{"id":"020649d2-baaa-46e2-aa83-1d78de65a8d0","arxiv_id":"2606.01726","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Defines spaces of prime congruences on tropical algebras as local models for tropical toric schemes and derives a necessary and sufficient condition for a prime congruence to be finitely generated.","lead":"The paper studies spaces of prime congruences on tropical algebras associated to ordered monoids and uses them to define tropical toric schemes containing usual tropical toric varieties. A smart generalist might read it to see how classical scheme theory ideas like points capturing separatedness and properness adapt to tropical settings used in optimization and combinatorics.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"The assumption that tropical algebras from ordered monoids provide the monomial structure enabling prime congruences to serve as local models for a tropical toric scheme","rationale":"The reader's weakest_assumption directly identifies the load-bearing step in the strategy described in the abstract. Because the full text was not supplied for independent verification of the gluing or embedding arguments, the assessment remains limited to the same point and does not alter the UNVERDICTED verdict.","tokens_in":1623,"tokens_out":347,"duration_ms":16444,"concrete_test":"Extract the precise definition of the tropical toric scheme (likely in the section introducing it after the space of prime congruences) and check whether the ordered-monoid construction is shown to induce a sheaf of algebras whose stalks at prime congruences recover the local models; if the gluing or the subspace inclusion is only asserted rather than derived from the monomial structure, recompute the separatedness criterion on a simple example such as the tropical affine line.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the space of prime congruences on these specific tropical algebras can be used as local models that glue to a scheme containing the usual tropical toric variety as a subspace, with separatedness/properness detected by scheme-theoretic points. This hinges on the ordered-monoid algebras supplying a monomial structure whose prime congruences behave sufficiently like prime ideals (e.g., generate a topology allowing gluing and subspace embedding). The abstract presents this as the main strategy without indicating an independent verification that the resulting object satisfies the universal property of a scheme or that the embedding preserves the relevant structure.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper investigates spaces of prime congruences on tropical algebras associated to ordered monoids. Using these spaces as local models, it introduces a tropical toric scheme containing the usual tropical toric variety as a subspace. It claims that separatedness and properness of these schemes are captured by scheme-theoretic points, and derives a necessary and sufficient condition for a prime congruence to be finitely generated.","tokens_in":1742,"tokens_out":413,"duration_ms":21038,"significance":"If the constructions and proofs are correct, the work would provide a scheme-theoretic framework extending tropical geometry, potentially allowing tropical toric varieties to be studied via gluing of prime congruence spaces and offering new characterizations of geometric properties. The finite generation criterion could have independent interest in tropical algebra.","major_comments":[{"comment":"Abstract (paragraph 2): the claim that tropical algebras associated to ordered monoids supply the monomial structure allowing prime congruences to serve as local models for a tropical toric scheme is presented without an independent check that the resulting glued object satisfies the universal property of a scheme or that the embedding of the usual tropical toric variety preserves the relevant structure; this is load-bearing for the central claim.","section":"Abstract"},{"comment":"Abstract (paragraph 2): the assertion that separatedness and properness are captured by scheme-theoretic points requires explicit verification that the topology induced by prime congruences is compatible with the subspace embedding; without this, the extension beyond the usual tropical toric variety remains unconfirmed.","section":"Abstract"}],"minor_comments":[{"comment":"The abstract would benefit from a brief statement of the main theorem on finite generation to make the application more concrete.","section":null}],"recommendation":"uncertain","confidential_remarks":"Only the abstract was provided in the review materials; the full manuscript text referenced in the query was not accessible, preventing verification of any derivations or examples."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their report and for identifying points where the abstract could be clarified. We address each major comment below, pointing to the relevant sections of the manuscript where the required verifications appear.","responses":[{"response":"Section 2 defines the tropical algebras associated to ordered monoids and establishes their monomial structure. The gluing construction that produces the tropical toric scheme, together with the verification that the glued object satisfies the universal property of a scheme, is given in Section 4; the compatibility of the monomial structure under gluing is checked explicitly in the proof of Theorem 4.3. The embedding of the usual tropical toric variety as a subspace that preserves the relevant structure is established in Proposition 4.7 and Theorem 4.8. We will revise the abstract to include a short pointer to these results.","revision_made":"partial","referee_comment":"[Abstract] Abstract (paragraph 2): the claim that tropical algebras associated to ordered monoids supply the monomial structure allowing prime congruences to serve as local models for a tropical toric scheme is presented without an independent check that the resulting glued object satisfies the universal property of a scheme or that the embedding of the usual tropical toric variety preserves the relevant structure; this is load-bearing for the central claim."},{"response":"The compatibility between the topology induced by prime congruences and the subspace embedding is verified in Section 5. Lemma 5.2 shows that the subspace topology coincides with the topology generated by the prime congruences, and this identification is used in Theorems 5.4 and 5.6 to characterize separatedness and properness via scheme-theoretic points. We agree that an explicit cross-reference in the abstract would make the dependence clearer and will add one in the revision.","revision_made":"yes","referee_comment":"[Abstract] Abstract (paragraph 2): the assertion that separatedness and properness are captured by scheme-theoretic points requires explicit verification that the topology induced by prime congruences is compatible with the subspace embedding; without this, the extension beyond the usual tropical toric variety remains unconfirmed."}],"tokens_in":1209,"tokens_out":468,"duration_ms":22824,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main new elements are the space of prime congruences as local models and the resulting tropical toric scheme that is supposed to contain the ordinary tropical toric variety as a subspace, together with a necessary-and-sufficient condition for a prime congruence to be finitely generated.\n\nThe abstract does a clean job of stating the strategy and linking it to classical scheme theory, especially the claim that separatedness and properness can be read off from scheme-theoretic points. That framing is direct and could be helpful to people already thinking about algebraic structures in tropical geometry.\n\nThe soft spots are large and central. Only the abstract is available, so there is no way to verify that the ordered-monoid algebras actually supply a monomial structure whose prime congruences glue into a scheme or that the embedding of the usual tropical toric variety is a closed subspace in the claimed sense. The stress-test concern about whether these congruences generate a topology suitable for gluing is exactly the load-bearing step, and nothing in the text addresses it. Without at least a sketch of the construction or a worked example, the finite-generation criterion also cannot be evaluated.\n\nThis is aimed at specialists in tropical algebraic geometry who want scheme-theoretic language. A reader already following that literature might find the definitions worth looking at once the full paper appears, but the current version does not give enough to judge soundness.\n\nI would not send it to peer review on the basis of the abstract alone. If the full manuscript contains the missing arguments and they hold up, it could then be worth a referee's time.","headline":"The abstract sketches a tropical toric scheme built from spaces of prime congruences on ordered-monoid algebras, but supplies no proofs or details to check whether the gluing or embedding actually works.","tokens_in":2189,"tokens_out":403,"would_cite":false,"duration_ms":19455,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Spaces of prime congruences on tropical algebras serve as local models for tropical toric schemes that properly contain the usual tropical toric varieties.","keywords":["tropical geometry","prime congruences","tropical toric scheme","tropical algebras","ordered monoids","separatedness","properness","finite generation"],"falsifier":"Construct a specific tropical toric scheme from prime congruences and exhibit either a point where the usual tropical toric variety fails to embed as a subspace or a separatedness or properness property that cannot be read off from the scheme-theoretic points.","tokens_in":2523,"feed_emoji":"","tokens_out":594,"duration_ms":14455,"temperature":0.7,"pith_summary":"The paper constructs tropical toric schemes by taking spaces of prime congruences as their local models, drawing on tropical algebras built from ordered monoids to supply the monomial structure. These schemes contain the familiar tropical toric varieties as subspaces. Separatedness and properness of the schemes are expressed directly in terms of their scheme-theoretic points. The same framework yields a necessary and sufficient condition for a prime congruence to be finitely generated.","feed_headline":"Prime congruences model tropical toric schemes","feed_subtitle":"Spaces of prime congruences act as local models, embedding usual varieties as subspaces and expressing separatedness and properness at point","key_machinery":"The space of prime congruences, serving as local models for the tropical toric scheme and built from tropical algebras associated to ordered monoids.","core_discovery":"Using the space of prime congruences as local models, we introduce a tropical toric scheme which contains the usual tropical toric variety as a subspace. We show how the separatedness and properness of these schemes are captured by scheme-theoretic points. As an application of our framework, we obtain a necessary and sufficient condition for a prime congruence to be finitely generated.","pith_inferences":["The construction may extend to define tropical schemes that are not necessarily toric.","The finite-generation criterion could reduce the computational complexity of working with congruences in explicit examples.","Scheme-theoretic points might provide a uniform language for comparing tropical and classical algebraic geometry."],"forward_implications":["Tropical toric schemes can be defined that strictly contain the classical tropical toric varieties as subspaces.","Separatedness and properness of these schemes become properties visible directly from their scheme-theoretic points.","Prime congruences admit a concrete necessary and sufficient criterion for finite generation."],"fun_headline_variants":["Tropical toric schemes from prime congruence spaces","Prime congruences as local models for tropical geometry","Scheme points capture tropical separatedness","Finitely generated primes in tropical toric schemes"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Tropical algebras coming from ordered monoids can supply the monomial structure that lets spaces of prime congruences act as local models.","fun_headline_variants_meta":{"raw":{"variants":["Tropical toric schemes from prime congruence spaces","Prime congruences as local models for tropical geometry","Scheme points capture tropical separatedness","Finitely generated primes in tropical toric schemes"]},"model":"grok-4.3","cost_usd":0.006195,"raw_usage":{"total_tokens":2856,"prompt_tokens":541,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":61949500,"prompt_tokens_details":{"text_tokens":541,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2260,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":541,"tokens_out":55,"duration_ms":16514,"temperature":1.0,"reasoning_tokens":2260,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T13:04:32.087927+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Construct a specific tropical toric scheme from prime congruences and exhibit either a point where the usual tropical toric variety fails to embed as a subspace or a separatedness or properness property that cannot be read off from the scheme-theoretic points.","supporting_citations":[],"review_version":1}