{"id":"79882ad5-7c4d-412a-b75a-12297bffa42d","arxiv_id":"2508.20935","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new dinv statistic on fall-decorated rectangular Dyck paths yields a proven q,t-generating function formula equal to the skewing operator applied to e_{m,n+km}, extending the rational shuffle theorem and the rise Delta theorem.","lead":"The authors prove a new fall-decorated rational shuffle theorem: a symmetric function formed by a skewing operator applied to e_{m,n+km} equals the q,t-weighted sum over fall-decorated rectangular Dyck paths. The result generalizes both the rational shuffle theorem and the rise Delta theorem, and extends to all rectangular paths only if an earlier conjecture of the same authors holds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 depends on Proposition 4.14, whose proof is deferred to 'same argument as [GGG24, Lemma 4.6]'; if that transfer fails for fall-decorated rectangular paths, the sign-reversing cancellation in Section 6 collapses.","rationale":"The reader correctly identified Proposition 4.14 as the weakest assumption. I agree that this is the load-bearing step, but I would make the verdict conditional rather than ACCEPT: as written, Theorem 1.1 is not fully proven because the critical invariance is asserted by reference to a lemma in a different decorated setting without demonstrating the transfer. The difference is substantive: the fall-decoration star word is ordered by vertical distance, and the relabeling induced by φ affects B-steps through complementation in horizontal blocks, so the 'same argument' is not merely cosmetic. A computational check on small parameters would immediately detect a counterexample if one exists. If the check passes, the paper should be accepted; if it fails, the main theorem's proof needs repair. The other flagged issue, the unproven LLT positivity assertion, is secondary because Proposition 4.2 for LRD follows from Mellit's rational shuffle theorem; for LRP it is only used in the explicitly conditional Theorem 1.2.","tokens_in":19872,"tokens_out":14348,"duration_ms":147429,"concrete_test":"Enumerate all small instances, e.g. (m,n,k) = (2,1,2), (2,2,2), (3,2,2), (2,3,3), (3,3,2), and for every π ∈ LRD(m+k,n+k)^{*k} and every allowable fall-labeling w ∈ FL(π), compute falldinv(π, w) via ψ^{-1} and compare it with falldinv(π, φ(w)) using the word involution in Definition 4.12. Also verify that φ(w) is a valid fall-labeling and that the content is allowable. If any equality fails, Proposition 4.14 is false and Theorem 1.1 does not follow from the presented argument; if all small cases pass, the remaining risk is a formal proof gap rather than a substantive counterexample.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central proof of Theorem 1.1 cancels all non-fixed fall-labelings using the sign-reversing involution φ. The step that makes this cancellation valid is Proposition 4.14: falldinv(π, w) = falldinv(π, φ(w)). The provided proof is a single sentence referring to [GGG24, Lemma 4.6]. This is not automatic in the present setting: falldinv is defined through attacking pairs involving the big vertical steps B of ψ^{-1}(π, w), and the labels of those B-steps are obtained from the fall-labeling w by complementation inside each horizontal block, not simply by reading the star word in vertical-distance order. A word-level involution preserving tied inversions in the star word does not, by itself, guarantee that the induced relabeling of the B-steps preserves the vertical-offset data counted by falldinv. If Proposition 4.14 fails for any path, the nonzero contributions from w and φ(w) will not cancel, and the equality in Theorem 1.1 is unsupported. This is the single most load-bearing gap in the paper; the rest of the proof is either direct adaptation or citation of proven results.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves a 'fall-decorated rational shuffle theorem' (Theorem 1.1): for any positive m and natural n, k, the skewing operator s^⊥_{(m-1)k} applied to e_{m,n+km} equals the q,t-generating function over labeled fall-decorated rectangular Dyck paths in LRD(m+k,n+k)^{*k}, using a newly defined dinv statistic and the area statistic. The proof combines Mellit's rational shuffle theorem, the Jacobi-Trudi expansion of s_{(m-1)k}, a bijection ψ from certain labeled rectangular paths to fall-decorated paths, a sign-reversing involution on fall-labelings, and a comparison (Theorem 5.1) between the dinv of a path and the dinv of its preimage under ψ. The paper also proves, conditional on [IPPVW23, Conjecture 4.2], an analogous formula for rectangular paths (Theorem 1.2), and discusses connections to Delta theorems, D_α operators, Theta operators, and the Delta square conjecture.","tokens_in":20206,"tokens_out":13179,"duration_ms":134008,"significance":"If fully established, Theorem 1.1 is a meaningful unification: k=0 reduces to Mellit's rational shuffle theorem, and m=n reduces to the rise Delta theorem, so the new dinv statistic genuinely interpolates between known settings. The proof is constructive and non-circular, relying on external proven results and a detailed bijection. The conditional Theorem 1.2 is clearly labeled, and the paper includes useful consistency checks and several new conjectures. However, there are two load-bearing gaps: the proof of Proposition 4.14 is deferred to a citation, and the use of Proposition 4.2 for non-Dyck rectangular paths is not justified. These issues should be fixed before the paper is accepted.","major_comments":[{"comment":"The proof is a single sentence: 'By the same argument as [GGG24, Lemma 4.6], the result follows.' This is load-bearing for the sign-reversing cancellation in Section 6: if falldinv is not preserved by φ, the contributions of w and φ(w) do not cancel, and Theorem 1.1 does not follow from the presented argument. The transfer is not automatic: falldinv is defined through attacking pairs involving the big vertical steps B of ψ^{-1}(π,w), whose labels are obtained from the fall-labeling w by complementation inside each horizontal block, not by reading the star word in vertical-distance order. An involution preserving tied inversions in the star word does not, by itself, obviously preserve the vertical-offset data counted by falldinv. Please provide a complete proof or a precise reduction to [GGG24, Lemma 4.6].","section":"Section 4.3, Proposition 4.14"},{"comment":"The preamble states that the generating function over LRP(m,n) is symmetric 'as it is a positive sum of LLT polynomials'. For rectangular paths without the Dyck condition, this is not a known theorem; it is essentially the content of the rectangular paths conjecture [IPPVW23, Conjecture 4.2], which the paper leaves open. The unconditional statement of Proposition 4.2 for LRP is therefore unsupported. The proof of Theorem 1.1 only needs the LRD case, but the proof of the conditional Theorem 1.2 uses the LRP case; if Proposition 4.2 for LRP is not available, the derivation of Theorem 1.2 from Conjecture 6.2 must be supplied explicitly, e.g., by invoking the assumed symmetry after Conjecture 6.2.","section":"Section 4.1, Proposition 4.2"}],"minor_comments":[{"comment":"Typo: 'the the success story' should be 'the success story'.","section":"Section 1"},{"comment":"Typo: 'symmetric functionf' should be 'symmetric function f'.","section":"Definition 2.4"},{"comment":"Author name spacing: 'GIOV ANNI PAOLINI' should be 'GIOVANNI PAOLINI'.","section":"Title page"},{"comment":"The statement is called a theorem but is conditional on an unproved conjecture. Consider relabeling it as 'conditional theorem' or explicitly 'Theorem (conditional on Conjecture 6.2)' to avoid ambiguity.","section":"Theorem 1.2"},{"comment":"The dashed projection lines and the distinction between the usual and ENS representations are hard to read in the small figures. Enlarging the figures or labeling the relevant steps would improve clarity.","section":"Figures 3 and 5"}],"recommendation":"major_revision","confidential_remarks":"The main theorem appears likely correct and the paper is a good fit for math.CO, but the proof of Proposition 4.14 is currently a citation to a lemma in a different setting, and the unconditional use of Proposition 4.2 for non-Dyck rectangular paths is unjustified. I would be willing to accept after these two points are addressed with a full proof or a precise reduction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper makes a real advance: it proves a fall-decorated rational shuffle theorem (Theorem 1.1) that simultaneously specializes to Mellit's rational shuffle theorem at k=0 and to the rise Delta theorem when m=n. The new dinv statistic on fall-decorated rectangular paths is genuinely new, and the ENS representation is a nice trick that simplifies the geometry. The proof strategy is sound: skewing operator via Jacobi-Trudi, bijection to fall-labeled paths, sign-reversing involution, and a careful dinv comparison. I checked the algebra in the main chain of equalities and found no error.\n\nThe soft spot is Proposition 4.14, where the proof is a single sentence deferring to 'the same argument as [GGG24, Lemma 4.6]'. This is the step that makes the sign-reversing cancellation work, so it's load-bearing. The transfer is not completely automatic, since falldinv involves big vertical steps whose labels are complements of the fall labels within each horizontal block, not just the star word. I don't have a counterexample, and the argument is likely adaptable, but a referee should ask the authors to spell out the details. Minor in comparison: Theorem 1.2 is explicitly conditional on [IPPVW23, Conjecture 4.2], and the paper is clear about that. The q=1 Theorem 7.3 is a reformulation of known results, so no issue.\n\nThe paper is honest, well structured, and the main theorem deserves to be in the literature. I'd send it to peer review and ask for a careful check of Proposition 4.14. The conditional theorem should be marked as such in the final version.","headline":"A genuine unification in shuffle theory, with one deferred proof step that deserves a referee's eye.","tokens_in":20656,"tokens_out":5745,"would_cite":true,"duration_ms":59565,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05A19","05A30","05E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fall-decorated paths satisfy a rational shuffle identity","keywords":["fall-decorated rectangular Dyck paths","rational shuffle theorem","Delta conjecture","Delta square conjecture","dinv statistic","Macdonald polynomials","skewing operator","lattice paths"],"falsifier":"Brute-force check Proposition 4.14 for all allowable fall-labelings of a small path, say m=2, n=3, k=3, by comparing falldinv(π,w) and falldinv(π,φ(w)) under the star-word ordering; a single mismatch would invalidate Theorem 1.1. Independently, compute both sides of Conjecture 7.10 at n=2,k=2 to test the proposed fall Delta square identity.","tokens_in":19803,"feed_emoji":"⭐","tokens_out":8440,"duration_ms":78615,"temperature":0.7,"pith_summary":"This paper establishes a rational analog of the fall Delta theorem: applying a certain symmetric-function operator (the Schur skewing operator s^⊥_{(m-1)k}) to e_{m,n+km} produces a weighted count of fall-decorated labeled rectangular Dyck paths. The new ingredient is a dinv statistic defined on paths whose decorated horizontal steps are marked falls; it simultaneously extends the dinv of ordinary rectangular Dyck paths and of rise-decorated square paths. The resulting identity unifies two previously separate results—the rational shuffle theorem and the rise Delta theorem—and also gives the combinatorial side of a proposed fall version of the Delta square conjecture. A companion formula for unrestricted rectangular paths is proved conditionally on a previously stated conjecture, and unconditionally when the rectangle side lengths are coprime.","feed_headline":"Fall-decorated paths satisfy a rational shuffle identity","feed_subtitle":"New q,t dinv statistic extends the rational shuffle theorem and the rise Delta theorem.","key_machinery":"The load-bearing identity is the skewing expansion s_{(m-1)k} = Σ_α (-1)^{sgn(α)} h_α̃ (mod h_j, j>m), obtained by Jacobi-Trudi. The proof then runs on three mechanisms: the bijection ψ that prunes the big vertical steps of an (m,n+km) labeled rectangular path to produce an (m+k,n+k) fall-decorated path with a fall-labeling; the ENS representation, in which decorated horizontal steps become South steps and which preserves vertical distances and area; and the lifted sign-reversing involution φ on fall-labelings, ordered by the 'star word' (labels of decorated falls read by decreasing vertical distance), whose unique fixed point is the word 1 2 … k. Proposition 4.14 asserts that φ preserves th","core_discovery":"The paper's main theorem states that for any positive integers m, n, k, the Schur skewing operator s^⊥_{(m-1)k} applied to e_{m,n+km} equals the sum over fall-decorated labeled rectangular Dyck paths π in LRD(m+k,n+k)^{*k} of q^{dinv(π)} t^{area(π)} x^π. Here the paths stay weakly above a broken diagonal, the marked falls are horizontal steps immediately followed by another horizontal step, and dinv is a new diagonal-inversion statistic defined as a temporary inversion count plus a correction term cdinv built from decorated falls. The proof proceeds through a bijection ψ from labeled rectangular paths with big vertical labels to fall-decorated paths carrying a fall-labeling, an ENS represent","pith_inferences":["The proof's deferral to [GGG24, Lemma 4.6] for Proposition 4.14 marks the transfer of the sign-reversing involution as the central technical risk; a reader wanting to extend the theorem to other decorations should verify that transfer first.","Because Remark 3.8 states the construction works for any broken slope with positive reals a,b, the proof likely adapts verbatim to non-integer slope parameters; testing m=3,n=2 with an offset would be a direct extension.","The fixed-point fall-labeling w*=1 2 … k defines a canonical linear order on decorated falls by vertical distance; this order may be the right replacement for the 'rise' order in the open rise-decorated rectangular problem and could yield the missing dinv statistic there.","The identity in Conjecture 7.5 relating scalar products with h_d suggests that Schröder-path analogues of these theorems would follow by taking the appropriate scalar product, which may be a simpler testbed than full labeled paths."],"forward_implications":["Setting k=0 recovers Mellit's rational shuffle theorem, so the new formula is a strict extension rather than a separate conjecture.","Setting m=n recovers the rise Delta theorem (proved by D'Adderio and Mellit), giving a single statistic that covers both rectangular and decorated-square cases.","If the rectangular paths conjecture [IPPVW23, Conjecture 4.2] holds, the same method proves the rectangular (non-Dyck) version with [m]_q/[d]_q p_{m,n+km} on the left; the d=1 case is already unconditional.","Section 7 gives a q=1 refinement in which the generating function splits according to the fall-composition β and equals D_{α+β}(1)|_{q=1}, connecting the path sum to known D_α operators.","The right side of the fall version of the Delta square conjecture (Conjecture 7.9) is now a proved path generating function, so the open content is exactly the symmetric-function identity Θ_{e_k} ∇ω(p_n) = s^⊥_{(n-1)k} p_{n,n(k+1)} (Conjecture 7.10)."],"supporting_citations":[{"why":"Supplies the rational shuffle theorem, the base k=0 case whose combinatorial interpretation of e_{m,n+km} is the starting point of the proof.","marker":"[Mel21]"},{"why":"Supplies the skewing expansion of s_{(m-1)k}, the sign-reversing involution φ on allowable words, and Lemma 4.6 invoked in Proposition 4.14.","marker":"[GGG24]"},{"why":"Introduces rectangular paths and the broken-diagonal framework, and states the rectangular paths conjecture on which Theorem 1.2 depends.","marker":"[IPPVW23]"},{"why":"Defines the Delta conjecture and the rise-decorated Dyck path objects; Theorem 1.1 reduces to the rise Delta theorem when m=n.","marker":"[HR W18]"},{"why":"Proves the compositional Delta conjecture, establishing the rise Delta theorem used as the square case of the present formula.","marker":"[DM22]"},{"why":"Defines the cdinv correction for Dyck and rectangular paths that the new dinv statistic extends.","marker":"[HS15]"},{"why":"States the Delta square conjecture, which the fall version in Conjecture 7.9 mirrors.","marker":"[DIVW19]"}],"fun_headline_variants":["Fall-decorated paths deliver a rational Delta theorem","New dinv statistic unifies fall and rectangular shuffle cases","Rational shuffle identity from fall-decorated Dyck paths","Skewing operator meets fall decoration: a Delta analog","Extending the Delta square conjecture to fall-decorated paths"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The main theorem rests on the assertion that the sign-reversing involution on fall-labelings preserves the fall-correction statistic falldinv; that assertion is deferred to a cited lemma, and if it fails the proof's cancellations do not go through, while the rectangular version additionally assumes a previously stated open conjecture.","fun_headline_variants_meta":{"raw":{"variants":["Fall-decorated paths deliver a rational Delta theorem","New dinv statistic unifies fall and rectangular shuffle cases","Rational shuffle identity from fall-decorated Dyck paths","Skewing operator meets fall decoration: a Delta analog","Extending the Delta square conjecture to fall-decorated paths"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000528,"raw_usage":{"total_tokens":2332,"prompt_tokens":645,"completion_tokens":1687,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":389,"completion_tokens_details":{"reasoning_tokens":1606}},"tokens_in":389,"tokens_out":1687,"duration_ms":12967,"temperature":1.0,"reasoning_tokens":1606,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:42:00.137670+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Brute-force check Proposition 4.14 for all allowable fall-labelings of a small path, say m=2, n=3, k=3, by comparing falldinv(π,w) and falldinv(π,φ(w)) under the star-word ordering; a single mismatch would invalidate Theorem 1.1. Independently, compute both sides of Conjecture 7.10 at n=2,k=2 to test the proposed fall Delta square identity.","supporting_citations":[],"review_version":1}