{"id":"2b997c6d-d46d-41aa-a436-3086957f8a0e","arxiv_id":"2512.20790","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"2-split amplitude factorization is extended to (DF)^2, R^2 and R^3 theories and recast as a product of shifted lower-point on-shell amplitudes.","lead":"This paper extends the recently discovered 2-split factorization of tree-level amplitudes to higher-derivative gauge and gravity theories, and rewrites the split factors as lower-point on-shell amplitudes instead of gauge-dependent currents. The payoff is a cleaner handle on a structure that may be a universal feature of the S-matrix.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"General-n on-shell 2-split representation is asserted from n=5 examples; the kinematic-shift identification (eqs. 4.11–4.16, A.16) needs a higher-point check before the central claim can be accepted as established.","rationale":"The paper is a serious extension of the 2-split program. The explicit five-point BA and (DF)^2 checks are internally consistent, and the splitting of W and the reduced Pfaffian in higher-derivative theories is a natural and credible generalization. The soft spot is exactly the step from examples to a general on-shell representation: eqs. (4.11)–(4.16) and (A.16) are stated for arbitrary n, but the only evidence is n=5 plus an argument about the structure of W and Ψ. The Naculich shift changes the scattering equations, not just the final rational expressions; the paper does not show that localizing the shifted CHY integrand reproduces the current for all multiplicities. The Appendix A.2 choice of independent Mandelstam sets is a prescription, not a proof of sufficiency or choice-independence. This is a gap in the central claim, so I would not move the verdict: CONDITIONAL remains appropriate pending an explicit higher-point test. I agree with the reader's weakest-assumption identification.","tokens_in":18300,"tokens_out":11941,"duration_ms":117195,"concrete_test":"Compute the six-point BA amplitude A^{φ3}(1,2,3,4,5,6) with I={1,3,6}, A={2}, B={4,5} under the 2-split conditions s_{2,4}=s_{2,5}=0. Directly evaluate the two currents J^{φ3}(1,2,3,κ), with pκ=-(p1+p2+p3)=p4+p5+p6, and J^{φ3}(1,3,4,5,κ′), with pκ′=-(p1+p3+p4+p5)=p2+p6, via Berends-Giele recursion or CHY. Then evaluate the lower-point shifted amplitudes A_4^{φ3}(1,2,3,κ)|_{Σ(K_L^6)} and A_5^{φ3}(1,3,4,5,κ′)|_{Σ(K_R^6)} with K sets from Appendix A.2. If the product differs from the direct split, eq. (4.11)/(A.16) fails at n=6; if it matches, repeat for the YM analogue (4.14) with the same split to test the Pfaffian/W-shift sector.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central new claim is that any 2-split amplitude can be written as a product of lower-point on-shell amplitudes with shifted kinematics: A(1,...,n) -> A^L(i,A,j,κ)|_{Σ(K_L^n)} × A^R(1,...,i,j,...,κ')|_{Σ(K_R^n)} (eq. A.16; cf. 4.11–4.14). This requires that for arbitrary n the CHY amputated current equals the massive-leg amplitude evaluated after the Naculich shift (4.8), and that this identification commutes with taking the split limit. What is actually demonstrated is n=5: the BA and (DF)^2 examples in Sec. 4.2 and Appendix A.1, plus integrand-level arguments for W and Ψ. The general statement in A.2 is an assertion: it prescribes a choice of independent Mandelstam sets K_L^n and K_R^n, but does not prove that these sets are sufficient, that the shifted lower-point CHY integrals reproduce the currents for all n, or that the result is independent of the ordering choice in (A.10). The integrand splitting of W and Ψ does not by itself establish the integrated statement, because Σ(κ) acts inside the scattering equations and changes which CHY solutions contribute; replacing s by s+Δ in the final rational amplitude is not automatically equivalent to integrating the shifted integrand. Since Sec. 5's transmuting-operator split identities (5.11)–(5.15) inherit this representation, the gap propagates to the paper's advertised generalization.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the recently discovered 2-split behavior of tree-level scattering amplitudes, in which an n-point amplitude factorizes into a product of lower-point amputated CHY currents when certain kinematic invariants and polarization dot products are set to zero. The authors make two main claims: (i) the 2-split behavior extends to higher-derivative theories, specifically (DF)^2 gauge theory, R^2 gravity, and R^3 gravity; and (ii) the split amplitudes can be recast in terms of genuine lower-point on-shell amplitudes with shifted kinematic data (the 'on-shell representation'), thereby avoiding gauge-dependent off-shell currents. The second claim is developed in Section 4, with explicit five-point checks for bi-adjoint scalars and (DF)^2, a component-wise argument for the shifted YM integrand, and a general formula (A.16) in Appendix A.2. Section 5 generalizes transmuting operators to the higher-derivative theories, and Section 6 extends part of the analysis to 3-splits and hidden zeros.","tokens_in":18713,"tokens_out":20679,"duration_ms":185525,"significance":"If the on-shell representation is valid for arbitrary multiplicity, it is a useful conceptual advance: it removes the gauge dependence of the currents appearing in the earlier 2-split literature and states the splitting as a factorization of physical amplitudes. The extension to (DF)^2, R^2, and R^3 strengthens the evidence for universality of the 2-split phenomenon. The paper contains explicit five-point checks for the BA and (DF)^2 cases and a transparent CHY integrand-level argument for the splitting of W and Pfaffian; the derivations are parameter-free and algebraic in nature. The main weakness is that the general-n identification is asserted rather than proved, and the proposed independent kinematic variable sets in Appendix A.2 are not proven to be independent. These points are load-bearing for the advertised on-shell representation.","major_comments":[{"comment":"The central claim that the 2-split amplitude equals a product of lower-point amplitudes evaluated with shifted kinematics is asserted for arbitrary n after explicit n=5 checks. The equality J^L(i,A,j,κ)=A^L(i,A,j,κ)|_{K_L^n} requires that the Naculich shift (4.8), applied inside the lower-point CHY integrand, reproduces the amputated current after integration, and that this identification commutes with the split limit used in Section 2. The component-wise split of W and Ψ is not sufficient to establish the integrated statement, because the shift acts on the scattering equations and changes the set of CHY solutions; replacing s by s+Δ in the final rational expression is not automatically the same as integrating the shifted integrand. Please provide a proof for general n (for example, by stating and applying the precise Naculich theorem for amplitudes with up to three massive legs) or, at","section":"Sec. 4.2; App. A.2, Eq. (A.16)"},{"comment":"The proposed independent Mandelstam sets K_L^n and K_R^n are not proven to be independent. The five-point matrices (A.1) are checked by inspection, but for general n the statement that deleting the i,j,k columns and one additional entry leaves n(n-3)/2 - 1 independent invariants is an assertion. Since Eq. (A.16) is the compact form of the paper's main result, this linear-algebra point is load-bearing. The authors should provide a proof of independence, or at least a constructive change-of-basis argument, and should show that the evaluation is independent of the ordering choice in (A.10).","section":"App. A.2, Eqs. (A.8)-(A.16)"},{"comment":"The splittings for R^2 and R^3 gravity are stated without any explicit verification. In particular, the mixed currents such as J^{R2+YM} and J^{R3+(DF)^2} are said to follow from [16], but the paper does not show how the relevant mixed CHY integrand splits under the simultaneous imposition of the two different polarization conditions (2.20) and (3.16). Please provide at least an n=4 or n=5 example for one of these mixed cases, or an integrand-level derivation, so that the claimed extension to higher-derivative theories is not purely an assertion.","section":"Sec. 3.2, Eqs. (3.21)-(3.25)"}],"minor_comments":[{"comment":"The summation 'X_{a∈a}' should read 'a∈A'.","section":"Eq. (2.8b)"},{"comment":"The reference to 'the splitting (6.10)' at n=4 should likely be Eq. (3.5), as Eq. (6.10) is the later 3-split formula.","section":"Sec. 3.1, text near Eq. (3.11)"},{"comment":"The third measure is labeled dµ_A but should be dµ_C (it depends on κ_C and the set C).","section":"Eq. (6.3)"},{"comment":"The notation p34 and p13 is used without definition; please define p_{IJ}=p_I+p_J or similar.","section":"Eq. (A.6)"},{"comment":"The argument of A^{DF2+φ3}(1_φ,...,i_φ,i+1,...,n) is unclear: if leg i is a scalar, it should not also appear as i+1 in the same ordering. Please fix the notation.","section":"Sec. 5, Eq. (5.10)"},{"comment":"The entry removed from the left/right matrices, s_{j,j+1} and s_{j,i+1}, may not be upper-triangular for all orderings in (A.10); please specify the rule in a way that is unambiguous for arbitrary i,j.","section":"App. A.2, text after Eq. (A.10)"}],"recommendation":"major_revision","confidential_remarks":"The central idea is promising and the explicit checks are useful, but the general-n on-shell representation is currently an assertion rather than a proven statement. The gap is likely fixable by invoking the Naculich massive CHY framework more explicitly and by proving the linear independence of the kinematic sets. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid paper and the main idea is probably right, but the headline claim — that any 2-split amplitude is a product of lower-point on-shell amplitudes with shifted kinematics — is demonstrated for n=5 and then asserted for all n. That gap should be addressed before the result is treated as established.\n\nWhat is actually new: Section 3 shows the W integrand (Lam-Yao cycles) splits under the same conditions as the reduced Pfaffian, so (DF)^2, R^2, and R^3 amplitudes inherit 2-split behavior. The explicit five-point checks for BA and (DF)^2 work and match the split product formulas. Section 4 recasts splits in terms of on-shell amplitudes using the Naculich shift; that is a genuine step beyond the gauge-dependent currents in [4,5]. Section 5 introduces transmuting operators for W and gives clean identities. The paper is written honestly, the derivations are explicit, parameter-free, and the split identities are algebraic consequences of CHY integrands.\n\nWhere it is soft: the general-n on-shell representation is not proven. The component-wise splitting of the integrands is fine, but the shift Σ(κ) acts inside the scattering equations, and the claim that the shifted CHY integrals reproduce the amputated currents for arbitrary n is an assertion. The appendix constructs kinematic sets K_L^n and K_R^n and states they are independent, but independence is not the same as sufficiency: you need to show the shifted lower-point amplitude, after taking the split limit, gives the current for all n. The n=5 checks are encouraging, but a six- or seven-point test (even numerical) would materially raise confidence. The 3-split discussion in Section 6 is even sketchier, though it is presented as an extension rather than the main result.\n\nCitation pattern: the paper leans on [14,17,19], all co-authored by the present authors, for the CHY representations and the shifting method. That is not by itself a flaw — those are published results — but it does mean the independent content is thinner than the bibliography suggests.\n\nBottom line: worth a serious referee. The core examples support the claim, and the on-shell representation is a natural and potentially important reformulation. But the referee should push for a higher-point check or a genuine argument that the shift commutes with the split limit for arbitrary multiplicity.","headline":"Useful extension of the 2-split program: the on-shell recasting is a real step forward, but the general-n claim is asserted from n=5 examples rather than proven.","tokens_in":19183,"tokens_out":2797,"would_cite":true,"duration_ms":30254,"reading_group":"maybe","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 amplitudes can be rewritten as products of lower-point on-shell amplitudes when certain kinematic conditions are imposed, a property now extended to higher-derivative gauge and gravity theories.","keywords":["2-split","tree-level amplitudes","on-shell representation","kinematic shift","higher-derivative gauge theory","higher-derivative gravity","transmuting operators","hidden zeros"],"falsifier":"Take a six-point amplitude in Yang-Mills (or (DF)^2) and impose the 2-split conditions with A={2,3} and B={4,5} (in the appropriate ordering). Compare the direct amplitude restricted to these conditions with the product of shifted lower-point amplitudes A^{YM+φ^3}_5(...|_{Σ}) × A^{YM}_5(...|_{Σ}) using the independent kinematic variables defined in the appendix. If the two disagree at any non-zero kinematic point, the shift does not commute with the split limit and the on-shell representation fails for n=6.","tokens_in":18154,"feed_emoji":"✂️","tokens_out":5989,"duration_ms":55412,"temperature":0.7,"pith_summary":"This paper aims to show that the 2-split behavior of tree-level amplitudes—the factorization of an amplitude into lower-point pieces when certain kinematic invariants and polarization entries vanish—can be stated entirely in terms of on-shell amplitudes, avoiding the gauge-dependent off-shell currents used previously. It further demonstrates that the 2-split holds in higher-derivative theories: the (DF)^2 gauge theory, as well as R^2 and R^3 gravity. The argument uses a specific kinematic shift that turns lower-point on-shell amplitude integrands into the off-shell current structures, so the split amplitude becomes a product of shifted lower-point amplitudes. As a byproduct, the paper extends transmuting differential operators to these higher-derivative theories and connects the 3-split behavior to hidden zeros.","feed_headline":"2-split amplitudes become products of on-shell ones","feed_subtitle":"Kinematic shifting removes gauge-dependent currents and extends the split behavior to higher-derivative gauge and gravity theories.","key_machinery":"The central object is the kinematic-shift operation Σ(κ) defined in eq. (4.8): it replaces each scalar product s_{a,b} of momenta by s_{a,b}+Δ_{a,b}, where Δ involves the off-shellness of the shifted leg and the on-shell partners (i, j). When applied to lower-point on-shell amplitudes, this shift reproduces the amputated off-shell currents that appear in the split factorization, so the split amplitude becomes a genuine product of on-shell amplitudes. For higher-derivative theories the additional ingredient is the W integrand—a product over each particle of sums of polarization–momentum contractions—which splits cleanly under the same conditions. The transmuting differential operators (built","core_discovery":"The paper's central claim is that, under the 2-split conditions (vanishing of cross-set invariants and polarization products), a tree-level amplitude in a broad class of theories factorizes as a product of lower-point amplitudes evaluated with shifted kinematics: for example, A^{YM}(1,...,n) → A^{YM+φ^3}(i_φ, A, j_φ, κ_φ) restricted to the shift Σ(κ), multiplied by A^{YM}(1,...,i,j,...,κ′) restricted to Σ(κ′) and with the polarization of κ′ replaced by that of n (eq. 4.14), with analogous formulas for (DF)^2, R^2, R^3, and gravity. The kinematic shift Σ(κ) modifies the momentum invariants involving the off-shell leg in a specific way so that the off-shell current from the splitting is reprod","pith_inferences":["Editorial extension: the on-shell product form suggests that a recursive 'split-based' recursion could be built by iterating the 2-split at lower points, possibly offering an alternative to momentum-shift recursion schemes for extracting all-multiplicity amplitudes in these theories.","Editorial extension: since the kinematic shift is defined using only invariants, the same device might be tested at one-loop level by applying the shift to integrands before integration, though commutation with the extra loop integration is not addressed in the paper.","Editorial extension: the paper's dependence on the choice of independent invariant sets K_L and K_R hints that different choices must satisfy consistency conditions; exploring these could yield new 'kinematic basis' identities for amplitudes."],"forward_implications":["If the 2-split is an on-shell property, gauge artifacts are removed from the factorization statements; split amplitudes can be computed by recursively feeding lower-point amplitudes into the shifted formula.","The extension to (DF)^2, R^2, and R^3 theories makes the 2-split a robust feature across gauge and gravity theories with higher-dimensional operators, reinforcing the universality claim.","The generalized transmuting operators give new amplitude relations among higher-derivative theories, e.g., expressing ϕ^3 amplitudes from (DF)^2 amplitudes and vice versa.","The 3-split analog in Section 6 implies that hidden zeros appear whenever one of the shifted lower-point amplitude factors vanishes on a sub-locus, providing a direct mechanism for zeros."],"fun_headline_variants":["On-shell form found for split amplitudes","2-split amplitudes now on-shell","Higher-derivative theories also show 2-split amplitudes","2-split amplitudes: on-shell and universal"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The rewritten 2-split formula assumes that the kinematic shift Σ(κ) derived for off-shell worldsheet currents can be applied to lower-point on-shell amplitude integrands and commutes with the split limit for arbitrary multiplicity; the paper verifies this at five points rather than proving it in general.","fun_headline_variants_meta":{"raw":{"variants":["On-shell form found for split amplitudes","2-split amplitudes now on-shell","Higher-derivative theories also show 2-split amplitudes","2-split amplitudes: on-shell and universal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00035,"raw_usage":{"total_tokens":1698,"prompt_tokens":642,"completion_tokens":1056,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":386,"completion_tokens_details":{"reasoning_tokens":999}},"tokens_in":386,"tokens_out":1056,"duration_ms":9533,"temperature":1.0,"reasoning_tokens":999,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T14:16:47.559791+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a six-point amplitude in Yang-Mills (or (DF)^2) and impose the 2-split conditions with A={2,3} and B={4,5} (in the appropriate ordering). Compare the direct amplitude restricted to these conditions with the product of shifted lower-point amplitudes A^{YM+φ^3}_5(...|_{Σ}) × A^{YM}_5(...|_{Σ}) using the independent kinematic variables defined in the appendix. If the two disagree at any non-zero kinematic point, the shift does not commute with the split limit and the on-shell representation fails for n=6.","supporting_citations":[],"review_version":1}