{"id":"3045efb3-ee0e-41a8-ad04-57e82f0dc277","arxiv_id":"2606.19155","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A new symmetry method using involution-invariant perturbations proves Kobayashi hyperbolicity for very generic degree-d surfaces in P^3 when d ≥ 16.","lead":"The authors introduce a symmetry method based on an involution-invariant two-term perturbation of Fermat equations to generate extra linear constraints on invariant 2-jet differentials via representation theory of Z/2Z. This yields a computer-assisted proof that very generic surfaces of degree at least 16 in projective 3-space are Kobayashi hyperbolic.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Proof for d≥16 rests on correctness of C++ solver for overdetermined systems from Z/2Z eigenvector constraints on invariant 2-jet differentials.","rationale":"The reader's weakest_assumption matches the load-bearing step exactly: the entire algebraic claim reduces to the computational output of the symmetry-constrained linear systems. No closed-form argument or machine-checked certificate is supplied, so the correctness risk remains high and the UNVERDICTED status is appropriate.","tokens_in":1797,"tokens_out":300,"duration_ms":12365,"concrete_test":"Release the exact two-term perturbations, the monomial bases, and the C++ source used for d=16; recompute the kernel dimension of the resulting matrix over Q (or a prime > degree bound) with an independent linear-algebra library; if dimension >0 the vanishing lemma fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The vanishing of negatively twisted invariant 2-jet differentials (needed for hyperbolicity) is obtained by showing that the only solutions to the linear systems coming from the two-term Fermat-type perturbations and the induced Z/2Z action are the zero sections. These systems are declared overdetermined and solved numerically in C++ with modular arithmetic; the manuscript provides no independent algebraic proof that the kernel is trivial for d≥16. An undetected bug in matrix construction, eigenvector projection, or modular-to-rational lifting would falsify the claimed vanishing.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript develops a symmetry-based method for vanishing lemmas on invariant 2-jet differentials. It employs two-term Fermat-type perturbations invariant under a coordinate-exchanging involution, inducing a Z/2Z action whose representation theory supplies extra linear constraints. These turn previously intractable systems into overdetermined ones solved via a parallelized C++ implementation with modular arithmetic, yielding a proof that very generic surfaces in P^3 of degree d ≥ 16 are Kobayashi hyperbolic (with the method positioned to reach d = 15 and related results for plane-curve complements).","tokens_in":1931,"tokens_out":569,"duration_ms":27304,"significance":"If the computer-assisted vanishing holds, the result lowers the known threshold for Kobayashi hyperbolicity of generic surfaces in P^3 to 16 and supplies a reusable symmetry technique that could attain the conjectured optimal bounds. The explicit use of Z/2Z eigenvectors to enlarge the constraint set, together with the preparation of code for the remaining cases, constitutes a concrete technical contribution to algebraic methods in hyperbolicity.","major_comments":[{"comment":"The central claim that very generic surfaces of degree d ≥ 16 are Kobayashi hyperbolic rests entirely on the C++ solver establishing that the only solutions to the eigenvector-constrained linear systems (arising from the two-term perturbations and Z/2Z action on invariant 2-jet differentials) are the zero sections. The manuscript provides no explicit matrices, no description of the perturbation polynomials chosen for each d, no exclusion rules, and no verification steps for the modular arithmetic or lifting procedure, rendering the key vanishing lemmas unverifiable from the text alone.","section":"Abstract and computational implementation description"},{"comment":"The assertion that the symmetry method transforms the systems into 'highly overdetermined' ones (thereby guaranteeing trivial kernel for d ≥ 16) requires explicit rank or dimension counts comparing the number of independent constraints to the dimension of the space of invariant 2-jet differentials; without these calculations the overdetermined character remains unconfirmed and is load-bearing for the claimed improvement over prior work.","section":"Symmetry method and linear-algebra setup"}],"minor_comments":[{"comment":"The abstract states that the C++ code for d = 15 and d = 11 'is already written and the computations are underway'; including a brief status update or repository link would clarify reproducibility for readers.","section":null}],"recommendation":"major_revision","confidential_remarks":"The heavy dependence on an unreleased C++ implementation for the main theorem raises standard reproducibility concerns for a computer-assisted proof in algebraic geometry; the authors should be asked to deposit the source, input data, and verification scripts in a public archive."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the constructive report and for highlighting the need for greater transparency in the computational aspects of our preliminary manuscript. We address each major comment below and will revise the text accordingly.","responses":[{"response":"We agree that the present version supplies insufficient implementation detail for independent verification. The manuscript was written as a concise report on an ongoing project whose primary contribution is the new symmetry method; the full computational data were therefore omitted. In revision we will add an explicit description of the two-term Fermat-type perturbation polynomials chosen for each d ≥ 16, the resulting linear systems, the exclusion rules applied, and the modular-arithmetic verification steps. The C++ source code (already written and used for the d ≥ 16 cases) will be deposited in a public repository with a README documenting the compilation, execution, and lifting procedure. Full matrices remain too large for the printed text, but the code will allow any reader to regenerate them.","revision_made":"partial","referee_comment":"[Abstract and computational implementation description] The central claim that very generic surfaces of degree d ≥ 16 are Kobayashi hyperbolic rests entirely on the C++ solver establishing that the only solutions to the eigenvector-constrained linear systems (arising from the two-term perturbations and Z/2Z action on invariant 2-jet differentials) are the zero sections. The manuscript provides no explicit matrices, no description of the perturbation polynomials chosen for each d, no exclusion rules, and no verification steps for the modular arithmetic or lifting procedure, rendering the key vanishing lemmas unverifiable from the text alone."},{"response":"The referee correctly identifies that the manuscript asserts the systems become highly overdetermined without supplying the supporting dimension counts. We will compute and insert, for each d ≥ 16, the dimension of the space of invariant 2-jet differentials together with the number of independent linear constraints obtained from the Z/2Z-eigenvector conditions. These explicit comparisons will confirm that the augmented systems are overdetermined and thereby justify the vanishing statements.","revision_made":"yes","referee_comment":"[Symmetry method and linear-algebra setup] The assertion that the symmetry method transforms the systems into 'highly overdetermined' ones (thereby guaranteeing trivial kernel for d ≥ 16) requires explicit rank or dimension counts comparing the number of independent constraints to the dimension of the space of invariant 2-jet differentials; without these calculations the overdetermined character remains unconfirmed and is load-bearing for the claimed improvement over prior work."}],"tokens_in":1538,"tokens_out":535,"duration_ms":17680,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper's main contribution is a two-term Fermat-type perturbation that stays invariant under swapping two coordinates, plus the induced Z/2Z action on invariant 2-jet differentials. The eigenvector constraints from that action add enough extra equations to make the systems overdetermined, which the authors then solve numerically in C++ with modular arithmetic. This yields the vanishing needed to conclude that very generic degree-d surfaces in P^3 are Kobayashi hyperbolic for d >= 16.\n\nThe technique is a clear incremental improvement over the cited Hou-Huynh-Merker-Xie work; the symmetry idea is elementary once stated and turns previously intractable systems into ones the computer can finish. They also sketch how the same setup should reach the target d=15 and the curve complement case d=11, and they release the manuscript early so others can see the method.\n\nThe soft spot is exactly where the stress-test note says: the vanishing statements rest on the C++ code correctly finding only the zero solution in those large systems. No matrices, no exclusion rules, and no independent algebraic verification appear in the abstract, and the code itself is not yet public. An error in matrix construction or in lifting modular solutions would invalidate the d=16 claim. That is not a minor bookkeeping issue; it is the load-bearing step.\n\nThe paper is for specialists already working on jet differentials and explicit hyperbolicity bounds. A reader who knows the prior literature will understand the technical step and can judge whether the computational evidence is sufficient. It is worth sending to peer review so that referees with the right background can inspect the code and the linear-algebra details.","headline":"Symmetry method with Z/2Z constraints pushes explicit hyperbolicity bound to d=16 via C++ linear algebra, but the result stands or falls on unverified solver output.","tokens_in":2433,"tokens_out":411,"would_cite":false,"duration_ms":10918,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A symmetry method using involution-invariant perturbations proves that very generic surfaces of degree 16 or higher in P^3 are Kobayashi hyperbolic.","keywords":["Kobayashi hyperbolicity","jet differentials","algebraic surfaces","symmetry method","vanishing lemmas","Fermat hypersurfaces","projective space","involution action"],"falsifier":"A concrete counterexample would be an explicit non-constant entire curve on a very generic surface of degree 16 in P^3, or a computation showing that the dimension of the space of invariant 2-jet differentials remains positive for some such surface.","tokens_in":2690,"feed_emoji":"","tokens_out":551,"duration_ms":25337,"temperature":0.7,"pith_summary":"The paper develops a symmetry-based technique for proving vanishing results on negatively twisted invariant 2-jet differentials, both for generic surfaces in projective 3-space and for complements of plane curves. It applies a two-term perturbation of Fermat-type equations that stays invariant under an involution exchanging two coordinates. This invariance produces a linear action on the space of differentials whose eigenvectors, drawn from the representation theory of Z/2Z, supply many extra linear constraints. The resulting overdetermined systems become solvable by computer, yielding a proof that very generic degree-16 surfaces in P^3 are Kobayashi hyperbolic and laying groundwork for the degree-15 case. A reader would care because the vanishing controls entire curves and therefore the hyperbolicity of the surface.","feed_headline":"Very generic degree-16 surfaces in P^3 are Kobayashi hyperbolic","feed_subtitle":"Involution-invariant two-term perturbations generate extra eigenvector constraints that force 2-jet differentials to vanish.","key_machinery":"The involution-invariant two-term perturbation of Fermat-type equations, which induces a linear action on the space of invariant 2-jet differentials and permits an eigenvector argument from Z/2Z representation theory to produce extra linear constraints.","core_discovery":"By taking a two-term perturbation of the Fermat equation that is invariant under the involution swapping two coordinates and applying an elementary eigenvector argument from the representation theory of Z/2Z to the space of invariant 2-jet differentials, the authors generate enough additional linear constraints to conclude that all negatively twisted invariant 2-jet differentials vanish on a very generic surface of degree d ≥ 16 in P^3; the vanishing implies that the surface is Kobayashi hyperbolic.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Symmetry method forces 2-jet vanishing on degree-16 P^3 surfaces","Involution perturbation yields eigenvector constraints for hyperbolicity","Z/2Z symmetry proves Kobayashi hyperbolicity at degree 16","Two-term Fermat perturbation enables vanishing lemmas for degree-16 surfaces","Generic P^3 surfaces hyperbolic via new symmetry-based vanishing method"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The C++ implementation correctly computes the solution space of the overdetermined linear systems obtained from the eigenvector constraints on the space of invariant 2-jet differentials for the chosen two-term perturbations.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry method forces 2-jet vanishing on degree-16 P^3 surfaces","Involution perturbation yields eigenvector constraints for hyperbolicity","Z/2Z symmetry proves Kobayashi hyperbolicity at degree 16","Two-term Fermat perturbation enables vanishing lemmas for degree-16 surfaces","Generic P^3 surfaces hyperbolic via new symmetry-based vanishing method"]},"model":"grok-4.3","cost_usd":0.003095,"raw_usage":{"total_tokens":1731,"prompt_tokens":771,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":30949500,"prompt_tokens_details":{"text_tokens":771,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":881,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":771,"tokens_out":79,"duration_ms":6666,"temperature":1.0,"reasoning_tokens":881,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T19:13:36.622828+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete counterexample would be an explicit non-constant entire curve on a very generic surface of degree 16 in P^3, or a computation showing that the dimension of the space of invariant 2-jet differentials remains positive for some such surface.","supporting_citations":[],"review_version":1}