{"id":"176b1dfc-84c5-4e6c-b4f0-2c63247bff62","arxiv_id":"2506.00451","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An alternative formula for BKP connected n-point functions is derived from the KP hierarchy and shown to be equivalent to the known Wang-Yang formula.","lead":"This mathematics paper derives a new expression for the connected n-point functions of BKP hierarchy tau-functions by embedding BKP into the KP hierarchy, then proves this expression equals a known formula by Wang and Yang. The result is a cross-check of two methods and a template for computing correlation functions in other integrable hierarchies.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The printed equivalence Theorem 1.2/4.2 fails its own one-point test: by (3.25), A_BKP(z,-z) = -z/4[A_KP(z,z)+A_KP(-z,-z)], while the identity (1.10) uses the positive expression, and the z1...zn factor present in (4.2) is missing in (4.8).","rationale":"The reader identified Lemma 4.1 as the weakest assumption and recommended a CAS check of that lemma. The present stress test instead finds a more basic, independently checkable inconsistency: the main equivalence theorem as stated contradicts the paper's own one-point reduction. The n=1 case is not a marginal edge case; it is explicitly covered by the theorem's notation and by the paper's discussion of one-point functions in (4.1). Because the theorem's statement is false under the paper's definitions, the proof in Section 4 cannot be a proof of the stated claim. This does not necessarily mean the intended formula is unfixable—it may be restored by a sign correction and the z1...zn factor—but the current manuscript's central claim is not internally consistent. A single symbolic computation with the simple affine coordinate choice a_{1,0}=1 settles the issue and is far easier to verify than the long induction in Appendix A. For this reason the verdict should move from CONDITIONAL to REJECT, with the expectation that a corrected statement could be resubmitted.","tokens_in":23498,"tokens_out":41666,"duration_ms":361326,"concrete_test":"Set all BKP affine coordinates to zero except a_{1,0}=1 and a_{0,1}=-1. Using definitions (2.26) and (3.23), compute A_BKP(z,-z)=z^{-1}, A_KP(z,z)+A_KP(-z,-z)=-4z^{-2}. Then the left side of (1.10) is z^{-1}, the right side as printed is 1/4(-4z^{-2})=-z^{-2}, and even with the z1...zn factor restored it is -z^{-1}. A symbolic evaluation of this n=1 case, using only the formulas stated in the paper, therefore falsifies the displayed equivalence and shows the failure is a sign error in addition to the missing z factor.","verdict_should_be":"REJECT","load_bearing_attack":"The central equivalence is not self-consistent as written. For n=1, the Wang-Yang sum defined in (2.29) and quoted in (1.8) equals ξ(z,-z)=A_BKP(z,-z). The embedding formula (3.26) for the same data gives the generating function in z^{-i-1} as 1/4[A_KP(z,z)+A_KP(-z,-z)], so the comparison for z^{-i} requires multiplying by z. But (3.25) implies A_BKP(-z,z)=z/4[A_KP(z,z)+A_KP(-z,-z)], and A_BKP is antisymmetric, so A_BKP(z,-z)=-z/4[...]. Thus (1.10) would require the positive and negative of the same expression to be equal. Independently, Section 4 begins with the correct-looking identity (4.2) containing an explicit z1...zn factor, but the final theorem (4.8) omits that factor. The proof of Section 4 therefore establishes a different identity from the theorem stated. This problem is prior to Lemma 4.1: even if Lemma 4.1 is true, it supports a corrected identity, not Theorem 1.2 as printed.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives a formula for the connected bosonic n-point functions of a BKP tau-function by embedding the BKP hierarchy into the KP hierarchy. Using the relation tau_KP(t1,0,t3,0,...) = tau_BKP(t1,t3,...)^2, the authors apply Zhou's KP formula to obtain a new expression (Theorem 3.1) in terms of the affine coordinates of the BKP tau-function, converted into a KP generating series A_KP. They then claim that this formula is equivalent to the Wang--Yang formula for BKP n-point functions, a statement formalized as Theorem 1.2 and restated as Theorem 4.2. The proof of equivalence relies on a technical combinatorial lemma (Lemma 4.1) whose proof occupies Appendix A.","tokens_in":23705,"tokens_out":13498,"duration_ms":131677,"significance":"The embedding approach is natural and, if correct, would provide a uniform method for transferring n-point function formulas from KP to its reductions without developing a separate boson-fermion correspondence. The paper contains an explicit and internally consistent computation of the induced KP affine coordinates from BKP coordinates (Theorem 3.1(1)-(2)), which is a useful contribution. However, the central claim of the paper, the equivalence of the new formula with the Wang--Yang formula, is false as stated: it fails the one-point test. Since the main theorem is load-bearing for the paper's purpose, the present manuscript cannot be recommended for publication.","major_comments":[{"comment":"The claimed equivalence fails already for n=1. For n=1, the Wang--Yang sum in (2.29) equals xi(z,-z) = A_BKP(z,-z). The embedding formula (3.26) gives the generating series in z^{-i-1} as 1/4[A_KP(z,z)+A_KP(-z,-z)]; after multiplying by z to compare with the z^{-i} series in (2.29), this becomes z/4[A_KP(z,z)+A_KP(-z,-z)]. By relation (3.25), this equals A_BKP(-z,z). Since A_BKP is antisymmetric, A_BKP(z,-z) = -A_BKP(-z,z). Thus (1.10) for n=1 asserts that two opposite quantities are equal. The remark in Section 4 that A_BKP(-z,z)=1/4(zA_KP(-z,-z)+zA_KP(z,z)) and therefore the one-point functions coincide compares with the wrong object: the WY one-point function is A_BKP(z,-z), not A_BKP(-z,z).","section":"Section 4, Eqs. (4.2) and (4.8)"},{"comment":"The proof in Section 4 is directed at identity (4.2), whose right-hand side contains an explicit factor z1...zn and is subsequently rewritten with factors epsilon_{sigma_i(1)} z_{sigma_i(1)} attached to each A_KP factor. The theorem as stated in (4.8), and likewise (1.10), omits this z1...zn factor. The proof therefore establishes at most a different identity from the announced theorem. This is not a cosmetic discrepancy: for n=1, (4.2) with the z1 factor gives A_BKP(-z,z), while (4.8) gives 1/4[A_KP(z,z)+A_KP(-z,-z)], neither of which equals the Wang--Yang value A_BKP(z,-z).","section":"Section 4, Eqs. (4.2) and (4.8)"},{"comment":"Lemma 4.1 is the technical engine of the equivalence proof, and its proof in Appendix A is a long induction with many sign-sensitive rewritings, for example the manipulations leading from (A.5a) to (A.5e) and the identity (A.6). No machine-checked or computer-algebra verification is supplied, so the lemma carries residual risk. This is secondary to the two points above, but if the authors revise the theorem, the lemma and its application in (4.6) must be re-examined together with the corrected statement.","section":"Lemma 4.1 / Appendix A"}],"minor_comments":[{"comment":"The text contains corrupted symbol sequences such as '⌟⟨rro⟪⟪⟩r⟪⌟...' that make many formulas unreadable; the manuscript needs a clean typesetting pass before any resubmission.","section":"Throughout, esp. Eqs. (1.3), (1.8), (2.23), (2.29)"},{"comment":"There are multiple typos: 'tau-funtion', 'bosoincn-piont' in the abstract, and the keywords should be proofread.","section":"Abstract and keywords"},{"comment":"The heading 'The Equivalence of Two Fomulae' and the phrase 'for convinience' should be corrected; in Theorem 3.1(1), 'sesne' should be 'sense'.","section":"Section 4 heading and Theorem 3.1(1)"},{"comment":"The notation i_{x,y} is used before it is defined; it should be defined at first use, for example in Section 1.","section":"Eqs. (1.5) and (2.25)"},{"comment":"In the definition of the map O, the notation z_i1...z_in is ambiguous; use z_{i_1}...z_{i_n} for clarity.","section":"Section 4, definition of the map O"}],"recommendation":"reject","confidential_remarks":"The n=1 failure is decisive: the main equivalence theorem is false as stated. The paper cannot be accepted without reformulating the main result and redoing the proof. The embedding computation in Theorem 3.1(1)-(2) appears to be a useful contribution, and a corrected version may be worth considering, but the current manuscript should be rejected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: the central equivalence, Theorem 1.2/4.2, is not correct as stated. For n=1, the claimed identity says A_BKP(z,-z) = 1/4(A_KP(z,z)+A_KP(-z,-z)). The paper's own (3.25), plus antisymmetry of A_BKP, gives A_BKP(z,-z) = -z/4(A_KP(z,z)+A_KP(-z,-z)). Those can only agree for zero tau functions. Independently, Section 4 starts with the correct-looking identity (4.2), which includes a z1...zn factor because the Wang-Yang formula uses z^{-i} while the embedding formula uses z^{-i-1}; the restated Theorem 4.2 drops that factor. So the proof establishes a different identity from the theorem, and this failure is prior to Lemma 4.1.\n\nWhat is genuinely useful: Section 3 works out the embedding of BKP into KP explicitly, giving A_KP in terms of BKP affine coordinates and a structurally new formula (3.26) for the n-point functions. This is a direct realization of Zhou's embedding proposal, and the method should transfer to other KP reductions. Lemma 4.1 is a bold combinatorial reduction; if it survives a machine check, it would be a nice result in its own right. The paper is honest about the technical difficulty and does not hide the gap behind vague statements.\n\nSoft spots: the errors in the theorem statements are the biggest problem, but there is another layer of risk. Lemma 4.1's proof in Appendix A is long and sign-sensitive, and the OCR-corrupted text makes line-by-line verification impractical; a CAS check of the lemma and of small-n cases would settle it. The reliance on the second author's arXiv preprints for the KP formula and embedding proposal is not itself a flaw, but external confirmation would help. No data or code, so the claims rest entirely on the written proofs.\n\nBottom line: the paper is for specialists in integrable hierarchies and tau-function correlators. The derivation idea has value, but the central claim as printed is broken. I would not cite it in its current form. A serious referee could help the authors fix the sign and factor issues, so the paper deserves peer review rather than desk rejection, but only with the expectation of substantial revision.","headline":"The paper's central equivalence theorem is not self-consistent as printed: the n=1 case contradicts the authors' own (3.25), and the z1...zn factor vanishes between (4.2) and (4.8).","tokens_in":24307,"tokens_out":14122,"would_cite":false,"duration_ms":127473,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The connected bosonic n-point functions of a BKP tau-function can be computed by embedding the BKP hierarchy into the KP hierarchy, and the resulting formula provably agrees with the existing BKP formula.","keywords":["KP hierarchy","BKP hierarchy","connected n-point functions","tau-functions","affine coordinates","boson-fermion correspondence","Sato Grassmannian","integrable hierarchies"],"falsifier":"Expand both sides of equation (1.10) for $n=3$ and $n=4$ with a truncated BKP affine-coordinate choice, keeping a few nonzero $a^{BKP}_{m,n}$ and $a^{BKP}_{m,0}$, and compare coefficients in the variables $z_i$ up to a fixed order. A single mismatching coefficient would refute Theorem 1.2. A cheaper check is to verify Lemma 4.1 for $k=2,3$ with generic truncated series $s$ and $t$.","tokens_in":23205,"feed_emoji":"📐","tokens_out":12093,"duration_ms":127775,"temperature":0.7,"pith_summary":"The paper establishes an embedding route to the connected bosonic $n$-point functions of BKP tau-functions: it exploits the classical identity $\\tau^{KP}(t_1,0,t_3,0,\\ldots)=\\tau^{BKP}(t_1,t_3,\\ldots)^2$ and rewrites the known KP formula for connected $n$-point functions in terms of BKP affine coordinates. The formula that comes out is visibly different from the one obtained earlier by the type-B boson-fermion correspondence, and the paper's main result is a proof that the two formulas are equivalent. A reader should care because the embedding method does not need the type-B boson-fermion correspondence at all; it uses only the KP machinery plus a coordinate conversion, and it is intended to transfer to other reductions of KP, where such equivalence checks are expected to be equally technical.","feed_headline":"BKP embedded in KP: n-point formulas match","feed_subtitle":"Same BKP correlation functions now derivable from KP, so either formula is usable.","key_machinery":"The machinery has three parts. The first is the embedding relation $\\tau^{KP}(t_1,0,t_3,0,\\ldots)=\\tau^{BKP}(t_1,t_3,\\ldots)^2$, which makes $F^{BKP}=\\frac12 F^{KP}$ after the even times are set to zero. Here the affine coordinates are the expansion coefficients specifying the point of the Sato Grassmannian carried by the tau-function. The second is an explicit conversion of affine coordinates: the paper expresses the generating series $A^{KP}(w,z)$ of the KP affine coordinates in terms of the BKP affine coordinates $a^{BKP}_{m,n}$ and the diagonal coefficients $a^{BKP}_{m,0}$ (Theorem 3.1(1)), and derives the antisymmetrization relation $A^{BKP}(w,z)=\\frac14(zA^{KP}(w,-z)-wA^{KP}(z,-w))$. The third is combinatorial: the KP $n$-point formula is a cycle sum of $\\hat A^{KP}$ factors, and a sign-averaging projection selects the odd time variables that BKP depends on. The equivalence of the two BKP formulas is ultimately decided by Lemma 4.1, an algebraic identity relating two products built from formal series $s$ and $t$; its proof in the appendix is a long induction, and it is the load-bearing step of the comparison.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.2: the new embedding-derived formula (1.3) for the connected bosonic $n$-point functions of a BKP tau-function is equivalent to the earlier type-B formula (1.8). Equivalently, Theorem 3.1(3) gives the generating series $$\\sum_{i_1,\\ldots,i_n\\ge 1,\\, i_k\\text{ odd}}\\frac{\\partial^n $F^{{BKP}}$}{\\partial t_{i_1}\\cdots\\partial t_{i_n}}\\Big|_{t=0} $z_1^{{-i_1-1}}$\\cdots $z_n^{{-i_n-1}}$$$ as a sum over $n$-cycles of products of $\\hat A^{KP}$ factors, with $A^{KP}$ built from BKP affine coordinates by (1.1), up to an explicit two-point correction. The proof shows that this expression coincides with the cycle sum built from $\\hat A^{BKP}$, and the final comparison is reduced to a purely algebraic identity, Lemma 4.1, for two formal series $s$ and $t$ with $s(y,x)=-s(x,y)$.","pith_inferences":["Because Lemma 4.1 is not machine-checked, a cheap way to test the paper's central equivalence is to verify the identity for small $k$ with random truncated series; the paper does not report such a check.","If the equivalence holds generally, the relation $A^{BKP}(w,z)=\\frac14(zA^{KP}(w,-z)-wA^{KP}(z,-w))$ may be a model for a general embedding dictionary between the affine coordinates of a KP reduction and those of KP itself.","One could test the same technique on the CKP or DKP reductions: if analogous antisymmetrization relations hold, the embedding approach would give a uniform derivation of $n$-point functions across all KP reductions, not just BKP."],"forward_implications":["The two known presentations of BKP $n$-point functions are interchangeable, so computations can use whichever form is easier to evaluate.","BKP $n$-point functions can be obtained without the type-B boson-fermion correspondence: the KP formula plus the $A^{KP}$-from-$A^{BKP}$ conversion is sufficient.","The two-point correction term $-\\delta_{n,2}\\frac{z_1^2+z_2^2}{2(z_1^2-z_2^2)^2}$ reappears naturally from the embedding, matching its role in KdV correlator formulas.","The same embedding strategy should produce $n$-point formulas for other KP reductions, and the paper states that checking these against the known formulas is a natural but technically heavy next step."],"supporting_citations":[{"why":"Supplies the KP connected n-point formula in affine coordinates that the paper embeds to get its BKP formula.","marker":"[Zho15a]"},{"why":"Provides the earlier BKP connected n-point formula via the type-B boson-fermion correspondence that the paper proves equivalent to its new formula.","marker":"[WY22]"},{"why":"Proposes the embedding of reductions into KP, the method the paper carries out for BKP.","marker":"[Zho15b]"},{"why":"Gives the classical relation between KP and BKP tau-functions used as the starting point of the embedding.","marker":"[DJKM82]"},{"why":"Introduces neutral free fermions and the BKP spin representation used to define BKP affine coordinates and the target formula.","marker":"[JM83]"}],"fun_headline_variants":["BKP in KP: two n-point formulas proven equivalent","Embedding BKP into KP unifies n-point formulas","Connected BKP n-point functions: new formula equals old","KP embedding yields equivalent BKP correlation formulas","Two BKP n-point formulas shown identical via KP"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The comparison rests entirely on Lemma 4.1, a purely algebraic identity about two formal series, whose proof in the appendix is a long sign-sensitive induction that the paper does not machine-check; if that identity is wrong, the equivalence theorem falls with it.","fun_headline_variants_meta":{"raw":{"variants":["BKP in KP: two n-point formulas proven equivalent","Embedding BKP into KP unifies n-point formulas","Connected BKP n-point functions: new formula equals old","KP embedding yields equivalent BKP correlation formulas","Two BKP n-point formulas shown identical via KP"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000142,"raw_usage":{"total_tokens":1098,"prompt_tokens":808,"completion_tokens":290,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":424,"completion_tokens_details":{"reasoning_tokens":213}},"tokens_in":424,"tokens_out":290,"duration_ms":3857,"temperature":1.0,"reasoning_tokens":213,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:06:20.825691+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Expand both sides of equation (1.10) for $n=3$ and $n=4$ with a truncated BKP affine-coordinate choice, keeping a few nonzero $a^{BKP}_{m,n}$ and $a^{BKP}_{m,0}$, and compare coefficients in the variables $z_i$ up to a fixed order. A single mismatching coefficient would refute Theorem 1.2. A cheaper check is to verify Lemma 4.1 for $k=2,3$ with generic truncated series $s$ and $t$.","supporting_citations":[],"review_version":1}