{"id":"41c0073b-1890-4479-84bf-7df597ce6da2","arxiv_id":"2412.14091","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the positive orthogonal Grassmannian, the authors describe boundaries in the simplest case, prove a duality between two special families, and show the standard cell decomposition fails when n>2k+1.","lead":"This paper maps out the positive orthogonal Grassmannian, a geometric object tied to particle-scattering formulas, for all dimensions k,n rather than only the previously studied n=2k case. It gives the boundary structure for small n, a duality between two special cases, and a counterexample showing the usual cell decomposition fails once n is large.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Section 5 obstruction for OGr+(2,6) is not valid as stated: the displayed edge e1 has a negative Plücker coordinate, and the sign-corrected edge is not in the closure of the square cell Cτ. The central claim is therefore not established.","rationale":"The reader's CONDITIONAL verdict targeted the sketched block-matrix extension, but the more serious issue is in the base case: the explicit (2,6) example contains a false statement about the edge e1 and the square cell Cτ. The displayed e1 has a negative Plücker coordinate, so it is not a point of the positive orthogonal Grassmannian; the sign-corrected boundary edge of Cσ has a support disjoint from that of Cτ, so it cannot be a diagonal of Cτ. Since this example is the sole evidence for the main negative theorem and the template for the claimed n>2k+1 generalization, the central claim is currently unsupported. The paper's other contributions may remain valid, but the headline claim should not be accepted without a corrected counterexample.","tokens_in":18366,"tokens_out":29058,"duration_ms":256100,"concrete_test":"Recompute by hand the 15 Plücker coordinates of e1 and of Mτ from Section 5. For e1, verify p13=1 and p35=−b; for the corrected y=0 boundary, verify p15=p16=p25=p26=0 and p35=p36=p45=p46=t. For Mτ, verify p35=p36=p45=p46=0 for all a,b,c. If these identities hold, the stated diagonal gluing is impossible. A stronger check is to search the paper's 99 orthopositroid cells for any pair with a triangle edge contained in the interior of a square cell, using the Plücker coordinates; if no such pair exists, the main theorem is not proved.","verdict_should_be":"REJECT","load_bearing_attack":"In Section 5, the proof of the main negative theorem rests on the claim that the 2-dimensional cell Cσ (a triangle) has an edge 'e1' that is one of the diagonals of the 2-dimensional square cell Cτ. This is the only explicit obstruction to a CW decomposition and the template for the n>2k+1 generalization. The displayed edge e1 = [[1,1,0,0,b,b],[0,0,1,1,0,0]] with b≥0 has Plücker coordinate p35 = det(columns 3 and 5) = 0·0 − b·1 = −b < 0, while p13 = 1. The Plücker vector therefore has mixed signs, so e1 is not in Gr+(2,6), hence not in OGr+(2,6); it cannot be an edge of the closure of a positive cell. If the intended edge is the actual y=0 boundary of Cσ, namely [[1,1,0,0,−t,−t],[0,0,1,1,0,0]] with t>0, then its support has p15=p16=p25=p26=0 and p35=p36=p45=p46=t. But every point of Cτ, as represented by Mτ, has p35=p36=p45=p46=0 identically; no point with t>0 lies in the closure of Cτ. Thus neither version of e1 can be 'one of the diagonals' of Cτ. The claimed triangle-square gluing is unsupported, and the argument does not establish failure of a CW decomposition even for k=2,n=6.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the positive orthogonal Grassmannian OGr_+(k,n) for the alternating quadratic form ω0. The authors give a Gröbner basis and degree formula for the orthogonal Grassmannian OGr(k,n), describe OGr_+(1,n) as a product of simplices and as a positive geometry, prove an isomorphism between OGr_+(k,2k+1) and OGr_+(k+1,2k+2) with a matching description, and claim that for n>2k+1 and k>1 the positroid cells of Gr_+(k,n) do not induce a CW cell decomposition of OGr_+(k,n). The negative claim is supported by an example in OGr_+(2,6) involving a triangular cell and a square cell whose edge is claimed to glue to a diagonal.","tokens_in":18701,"tokens_out":19729,"duration_ms":171045,"significance":"If fully correct, the paper would make several useful contributions: a concrete positive-geometry structure for OGr_+(1,n), a bridge between OGr_+(k,2k+1) and OGr_+(k+1,2k+2), and a motivation for the new notion of orthopositroid. The exhaustive enumeration of 99 realizable orthopositroids in Table 1 and the representation-theoretic degree computation are valuable. However, the central negative theorem is not established as written: the only explicit obstruction in Section 5 is invalid, and the generalization to all k,n is only asserted. The remaining results may survive revision, but the main novelty needs a corrected argument or a replacement counterexample.","major_comments":[{"comment":"The edge e1 is not a boundary edge of the closure of Cσ in the positive orthogonal Grassmannian. For b>0, the Plücker coordinate p35 = det[[0,b],[1,0]] = -b is negative while p13 = 1, so the row span is not in Gr_+(2,6); for b=0, e1 is just the single vertex also lying on e3. If one instead takes the actual y=0 boundary of Cσ, represented by [[1,1,0,0,-x,-x],[0,0,1,1,0,0]] with x>0, then p35=p36=p45=p46=x>0, while every point of Cτ has p35=p36=p45=p46=0 identically, so this boundary is not in the closure of Cτ. The claimed gluing of a triangle edge to a diagonal of the square is therefore not established, and the only explicit obstruction to a CW decomposition in the paper disappears.","section":"§5, displayed matrices after Mσ and Mτ"},{"comment":"The extension from OGr_+(2,6) to arbitrary (k,n) with n>2k+1 is asserted without proof. The displayed block extension is not shown to satisfy the orthogonality equations (5), nor is it shown that the embedded copies of Cσ and Cτ remain positive and preserve the claimed incidence. Even if the k=2,n=6 obstruction were valid, this paragraph would not establish the theorem for all k>1 and n>2k+1.","section":"§5, final paragraph starting 'In general this problem arises...'"},{"comment":"The positivity statement is justified by 'It is not so difficult to see' without a verification. Since the claim depends on the sign conventions in equation (5) and on which connected component is selected in OGr_+(k+1,2k+2), please provide an explicit check that the linear isomorphism sends the positive locus to the positive locus, or give a precise reference that contains this statement.","section":"§4, proof of Theorem 4.5"},{"comment":"The passage from a Gröbner basis of I_{k,n,0} to one of I_{k,n} under the substitution φ is not justified. The map φ identifies q_J with ±p_{[n]\\J}, so it is not injective on monomials: q_J and p_{[n]\\J} have the same image, and cancellations can occur. The claim that an inequality between monomials 'continues to hold' after applying φ requires proof; please supply a direct argument or a standard reference showing that the images of the initial monomials generate in(I_{k,n}).","section":"§2, proof of Theorem 2.6"}],"minor_comments":[{"comment":"The text refers to '(8) and (8)' where two different displayed equations are evidently intended; please renumber them.","section":"§2, equation numbering"},{"comment":"The block matrix used for the claimed generalization is difficult to parse; please rewrite it with clearly labeled blocks and explicitly state which entries are zero, which are free parameters, and how the blocks are chosen to satisfy the isotropy equations.","section":"§5, block matrix display"},{"comment":"The proof of the positive-geometry property is abbreviated; in particular, the verification on all boundary strata and the treatment of the u_n=0 boundary are only sketched. Please expand the residue computations or indicate which steps are standard.","section":"§3, proof of Theorem 3.3"},{"comment":"The enumeration of the 99 orthopositroids is presented as the output of 'an exhaustive computation'; please specify the computational method used and state whether code or a verification script is available.","section":"Table 1"},{"comment":"Reference [6] lacks journal or preprint data; please complete the bibliographic information. Also, 'A exhaustive computation' on page 20 should be 'An exhaustive computation'.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is Section 5: the explicit counterexample is incorrect, and the general claim is unsupported. I do not recommend rejection outright because the other sections contain substantial material and the negative result might be repairable with a valid counterexample. However, if a correct obstruction cannot be supplied, the authors should remove or sharply weaken the claim that positroid cells fail to induce a CW decomposition, since that claim is central to the paper's stated novelty."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper has two solid sections and one load-bearing section that does not work as written. The stress-test note is correct. The displayed edge e1 in Section 5 has Plücker coordinate p35 = -b, so for b > 0 it is not in Gr+(2,6); if you fix the sign as the text seems to intend, that edge lies outside the closure of Cτ and cannot be one of its diagonals. Either way, the claimed triangle-square gluing is not established. Since this example is the only explicit obstruction to a CW decomposition and the template for the n > 2k+1 generalization, the paper's headline negative result is unsupported as it stands.\n\nWhat is actually new and good: Section 3 gives a clean face decomposition of OGr+(1,n) as a product of simplices and provides a canonical form, and that part looks correct and useful. Section 4's isomorphism between OGr+(k,2k+1) and OGr+(k+1,2k+2) is a real result; the matchings-to-permutations dictionary is a nice contribution. The Gröbner basis material in Section 2 is mostly a repackaging of the prior spinor-helicity work [8], but the presentation is coherent and the connection to standard monomial theory is worth having.\n\nSoft spots, in proportion: aside from the Section 5 failure, the proof of Theorem 4.5 handwaves at the crucial positivity step with \"It is not so difficult to see.\" That is a minor exposition gap—the claim is plausibly true and easily checked—but it should be written out. More substantively, the enumeration of 99 orthopositroids in OGr+(2,6) is presented as an \"exhaustive computation\" with no code, data, or algorithmic description, so it is not independently reproducible. The block-matrix extension to all k,n in Section 5 also does not prove that isotropy with respect to ω0 and nonnegativity of all Plücker coordinates are preserved; it is a sketch. None of these would be fatal if Section 5's central example were valid, but it is not.\n\nWho gets value: readers working on the positive orthogonal Grassmannian, positive geometries, or the combinatorics of Gr+(k,n). Sections 3 and 4 deserve a serious referee. The negative result does not, as written.\n\nRecommendation: send it to peer review, but the referee should demand a repaired Section 5—either a corrected obstruction that actually lies in the positive region, or a downgrade of the CW claim to a conjecture. If the example cannot be fixed, the paper's frame changes substantially.","headline":"Sections 3-4 are genuinely useful, but the Section 5 obstruction for OGr+(2,6) is not valid as stated, so the advertised negative result is unproven.","tokens_in":19269,"tokens_out":2365,"would_cite":false,"duration_ms":22844,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","15B48","05E14"],"pacs":[],"model":"deepseek-v4-flash","headline":"Positroid cells of Gr_+(k,n) do not induce a CW cell decomposition of OGr_+(k,n) for n>2k+1; a triangle cell glues to the diagonal of a square cell.","keywords":["Orthogonal Grassmannian","Positive geometry","Positroid cells","CW cell decomposition","Matchings","Gröbner basis","Straightening laws","Plücker coordinates"],"falsifier":"Compute $P\\Omega P^T$ for the $k\\times n$ block matrix displayed in Section 5 with a generic positive $(k-2)\\times(n-6)$ block: if the isotropy equations (5) force some Plücker coordinate to change sign or force the block to vanish, then the extension to all $n>2k+1$ and $k>1$ fails, leaving only the $(2,6)$ obstruction. A positive result would confirm the intended embedding and support the non-CW claim in the general regime.","tokens_in":18155,"feed_emoji":"📐","tokens_out":10670,"duration_ms":81217,"temperature":0.7,"pith_summary":"The paper aims to map out the positive orthogonal Grassmannian $\\mathrm{OGr}_+(k,n)$---the isotropic $k$-planes in $\\mathbb{C}^n$ with all Plücker coordinates nonnegative---for general $k,n$, beyond the well-studied $n=2k$ case. It proves that for $n=2k+1$ the space is linearly isomorphic to $\\mathrm{OGr}_+(k+1,2k+2)$, so its boundary combinatorics are governed by matchings on $[2k+2]$. It also proves $\\mathrm{OGr}_+(1,n)$ is a positive geometry combinatorially equivalent to a product of two simplices. The paper's main negative claim is that for $n>2k+1$ and $k>1$, the positroid cells of $\\mathrm{Gr}_+(k,n)$ do not induce a CW cell decomposition of $\\mathrm{OGr}_+(k,n)$, because in $\\mathrm{OGr}_+(2,6)$ a triangular cell's edge is glued to the diagonal of a square cell. If this claim holds, the standard positroid combinatorics must be replaced by a new framework, and the paper introduces orthopositroids as the candidate language.","feed_headline":"Positroid cells fail to CW-decompose OGr_+(k,n) for n>2k+1","feed_subtitle":"A triangular cell glues to a square diagonal, so the usual positroid boundaries cannot work.","key_machinery":"The mechanism behind the positive results and the obstruction is the sign-alternating quadratic form $\\omega_0(x,y)=x_1y_1-x_2y_2+\\cdots+(-1)^{n-1}x_ny_n$ and the equations it imposes on Plücker coordinates, $P\\Omega P^T=0$ in the notation of Remark 2.3. These equations force a compatibility condition on positroids: a positroid $M$ can meet $\\mathrm{OGr}_+(k,n)$ only if, for every pair of $(k-1)$-element index sets $I,J$, the sets $A^+_{IJ}(M)$ and $A^-_{IJ}(M)$ are either both nonempty or both empty; such $M$ are called orthopositroids. The paper's counterexample is carried by two explicit matrices in $\\mathrm{OGr}_+(2,6)$, one giving a triangular cell and one giving a square cell whose closure is isomorphic to $\\mathrm{OGr}_+(1,4)$, with the triangle edge glued to the square diagonal. A displayed $(k-2)\\times(n-6)$ block extension is then used to transport this glued configuration to all $n>2k+1$ and $k>1$.","core_discovery":"The central discovery is that the positroid stratification of the positive Grassmannian does not descend to a CW decomposition of the positive orthogonal Grassmannian in the regime $n>2k+1$, $k>1$. The obstruction is explicit: in $\\mathrm{OGr}_+(2,6)$, two two-dimensional cells, one triangular and one square, have closures whose intersection is a triangle edge that is also a diagonal of the square, so the cell poset cannot be a regular CW complex. Alongside this, the paper establishes positive structural results: $\\mathrm{OGr}_+(1,n)$ is combinatorially the product of simplices $\\Delta^{\\lceil n/2\\rceil-1} \\times \\Delta^{\\lfloor n/2\\rfloor-1}$ and admits the canonical form given in Theorem 3.3, and $\\mathrm{OGr}_+(k,2k+1)$ is linearly isomorphic to $\\mathrm{OGr}_+(k+1,2k+2)$, with the faces of the former indexed by matchings on $[2k+2]$. The paper further defines orthopositroids---positroids for which the two sign sets $A^+_{IJ}$ and $A^-_{IJ}$ are simultaneously empty or nonempty---shows that every point of $\\mathrm{OGr}_+(k,n)$ lies in an orthopositroid cell, and conjectures that every orthopositroid is realized by such a point.","pith_inferences":["A testable extension is to compute the Euler characteristic of the cell complex assembled from the 99 orthopositroid cells of $\\mathrm{OGr}_+(2,6)$ and compare it with the topological Euler characteristic of the variety; equality would support Conjecture 5.3, while failure would show that realizability alone does not guarantee a CW structure.","The triangle-square gluing suggests that the correct boundary complex for $n>2k+1$ may require cells that are not convex polytopes, or may require subdividing positroid cells, since a single positroid cell closure already exhibits non-regular incidence.","The same obstruction likely appears in the Plücker-positive flag variety $F_+(k,n)$, because $\\mathrm{OGr}_+(k,n)$ is cut out from it by the diagonal condition; if so, a new cell decomposition for $F_+(k,n)$ in the $n>2k+1$ range would be a prerequisite for $\\mathrm{OGr}_+(k,n)$."],"forward_implications":["For $n=2k+1$, the linear isomorphism between $\\mathrm{OGr}_+(k,2k+1)$ and $\\mathrm{OGr}_+(k+1,2k+2)$ reduces boundary computations to the known $n=2k$ case, where matchings on $[2k+2]$ index the cells.","For $k=1$, the boundary structure is a product of two simplices, and the canonical form from Theorem 3.3 gives an explicit logarithmic-form description of $\\mathrm{OGr}_+(1,n)$ as a positive geometry.","For $n>2k+1$ and $k>1$, any future CW decomposition of $\\mathrm{OGr}_+(k,n)$ cannot use the positroid cells as its cells; the paper's orthopositroid condition is the necessary first constraint on which positroid cells survive.","The Gröbner basis and primeness results imply that, for $n>2k$, the homogeneous coordinate ring of $\\mathrm{OGr}(k,n)$ is described by the straightening-law quadrics and has the explicit degree from Proposition 2.7.","If Conjecture 5.3 holds, the orthopositroids of type $(k,n)$ give exactly the cells that meet $\\mathrm{OGr}_+(k,n)$, providing the raw material for the missing cell decomposition."],"supporting_citations":[{"why":"introduced the positive orthogonal Grassmannian $\\mathrm{OGr}_+(k,2k)$ in connection with scattering amplitudes, motivating the study of general $(k,n)$.","marker":"[13]"},{"why":"gives the face structure and cell parametrization of $\\mathrm{OGr}_+(k,2k)$, used here for the isomorphism between $\\mathrm{OGr}_+(2,5)$ and $\\mathrm{OGr}_+(3,6)$ and for matchings on $[2k+2]$.","marker":"[10]"},{"why":"supplies the Gröbner basis and the $P\\Omega P^T=0$ equations for spinor-helicity varieties, which the paper adapts to prove the ideal and degree statements for $\\mathrm{OGr}(k,n)$.","marker":"[8]"},{"why":"defines positive geometries and canonical forms, the framework in which $\\mathrm{OGr}_+(1,n)$ is shown to be a positive geometry.","marker":"[2]"},{"why":"introduces positroid cells and their combinatorial indexing, the objects whose failure to CW-decompose $\\mathrm{OGr}_+(k,n)$ is the paper's main negative result.","marker":"[16]"},{"why":"provides the classical fact that $\\mathrm{OGr}(k,n)$ is empty for $n<2k$ and has two components for $n=2k$, used in Proposition 2.4.","marker":"[12]"},{"why":"the matroid stratification of the Grassmannian from which positroid cells derive, cited when defining the decomposition of $\\mathrm{Gr}_+(k,n)$.","marker":"[11]"}],"fun_headline_variants":["Positroid cells trip on a triangle-square conflict in OGr_+(k,n)","OGr_+(k,n): positroid cells stuck on a diagonal for n>2k+1","Triangular cell glues to square diagonal: OGr_+ resists CW decomposition","n>2k+1: positroid cells fail to CW-decompose OGr_+"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The general claim for all $k>1$ and $n>2k+1$ rests on the assertion that the displayed block-matrix extension embeds the $(2,6)$ triangle-square configuration into $\\mathrm{OGr}_+(k,n)$ while preserving isotropy with respect to $\\omega_0$, nonnegativity of all Plücker coordinates, and the offending gluing; the paper states this extension but does not prove that the appended $(k-2)\\times(n-6)$ block satisfies the defining equations (5), so if that embedding fails, only the $k=2$, $n=6$ failure is established.","fun_headline_variants_meta":{"raw":{"variants":["Positroid cells trip on a triangle-square conflict in OGr_+(k,n)","OGr_+(k,n): positroid cells stuck on a diagonal for n>2k+1","Triangular cell glues to square diagonal: OGr_+ resists CW decomposition","n>2k+1: positroid cells fail to CW-decompose OGr_+"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000846,"raw_usage":{"total_tokens":3725,"prompt_tokens":1033,"completion_tokens":2692,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":649,"completion_tokens_details":{"reasoning_tokens":2594}},"tokens_in":649,"tokens_out":2692,"duration_ms":17452,"temperature":1.0,"reasoning_tokens":2594,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:31:08.035932+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $P\\Omega P^T$ for the $k\\times n$ block matrix displayed in Section 5 with a generic positive $(k-2)\\times(n-6)$ block: if the isotropy equations (5) force some Plücker coordinate to change sign or force the block to vanish, then the extension to all $n>2k+1$ and $k>1$ fails, leaving only the $(2,6)$ obstruction. A positive result would confirm the intended embedding and support the non-CW claim in the general regime.","supporting_citations":[{"cited_title":"The positive orthogonal Grassmannian and loop amplitudes of ABJM","cited_arxiv_id":null,"evidence_quote":"introduced the positive orthogonal Grassmannian $\\mathrm{OGr}_+(k,2k)$ in connection with scattering amplitudes, motivating the study of general $(k,n)$."},{"cited_title":"Ising model and the positive orthogonal Grassmannian","cited_arxiv_id":null,"evidence_quote":"gives the face structure and cell parametrization of $\\mathrm{OGr}_+(k,2k)$, used here for the isomorphism between $\\mathrm{OGr}_+(2,5)$ and $\\mathrm{OGr}_+(3,6)$ and for matchings on $[2k+2]$."},{"cited_title":"Positive geometries and canonical forms","cited_arxiv_id":null,"evidence_quote":"defines positive geometries and canonical forms, the framework in which $\\mathrm{OGr}_+(1,n)$ is shown to be a positive geometry."},{"cited_title":"Total positivity, grassmannians, and networks","cited_arxiv_id":null,"evidence_quote":"introduces positroid cells and their combinatorial indexing, the objects whose failure to CW-decompose $\\mathrm{OGr}_+(k,n)$ is the paper's main negative result."},{"cited_title":"Principles of algebraic geometry","cited_arxiv_id":null,"evidence_quote":"provides the classical fact that $\\mathrm{OGr}(k,n)$ is empty for $n<2k$ and has two components for $n=2k$, used in Proposition 2.4."},{"cited_title":"Com- binatorial geometries, convex polyhedra, and schubert cells","cited_arxiv_id":null,"evidence_quote":"the matroid stratification of the Grassmannian from which positroid cells derive, cited when defining the decomposition of $\\mathrm{Gr}_+(k,n)$."}],"review_version":1}