertex. The paper proves the unimodular Gauss-Bonnet formula $\\mathbb{E}[L_f(P)] = 2\\pi - (\\pi/3)\\mathbb{E}[\\deg(f)]$, so the sign of the expectation is equivalent to whether the expected face degree is $6$ or larger. Under the paper's tameness and admissibility conditions this yields the dichotomy $\\mathbb{E}[L_f(P)]=0$ exactly in the parabolic case and $\\mathbb{E}[L_f(P)]<0$ exactly in the hyperbolic case, and also shows that hyperbolic examples cannot be approximated by finite trivalent hyperbolic polyhedra. If the argument is right, a statistician or geometer who can estimate the expected degree of one face knows the conformal type, boundary behaviour, and escape rate of the entire random polyhedron.","feed_headline":"Expected degree 6 sets the parabolic-hyperbolic barrier","feed_subtitle":"One local Gauss-Bonnet identity turns a face's angle defect into the global conformal type of the whole random polyhedron.","key_machinery":"The load-bearing object is the geometric characteristic number $$L_f(P)=2\\pi-\\sum_{v\\in f}\\theta_v^f,$$ where $\\theta_v^f$ is the angle at vertex $v$ in an auxiliary Euclidean triangle whose side lengths come from the three dihedral angles at that trivalent vertex; the defining identity is that the three angles at each vertex sum to $\\pi$. The argument's mechanism is the mass transport principle applied to a specifically chosen transport on the dual angled disk triangulation: the mass sent from a root face is $6\\pi-3L_f(P)$, the mass received is $\\pi\\deg(f)$, and unimodularity equates the expectations, giving the Gauss-Bonnet identity. For the stochastic half, the key tool is the refined ring lemma, an exponential radius-ratio bound $r(v)/r(u)>e^{-C S(u)}$ where $S(u)=\\sum_{w\\sim u}\\deg(w)$ is the flower degree of $u$; this replaces the uniform bounds of classical ring lemmas in the presence of unbounded degrees and supplies the integrability needed for boundary convergence and positive speed.","core_discovery":"On the paper's own terms, the central discovery is a three-way rigidity: local dihedral geometry, global conformal type, and stochastic asymptotics are locked together by one expectation. Theorem 1.4 states that for a tame, infinite, ergodic, unimodular random trivalent hyperbolic polyhedron satisfying (Z3) and (Z4) with $\\mathbb{E}[\\deg(f)]<\\infty$, the root-face characteristic satisfies $\\mathbb{E}[L_f(P)]=0$ if and only if the polyhedron is almost surely parabolic, and $\\mathbb{E}[L_f(P)]<0$ if and only if it is almost surely hyperbolic. The engine is the mass transport identity $\\mathbb{E}[L_f(P)] = 2\\pi - (\\pi/3)\\mathbb{E}[\\deg(f)]$, whose angular dependence cancels because the three auxiliary Euclidean angles at each trivalent vertex sum to $\\pi$. The paper further proves that every admissible Benjamini-Schramm limit of uniformly face-rooted finite trivalent hyperbolic polyhedra is parabolic, so genuinely hyperbolic structures are not polyhedrally sofic; and, under a strengthened cycle condition and a third-moment assumption, that for hyperbolic examples the face random walk converges to the circle at infinity, that the exit measure is non-atomic with full support and realizes the Poisson boundary, and that the walk has positive linear speed.","pith_inferences":["A consequence the paper leaves implicit is that the classification is insensitive to the angle distribution itself: any two admissible angle laws on the same random combinatorial model should give the same conformal type whenever they share the same expected degree.","The strengthened cycle condition is the bottleneck for making the stochastic conclusions unconditional; proving it almost surely for natural random polyhedral models (for instance, limits of finite polyhedra with compactly supported angle data) would be the direct next step.","The no-hyperbolic-limit theorem is special to spherical approximations; the paper's own remarks indicate that approximating the same graph by higher-genus surfaces bypasses the finite-spherical Gauss-Bonnet obstruction, so polyhedral soficity and ordinary graph soficity are plausibly different properties.","The refined ring lemma's exponential control in flower degree is a general analytic tool; it should transfer to other angle-prescribed discrete conformal structures (square tilings, hyperideal circle patterns), carrying the Poisson-boundary and positive-speed results with it."],"forward_implications":["The conformal type of the whole random polyhedron is controlled by one local observable: the expected geometric characteristic of the root face, with the angle configuration otherwise cancelling in expectation.","The degree threshold is universal: expected face degree $6$ forces parabolicity, expected degree $>6$ forces hyperbolicity, independently of the numerical values of the dihedral angles (within the admissibility conditions).","Every admissible Benjamini-Schramm limit of uniformly face-rooted finite trivalent hyperbolic polyhedra is parabolic, and every hyperbolic unimodular trivalent hyperbolic polyhedron satisfying the standing assumptions is not polyhedrally sofic.","In the hyperbolic regime, under the strengthened cycle condition and finite third moment, the face random walk converges almost surely to a point of the circle at infinity, the exit measure is atom-free with full support and realizes the Poisson boundary, and the walk escapes with positive linear speed.","On trivalent ideal polyhedra the new invariant agrees with the previously studied angle defect, so the dichotomy extends the ideal theory to ordinary, ideal, and hyperideal vertices without changing the answer."],"supporting_citations":[{"why":"gives existence and rigidity of regular circle patterns realizing an ADT and the RCP-THP correspondence, so the geometric realization used throughout is available.","marker":"[26]"},{"why":"supplies the ideal-angle-defect invariant and the boundary argument template; its Lemma 4.1 is used inside the refined ring lemma.","marker":"[24]"},{"why":"gives the unimodular planar-map dichotomy relating expected degree, VEL type, amenability, and transience, which carries Theorem 3.2 across the five criteria.","marker":"[8]"},{"why":"provides the circle-packing boundary identification and positive-speed strategy that Section 6 adapts to regular circle patterns.","marker":"[7]"},{"why":"defines unimodular random networks and Benjamini-Schramm convergence, providing the probabilistic framework and the notion of soficity used in Section 4.","marker":"[1]"},{"why":"establishes the equivalence between vertex extremal length and circle-packing type for disk triangulations, the VEL/RCP bridge.","marker":"[35]"},{"why":"introduced the geometric characteristic number as a discrete curvature of circle packings and is the source of the invariant's definition.","marker":"[23]"}],"fun_headline_variants":["Local angle defect sets global conformal type","Zero expected angle defect means parabolic","One local expectation decides parabolic vs hyperbolic","Expected face type determines whole polyhedron's conformal type","Angle defect expectation is the parabolic-hyperbolic switch"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every simple closed curve in the dual triangulation that is not the boundary of a face has total angle at most $(s-2)\\pi - \\varepsilon_0$ for one fixed deterministic $\\varepsilon_0>0$, and the paper gives no argument that a typical unimodular random trivalent hyperbolic polyhedron satisfies this strengthened cycle condition.","fun_headline_variants_meta":{"raw":{"variants":["Local angle defect sets global conformal type","Zero expected angle defect means parabolic","One local expectation decides parabolic vs hyperbolic","Expected face type determines whole polyhedron's conformal type","Angle defect expectation is the parabolic-hyperbolic switch"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001202,"raw_usage":{"total_tokens":5062,"prompt_tokens":1161,"completion_tokens":3901,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":777,"completion_tokens_details":{"reasoning_tokens":3833}},"tokens_in":777,"tokens_out":3901,"duration_ms":23306,"temperature":1.0,"reasoning_tokens":3833,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:49:34.507539+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Sample an explicit ergodic unimodular random angled disk triangulation (for instance, a random disk triangulation with admissible i.i.d. angle marks), solve its regular circle pattern numerically, and compare root expected degree with the computed VEL or circle-packing type: a realization with $\\mathbb{E}[\\deg(\\rho)]=6$ that is RCP-hyperbolic, or with $\\mathbb{E}[\\deg(\\rho)]>6$ that is RCP-parabolic, would falsify Theorem 3.2 and the dichotomy it drives.","supporting_citations":[{"cited_title":"Characterizations of infinite circle patterns and convex polyhedra in hyperbolic 3-space","cited_arxiv_id":"2511.09368","evidence_quote":"gives existence and rigidity of regular circle patterns realizing an ADT and the RCP-THP correspondence, so the geometric realization used throughout is available."},{"cited_title":"Random infinite ideal angled graphs and ideal hyperbolic polyhedra","cited_arxiv_id":"2601.14909","evidence_quote":"supplies the ideal-angle-defect invariant and the boundary argument template; its Lemma 4.1 is used inside the refined ring lemma."},{"cited_title":"Angel, T","cited_arxiv_id":null,"evidence_quote":"gives the unimodular planar-map dichotomy relating expected degree, VEL type, amenability, and transience, which carries Theorem 3.2 across the five criteria."},{"cited_title":"Angel, T","cited_arxiv_id":null,"evidence_quote":"provides the circle-packing boundary identification and positive-speed strategy that Section 6 adapts to regular circle patterns."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines unimodular random networks and Benjamini-Schramm convergence, providing the probabilistic framework and the notion of soficity used in Section 4."},{"cited_title":"He and O","cited_arxiv_id":null,"evidence_quote":"establishes the equivalence between vertex extremal length and circle-packing type for disk triangulations, the VEL/RCP bridge."},{"cited_title":"Ge and A.-J","cited_arxiv_id":null,"evidence_quote":"introduced the geometric characteristic number as a discrete curvature of circle packings and is the source of the invariant's definition."}],"review_version":2}