{"id":"04cbfda8-187d-4d61-b5d4-53cad2cbfee3","arxiv_id":"2608.11109","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The type C Richardson boundary of P^{2n-1} and its Rietsch mirror superpotential yield a Newton-Okounkov degeneration whose value semigroup equals the lattice points of the polar dual of the mirror's Newton polytope.","lead":"For every odd-dimensional projective space, the paper computes the type C boundary divisor and the mirror Laurent polynomial coming from the symplectic group presentation, and shows that both lead to a toric degeneration via Newton-Okounkov bodies. It also degenerates the symplectic boundary into the standard toric boundary while keeping the ambient space fixed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest-assumption pick (admissibility of the flag) is the most compressed point, but after checking the local equations it holds: at p∞ each F_m contributes a linear term ±p0 and the imposed coordinate hyperplanes contribute distinct coordinate differentials, so every stratum is smooth there. The irreducibility argument via cones over smooth quadrics is also valid. The other potentially delicate point, the converse direction of Proposition 4.6, is not explicitly flagged by the reader; I examined it and found no gap, since monomial lifts in the final four variables avoid divisibility by F_{n-1} and by the intermediate hyperplanes. The Lusztig superpotential computation is tedious but passes the n=2 consistency check and the algebraic identities in Lemma 3.10. I therefore see no change to the ACCEPT verdict, despite the compressed exposition in Section 4.4.","tokens_in":99,"tokens_out":56155,"duration_ms":502636,"concrete_test":"Verify Lemma 4.4 computationally for n=3 and n=4: in the affine chart p_N=1, write the explicit defining equations of every flag member Y_i near p∞, compute the Jacobian matrix at p∞, and check that its rank equals the codimension of Y_i. If all ranks are full, the compressed local-parameter argument is confirmed; if any rank is deficient, admissibility fails and Theorem 1.2 would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central claim. Theorem 1.2 depends on two compressed steps: Lemma 4.4 (admissibility of the flag) and Proposition 4.6 (exact value-semigroup equality). Both appear sound on close reading. For Lemma 4.4, at p∞ the quadratic F_m has linear part ±p0, while the coordinate hyperplanes p_m, p_{N-m} have independent linear parts; the strata are smooth quadrics or cones over smooth quadrics, with p∞ away from the cone vertices. For Proposition 4.6, the converse construction works because a lift eσ can be chosen as a monomial in p0, p1, p_{N-1}, p_N; such a lift is not divisible by F_{n-1} for n>2, and the n=2 case is handled by the nonvanishing of its restriction to Q1. The intermediate orders are exact because the surviving factors p_{N-m} and lower-index monomials are nonzero at the relevant generic points. The Lusztig superpotential computation is lengthy but internally consistent, and the n=2 toy example matches the mutation picture. I therefore cannot identify a concrete failure mode that would invalidate Theorem 1.2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper compares two mirror-theoretic degeneration pictures attached to odd-dimensional projective space P^{2n-1}. The first is the classical toric mirror; the second is the Rietsch mirror of Sp_{2n}/P_1, whose dual side is compactified by the odd quadric SO_{2n+1}/P_1^∨. The author computes the type C_n Richardson boundary D_C on P^{2n-1} (Theorem 1.1) and the Lusztig-torus Laurent superpotential W_Lus^{(2n-1)} (Theorem 3.11). The central result (Theorem 1.2) constructs an admissible flag Y• whose flag valuation satisfies σ({ν_flag(s)−k1}) = kP_n^∨ ∩ Z^{2n-1} for every k, where P_n^∨ is the polar dual of the Newton polytope of W_Lus. Consequently the degree-augmented value semigroup is identified with the lattice-point semigroup of the cone over P_n^∨, giving a flat toric degeneration P^{2n-1} ⇝ Proj C[S_n] whose fan is the normal fan of P_n^∨. A rank-one weight degeneration connects (P^{2n-1}, D_C) to the standard toric boundary (Theorem 4.7 and Section 4.5).","tokens_in":27034,"tokens_out":36374,"duration_ms":286688,"significance":"The paper gives a fully explicit, parameter-free comparison of two degeneration pictures attached to the same Fano variety: the toric boundary mirror and the Lie-theoretic Rietsch mirror. The type C boundary and the dual-side superpotential are computed independently from representation-theoretic data, and the equality of the flag-valuation polytope with the polar dual of the Newton polytope is derived rather than fitted. The matrix computations in Section 3 are detailed and checkable, and the n=2 toy example with the Laurent mutation is a useful sanity check. If the construction extends to other homogeneous spaces it would be a substantial contribution; already for projective space it is a clean illustration of the Gross–Siebert/Rietsch mirror philosophy.","major_comments":[],"minor_comments":[{"comment":"The admissibility proof is compressed, especially the claim that the defining equations form a regular system of parameters at p∞. Please expand the check: state explicitly that at p∞ the linear part of F_m is (−1)^m p0, that the previously imposed coordinate hyperplanes are distinct from p0 and from one another, and that therefore the displayed equations have independent linear parts at every stratum.","section":"Section 4.4, Lemma 4.4"},{"comment":"In the converse direction, the claim that the chosen lift eσ introduces no additional vanishing along the intermediate branches is load-bearing but only asserted. Please add the missing argument: a nonzero polynomial in p0, p1, p_{N−1}, p_N cannot be divisible by F_m for m ≥ 2 because F_m contains a monomial involving p_m or p_{N−m}; together with the fact that eσ restricts to the chosen nonzero section on Q1, this shows it does not vanish on any intermediate stratum Z_m.","section":"Section 4.4, Proposition 4.6"},{"comment":"The flatness argument for H_{N,t} states that H_{N,t} has a coefficient equal to 1 and hence no nonzero polynomial in t divides it. Please cite or briefly sketch the primitive-polynomial criterion for flatness over C[t], so that the conclusion that the quotient is C[t]-flat is fully transparent.","section":"Section 4.5"},{"comment":"The statement that the fan of the special fibre is the face fan of Newt(W_n) relies on 0 being in the interior of Newt(W_n). This follows from the presence of the positive coordinate vertices and the negative monomials, but it is not stated; please add a sentence making this explicit.","section":"Section 4.8, Corollary 4.8"}],"recommendation":"minor_revision","confidential_remarks":"The central computations appear sound. The only external dependency is the unpublished manuscript of Tillmann-Morris for the n=2 case; the general derivation in Sections 3.2 and 4 is independent, so I do not regard this as a blocker. I would be happy to see the paper accepted after the requested clarifications."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious explicit computation. The genuinely new part is not the Lusztig Laurent polynomial—the paper credits [22] for that—but the type C flag valuation on P^{2n-1} whose degree-augmented value semigroup is exactly the lattice points of kP_n^vee, and the two consequences: a flat toric degeneration via Anderson, and a rank-one boundary degeneration from D_C to D_tor. I went through the main line and it holds.\n\nWhat the paper does well: the boundary computation (Thm 3.3) is a genuine piece of matrix work; the Gauss-factorization criterion and Jacobi minor computation are detailed and correct. Lemma 3.10, the triangular-factorization calculation giving the superpotential, is heavy but internally consistent. The proof of Prop 4.6 is the heart. The converse direction is a real construction: for each lattice point it builds an explicit section and checks the successive vanishing orders. I did not find a gap. Cor 4.8 correctly identifies the polar dual. The n=2 toy example is presented as motivation and matches the mutation picture.\n\nWhere it is soft: Lemma 4.4, the admissibility of the flag, is compressed—it says the defining equations form a regular system of parameters without writing out each independence check. I checked the local linear parts; the claim is correct. The divisibility argument in Prop 4.6 is also terse at the point where e_sigma is shown not to be divisible by F_{n-1}; the n>2 argument is fine, and the n=2 case is disposed of by nonvanishing of the restriction to Q_1. Both are places a referee should ask for one extra sentence, not places where the proof collapses. The dependence on [22] for the Laurent polynomial formula is a provenance issue only: the formula is re-proved here, so the unpublished status of [22] does not create a correctness hole. Minor: the cluster question is left open, but it is honestly labeled a question.\n\nThe paper is for people working on Rietsch mirrors, Newton-Okounkov degenerations, and toric mirror symmetry for homogeneous spaces. It is an extension of Rietsch-Williams and Wang to type C projective space, not a paradigm shift. It deserves a serious referee. Send it out, with the request that the referee check Lemma 4.4 and the converse of Prop 4.6 carefully; I expect minor revisions on exposition rather than substantive corrections.","headline":"Serious explicit computation in the Rietsch-mirror framework: the flag-valuation semigroup identification is new and holds up, the Lusztig superpotential is honestly credited to [22], and the paper deserves a real referee.","tokens_in":27579,"tokens_out":3429,"would_cite":true,"duration_ms":31131,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","14M25","14J33"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that an admissible flag on $P^{2n-1}$, built from the type $C_n$ boundary, yields a flag valuation whose degree-augmented value semigroup coincides with the lattice points of the cone over the polar dual of the Newton…","keywords":["odd-dimensional projective space","Rietsch mirror","Newton-Okounkov bodies","toric degenerations","Lusztig tori","Landau-Ginzburg models","flag valuations","type C homogeneous space"],"falsifier":"Check whether the local equations defining each flag stratum at $p_\\infty$ have independent linear parts; if for some $m$ the linear part of $F_m$ is a combination of the linear parts already imposed, smoothness fails and the value semigroup could contain extra points. Alternatively, search for a section $s\\in H^0(P^N,\\mathcal{O}((N+1)k))$ whose normalized flag value lies outside $kP_n^\\vee\\cap \\mathbb{Z}^{2n-1}$, or for a lattice point of $kP_n^\\vee$ that no section attains.","tokens_in":26629,"feed_emoji":"📐","tokens_out":9424,"duration_ms":70305,"temperature":0.7,"pith_summary":"The paper aims to establish that the two mirror descriptions of odd-dimensional projective space—the classical toric mirror and Rietsch's Lie-theoretic mirror for $P^{2n-1}=Sp_{2n}/P_1$—are two faces of one explicit Newton–Okounkov degeneration. It constructs an admissible flag whose flag valuation records orders of vanishing along quadrics and coordinate hyperplanes, and proves that the degree-augmented value semigroup equals the lattice-point semigroup of the cone over the polar dual of the Newton polytope of the dual-side Laurent polynomial. This yields a flat degeneration of $P^{2n-1}$ to a toric variety whose fan is the normal fan of that polar dual. Along the way the paper computes the type $C_n$ boundary divisor and the Lusztig-torus superpotential explicitly, and exhibits a rank-one degeneration of the type $C_n$ boundary to the standard toric boundary while the ambient projective space stays fixed. A sympathetic reader would care because the representation-theoretic mirror is converted into concrete polytopal data that also links the two boundary structures on the same Fano variety.","feed_headline":"A mirror flag degenerates odd projective space to a toric variety","feed_subtitle":"Rietsch's mirror and the toric mirror yield the same polytope, and a flat degeneration follows.","key_machinery":"The carrying object is the admissible flag $Y_\\bullet$ built from the quadrics $Q_r=\\{F_r=0\\}$, where $F_r=\\sum_{j=0}^r(-1)^j p_{r-j}p_{N-r+j}$, with the final surface $Q_1$ identified with $\\mathbb{P}^1\\times\\mathbb{P}^1$ and a toric flag on it. The flag valuation $\\nu_{\\mathrm{flag}}$ records successive orders of vanishing along these strata; subtracting $k\\mathbf{1}$ and applying the lattice permutation $\\sigma$ re-labels the values as exponent coordinates of the dual Laurent polynomial. Anderson's Newton–Okounkov theorem converts the finitely generated value semigroup into a flat toric degeneration, and the polar-dual relation together with the inequalities defining $P_n^\\vee$ encode exactly which lattice points occur in each degree.","core_discovery":"The central claim is Theorem 1.2: there is an admissible flag $Y_\\bullet$ on $P^N$, $N=2n-1$, with flag valuation $\\nu_{\\mathrm{flag}}$ such that for every $k\\ge 0$, $$\\$\\sigma$\\left(\\{\\nu_{\\mathrm{flag}}(s)-k\\mathbf{1}: 0\\ne s\\in $H^{0}$(P^N,\\mathcal{O}((N+1)k))\\}\\right)=kP_n^\\vee\\cap \\mathbb{Z}^{2n-1},$$ where $H_N=p_0p_N\\prod_{r=1}^{n-1}F_r$ satisfies $\\nu_{\\mathrm{flag}}(H_N)=\\mathbf{1}$ and $P_n^\\vee$ is the polar dual of the Newton polytope of the Lusztig-torus superpotential (with $q=1$). Corollary 1.3 then identifies the degree-augmented value semigroup, under the unimodular map $\\Sigma$, with $S_n=\\{(k,m):k\\ge 0,\\ m\\in kP_n^\\vee\\cap \\mathbb{Z}^{2n-1}\\}$, giving a flat degeneration $P^{2n-1}\\rightsquigarrow \\mathrm{Proj}\\,\\mathbb{C}[S_n]$; the special fibre is the toric variety whose fan is the normal fan of $P_n^\\vee$, equivalently the face fan of $\\mathrm{Newt}(W_{\\mathrm{Lus}})$. The supporting explicit formulae are the type $C_n$ boundary $D_C=\\mathrm{div}(p_0p_N F_1\\cdots F_{n-1})$ and the superpotential $W_{\\mathrm{Lus}}=a_1+\\cdots+a_{n-1}+b+c_{n-1}+\\cdots+c_1+q(a_1+c_1)/(a_1\\cdots a_{n-1}b^2c_{n-1}\\cdots c_1)$ on the Lusztig torus.","pith_inferences":["Inference: The flag valuation's Newton–Okounkov body should be exactly $P_n^\\vee$; if so, its Euclidean volume must equal the leading coefficient of the Hilbert polynomial of the anticanonical ring, giving a concrete numerical check of the construction.","Inference: By analogy with the Grassmannian case, the flag valuation or its value semigroup likely admits a cluster-theoretic description; the paper raises this as a question but does not prove it.","Inference: The rank-one degeneration connecting $D_C$ to $D_{\\mathrm{tor}}$ suggests there should be a compatible degeneration of the mirror Laurent polynomials from $W_{\\mathrm{Lus}}$ to $W_{\\mathrm{tor}}$, possibly through mutations, which the paper does not construct."],"forward_implications":["The Rees algebra of the flag-valuation filtration is $\\mathbb{C}[S_n]$, so $P^{2n-1}$ admits a flat degeneration to the toric variety whose fan is the normal fan of $P_n^\\vee$.","For every $k$, the normalized value set of sections of degree $(N+1)k$ is exactly the lattice points of $kP_n^\\vee$, so the Newton–Okounkov body of the flag is $P_n^\\vee$ itself.","The degree-augmented value semigroup is finitely generated by Gordan's lemma, so the degeneration is a genuine flat Rees degeneration rather than a formal construction.","The toric special fibre is also described by the face fan of $\\mathrm{Newt}(W_{\\mathrm{Lus}})$, directly linking the Lie-theoretic Laurent polynomial to the degeneration combinatorics.","The rank-one weight degeneration gives a flat family $(P^{2n-1},D_C)\\rightsquigarrow(P^{2n-1},D_{\\mathrm{tor}})$ with the ambient projective space fixed."],"supporting_citations":[{"why":"Supplies Anderson's Newton–Okounkov toric degeneration theorem, which turns the finitely generated value semigroup into a flat degeneration to a toric variety.","marker":"[2]"},{"why":"Supplies Rietsch's Lie-theoretic mirror construction and the definition of the superpotential on the open Richardson variety.","marker":"[20]"},{"why":"Supplies the Lusztig torus parametrizations from reduced words, used for the explicit coordinates on the dual side.","marker":"[15, 16]"},{"why":"Supplies the unpublished $P^3$ mini-project calculation used as a toy model and the odd-dimensional formula that appears in Theorem 1.1.","marker":"[22]"},{"why":"Provides the foundational theory of Newton–Okounkov bodies used together with Anderson's degeneration theorem.","marker":"[11, 14]"},{"why":"Sets the polar-dual convention that identifies the degree-one valuation polytope with the polar dual of the Newton polytope of the superpotential.","marker":"[4]"}],"fun_headline_variants":["Type C mirror yields toric degeneration of odd projective space","Two mirror models flatten odd projective space to one toric variety","Newton-Okounkov bodies align mirror boundaries on odd projective space","Odd projective space: two mirror pictures, one flat degeneration"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the flag $Y_\\bullet$ constructed in Section 4.4 is admissible: every flag member is irreducible and smooth at the point $p_\\infty$, so the flag valuation has one-dimensional leaves and Anderson's theorem applies. Lemma 4.4 asserts this through a local-equation argument, but the independence check for every stratum is compressed.","fun_headline_variants_meta":{"raw":{"variants":["Type C mirror yields toric degeneration of odd projective space","Two mirror models flatten odd projective space to one toric variety","Newton-Okounkov bodies align mirror boundaries on odd projective space","Odd projective space: two mirror pictures, one flat degeneration"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00125,"raw_usage":{"total_tokens":5221,"prompt_tokens":1138,"completion_tokens":4083,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":754,"completion_tokens_details":{"reasoning_tokens":4013}},"tokens_in":754,"tokens_out":4083,"duration_ms":26120,"temperature":1.0,"reasoning_tokens":4013,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:04:26.524763+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the local equations defining each flag stratum at $p_\\infty$ have independent linear parts; if for some $m$ the linear part of $F_m$ is a combination of the linear parts already imposed, smoothness fails and the value semigroup could contain extra points. Alternatively, search for a section $s\\in H^0(P^N,\\mathcal{O}((N+1)k))$ whose normalized flag value lies outside $kP_n^\\vee\\cap \\mathbb{Z}^{2n-1}$, or for a lattice point of $kP_n^\\vee$ that no section attains.","supporting_citations":[{"cited_title":"Anderson,Okounkov bodies and toric degenerations, Math","cited_arxiv_id":null,"evidence_quote":"Supplies Anderson's Newton–Okounkov toric degeneration theorem, which turns the finitely generated value semigroup into a flat degeneration to a toric variety."},{"cited_title":"Rietsch,A mirror symmetric construction ofqH ∗ T (G/P) (q), Adv","cited_arxiv_id":null,"evidence_quote":"Supplies Rietsch's Lie-theoretic mirror construction and the definition of the superpotential on the open Richardson variety."},{"cited_title":"Tillmann-Morris,An alternative Landau–Ginzburg mirror toCP 3, 2020, unpublished manuscript","cited_arxiv_id":null,"evidence_quote":"Supplies the unpublished $P^3$ mini-project calculation used as a toy model and the odd-dimensional formula that appears in Theorem 1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Sets the polar-dual convention that identifies the degree-one valuation polytope with the polar dual of the Newton polytope of the superpotential."}],"review_version":1}