{"id":"54001f85-276b-49a2-86f1-c96c4eae4ce0","arxiv_id":"2606.30184","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Binom(n+r,r) basic r-symmetric tropical polynomials of degree <=n separate S_n orbits on R^{nr} for all n,r.","lead":"The paper constructs binom(n+r, r) basic r-symmetric tropical polynomials of degree at most n that serve as complete invariants separating orbits of the symmetric group on multisets of points in R^r. This provides explicit stable coordinates for point clouds and persistence barcodes that are smaller and lower degree than general constructions from tropical invariant theory.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Recovery of the multiset from the dual of the identified transportation problem is the load-bearing step for orbit separation.","rationale":"The reader's weakest assumption directly matches the constructive core of the separation argument. Because the proof is claimed to be elementary and algorithmic rather than existential, verifying the algorithm on degenerate inputs is the single concrete check that would confirm or refute the central claim. No other part of the argument (bi-Lipschitz property via max filters, necessity results) is load-bearing for the separation statement itself.","tokens_in":1910,"tokens_out":343,"duration_ms":34902,"concrete_test":"Extract the explicit recovery algorithm from the manuscript (likely in the section following the transportation identification) and apply it to a test multiset with repeated points or a non-unique optimal transport plan for n=4, r=3; check whether two distinct multisets produce identical basic values but the algorithm outputs only one.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The separation claim rests on identifying the \binom{n+r}{r} basic r-symmetric tropical polynomial values with the costs of a transportation problem on the multiset, then recovering the multiset uniquely from an optimal dual solution via an explicit algorithm. If this identification or recovery step fails to be injective for some configurations (e.g., when the support of the optimal transport plan is non-unique or when points coincide), distinct orbits could map to the same basic values. The abstract presents this identification and algorithm as the basis of the constructive proof; any gap in handling degenerate cases or in the explicit reconstruction for r>2 would falsify the claim that these specific polynomials separate all S_n orbits.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims that the \binom{n+r}{r} basic r-symmetric tropical polynomials of degree at most n form a complete separating family for the orbits of S_n acting on \\mathbb{R}^{nr} (i.e., on unordered n-point multisets in \\mathbb{R}^r) for all n,r \\geq 1. The proof is elementary and constructive: the polynomial values are identified with the costs of a transportation problem whose supply/demand data encode the multiset, after which the multiset is recovered uniquely from an optimal dual solution by an explicit algorithm. The coordinate map is further shown to be bi-Lipschitz (via max-filtering theory), with an explicit forward Lipschitz constant and a dimension-free distortion bound when r=1. The paper also determines that pairwise values suffice precisely when n \\leq 3 and exhibits a discrete-tomography obstruction showing that degree <n or fewer than three columns cannot separate in general.","tokens_in":2039,"tokens_out":471,"duration_ms":32861,"significance":"If the central claims hold, the work supplies an explicit, small, and stable complete coordinate system for multisets that is dramatically smaller and lower-degree than the general separating sets furnished by Derksen's tropical invariant theory. The constructive transportation-duality argument, the bi-Lipschitz embedding, and the necessity results constitute concrete strengths; the explicit algorithm and dimension-free distortion for r=1 are particularly useful for applications such as persistence barcodes.","major_comments":[],"minor_comments":[{"comment":"§2 (or wherever the transportation identification is stated): the precise correspondence between the basic r-symmetric tropical polynomial evaluations and the transportation costs should be written as an explicit equality (rather than left implicit) so that the subsequent dual-recovery algorithm can be checked line-by-line.","section":null},{"comment":"The bi-Lipschitz section invokes the general max-filtering theory; a short self-contained paragraph recalling the precise statement of the forward and reverse Lipschitz bounds used would improve readability for readers outside that subfield.","section":null},{"comment":"The necessity argument via the discrete-tomography configuration is convincing, but the explicit counter-example matrices for r=2, n=4 should be displayed in a small table to make the obstruction immediately verifiable.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so there are no individual points requiring response or manuscript changes.","responses":[],"tokens_in":1508,"tokens_out":58,"duration_ms":18750,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper settles the open problem from Kubo 2019 by giving binom(n+r,r) basic r-symmetric tropical polynomials of degree at most n that separate all S_n orbits on R^{nr} for any n and r. The family is small and the proof is constructive.\n\nIt does well on the comparison to prior work: the size and degree beat the general tropical invariants from Derksen. The identification with a transportation problem and the recovery from the dual make the separation explicit. The bi-Lipschitz embedding follows from max filtering, which adds stability for applications like point clouds and barcodes.\n\nThe soft spot is the recovery step. If the explicit algorithm fails to distinguish some configurations with repeated points or non-unique duals, the separation would not hold. The abstract presents it as working, but that part carries the load. One would want to see how it handles the cases where the support of the optimal transport plan is not unique. The necessity results on degree and columns look fine, based on the tomography obstruction.\n\nReaders working on invariants for unordered data or tropical methods in data analysis will get the most from this. It is worth a serious referee to verify the algorithm and the general r case in detail. The bi-Lipschitz constants are also worth checking for r greater than 1.\n\nI would recommend sending it for peer review.","headline":"Kubo closes the separation problem for point multisets using binom(n+r,r) tropical polynomials of low degree.","tokens_in":2526,"tokens_out":345,"would_cite":true,"duration_ms":60535,"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":"Binomial(n+r,r) basic r-symmetric tropical polynomials separate all S_n orbits on R^{nr} for every n and r.","keywords":["tropical polynomials","symmetric group orbits","multisets of points","point clouds","transportation problems","bi-Lipschitz embeddings","discrete tomography","persistence barcodes"],"falsifier":"Two different multisets of n points whose corresponding basic r-symmetric tropical polynomials evaluate to the same values would disprove the separation result.","tokens_in":2791,"feed_emoji":"","tokens_out":611,"duration_ms":46172,"temperature":0.7,"pith_summary":"For any number n of points in any dimension r, a family of binomial(n+r,r) tropical polynomials provides complete coordinates on the space of unordered point sets. These are the basic r-symmetric ones of degree at most n, and they are stable because they arise in the max-plus semiring. The construction is smaller and lower-degree than general methods from tropical invariant theory. The proof works by linking the polynomial values to a transportation problem whose dual recovers the multiset via an explicit procedure. The resulting coordinate map is shown to be bi-Lipschitz.","feed_headline":"Tropical polynomials coordinate multisets of any n points in r dimensions","feed_subtitle":"Binomial(n+r,r) basic r-symmetric ones of degree at most n separate all S_n orbits and form a bi-Lipschitz map.","key_machinery":"the binomial(n+r,r) basic r-symmetric tropical polynomials of degree at most n, which serve as invariants that separate orbits by functioning as an injective max filter bank","core_discovery":"The central claim is that for all n ≥ 1 and r ≥ 1, the binomial(n+r,r) basic r-symmetric tropical polynomials of degree at most n separate the orbits of the symmetric group S_n on R^{nr}. This means they form a complete set of permutation-independent coordinates for multisets of n points in R^r. The proof is elementary and constructive, identifying the values with a transportation problem and recovering the multiset from the dual by an explicit algorithm. The map is moreover a bi-Lipschitz embedding.","pith_inferences":["These coordinates could supply stable and complete distances between persistence diagrams.","The transportation-problem view may permit efficient numerical recovery of the multiset via linear programming.","For r=1 the distortion bound being dimension-free hints at possible extensions to other semirings or algebraic settings.","Running the recovery algorithm on sampled point clouds would provide direct computational checks of separation."],"forward_implications":["The coordinates are stable and explicit for point clouds and for persistence barcodes when r=2.","The embedding has an explicit Lipschitz constant for the forward bound and a fully explicit dimension-free distortion when r=1.","Pairwise values suffice precisely when n ≤ 3.","In general, invariants involving at least three columns and of degree n are necessary due to non-uniqueness configurations from discrete tomography."],"fun_headline_variants":["Tropical polys separate all S_n orbits of n-point multisets in R^r","Binomial(n+r,r) tropical polys provide complete coordinates for multisets","Basic symmetric tropical polynomials coordinate any multiset of points stably","r-symmetric tropical polynomials separate orbits for all n r dimensions"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The polynomial values can be identified with a transportation problem whose solution dual yields an explicit algorithm that recovers the original multiset uniquely.","fun_headline_variants_meta":{"raw":{"variants":["Tropical polys separate all S_n orbits of n-point multisets in R^r","Binomial(n+r,r) tropical polys provide complete coordinates for multisets","Basic symmetric tropical polynomials coordinate any multiset of points stably","r-symmetric tropical polynomials separate orbits for all n r dimensions"]},"model":"grok-4.3","cost_usd":0.004492,"raw_usage":{"total_tokens":2347,"prompt_tokens":885,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":44924500,"prompt_tokens_details":{"text_tokens":885,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1389,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":885,"tokens_out":73,"duration_ms":22671,"temperature":1.0,"reasoning_tokens":1389,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T03:23:21.281039+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Two different multisets of n points whose corresponding basic r-symmetric tropical polynomials evaluate to the same values would disprove the separation result.","supporting_citations":[],"review_version":1}