{"id":"6c254ce1-6218-40bd-b743-33cf4955355d","arxiv_id":"2412.15198","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Tree-level Yang-Mills amplitudes factorize into gluings of lower-point amplitudes when a rectangular set of Mandelstam variables vanishes, and this paper gives a rigorous CHY-based proof.","lead":"This paper proves a rule for splitting tree-level Yang-Mills scattering amplitudes into sums of smaller amplitudes when a block of momentum invariants vanishes. The proof uses the CHY worldsheet formalism and makes the recently discovered hidden zeros of amplitudes explicit.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The factorization (2.2) hinges on X(s,ρ) being defined via the inverse of g[ρ,π] in (2.5); the paper never proves g is invertible on the hm=0 subspace, so the RHS of (2.2) could be ill-defined where det g=0.","rationale":"The reader's verdict of CONDITIONAL is appropriate because the proof has assertive technical steps. I partially agree with the reader: the pinch classification in §3.2 is actually well-supported (triple pinches would force non-rectangular Mandelstams to scale as τ, contradicting (3.1)), so I would not rank it as the weakest assumption. The more load-bearing gap is the unproven invertibility of g[ρ,π], which is required to define X(s,ρ) in (2.5). Without invertibility, the linear system (5.28) may have no solution, breaking the reduction of the spurious Jacobian in (5.31). The paper's numerical checks on X's pole structure are suggestive but not a proof. A direct determinant test would settle whether this is a removable technicality or a genuine obstruction. Therefore I keep the verdict at CONDITIONAL (UNCHANGED).","tokens_in":41030,"tokens_out":26844,"duration_ms":194241,"concrete_test":"Compute det(g[ρ,π]) for m=3,4,5 with randomly generated massless momenta satisfying hm=0 (using the explicit B coefficients in Appendix A), and check whether the determinant vanishes for a generic sample. If it is nonzero on a dense open set, the definition (2.5) is generically valid and the gap is a missing proof rather than a flaw; if it vanishes on an open set of the constraint subspace, the factorization (2.2) is ill-defined and the proof fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (2.2) is an equality between a finite YM amplitude and a sum of gluings involving the coefficients X(s,ρ). These coefficients are defined in (2.5) as g^{-1}[ρ,π] times a B-term, where the square matrix g[ρ,π] is given in (2.6). The derivation in Section 5.2 determines X(s,ρ) by solving the linear system (5.28); a solution exists only if det g ≠ 0. The paper does not prove this invertibility, and it is not evident from the definition: g involves sums of products of BCJ coefficients B[...|π], which are rational functions of the Mandelstam invariants. If det g vanishes on any open region of the constraint subspace hm=0, then X(s,ρ) has a spurious pole and the RHS of (2.2) is not a well-defined function of the kinematics, while the LHS is finite. The paper's observation in §5.2.3 that X contains only physical poles up to m=7 is numerical evidence that the spurious poles cancel after summing over ρ, but this does not establish invertibility. The reader's weakest assumption—the pairwise-only pinch classification in §3.2—is on firmer ground: a triple pinch σa≈σb≈σb' would introduce a term sbb'/σbb' ∼ O(τ^{-1}) in ˚Eb with sbb' generically O(1), so no solution exists; this is standard CHY reasoning. Thus the least secure step is the linear algebra defining X.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to prove, within the CHY formalism, the new factorization of tree-level Yang-Mills amplitudes conjectured in the author's companion paper: when the rectangular Mandelstam matrix hm of (2.1) vanishes, the n-point amplitude decomposes into a sum of gluings of a three-point amplitude and linear combinations of (n-1)-point amplitudes, as written in (2.2). The proof proceeds by analyzing singular solutions of the scattering equations under the scaling sab=τ ŝab, showing that only pairwise pinches contribute, reducing the Pfaffian and Parke-Taylor factors to lower-point objects, and introducing coefficients X(s,ρ) defined through the inverse of a matrix g[ρ,π] in (2.5)-(2.6). The paper works out the cases of one, two, three, and then an arbitrary number of entries of the polarization matrix Ĥm turned on, with the key cancellations encoded in identities for t-coefficients, proved in Appendix C and verified by explicit examples for m=2 and m=3.","tokens_in":41428,"tokens_out":4124,"duration_ms":33443,"significance":"If the proof is completed, this would be a significant result: it would give a first-principles CHY derivation of a new recursive structure in Yang-Mills amplitudes, making the hidden zeros of [13] manifest and connecting them to a concrete factorization formula. The manuscript has several strengths: it starts from the standard CHY representation rather than assuming the factorization; it provides explicit m=2 and m=3 examples in Appendices B and C; it includes numerical checks for the coefficients X(s,ρ) up to m=7; and it develops a general combinatorial cancellation scheme that reduces the problem to identities among purely rational functions of Mandelstam invariants. These are concrete, checkable contributions. However, two load-bearing technical steps are not fully proven: the invertibility of the matrix g defining X(s,ρ), and the classification of all singular solutions as pairwise pinches. As they stand, these gaps prevent the argument from being a complete proof for arbitrary multiplicity.","major_comments":[{"comment":"The coefficients X(s,ρ) are defined in (2.5) as g^{-1}[ρ,π] times a B-term, where g[ρ,π] is the square matrix in (2.6). The derivation in §5.2.2 solves the linear system (5.28) for X(s,ρ); a unique solution exists only if the matrix g is invertible on the subspace hm=0. The paper does not prove this invertibility. The matrix entries are sums of products of BCJ coefficients B[...|π], which are rational functions of Mandelstam invariants, so det g could vanish on some open region of the hm=0 subspace without any obvious contradiction. If det g=0 on such a region, the right-hand side of (2.2) is ill-defined while the left-hand side is a finite amplitude. The observation in §5.2.3 that X(s,ρ) contains only physical poles up to m=7 is numerical evidence that spurious poles cancel in the sum over ρ, but it does not establish invertibility of g for arbitrary m. Since (2.2) is claimed for all n and all 1≤m≤n−3, this is a load-bearing gap and must be addressed by a proof or a precise condition on the kinematics.","section":"§5.2.2–5.2.3, Eqs. (2.5)–(2.6), (5.28)"},{"comment":"The proof relies on the classification of singular solutions of the scattering equations as containing only pairwise pinches of the form (3.4). The manuscript states 'we claim' that no three punctures pinch simultaneously and cites [30,31,40], but does not give a self-contained proof. This classification is essential: if a triple pinch σa≈σb≈σb′ occurred at leading order, additional terms would appear in the leading scattering equations and the enumeration of contributing solutions underlying every subsequent reduction would be incomplete. The τ-parameterization argument in the text is plausible and consistent with standard CHY lore, but the paper should either provide a complete proof or state this as a precise lemma with a rigorous derivation, since the factorization theorem depends on it.","section":"§3.2, Eq. (3.4)"},{"comment":"The proof of the key cancellation identity (8.7) contains several steps that are asserted rather than demonstrated. In C.2.5, the text says 'We claim that A vanishes algebraically' and then concludes that the coefficients must vanish without providing the actual computation. In C.2.6, it says 'we assert that X⋆(s,ρv′) = X(s,ρv′)' and 'this follows directly' from the definition, but the identification involves a nontrivial step where the boundary terms in (C.13) are absorbed. The ansatz in C.2.2 for the boundary terms with undetermined functions Z⋆ is introduced without showing that it spans all possible contributions. Since (8.7) is the mechanism that cancels all (n−u)-point CHY integrals for u≥2, these gaps are load-bearing; the proof needs to be completed with explicit algebraic identities, not just assertions.","section":"Appendix C, Eqs. (8.7), (C.14)–(C.20)"}],"minor_comments":[{"comment":"The text says 'Plugging (5.26) and (5.26) into (5.25)' but the second reference should be (5.27).","section":"§5.2.2, sentence after Eq. (5.25)"},{"comment":"In the sentence after (C.35), the text writes 'subtracting s123PT(123)' but the identity being proved, (C.31), involves s1234PT(1234); this appears to be a typographical error.","section":"Appendix C.3.2, around Eq. (C.36)"},{"comment":"The notation s2n in (5.30) is used without definition in the main text; it is later explained in Appendix C.3.2 as −s12−s23−s24 on the support of hm=0, but the reader would benefit from an explicit definition near (5.30).","section":"Eq. (2.4) and Eq. (5.30)"},{"comment":"The term −si1i2/σj1j2 in (6.10) is not self-explanatory; the explanation in the following paragraph is helpful but should be moved closer to the equation or made more explicit in the display.","section":"Eq. (6.10)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript presents an interesting and potentially important derivation, but the proof as written is not fully complete: the invertibility of g and the pinch classification are load-bearing assumptions, and the proof of the key identity (8.7) in Appendix C contains several unproven assertions. These are fixable within the manuscript's scope, so I recommend major revision rather than rejection. I would also note that the target formula (2.2) is taken from the author's own companion paper; while the derivation here does not assume (2.2), the lack of an independent check for the general structure makes the numerical and small-m checks even more important. The paper would be strengthened by a self-contained proof of invertibility of g on the hm=0 subspace (or a counterexample showing where the present definition fails) and by a rigorous treatment of the singular-solution classification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it delivers what it promises: a real CHY-based proof of the factorization conjecture from the companion paper [1]. The central machinery—classifying singular solutions, deriving induced lower-point scattering equations, and canceling all lower-point integrals through the t-coefficient identities—is a substantial piece of work. Second, the proof has a genuine technical gap that needs addressing before I would call it airtight, and it is not the one flagged in the reader's report.\n\nThe strongest parts are the explicit reductions. Section 5 works out the one-entry case and produces the factorization formula including the X(s,ρ) coefficients; the examples for m=2 and m=3 in Section 6 and Appendix B are concrete and checkable. Appendix C is a serious attempt at proving the key identity (8.7), and the constructive argument for decomposing PT factors with boundary terms is the most original part of the paper. If that identity is correct, and the examples strongly suggest it is, the cancellation argument is convincing.\n\nThe reader's weakest assumption—pairwise-only pinches—is actually fine. The stress-test note is right: a triple pinch would force s_{abb'} ~ O(τ), inconsistent with the CHY parameterization. That step is standard and I would not worry about it.\n\nThe real issue is the invertibility of g[ρ,π]. The X(s,ρ) coefficients are defined in (2.5) through g^{-1}, and Section 5.2 solves a linear system (5.28) to determine them. The paper never proves det g ≠ 0 on the hm=0 subspace. If the determinant vanishes on any open region, the right-hand side of (2.2) is not well-defined, while the left-hand side is. The numerical check up to m=7 that X has only physical poles is suggestive, but it does not establish invertibility. This is a load-bearing step, and it is fixable: one either proves generic invertibility or defines X(s,ρ) through the linear system with a regulator and shows the final sum is regulator-independent. The paper should do one of those.\n\nMinor issues: Appendix C is dense and hard to verify line-by-line; some typos like 'descrapency' and the notation ˚E_i / PT reductions need cleanup. None of that affects the logic.\n\nWho is this for? Amplitude practitioners working on hidden zeros, CHY formalism, or recursive structures. It is a serious proof paper that deserves a referee. I would send it to review, with the request that the referee focus on the invertibility gap and verify the t-coefficient identity for at least one nontrivial case beyond m=3.\n\nBottom line: solid work, one real gap, worth engaging with seriously.","headline":"Genuine CHY proof of the companion-paper factorization, with a load-bearing but likely fixable gap around invertibility of the g-matrix.","tokens_in":41874,"tokens_out":1798,"would_cite":true,"duration_ms":18983,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Tree-level Yang-Mills amplitudes decompose into glued three-point and $(n-1)$-point amplitudes whenever a rectangular block of Mandelstam invariants vanishes, making hidden zeros manifest.","keywords":["Yang-Mills amplitudes","hidden zeros","CHY formalism","scattering equations","tree-level factorization","BCJ relations","Mandelstam matrix","gluon amplitudes"],"falsifier":"Evaluate a concrete case, such as the $n=7$, $m=3$ color-ordered YM amplitude with $h_3=0$ and only $x_{1,5}$, $x_{2,6}$ turned on, using an independent numerical method, and compare with $F_{3,7}(1,5)+F_{3,7}(2,6)$ from (2.3).","tokens_in":40816,"feed_emoji":"⚛️","tokens_out":10062,"duration_ms":77335,"temperature":0.7,"pith_summary":"This paper proves a factorization pattern for tree-level Yang-Mills (YM) amplitudes that was proposed in the companion paper. The central claim is equation (2.2): when the rectangular matrix of Mandelstam invariants $h_m$ vanishes, an $n$-point color-ordered YM amplitude equals a sum over gluon pairs $(i,j)$ of $F_{m,n}(i,j)$, each term being a gluing of a three-point amplitude $A(ij\\,-\\hat j)$ with an $(n-1)$-point YM amplitude weighted by rational coefficients $X(s,\\rho)$. The proof uses the CHY formalism, an integral representation over puncture positions on a sphere, and analyzes the singular solutions of the scattering equations. If the claim is right, the previously conjectured hidden zeros of YM amplitudes become manifest, and higher-point amplitudes can be built recursively from lower-point ones under specific kinematic constraints.","feed_headline":"Tree-level YM amplitudes factorize under rectangular kinematic zeros","feed_subtitle":"Vanishing Mandelstam block decomposes n-gluon amplitudes into three-point and (n-1)-point pieces.","key_machinery":"The load-bearing object is the CHY integral, which represents a YM amplitude as a sum over solutions of the scattering equations $\\sum_b s_{ab}/(\\sigma_a-\\sigma_b)=0$ on a punctured sphere, with a reduced Pfaffian encoding polarizations. Under $h_m=0$ the equations have no regular solutions; the paper classifies the singular ones as $r$-pinch solutions in which up to $\\min(m,n-m-2)$ pairs of punctures coincide at order $\\tau$. Each $r$-pinch solution induces $(n-r)$-point scattering equations, and the reduced Pfaffian factorizes into three-point amplitudes $A(ij\\,-\\hat j)$ times an $(n-r)$-point Pfaffian; the PT factor is reduced with the help of BCJ relations, and the identity (8.7) among the resulting t-coefficients cancels all unwanted lower-point integrals.","core_discovery":"The paper establishes that for $1\\le m\\le n-3$, once the rectangular block $h_m$ of Mandelstam variables $s_{ab}=k_a\\cdot k_b$ with $a\\le m$ and $m+2\\le b\\le n-1$ is set to zero, the tree-level color-ordered YM amplitude $A(I_n)$ satisfies condition (2.2); the right-hand side is a sum over $1\\le i\\le m$, $m+2\\le j\\le n-1$ of $F_{m,n}(i,j)$, where each $F_{m,n}(i,j)$ is a three-point amplitude $A(ij\\,-\\hat j)$ multiplied by a linear combination of $(n-1)$-point YM amplitudes with rational coefficients $X(s,\\rho)$ built from Mandelstam variables through BCJ relations. The proof derives this factorization from the CHY formula by showing that all solutions of the scattering equations become singular, classifying them as $r$-pinch configurations, reducing each to induced $(n-r)$-point scattering equations, factorizing the reduced Pfaffian into three-point amplitudes times lower-point Pfaffians, and using the t-coefficient identity (8.7) to cancel every $(n-2)$-point and lower-point CHY integral.","pith_inferences":["A natural extension, not pursued by the paper, is to turn the factorization into a practical bootstrap: if the $X(s,\\rho)$ coefficients can be generated efficiently, the formula recursively determines all tree-level YM amplitudes from three-point input, offering an alternative to pole-based recursion.","The same cancellation mechanism may transfer to gravity amplitudes through the double copy, but the second reduced Pfaffian brings additional structure; testing the analogous decomposition for GR is a concrete next step the paper leaves open.","The t-coefficient identity depends only on the $m+1$ punctures, so it may survive as a combinatorial statement in string-theory settings, suggesting the factorization could extend to open-string disk amplitudes."],"forward_implications":["The hidden zeros of YM amplitudes are recovered as the special case where the polarization block $\\hat H_m$ also vanishes: the right-hand side of (2.2) becomes zero.","The factorization yields a recursive construction of higher-point YM amplitudes from three-point and $(n-1)$-point data, with the kinematic coefficients $X(s,\\rho)$ fixed by BCJ relations.","The result holds for arbitrary subsets of turned-on entries: one entry, whole rows or columns, and general non-aligned configurations all produce sums of the same $F_{m,n}(i,j)$ building blocks.","Because the decomposition is an equality on the support of $h_m=0$, it upgrades the hidden-zero statement from a vanishing condition to a complete formula for the amplitude in that kinematic subspace."],"supporting_citations":[{"why":"proposes the new factorization pattern (2.2)-(2.3) that this paper sets out to prove.","marker":"[1]"},{"why":"supplies the CHY integral representation of massless scattering amplitudes used as the proof's starting point.","marker":"[30]"},{"why":"provides the CHY formulation of Yang-Mills amplitudes with the reduced Pfaffian integrand.","marker":"[31]"},{"why":"introduces hidden zeros and the splitting framework whose vanishing case the factorization recovers.","marker":"[13]"},{"why":"gives the BCJ relations used to define the coefficients $B_{m+1,i}$ and hence $X(s,\\rho)$.","marker":"[27]"},{"why":"supplies the BCJ/color-kinematics review whose relations also enter the definition of $X(s,\\rho)$.","marker":"[32]"},{"why":"supports the parameterization of singular scattering-equation solutions used to justify the pairwise-pinch classification.","marker":"[40]"}],"fun_headline_variants":["YM amplitudes factorize under vanishing rectangular Mandelstam blocks","New CHY proof: YM amplitudes split under rectangular kinematic zeros","Tree-level YM amplitudes decompose via rectangular Mandelstam zeros","Rectangular zeros expose recursive decomposition of YM amplitudes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that when the Mandelstam block $h_m$ is scaled by a small parameter $\\tau$, every singular solution of the scattering equations consists of pairwise pinches of punctures, with no simultaneous three-puncture pinches; if a singular solution outside this classification exists, the factorization proof would be incomplete.","fun_headline_variants_meta":{"raw":{"variants":["YM amplitudes factorize under vanishing rectangular Mandelstam blocks","New CHY proof: YM amplitudes split under rectangular kinematic zeros","Tree-level YM amplitudes decompose via rectangular Mandelstam zeros","Rectangular zeros expose recursive decomposition of YM amplitudes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00069,"raw_usage":{"total_tokens":3126,"prompt_tokens":947,"completion_tokens":2179,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":2108}},"tokens_in":563,"tokens_out":2179,"duration_ms":12306,"temperature":1.0,"reasoning_tokens":2108,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:32:26.283625+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate a concrete case, such as the $n=7$, $m=3$ color-ordered YM amplitude with $h_3=0$ and only $x_{1,5}$, $x_{2,6}$ turned on, using an independent numerical method, and compare with $F_{3,7}(1,5)+F_{3,7}(2,6)$ from (2.3).","supporting_citations":[],"review_version":1}