{"id":"a8e8b49b-5ac7-4925-aa85-7bab6e2c1979","arxiv_id":"2504.17734","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A signed tiling rule computes every Schubert coefficient, and the rule implies a new polynomiality result for coefficient sums with bounded inversions.","lead":"This paper introduces a signed puzzle rule that computes Schubert coefficients for all permutations, and uses the rule to prove that the sums of these coefficients over permutations with a fixed number of inversions are polynomials in the permutation size.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Section 5 row-by-row encoding of Knutson's recurrence, especially the non-local transmuter step, is asserted rather than formally verified; an exhaustive small-n enumeration would settle whether the signed puzzle count matches c^w_{u,v}.","rationale":"I read the paper as aiming to establish Theorem 1.1 as an exact signed puzzle rule, with Theorem 1.2 as an application. The strongest point is that the construction is explicit and falsifiable: the puzzle pieces are finite, the region is concrete, and Knutson's recurrence is an established identity. The weakest point, as the reader also identified, is that Section 5 gives a narrative rather than a formal bijection between rows of a puzzle and steps of the recurrence. The non-local transpositions in case (3) are the natural place for an edge case: a label may be transmitted across long distances, and the inversion condition (⋄) is justified only by counting rows, not by a local check. The exhaustive small-n enumeration I propose would settle this concretely because any mismatch would appear as a small counterexample. I also note a separate concern in the proof of Theorem 1.2: the rewrite in Section 7.1 appears to contain an identity that fails dimension checks as written, which would require a corrected duality statement before the polynomiality proof is complete. This is a second issue, but it does not change the verdict from conditional; the paper should be accepted only after those details are either fully written out or verified computationally.","tokens_in":14300,"tokens_out":15070,"duration_ms":159558,"concrete_test":"Write an exact enumerator for the puzzle set T_n and the region Γ from Section 4, and for n≤4 (or n≤5 if feasible) enumerate all signed puzzles for every u,v,w∈S_n satisfying inv(u)+inv(v)=inv(w). Compare the signed sum t_+−t_- with c^w_{u,v} computed independently (e.g., via Knutson's recurrence or a Schubert calculus package). If every triple matches, the row-by-row correspondence is confirmed on a nontrivial set; if any mismatch occurs, locate the first row whose top/bottom labels deviate from Lemma 3.1. Pay special attention to dark red docket 2 blank-transmuter configurations with j=i+1 and k distant, since these are the least specified in Section 5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 rests entirely on the row-by-row claim in Section 5 that each puzzle row encodes one application of Lemma 3.1. The most delicate part is case (3): dark blue and red docket 1/2 triangles must implement transpositions t_{jk} with |{j,k}∩{i,i+1}|=1. Section 5 does not give a formal specification of how transmuter labels select k or enforce the inversion-increase condition (⋄); the text says these are \"immediate translations\" and \"The details are straightforward\", and the blank-transmuter dark red docket 2 case is dismissed in one sentence. The condition (⋄) is not checked locally: it is argued globally from the total number of rows ℓ, which cannot catch a single row in which a transmuter configuration realizes the wrong transposition. Additionally, Section 5 asserts that the dark-triangle index i is forced to be the first ascent of u, whereas Lemma 3.1 applies at every ascent; this would need a separate justification if it is a deliberate choice of recurrence path. If any family of transmuter tiles realizes the wrong transposition, or if the first-ascent mechanism is not a valid evaluation of the recurrence, the signed puzzle sum deviates from c^w_{u,v} and Theorem 1.1 fails. Since Theorem 1.2 and the polynomiality application depend on this exact equality, the unverified correspondence is the load-bearing concern.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a signed puzzle rule for Schubert coefficients: for every n and every triple u,v,w in S_n satisfying inv(u)+inv(v)=inv(w), the signed count of tilings of an n-by-(binom(n,2)-inv(u)) parallelogram region by a set T_n of O(n^9) labeled triangular pieces is claimed to equal the Schubert coefficient c^w_{u,v}. The proof is by induction on the number of rows, reducing to a row-by-row simulation of Knutson's recurrence. As an application, the paper proves that the sum gamma_k(n) of Schubert coefficients over permutations w with inv(w)=k is a polynomial in n, using an Ehrhart-theoretic argument on unions of rational polyhedra associated to relative placements of the special tiles.","tokens_in":14605,"tokens_out":27444,"duration_ms":242280,"significance":"If correct, this is the first signed puzzle rule that applies to all Schubert coefficients, extending earlier special cases and providing a new type of combinatorial rule in an area where even signed interpretations are scarce. The polynomiality theorem for gamma_k(n) is a genuinely new structural result, and the proof strategy connecting tile placements to Ehrhart theory of unimodular polyhedra is interesting in its own right. The construction is explicit, richly illustrated, and the authors are transparent about the role of Knutson's recurrence. The main weakness is that the central verification in Section 5 is incomplete at exactly the point where non-local transpositions are implemented; without that verification, the equality in Theorem 1.1 is not established.","major_comments":[{"comment":"The proof of Theorem 1.1 rests on the claim that each puzzle row simulates one application of Lemma 3.1, but the implementation of the non-local transpositions t_{jk} in case (3) via transmuter labels is not verified. The text states that the translation is 'immediate' and that 'The details are straightforward', yet it does not specify how the transmuter labels determine the index k, how the condition |{j,k}∩{i,i+1}|=1 is enforced, or why the bottom permutation labels are exactly u t_{jk}. This is the load-bearing step of the entire paper. Please provide a complete, case-by-case proof of the transmuter mechanism, or supply a machine-checked enumeration for small n and k that confirms the signed puzzle count agrees with c^w_{u,v}.","section":"Section 5, paragraphs on dark blue and dark red triangles"},{"comment":"The identity c^w_{u,v} = c^u_{w,v}, stated with the ambiguous clause 'where w = w·w◦', is used to reduce the parallelogram height to k in the proof of Theorem 1.2, but the precise symmetry and its proof are not given. The notation uses w for two different permutations in the same sentence, making the statement impossible to verify as written. Please restate the duality with distinct variables, prove it or give a precise reference, and then derive the correct summation condition for the new coefficient.","section":"Section 7.1, duality identity"},{"comment":"The text says 'we will work with a modified set T′n' and then writes 'gamma_k(n) = sum_{puzzle T of Γ with Tn} s(T)', using T_n rather than T′_n. This is either a typo or an unproven assertion. If the intended equality is with T′_n, then Theorem 1.1 must be proved for T′_n, which is not done; Remark 5.1 only sketches the modification. Please correct the displayed equation and clarify whether the equality in Theorem 1.1 holds for the modified tile set.","section":"Section 7.1, displayed equality for gamma_k(n)"},{"comment":"The claim that the condition that the labels in each row form permutations 'gives inequalities relating differences between the labels and distances between the columns' is not derived. In particular, the text asserts inequalities of the form alpha_i - alpha_j > m with m depending on the distance between columns, but a permutation only imposes distinctness of labels, not a lower bound on their difference that grows with column distance. Since the total unimodularity argument in Section 7.5 depends on the inequalities having the form alpha_i - alpha_j <= b, please provide a precise derivation of these inequalities.","section":"Section 7.4, derivation of distance-dependent inequalities"}],"minor_comments":[{"comment":"The notation 'inv(w) := {(i,j): i<j, w(i)>w(j)} denote the number of inversions' confuses the set of inversion pairs with its cardinality; use |inv(w)| or define Inv(w) for the set and inv(w) for its size.","section":"Section 2"},{"comment":"In the paragraph on shaded triangles, 'eight of fewer distinct labels' should read 'eight or fewer distinct labels'.","section":"Section 4.4"},{"comment":"The proof that the dark triangle's index i is forced to be the first ascent of u is compressed. It would help to add a short explanation of why no feedback or transmuter labels can affect the shaded triangles to the left of the dark triangle in the first row, so that the permutation labels transmitted there are exactly u(1),...,u(i-1).","section":"Section 5, first-ascent argument"},{"comment":"The phrase 'where w = w·w◦' uses the symbol w for both the original and the transformed permutation; use a different letter such as \\tilde{w} for clarity.","section":"Section 7.1"},{"comment":"The terminology 'separated shaded triangles' is introduced after a confusing count of 'at most 2k such shaded triangles, where at most k are not immediately following the dark triangles'. Please clarify exactly which shaded triangles are counted and which are called separated.","section":"Section 7.2"},{"comment":"There are several typos: the header of the paper reads 'SCHUBERT T COEFFICIENTS' instead of 'SCHUBERT COEFFICIENTS', and the displayed formula (⊛) has a formatting issue ('the form a,p /∈ [g,h]' should be 'a,p notin [g,h]').","section":"Global"}],"recommendation":"major_revision","confidential_remarks":"The central gap is the informal verification in Section 5 of the transmuter mechanism for non-local transpositions. This is not a presentation issue: Theorem 1.1 and the application in Theorem 1.2 both depend on that correspondence. I believe the gap is fixable, either by supplying a complete local verification or by including a rigorous computational check for small n and k, but the current version is not yet acceptable. The duality identity used in Section 7.1 also needs to be stated clearly and proved. I do not see grounds for rejection, provided the authors address these issues in a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nShort version: this is a serious paper with a real new result, but the proof as written has a gap in the central construction. I'd send it to a referee, with the expectation of major revision.\n\nWhat's new: first signed puzzle rule for Schubert coefficients in full generality, based on Knutson's recurrence, and a new polynomiality theorem for sums of Schubert coefficients with bounded inversions. The Ehrhart argument in Theorem 1.2 is elegant and the application of the puzzle rule to get a structural result is a good payoff.\n\nThe paper is well written and does a lot of work to make the construction concrete: many figures, a worked example, explicit piece counts. The authors are careful to situate it relative to existing signed rules and to the open positivity problem.\n\nThe soft spot is Section 5. Theorem 1.1 rests on the claim that each row of the puzzle simulates one application of Lemma 3.1. That is the load-bearing step, and it is not fully formalized. The non-local transpositions in case (3) are handled by transmuter labels, but the text says the conditions are 'immediate translations' and 'the details are straightforward' — for a tiling rule this intricate, that is not enough. In particular, the argument that condition (⋄) holds globally by counting rows only rules out transpositions with the wrong inversion increase; it does not identify exactly which transposition a transmuter configuration performs. A wrong transposition with the same inversion increase would not be caught, and the puzzle sum would drift from the recurrence. I would want either a complete formal specification of the transmuter mechanism, or a computational verification for all permutations up to, say, n=5 or 6 that the signed puzzle count equals the Schubert coefficient.\n\nThere are also a few apparent notational slips in the setup of Theorem 1.2, but they don't seem to affect the argument.\n\nThe positivity question remains open, and the authors are honest about that. The paper is for people working in Schubert calculus and combinatorial tilings; it deserves peer review, but in its current form I would not accept it. The authors should be asked to fill the gap or add the verification, and to tidy the notation.","headline":"A plausible and likely correct signed puzzle rule for Schubert coefficients, but the proof of the central row-by-row correspondence is under-specified and needs either a filled-in proof or a computational check.","tokens_in":15136,"tokens_out":4308,"would_cite":true,"duration_ms":41045,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N15","05B45"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a signed puzzle rule that computes every Schubert coefficient, and uses it to prove that fixed-inversion coefficient sums are polynomial.","keywords":["Schubert coefficients","signed puzzle rule","Knutson recurrence","Bruhat order","Ehrhart theory","polynomiality","flag variety cohomology"],"falsifier":"Enumerate by brute force all tilings of the region $\\Gamma$ for a small triple with a known coefficient (for example $n=4$, $u=2143$, $v=3412$, $w=4321$) and compare the signed sum to $c^w_{u,v}$ computed from Schubert polynomials; any mismatch falsifies the theorem.","tokens_in":14094,"feed_emoji":"🧩","tokens_out":14112,"duration_ms":128314,"temperature":0.7,"pith_summary":"Schubert coefficients are the integers that record products in the cohomology of flag varieties, and finding combinatorial formulas for them is a long-standing open problem. This paper proves that every Schubert coefficient $c^w_{u,v}$ satisfying the dimension equation $\\operatorname{inv}(u)+\\operatorname{inv}(v)=\\operatorname{inv}(w)$ equals the signed count of tilings of a finite parallelogram by labeled triangles. The construction is the first signed puzzle rule that works for all Schubert coefficients, rather than for special families. The rule is built directly from a recurrence: one row of the tiling encodes one step of the recurrence, so iterating over $\\binom n2-\\operatorname{inv}(u)$ rows reproduces the coefficient. As an application, the paper proves that the sum $\\gamma_k(n)$ of all coefficients with $\\operatorname{inv}(w)=k$ is a polynomial in $n$, a structural conclusion that follows from an Ehrhart-theoretic count of the puzzles.","feed_headline":"Signed puzzle rule computes every Schubert coefficient","feed_subtitle":"Tilings with O(n^9) labels reproduce the recurrence and show fixed-inversion sums are polynomial.","key_machinery":"The central object is the signed puzzle itself: a parallelogram region tiled by $O(n^9)$ types of unit equilateral triangles, colored white, shaded, and dark (the dark divided into yellow, blue, and red), carrying three layers of labels—permutation labels $(a,b,c)$, feedback labels with blank entries, and transmuter labels $(g,h)$ with $g<h$—together with edge indicators $\\circ$ and $\\ast$ that force exactly one dark triangle per row. The load-bearing mechanism is the row-by-row correspondence with Knutson's recurrence: in each row the unique dark triangle marks the index $i$ of the recurrence, and the labels record how the three permutations change. The overall sign of a puzzle is the parity of the number of red dark triangles, which matches the signs of the positive and negative terms in the final case of the recurrence.","core_discovery":"On its own terms, the central discovery is a complete signed puzzle rule for Schubert coefficients. Given permutations $u,v,w\\in S_n$ satisfying the dimension equation, one draws an $n\\times \\ell$ parallelogram $\\Gamma$ whose top boundary carries the triples $(u(i),v(i),w(i))$ and whose bottom boundary carries the reversed triples $(n-i+1,i,n-i+1)$; a puzzle is a tiling by the finite set $T_n$ of white, shaded, and dark labeled unit triangles, and the sign of a tiling is $(-1)^p$, where $p$ is the number of red triangles in it. The paper proves that summing these signs over all tilings returns $c^w_{u,v}$. The proof works row by row: each row of the tiling realizes exactly one of the cases of Knutson's recurrence, the unique dark triangle in the row selects the index $i$ where the recurrence is applied, and the feedback and transmuter labels implement the transpositions that appear in the recurrence, so the whole tiling is a geometric trace of the recurrence expansion.","pith_inferences":["For special families of permutations, the signed contributions may cancel in pairs, so the same construction could potentially be refined to a manifestly positive puzzle rule; the paper does not attempt this.","The architecture of encoding a recurrence as labeled tile rows and counting fixed-height regions by Ehrhart theory should transfer to equivariant or K-theoretic Schubert coefficients if a suitable recurrence exists; the paper notes this possibility without proving it.","For small $n$, the signed puzzle count can serve as a computational certificate against independent pipe-dream or Schubert polynomial calculations, giving a quick consistency check for any proposed simplification of the piece set.","The $O(n^9)$ piece count is a proof-of-concept rather than an efficient algorithm; allowing rotations or quotienting by symmetries would be a natural next step, but the paper uses only parallel translations."],"forward_implications":["For every triple satisfying the dimension equation, $c^w_{u,v}$ equals the signed count of a finite tiling, so Schubert coefficients have an explicit GapP-type formula with a piece set of size $O(n^9)$.","For each fixed $k$, the total $\\gamma_k(n)$ is a polynomial in $n$; the proof gives degree $O(k^2)$, which is weaker than the elementary $6k$ bound, so the polynomiality itself is the new content.","A puzzle for $c^w_{u,v}$ has exactly $\\binom n2-\\operatorname{inv}(u)$ rows, so the signed count is a linear-depth trace of the recurrence rather than a sum over independent cases.","The modified piece set $T'_n$ described in Remark 5.1, which enforces the non-local constraint through an added label inequality, also computes the same coefficients and is the version needed for the polynomiality argument."],"supporting_citations":[{"why":"Provides the recurrence (Lemma 3.1) whose row-by-row iteration the puzzle rule encodes.","marker":"[Knu03]"},{"why":"Supplies the order-polynomial and Ehrhart quasi-polynomial machinery used to count labelings in the proof of Theorem 1.2.","marker":"[Sta99]"},{"why":"Provides the unimodularity criterion converting Ehrhart quasi-polynomials to polynomials.","marker":"[Bar97]"}],"fun_headline_variants":["Signed tilings compute all Schubert coefficients","Puzzle signs reproduce Knutson's recurrence exactly","Signed puzzle tiling: all Schubert coefficients exactly","Signed tilings yield exact Schubert coefficients"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction assumes that every valid row of tiles corresponds faithfully to one step of the recurrence, including the non-local transpositions implemented by transmuter labels, and that this translation has no exceptional edge case.","fun_headline_variants_meta":{"raw":{"variants":["Signed tilings compute all Schubert coefficients","Puzzle signs reproduce Knutson's recurrence exactly","Signed puzzle tiling: all Schubert coefficients exactly","Signed tilings yield exact Schubert coefficients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000909,"raw_usage":{"total_tokens":3828,"prompt_tokens":790,"completion_tokens":3038,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":406,"completion_tokens_details":{"reasoning_tokens":2979}},"tokens_in":406,"tokens_out":3038,"duration_ms":22781,"temperature":1.0,"reasoning_tokens":2979,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:32:42.844720+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate by brute force all tilings of the region $\\Gamma$ for a small triple with a known coefficient (for example $n=4$, $u=2143$, $v=3412$, $w=4321$) and compare the signed sum to $c^w_{u,v}$ computed from Schubert polynomials; any mismatch falsifies the theorem.","supporting_citations":[],"review_version":1}