{"id":"ecec6657-ddcf-44e6-9798-bfac6e7b0e6a","arxiv_id":"2608.06857","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Basic r-symmetric tropical polynomials generate the semifield of S_n-invariant tropical rational functions, and arbitrary permutation groups admit a quadratic degree bound that is optimal for alternating groups.","lead":"A mathematics paper proves that a specific list of symmetric tropical rules can build every symmetric tropical rational rule on matrices, and that a similar statement holds for every permutation group. The result settles an open question in tropical invariant theory and gives an optimal quadratic degree bound, which matters to researchers studying symmetries of max-plus geometry and multiset invariants.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The penalty argument of Theorem 3.5 rests entirely on the unproven, qualitative bi-Lipschitz constant c_{n,r} imported from the author's preprint [2]; without an independent positive lower bound, the central claim is conditional.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: the bi-Lipschitz constant c_{n,r} from [2] is unquantified and not independently proven, and the penalty argument in Theorem 3.5 needs a uniform positive lower bound to work. My reading of the full text confirms this is the only true soft spot. The internal structure — Lemma 3.1 (sorted columns as differences of basic values), Lemma 3.2 (re-assembly by couplings), Lemma 3.3 (recoupling discriminants), and Lemma 3.4 (Lipschitz constant for tropical rational functions) — is coherent, and the final formula (3) is explicit. The worked (2,2) example in Remark 3.11 checks out: the two couplings give the stated discriminants, and the penalty terms behave as claimed. The axiomatic Section 4 is a clean abstraction that would be valid if the cited hypotheses (S) and (T) hold. The only reason the theorem is not established unconditionally within this paper is the reliance on [2] for (2). Since the paper itself flags the qualitative nature of c_{n,r} in Remark 3.10 and offers no independent proof, the appropriate outcome is to keep the reader's CONDITIONAL verdict: accept if the constant is supplied or independently verified, otherwise wait. I do not see grounds to reject outright, because the cited inequality is plausible and the rest of the argument is sound; nor do I see grounds to accept unconditionally, because the central proof depends on an external unverified statement. Hence UNCHANGED from the reader's CONDITIONAL.","tokens_in":13726,"tokens_out":17359,"duration_ms":163154,"concrete_test":"Independently re-derive the bi-Lipschitz lower bound (2) directly from the definition of the basic values b_c in (1), without citing [2, Corollary 3.8]. Specifically, for n=2,r=2, express Φ and the quotient metric d explicitly in terms of the sorted column entries x_1≥x_2, y_1≥y_2 and the pairing invariant w_11, and compute the infimum of ∥Φ(M)−Φ(M')∥_2 / d([M],[M']) over pairs on the two image sheets with identical sorted columns. If this infimum is 0, (2) fails and the proof of Theorem 3.5 collapses; if it is positive, attempt the same derivation for n=3,r=2 via the reconstruction algorithm of [2]. A successful closed-form positive lower bound for all n,r would remove the dependency; failure to produce one would confirm that the paper's central theorem is conditional on an unverified constant.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.5 proves that every S_n-invariant tropical rational function f on R^{n×r} equals g∘Φ for a tropical rational g on R^K, via the formula g(w)=min_σ [f(R_σ(s(w)))+κ_f D_σ(w)]. The proof that this minimum returns f(M) for w=Φ(M) uses two inequalities: (i) the recoupling discriminant D_σ(Φ(M)) ≥ (c_{n,r}/√K) d_∞([M],[R_σ(s(Φ(M)))]), and (ii) κ_f ≥ L_f/c'_{n,r}. Both rely on the bi-Lipschitz lower bound ∥Φ(M)−Φ(M')∥_2 ≥ c_{n,r} d([M],[M']) from [2, Corollary 3.8]. This constant is not proved in the present paper; Remark 3.10 explicitly concedes that its existence in [2] is qualitative. If c_{n,r} were zero, or if the proof of [2, Corollary 3.8] contained a gap, then D_σ could be arbitrarily small relative to the quotient distance for wrong couplings, and no finite κ_f would dominate the Lipschitz loss L_f d_∞ in f. The doubling-loop algorithm in §3.3 does not remove the dependency: its termination is guaranteed only by Theorem 3.5, which in turn assumes (2). Thus the central claim stands or falls with an inequality that is cited, not demonstrated, and that inequality is the single load-bearing external input. All other steps — sorted-column recovery, re-assembly by couplings, Lipschitz bound, and the axiomatic generalization — are internally coherent and are consistent with the (2,2) worked example.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that, for the action of S_n permuting the rows of n × r real matrices, the basic r-symmetric tropical polynomials b_c (|c| ≤ n) generate the semifield of S_n-invariant tropical rational functions. The main theorem (Theorem 3.5) gives an explicit expression for any invariant tropical rational f as g∘Φ, where Φ is the basic-coordinate map and g is a minimum over the (n!)^{r-1} couplings of the sorted columns, each term being f evaluated on the re-assembled matrix plus a penalty built from the basic values. The proof combines sorted-column recovery, a re-assembly lemma, a recoupling-discriminant lower bound obtained from the bi-Lipschitz embedding of the orbit space, and the Lipschitz property of tropical rational functions. The paper also gives an image description of Φ, a subfamily generation criterion, an expression algorithm, and a generalization to arbitrary permutation groups with a quadratic degree bound, together with an optimality statement for the alternating group.","tokens_in":13990,"tokens_out":14488,"duration_ms":150129,"significance":"If the result is correct, it is a substantial advance in tropical invariant theory: it replaces Derksen's primorial degree bound for the row-permutation action by generators of degree at most n, answering a question explicitly raised in the author's earlier work. The construction is explicit and algorithmic, and the axiomatic formulation in Section 4 isolates the two structural ingredients, yielding a quadratic degree bound for all permutation groups and a sharp A_N rigidity theorem. The paper is carefully written and contains a useful worked example for (n,r)=(2,2). The central caveat is that the main theorem is conditional on a bi-Lipschitz constant imported from an unpublished preprint, as detailed below.","major_comments":[{"comment":"The load-bearing inequality in the proof of Theorem 3.5 is the discriminant lower bound D_σ(Φ(M)) ≥ c'_{n,r} d∞([M],[R_σ(s(Φ(M)))]), which is derived from the bi-Lipschitz estimate ∥Φ(M)-Φ(M')∥_2 ≥ c_{n,r} d([M],[M']) cited as [2, Corollary 3.8]. The paper gives no proof of this inequality and no quantitative value for c_{n,r}; Remark 3.10 explicitly concedes that the existence of the constant in [2] is qualitative. The lower-bound half of the proof of Theorem 3.5, namely g(w) ≥ f(M) for w=Φ(M), relies on this positive constant for every coupling: without c_{n,r}>0, no finite κ_f can be guaranteed to dominate the Lipschitz loss L_f, and formula (3) may fail. Since this is the single external input on which the main theorem rests, the central claim is conditional on [2] being correct and, in particular, on its Corollary 3.8. Please either prove a positive lower bound for c_{n,r} in this paper, give an explicit bound, or state the main theorem as conditional on that preprint.","section":"Section 3.2, Eq. (2) and Lemma 3.3"},{"comment":"The doubling-loop algorithm is presented as an alternative that does not require knowledge of c_{n,r}. This does not circumvent the dependence: the proof of Proposition 3.12 shows termination by invoking Theorem 3.5 ('By Theorem 3.5 the test passes as soon as κ_f ≥ L_f/c'_{n,r}'), and Theorem 3.5 itself uses inequality (2). If c_{n,r} were zero, or if the proof of [2, Corollary 3.8] contained a gap, the loop would have no guaranteed halting time. Thus the algorithm's correctness and termination inherit the same external assumption, and it cannot serve as an independent effectivity argument unless termination is proved using a separately established lower bound.","section":"Section 3.3, Proposition 3.12"}],"minor_comments":[{"comment":"The notation s(w)_{iα} := w_{i eα} - w_{(i-1)eα} uses the symbol w_0, which is not a coordinate of w; please define the convention w_{0 eα} := 0 explicitly, both here and in the algorithm of Section 3.3 where σ_α(i)-1 can be zero.","section":"Section 3.1, Lemma 3.1"},{"comment":"The paper calls the structure a semifield while also using subtraction (⊘=−) as a total operation on functions. Since a semifield is normally understood without additive inverses, please add a clarifying sentence that 'tropical rational functions' are identified as functions and that the algebraic structure considered is the set of such functions under pointwise max, plus, and subtraction.","section":"Section 2.2"},{"comment":"The proof asserts that each member of F is a max filter but does not give the template construction. Since the claim is correct, please include one sentence explaining that b_c is the max filter whose template has blocks of 1s in disjoint template rows: rows 1 through c_1 in column 1, rows c_1+1 through c_1+c_2 in column 2, and so on.","section":"Corollary 3.9(ii)"},{"comment":"Reference [2] is a submitted preprint and [3] is an arXiv preprint; please update both if they are accepted during revision, and state clearly in the text which results of the paper depend on each of them.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically strong and the derivation is internally coherent; the use of f inside the expression (3) is a legitimate substitution and not circular. The sole blocking issue is the reliance on the qualitative, unquantified bi-Lipschitz constant from the author's own submitted preprint [2]. If that result is accepted or if the author supplies an independent positive lower bound in this paper, I would be inclined to accept. I would ask the editor to ensure that the dependency on [2] is made explicit and, if possible, that the proof of [2, Corollary 3.8] is included or referenced in a form accessible to the reader."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is real and worth refereeing: the basic r-symmetric tropical polynomials generate the semifield of S_n-invariant tropical rational functions, improving Derksen's primorial degree bound to degree at most n. Section 4's quadratic bound for arbitrary permutation groups and the A_N optimality theorem are new, internally coherent, and give the paper reach well beyond the row-permutation case. The recoupling-discriminant expression is explicit, the sorted-columns and re-assembly lemmas are correct, and the (2,2) worked example makes the mechanism transparent. Lemma 4.7 and Theorem 4.8 (trace normal form and A_N rigidity) are clever and the proofs check out. The doubling-loop algorithm in Section 3.3 is a nice touch: since the identity test is decidable, the constant kappa_f can be found without knowing c_{n,r} explicitly.\n\nThe soft spot is real and localized: the proof of Theorem 3.5 uses the lower Lipschitz bound ||Phi(M) - Phi(M')||_2 >= c_{n,r} d([M],[M']) from the author's own submitted preprint [2], and Remark 3.10 concedes the constant is qualitative. This is load-bearing for both the penalty lower bound and the choice of kappa_f. The doubling-loop algorithm mitigates the practical issue but does not remove the logical dependency: its termination is guaranteed only by the theorem it is trying to instantiate. So the central claim stands or falls with an inequality that is cited, not demonstrated here. That is not a fatal flaw—papers cite preprints all the time—but it is the one point a referee should press. The solution is straightforward: either include a proof of the bound in this manuscript, give an explicit c_{n,r}, or ensure [2] is publicly available and verified before final acceptance. The dependence on [3], [4], and [9] is less concerning; those are either published or have detailed proofs.\n\nI agree with the reader's conditional verdict. The structure is sound, the exposition is clear, and the honest admission in Remark 3.10 actually increases confidence that the author knows where the weak point is. This paper deserves serious peer review, but the referee should request that the bi-Lipschitz constant be made explicit or at least independently checkable. I would bring it to a reading group focused on tropical or invariant theory, and I would cite Section 4 for the quadratic bound and A_N optimality even while the Section 3 result remains conditional.","headline":"A strong, mostly self-contained generation theorem for tropical symmetric rational functions, held back only by an unquantified bi-Lipschitz constant imported from the author's own preprint.","tokens_in":14595,"tokens_out":1991,"would_cite":true,"duration_ms":22934,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14T10","13A50","12K10","05E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the basic $r$-symmetric tropical polynomials generate the semifield of all $S_n$-invariant tropical rational functions on $\\mathbb{R}^{n\\times r}$, with explicit penalty-based expressions.","keywords":["tropical polynomial","max-plus algebra","symmetric functions","generating sets","separating invariants","semifield","piecewise linear function"],"falsifier":"For a specific small case such as $(n,r)=(2,2)$, compute the infimum of $\\|\\Phi(M)-\\Phi(M')\\|_2/d([M],[M'])$ over pairs of matrices with distinct row-multisets; if the infimum is $0$ — equivalently, if two distinct orbits share all basic values $b_{(1,0)}$, $b_{(0,1)}$, $b_{(1,1)}$ — then the separating and bi-Lipschitz premise fails and Theorem 3.5 collapses. Alternatively, test the paper's explicit $(2,2)$ formula for $f(M)=\\max_i(m_{i1}+m_{i2})$ with the doubling loop for $\\kappa_f$ on a dense grid of matrices; a matrix for which no tested $\\kappa_f$ reproduces $f(M)$ would contradict the generation claim.","tokens_in":13431,"feed_emoji":"➕","tokens_out":21165,"duration_ms":175500,"temperature":0.7,"pith_summary":"Let $S_n$ permute the rows of an $n\\times r$ matrix, so orbits are multisets of $n$ points in $\\mathbb{R}^r$. The paper proves that the finite family of basic $r$-symmetric tropical polynomials indexed by multi-indices $c\\in\\mathbb{N}^r$ with $|c|\\le n$ generates the semifield of $S_n$-invariant tropical rational functions on $\\mathbb{R}^{n\\times r}$, using only max, plus, and subtraction. The generating expression is a finite minimum over all ways of re-assembling a multiset from its sorted columns, with a penalty term that vanishes on the correct re-assembly and, by a bi-Lipschitz inequality, dominates every wrong one. This settles a question left open in earlier work and replaces the prior group-order-dependent degree bound with generators of degree at most $n$; along the way the same construction describes the image of the basic coordinate map as the zero set of a single tropical rational function and yields a quadratic degree bound for every permutation group.","feed_headline":"Basic tropical polynomials generate all r-symmetric rational functions","feed_subtitle":"Explicit generators of degree at most n replace a bound using the first n! primes.","key_machinery":"The load-bearing identity is the sorted-column recovery $b_{i e_\\alpha}(M)-b_{(i-1)e_\\alpha}(M)$ equal to the $i$-th largest entry of column $\\alpha$, which makes the map $s(w)$ that reconstructs sorted columns from basic values tropical rational. The re-assembly maps $R_\\sigma$ join the $r$ sorted columns through permutations $\\sigma_2,\\dots,\\sigma_r$; for the true orbit some $\\sigma$ realizes $[M]$ itself. The recoupling discriminants $D_\\sigma(w)=\\max_c |w_c-b_c(R_\\sigma(s(w)))|$ are tropical rational, vanish on a correct re-assembly, and by the bi-Lipschitz inequality $\\|\\Phi(M)-\\Phi(M')\\|_2\\ge c_{n,r}\\,d([M],[M'])$ are bounded below by a positive multiple of the distance to any wrong re-assembly. That lower bound calibrates the integer multiplier $\\kappa_f$ that makes every incorrect term in the minimum dominate $f(M)$, forcing $g\\circ\\Phi=f$.","core_discovery":"The central claim, Theorem 3.5, is that the basic-coordinate map $\\Phi$ that records all values $b_c$ is a generating coordinate system for the invariant semifield: every $S_n$-invariant tropical rational function $f$ on $\\mathbb{R}^{n\\times r}$ equals $g\\circ\\Phi$ for some tropical rational $g$ on $\\mathbb{R}^K$, $K=\\binom{n+r}{r}-1$. The proof is constructive: the formula $g(w)=\\min_{\\sigma}[f(R_\\sigma(s(w)))+\\kappa_f D_\\sigma(w)]$ ranges over the $(n!)^{r-1}$ couplings $\\sigma$ of the sorted columns recovered from $w$, and each term adds a recoupling discriminant $D_\\sigma(w)$ — the maximal deviation between the basic values of $w$ and those of the re-assembled matrix — multiplied by an integer $\\kappa_f$ chosen large enough that, by the bi-Lipschitz inequality, any wrong re-assembly pays a penalty exceeding the possible decrease of $f$. The correct coupling has zero penalty and reproduces $f(M)$, so the minimum equals $f(M)$. Hence the invariant semifield is finitely generated in degree at most $n$, and the same mechanism gives a quadratic degree bound for all permutation groups that is optimal for the alternating group.","pith_inferences":["The penalty construction suggests a general recipe: any separating bi-Lipschitz invariant coordinate system that admits a tropical rational 'sorting' recovery map generates the invariant semifield; a natural next test is whether such recovery maps exist, with low degree, for linear actions beyond coordinate permutations, where the quadratic bound need not be tight.","The paper leaves the minimal separating subfamilies and the minimal size of $\\kappa_f$ open; a computational scan of the $(n,r)=(2,2)$ and $(3,2)$ cases could reveal how far $\\kappa_f$ must exceed the Lipschitz-to-bi-Lipschitz ratio in practice and whether minimal generating expressions correspond to the minimal separating subfamilies.","Corollary 3.7's description of the image as a finite union of polyhedra is a tropical analogue of the relations in the ring of multisymmetric functions; this invites a normal-form or elimination algorithm for the invariant semifield, in the spirit of classical invariant theory, which the paper does not develop.","The trace-normal-form proof that low-degree alternating invariants are fully symmetric is carried out with explicit templates; the same mechanism could yield degree lower bounds for other permutation groups generated with a normalizing transposition, and the paper's order-20 group example in Remark 4.10 suggests that low-degree blindness can occur even without such normalizers, pointing toward a g"],"forward_implications":["The semifield of $S_n$-invariant tropical rational functions on $\\mathbb{R}^{n\\times r}$ is generated by $\\binom{n+r}{r}-1$ explicit polynomials of degree at most $n$, replacing the primorial degree bound of prior work.","The image of the basic coordinate map $\\Phi$ is exactly the zero set of the tropical rational function $\\min_\\sigma D_\\sigma$, a finite union of polyhedra, giving a geometric form of the relations among generators.","For every permutation group $G\\le S_N$, the same mechanism generates the invariant semifield in degree at most $\\max\\{N,\\binom{N}{2}\\}$, independent of $|G|$; there exist $2N+1$ invariant tropical polynomials that separate orbits and $3N$ that generate, with at least $N$ necessary for each task.","The quadratic degree bound is optimal: every $A_N$-invariant tropical polynomial of degree below $\\binom{N}{2}$ is $S_N$-invariant, so any separating family for $A_N$ must contain a member of degree at least $\\binom{N}{2}$.","For subfamilies of the basic family that contain the single-column values, generating the invariant semifield and separating orbits are equivalent."],"supporting_citations":[{"why":"Introduces the basic $r$-symmetric tropical polynomials and raises the generation question answered by Theorem 3.5.","marker":"[1]"},{"why":"Supplies the separating and bi-Lipschitz properties of the basic coordinate map, the load-bearing inequality that calibrates the penalty terms.","marker":"[2]"},{"why":"Provides the tropical invariant-theory framework, the prior primorial degree bound that this paper improves, and the separating bi-Lipschitz family used for the general permutation-group bound.","marker":"[3]"},{"why":"The genericity theorem for group-invariant max filters that yields the $2N+1$ separating and $3N$ generating invariant tropical polynomials.","marker":"[4]"},{"why":"The max-min representation used to characterize tropical rational functions as continuous piecewise linear functions with finitely many integer-slope affine pieces.","marker":"[8]"},{"why":"Gives the bi-Lipschitz property of injective max filter banks, used to turn the generic separating family into a bi-Lipschitz one.","marker":"[9]"}],"fun_headline_variants":["Degree-n generators for symmetric tropical rationals","Quadratic bound for permutation invariant semifields","Basic invariants generate all r-symmetric tropical functions","Shrinking generator degree from exponential to n","Optimal quadratic degree for alternating groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem rests on the bi-Lipschitz lower bound for the basic coordinate map, taken as a black box from earlier work with a constant whose existence is asserted but not quantified; if that constant were zero, or that earlier proof had a gap, the penalty terms could no longer be guaranteed to outweigh wrong re-assemblies and the generating formula would fail.","fun_headline_variants_meta":{"raw":{"variants":["Degree-n generators for symmetric tropical rationals","Quadratic bound for permutation invariant semifields","Basic invariants generate all r-symmetric tropical functions","Shrinking generator degree from exponential to n","Optimal quadratic degree for alternating groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000283,"raw_usage":{"total_tokens":1829,"prompt_tokens":1257,"completion_tokens":572,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":873,"completion_tokens_details":{"reasoning_tokens":504}},"tokens_in":873,"tokens_out":572,"duration_ms":6199,"temperature":1.0,"reasoning_tokens":504,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:25:01.280599+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a specific small case such as $(n,r)=(2,2)$, compute the infimum of $\\|\\Phi(M)-\\Phi(M')\\|_2/d([M],[M'])$ over pairs of matrices with distinct row-multisets; if the infimum is $0$ — equivalently, if two distinct orbits share all basic values $b_{(1,0)}$, $b_{(0,1)}$, $b_{(1,1)}$ — then the separating and bi-Lipschitz premise fails and Theorem 3.5 collapses. Alternatively, test the paper's explicit $(2,2)$ formula for $f(M)=\\max_i(m_{i1}+m_{i2})$ with the doubling loop for $\\kappa_f$ on a dense grid of matrices; a matrix for which no tested $\\kappa_f$ reproduces $f(M)$ would contradict the generation claim.","supporting_citations":[{"cited_title":"Kubo,Basicr-symmetric tropical polynomials, J","cited_arxiv_id":null,"evidence_quote":"Introduces the basic $r$-symmetric tropical polynomials and raises the generation question answered by Theorem 3.5."},{"cited_title":"Stable complete coordinates for multisets of points via basic $r$-symmetric tropical polynomials","cited_arxiv_id":"2606.30184","evidence_quote":"Supplies the separating and bi-Lipschitz properties of the basic coordinate map, the load-bearing inequality that calibrates the penalty terms."},{"cited_title":"Derksen,Tropical invariants for permutation group actions, arXiv:2512.13452 (2025)","cited_arxiv_id":null,"evidence_quote":"Provides the tropical invariant-theory framework, the prior primorial degree bound that this paper improves, and the separating bi-Lipschitz family used for the general permutation-group bound."},{"cited_title":"Cahill, J","cited_arxiv_id":null,"evidence_quote":"The genericity theorem for group-invariant max filters that yields the $2N+1$ separating and $3N$ generating invariant tropical polynomials."},{"cited_title":"Ovchinnikov,Max-min representation of piecewise linear functions, Beitr¨ age Algebra Geom","cited_arxiv_id":null,"evidence_quote":"The max-min representation used to characterize tropical rational functions as continuous piecewise linear functions with finitely many integer-slope affine pieces."}],"review_version":1}