{"id":"0f030296-b8c9-4619-b025-f438c382d9ed","arxiv_id":"2607.26363","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Every 6-net with at most 50 vertices has an undented Euclidean realization from unit equilateral triangles, and prime 6-nets have unique convex ideal hyperbolic realizations.","lead":"The paper proves, with computer assistance, that all 10,412,340 sphere triangulations with at most 50 vertices and no vertex meeting more than 6 triangles can be built as dent-free polyhedra from unit equilateral triangles. It also proves uniqueness of the prime cases as ideal hyperbolic polyhedra and lays out conjectures for the whole infinite family.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main finite theorem depends on computer-assisted certificates that the paper does not make independently auditable; a clean re-run of the archived pipeline is the decisive test.","rationale":"After reading the paper carefully, I find no internal mathematical contradiction in the counts or in the structural arguments. The isoperimetric undentedness estimate in §3 (giving 2π−A ≥ (π/3)√(36−d²) and E=3 arccos(1/3)−π) is sound for the attachment argument, and the class 1+2 counts (50+95=145) are exact rational constructions. The total counts cross-check against the independent census [7], which is good evidence the enumeration is complete. However, the existence half of the theorem is not independently supported: the effective IFT, flat-limit handling, Perron brackets for Proposition 3, and interval embeddedness checks are all delegated to software and 'certificate formats' that are not described in enough detail for a reader to audit. This is not a criticism of the authors' integrity; it is a structural limitation of the paper's evidence. The reader's weakest_assumption identified exactly this, so I agree. The appropriate verdict remains CONDITIONAL: if the software is correct, the result is very plausible and well-scoped, but the paper alone does not make the proof checkable. A clean re-run of the archived pipeline would settle it.","tokens_in":8541,"tokens_out":18864,"duration_ms":168843,"concrete_test":"Download the Zenodo archive [10] and the two GitHub repositories, pin to the archived versions, and run the full verification pipeline from a clean environment (e.g., Docker with a fixed OS/toolchain). Confirm that: (a) the prime census for v≤50 sums to 8,239,684 and the non-prime class-3 count is 2,172,511; (b) every certificate contains an interval enclosure satisfying the edge-length equations (with coplanarity substitutions where claimed) plus an interval proof of embeddedness/undentedness; and (c) the certificate verifier is independent of the proof-generating code. If any certificate fails or the pipeline cannot be executed with the archived files, Corollary 2 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Corollary 2 (the central finite claim) is established by enumerating 8,239,684 prime 6-nets (Appendix A, Proposition 1) and then realizing non-prime nets via a three-class decomposition (§3). The proof of Proposition 1 is described only as: generate nets by a variant of Goedgebeur [14]; compute an approximate neoplatonic by numerical homotopy from ideal realizations; apply an effective inverse-function theorem with guaranteed error bounds; and at flat degree-6 vertices replace some edge-length equations by coplanarity conditions. None of these steps is specified at a level permitting independent audit, and the paper does not state the interval radii, the certificate format, or how embeddedness/undentedness is certified for the prime case. §3's class-3 argument ('extend the certified intervals ... An interval embeddedness check proves ...') is similarly one sentence. The only external check is the combinatorial count cross-reference in Corollary 2's proof to [7, Tables 1 and 2], which validates the enumeration but not existence. Thus every existence assertion in Corollary 2 depends on the correctness of the Zenodo/GitHub software, which a reader cannot verify from the paper. A bug in the IFT radii, flat-limit substitution, or interval embeddedness predicate would invalidate the theorem even if the underlying conjectures are true.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines 6-nets as simplicial triangulations of the 2-sphere with maximum degree at most 6 and studies two conjectured realizations: an 'undented' Euclidean polyhedron built from unit equilateral triangles (a neoplatonic solid) and an equilateral ideal hyperbolic polyhedron. The main finite result is Corollary 2: every 6-net with v ≤ 50 has a Euclidean neoplatonic realization; there are 8,239,684 prime and 2,172,656 non-prime nets, totaling 10,412,340. Proposition 1 asserts existence of Euclidean realizations for all prime 6-nets with v ≤ 50, and Proposition 3 asserts existence and uniqueness of convex ideal neoplatonic realizations for the same range. The proofs combine exact rational constructions for two simple non-prime classes, a Perron-type super/subsolution argument for ideal existence, and a large computer-assisted certification using archived software for the prime Euclidean case.","tokens_in":8866,"tokens_out":4967,"duration_ms":52272,"significance":"If the computational certificates are sound, the paper establishes a substantial finite case of the neoplatonic conjecture and provides a strong test of the broader conjectural framework. The exact rational constructions for the 145 nets in the first two non-prime classes, the independent count cross-checks against the census [7], and the use of Rivin's uniqueness theorem for ideal convex realizations are genuine strengths. However, the central Euclidean existence proof is currently not independently auditable from the text, and several load-bearing computational steps are described in only one or two sentences.","major_comments":[{"comment":"The proof of Proposition 1 is a black-box reference to [12] and [10]. It does not state the nonlinear system being solved, the dimension/degrees of freedom, the norm used in the effective inverse function theorem, the interval-arithmetic library or radii, or the certificate format. The key sentence 'In cases where the limit lies flat at one or more degree-6 vertices, some of the equations specifying edge lengths get replaced by coplanarity conditions' is not expanded into a precise mathematical statement: which equations, exactly how the substitution is performed, and why the resulting system implies existence of an actual neoplatonic realization. Since Corollary 2's prime case relies entirely on this proposition, this is a load-bearing gap in auditability.","section":"2, Proposition 1"},{"comment":"The class-3 argument, which handles 2,172,511 nets, is compressed into two sentences: 'After identifying combinatorial duplicates...' and 'An interval embeddedness check proves that each has an exact embedded realization.' The paper does not explain how the certified intervals from the prime case are extended across attached tetrahedra, what predicate is used for combinatorial duplicate identification, or what the interval embeddedness check verifies (e.g., no self-intersections, positive dihedral angles, local support planes). These details are essential because the class-3 count and the existence assertions for all non-prime v ≤ 50 depend on them.","section":"3, third class"},{"comment":"The ideal-existence proof uses a Perron-type method, but the construction of the super- and subsolutions u_0, u_1 is described only by example and by the assertion 'we chose super- and subsolutions with all defects equal to ±1/500... ±1/4000.' The paper does not specify how u_0 and u_1 are generated for each of the 8.2 million prime nets, how the descent algorithm is proved to terminate, or what certificates verify the bracketed triangle, Delaunay, and boundary inequalities. The statement 'we can check that as long as u0 ≤ u ≤ u1 the triangle inequalities hold' is a computational claim, but no check program or certificate format is described at a level that a reader can verify. Proposition 3's existence portion therefore rests on an unstated computational protocol.","section":"6"}],"minor_comments":[{"comment":"The three-class decomposition is clear in broad strokes, but the relation between the number of tetrahedral/octahedral factors and the number of vertices is not explicitly derived; equations such as v = 3 + 3m + t appear without supporting definitions of m and t. A short derivation would improve readability.","section":"3"},{"comment":"The term 'algebraic volume' for an oriented ideal cycle is not defined or referenced. Since it is used to state Conjecture (Volume max), a definition or reference is needed.","section":"4"},{"comment":"Reference [3] is given as 'Seek: Atlas of neoplatonic solids' with no URL or DOI; this is not a usable citation. Also, the acknowledgments mention that computations were carried out by Claude Code and ChatGPT; if the journal has a policy on AI-assisted computation, this should be stated more explicitly.","section":"8 and References"},{"comment":"The introductory characterization of prime nets states 'A 6-net is prime just if it is the tetrahedron, or if it has no vertex of degree 3 and is not a stack of two or more octahedra.' This is asserted without proof; a short justification or reference would be helpful.","section":"2"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is a computer-assisted proof, and the manuscript's current level of detail is insufficient for a reader to audit the computation. This is not a fundamental mathematical error in the arguments I can see; rather, the missing certificate specifications and algorithmic descriptions are load-bearing. I would recommend that the editors require the authors to provide a detailed appendix (or an auxiliary document) listing the exact systems, interval-arithmetic packages, certificate formats, and per-net data, along with a reproducible command line for the archived Zenodo software. Without that, the finite theorem cannot be independently validated. The exact rational constructions for the non-prime classes are a nice feature and should be highlighted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe thing to know: this is a serious computer-assisted paper, not a numerological exercise. Doyle and Ellison prove that every 6-net on at most 50 vertices has an 'undented' Euclidean realization built from unit equilateral triangles, and that every prime such net has a unique convex ideal hyperbolic realization. The census of 10,412,340 nets is new, as is the separating-triangle decomposition into prime factors and the exact rational constructions for the first two non-prime classes. If the computer-assisted part holds up, this is a solid contribution to the study of equilateral polyhedra.\n\nWhat is good: the paper is honest about what is proved and what is conjectured. The ideal existence proof uses a Perron super/subsolution method that is at least sketched, with explicit defect parameters, and uniqueness comes from Rivin's volume-maximization theorem. The Euclidean existence is obtained from a numerical seed and an effective inverse-function theorem, and the enumeration counts are cross-checked against an independent census. The spherical-isoperimetry argument showing attachments preserve undentedness is clean and rigorous.\n\nThe soft spot is exactly where your reader pointed: Proposition 1 and Corollary 2 rest on a computer-assisted pipeline that the text does not make auditable. The proof of Proposition 1 is one paragraph. We are not told the interval radii, the certificate format, how embeddedness is certified, or how the flat-limit substitution at degree-6 vertices works in practice. The class-3 argument in §3 is one sentence. The code is archived on Zenodo and GitHub, but with no commit hashes and no one-command reproduction, a referee cannot check it without substantial work. That does not make the theorem false, but it does make the paper's central claim conditional on the software being right. The authors themselves say 'the proof programs and certificate formats are distributed'—they do not claim the paper alone is sufficient.\n\nIs that a load-bearing flaw? Yes and no. It is a flaw in exposition and reproducibility, not in the mathematical conception. Computer-assisted proofs of this scale (10M cases) rarely put every certificate in the paper; the norm is to archive software and have referees run it. Here the archive is present, but the interface is not described well enough to trust quickly.\n\nWho should read this: anyone working on equilateral polyhedra, ideal hyperbolic polyhedra, or triangulations of the sphere. It deserves peer review, with the explicit instruction that a referee run the code or at least verify the pipeline end-to-end. I would also like to see the authors provide a manifest with hashes and a minimal repro script.\n\nMy recommendation: engage with it. Send it out. Treat the finite results as provisional until the software is independently checked, but the ideas are worth the referee time.","headline":"A genuinely new finite census with a plausible but not fully auditable computer-assisted proof; worth engaging, but the central existence theorem hangs on software the paper doesn't let you check.","tokens_in":9311,"tokens_out":4022,"would_cite":true,"duration_ms":39802,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52B70","51M10","52C26"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every 6-net with up to 50 vertices admits an undented Euclidean realization built from unit equilateral triangles, and every prime 6-net in this range has a unique convex ideal hyperbolic realization.","keywords":["6-net","neoplatonic solid","ideal hyperbolic polyhedron","equilateral triangle","simplicial triangulation","undented polyhedron","computer-assisted proof","sphere triangulation"],"falsifier":"Run an independent implementation of the enumeration and the interval-arithmetic certification on every 6-net with v≤50 and check that the prime/non-prime counts agree with 8,239,684 and 2,172,656 and that every net passes the embeddedness and undentedness checks; any mismatch or any net without a valid certificate would disprove the claim.","tokens_in":8427,"feed_emoji":"🔺","tokens_out":10441,"duration_ms":89951,"temperature":0.7,"pith_summary":"This paper works toward a conjecture that every 6-net — a simplicial triangulation of the 2-sphere in which no vertex touches more than six triangles — can be realized as an undented Euclidean polyhedron made of unit equilateral triangles, and also as an ideal hyperbolic polyhedron with equilateral ideal-triangle faces. The authors prove the Euclidean part of this conjecture for all 10,412,340 combinatorially distinct 6-nets with at most 50 vertices, and they prove the ideal-hyperbolic part for every prime 6-net in the same range, including uniqueness of the convex ideal realization. The proof is computer-assisted: it combines complete enumeration of prime nets, numerical homotopy from ideal to Euclidean realizations, and interval-arithmetic certification that an exact solution lies near each approximate one, with a separating-triangle decomposition that reduces non-prime nets to prime factors. If the conjecture holds in general, these 'neoplatonic solids' form an infinite family that extends the tetrahedron, octahedron, and icosahedron, and the methods here demonstrate a template for rigorous computer-assisted existence proofs in polyhedral geometry.","feed_headline":"10,412,340 triangulations become unit equilateral polyhedra","feed_subtitle":"A computer-assisted proof certifies every sphere triangulation with degree at most 6 and up to 50 vertices.","key_machinery":"The central objects are 6-nets — simplicial triangulations of the 2-sphere with maximum vertex degree at most 6 — with a net called prime if every 3-cycle bounds a face. An undented polyhedron is one where every vertex has a local support plane, so the exterior dihedral angles around each vertex sum to at least 0. The argument rests on three mechanisms: (1) an effective inverse function theorem with interval-arithmetic error bounds, which certifies that an exact Euclidean realization lies near a numerically computed approximate one, with coplanarity conditions substituted at degree-6 vertices that flatten; (2) a bracket method of super- and subsolutions for the equations defining ideal hyper","core_discovery":"The central claim is an existence theorem: every 6-net with v≤50 has an undented realization in Euclidean 3-space whose faces are unit equilateral triangles, and every prime 6-net with v≤50 has a unique convex ideal realization in hyperbolic 3-space. The Euclidean proof is computer-assisted: approximate realizations are obtained by numerical homotopy from the ideal ones, and an effective inverse function theorem with interval arithmetic proves that an exact realization lies nearby; when a degree-6 vertex flattens, coplanarity conditions replace some edge-length equations. Uniqueness of the Euclidean realization is not proved. Non-prime nets are handled by cutting along separating triangles,","pith_inferences":["The certification recipe — find a numerical solution, then close the gap with interval arithmetic and an effective inverse function theorem — is a template that could be applied to other equilateral polyhedral realization problems, for example triangulations with maximum degree 7 or with prescribed non-unit edge lengths, whenever a homotopy from a hyperbolic or other known solution exists.","If the general neoplatonic conjecture is true, the family of neoplatonic solids provides a discrete uniformization statement: every sphere triangulation of degree at most 6 has a canonical Euclidean and hyperbolic equilateral representative. Proving the paper's neoconvex rigidity hypothesis would imply uniqueness of these representatives and yield a rigidity theorem that permits local nonconvexity","The super- and subsolution bracketing for ideal realizations closely parallels circle-packing existence proofs; the authors highlight that the edge-length rule here is a product rather than a sum. A plausible route to the full ideal conjecture is to construct bracketing functions for all 6-nets, which the computations suggest must become extremely tight as the vertex count grows.","The volume-maximization conjecture for ideal neoplatonics is computationally testable within the current census: one could compute the algebraic volume of each certified ideal realization and check whether any alternative ideal geodesic 2-cycle with the same combinatorics has larger volume, which would either support or falsify the conjecture in the tested range."],"forward_implications":["Every 6-net with v≤50 has a neoplatonic realization; up to combinatorial isomorphism there are 10,412,340 such nets, of which 8,239,684 are prime and 2,172,656 are non-prime.","Every prime 6-net with v≤50 has a convex ideal hyperbolic realization, unique up to isometry; this provides a large certified family of ideal equilateral hyperbolic polyhedra.","The non-prime nets with v≤50 are completely classified into three families, all realized by gluing regular tetrahedra and octahedra; these gluings are checked exactly with rational coordinates or interval-extended angle bounds, and undentedness is proven by a spherical isoperimetric inequality.","The volume of the ideal icosahedron is computed to high precision, a numerical invariant that can serve as a check on future constructions.","The numerical homotopy from ideal to Euclidean realizations offers experimental evidence for the neoplatonic homotopy conjecture: that Euclidean neoplatonics are the zero-length limits of a one-parameter family of undented hyperbolic polyhedra."],"fun_headline_variants":["Every 6-net with ≤50 vertices becomes a unit equilateral polyhedron","10,412,340 sphere triangulations get equilateral polyhedron realizations","Computer proves: all degree-≤6 sphere triangulations up to 50 vertices are polyhedral","Neoplatonic solids: existence proven for all 6-nets up to 50 vertices","10,412,340 unit equilateral polyhedra from sphere triangulations"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole theorem rests on the unverified correctness of the archived computer programs that enumerate the nets and certify each existence claim; there is no formal machine-checked proof of the software, so any undetected bug in the enumeration, the interval bounds, or the homotopy could invalidate the result even if the underlying conjecture is true.","fun_headline_variants_meta":{"raw":{"variants":["Every 6-net with ≤50 vertices becomes a unit equilateral polyhedron","10,412,340 sphere triangulations get equilateral polyhedron realizations","Computer proves: all degree-≤6 sphere triangulations up to 50 vertices are polyhedral","Neoplatonic solids: existence proven for all 6-nets up to 50 vertices","10,412,340 unit equilateral polyhedra from sphere triangulations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000837,"raw_usage":{"total_tokens":3470,"prompt_tokens":712,"completion_tokens":2758,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":456,"completion_tokens_details":{"reasoning_tokens":2649}},"tokens_in":456,"tokens_out":2758,"duration_ms":17626,"temperature":1.0,"reasoning_tokens":2649,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T17:24:48.369159+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an independent implementation of the enumeration and the interval-arithmetic certification on every 6-net with v≤50 and check that the prime/non-prime counts agree with 8,239,684 and 2,172,656 and that every net passes the embeddedness and undentedness checks; any mismatch or any net without a valid certificate would disprove the claim.","supporting_citations":[],"review_version":1}