{"id":"90137442-b99d-4aea-b92a-3d86d2c303e7","arxiv_id":"2506.22538","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Imposing 5-point split conditions and unitarity bounds selects the string beta function as the unique 4-point amplitude, up to equal-mass infinite spin towers.","lead":"This paper shows that requiring five-particle amplitudes to obey hidden zero and split conditions imposes nonlinear constraints on four-particle interactions, and feeding those constraints into the numerical S-matrix bootstrap yields a non-convex allowed region. Under a spectral assumption, the region shrinks to a tiny island around the string beta function amplitude, indicating the string is the unique amplitude in this class.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uniqueness claim rests on the non-generic improved Regge assumption (1.10); Appendix C shows bifurcation and the shrinking island are not established under the standard Froissart bound, so the central conjecture is conditional.","rationale":"The reader identified exactly the same weak spot: the improved Regge behavior (1.10) is non-generic, once-subtracted dispersion relations (3.7) depend on it, and Appendix C shows that with standard Regge behavior the bifurcation and shrinking island cannot be cleanly established. My reading of the full text confirms this is the correct load-bearing concern. The analytic derivation of the nonlinear split constraints (Table 1) is self-contained and appears internally consistent; the example amplitudes (beta function and IST) satisfy the constraints, and the scalar-vector exchange is correctly excluded. The numerics are carefully described: SDPB with kmax up to 10, d=10 and d=4, Y-scan and double X-Y scans, with caveats about which nonlinear constraints can be linearized (k=7,9,10 cannot). The shrinking-island evidence is real finite-kmax evidence, and the paper is appropriately careful in labeling the uniqueness statement a conjecture. However, the central claim is precisely the extrapolation to infinite kmax, and that extrapolation is demonstrated only under (1.10). Appendix C's honest discussion shows the argument does not go through under the standard Froissart bound, so the verdict should remain CONDITIONAL rather than being upgraded to a claim about standard assumptions. No code or SDPB input files are released, which further limits reproducibility of the island bounds, though this is secondary to the Regge concern. I agree with the reader's verdict CONDITIONAL, and no change is needed beyond what the reader already stated.","tokens_in":37058,"tokens_out":1656,"duration_ms":16242,"concrete_test":"Repeat the d=10, mu_c=2 splitting bootstrap using twice-subtracted dispersion relations as in Appendix C, but replace the finite grid scan with the Skydiving boundary-finding algorithm to resolve whether the allowed region actually pinches off as kmax increases beyond 8. If Skydiving finds no bifurcation, or finds a bridge connecting the string neighborhood to the bulk region at arbitrarily high kmax, the uniqueness claim fails under standard Regge behavior.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central claim is that, absent an infinite spin tower at the mass gap, the string beta function amplitude is the unique 4-point amplitude compatible with hidden zero and the 5-point splits. The bootstrap evidence for this claim depends critically on the improved Regge behavior (1.10), i.e. A4(s,u)/s → 0 at fixed u and fixed t. The paper explicitly acknowledges this assumption is not generic and is stronger than the Froissart-Martin bound. Appendix C shows that with the standard twice-subtracted behavior, the relevant dispersion relations lose a0,0 = c1,0, the k=2,3 splitting constraints must be re-solved via (C.6), the Y-scan linearization is unavailable, and the allowed region does not cleanly bifurcate up to kmax=8; the paper states the island could 'escape' between grid points. Since uniqueness is inferred from the existence of a shrinking island that bifurcates off the trivial region, and since that inference fails precisely when the Regge assumption is relaxed, the uniqueness conclusion is not a property of unitarity plus splits per se, but of unitarity plus splits plus the non-generic Regge assumption. This is the load-bearing weak point: the conjecture's empirical evidence does not reach the standard assumptions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies how hidden-zero and split conditions on higher-point amplitudes constrain the 2-to-2 low-energy Wilson coefficients in weakly coupled adjoint scalar EFTs. The authors derive order-by-order nonlinear relations among the 4-point coefficients from the 5-point split conditions (e.g., Eq. (1.8) and Table 1), eliminate the cubic coupling using the k=2 constraint, and implement linearized versions of these constraints in an SDPB-based bootstrap with once-subtracted dispersion relations under the improved Regge assumption (1.10). The main numerical results are that the allowed region becomes non-convex, that assuming a finite number of states at the mass gap and a cutoff leads to bifurcation of the allowed region, and that an island around the string beta-function amplitude shrinks as more constraints are included. The paper concludes with the conjecture that the beta-function amplitude is the unique 4-point amplitude compatible with hidden zero, split, unitarity, and absence of an infinite spin tower at the mass gap.","tokens_in":37211,"tokens_out":8407,"duration_ms":92281,"significance":"If the uniqueness conjecture is correct, this is a striking demonstration that higher-point factorization properties can identify a specific 4-point amplitude without invoking string-theoretic input. The analytical derivation of the nonlinear constraints is explicit and is checked against the beta-function and infinite-spin-tower amplitudes; the numerical implementation using standard null constraints and SDPB is transparent; and the comparison with the monodromy-based bootstrap island places the result in a useful context. The main caveat is that the numerical evidence for uniqueness is conditional on the non-generic improved Regge behavior (1.10), as the authors themselves acknowledge in Appendix C, so the strength of the claim in the abstract and in Section 4.4 needs to be aligned with that condition.","major_comments":[{"comment":"The central uniqueness claim is conditional on the improved Regge behavior (1.10), which the paper explicitly states is stronger than the Froissart-Martin bound. The abstract and Section 4.4 state that, absent an infinite spin tower at the mass gap, the string beta function is unique among amplitudes satisfying hidden zero, the 5-point splits, and unitarity. All numerical evidence for this claim uses (1.10). Appendix C shows that under the standard twice-subtracted behavior (5.2) the k=2 and k=3 constraints must be re-solved via (C.6), the single-variable Y-scan linearization is unavailable, and up to kmax=8 the allowed region does not cleanly bifurcate; the paper notes that a thin island could 'escape between probed points' on the finite grid. Since the uniqueness inference relies on the bifurcation and the shrinking island, the conjecture is not established under standard Regge assumptions. Please either qualify the abstract and Section 4.4 explicitly, for example by stating the conjecture under the improved Regge assumption (1.10), or provide additional evidence for the standard-Regge case, such as a higher-kmax two-variable scan or a boundary-following method.","section":"Sec. 1, Eq. (1.10), App. C"},{"comment":"The 'unique' conclusion is an extrapolation from finite truncation. The d=10, cutoff-2 island at kmax=10 has linear size of order 4e-6, and the d=4, cutoff-1.2 island at kmax=8 has linear size of order 3e-3; there is no theorem that these intervals converge to the string point. Moreover, the double-scan feasibility method in Appendix B.2 samples a grid of (X,Y) points and then takes the convex hull of the feasible points, so thin allowed regions could be missed in the improved-Regge analysis as well. The abstract's 'numerics indicate' is appropriately cautious, but the phrasing near the end of Section 4.4, which says that 'the only theory without an infinite spin tower' is the string amplitude, goes beyond what finite-kmax numerics can establish. I request that this be presented strictly as a conjecture and that the finite-truncation extrapolation be explicitly flagged.","section":"Secs. 4.2-4.4, Table 2, App. B.2"},{"comment":"The recursive argument that a theory with only contributions above the cutoff must, by the same analysis, either have a spin tower at the cutoff or be the string amplitude with cutoff 1/(alpha' M_gap^2) assumes that the bifurcation and island-shrinking behavior persists at all scales and in the kmax-to-infinity limit. The numerical bifurcation is demonstrated only for specific choices of d, cutoff, and finite kmax, and Appendix C shows that the behavior is qualitatively different when the Regge assumption is relaxed. Please mark this recursive step as an extrapolation from the numerical evidence rather than as an established consequence.","section":"Sec. 4.4, final paragraph"}],"minor_comments":[{"comment":"The line 'Y a1,0 = 2a3,0 - a3,1 = 0' appears to contain an erroneous trailing '= 0'; it should be 'Y a1,0 - (2a3,0 - a3,1) = 0' or simply 'Y a1,0 = 2a3,0 - a3,1'.","section":"Eq. (B.4), k=3 line"},{"comment":"The phrase 'S-matrix withn-point scattering' has a missing space and should read 'S-matrix with n-point scattering'.","section":"Sec. 1, first paragraph"},{"comment":"The text says the hidden-zero scalar-vector exchange amplitude is 'shown in blue in Figure 1', but the figure caption identifies it as the 'purple dashed line'; please make the color references consistent.","section":"Sec. 3.4, Fig. 1"},{"comment":"The phrase 'the upper corner is close to the string amplitude with 1/alpha' = 2 M_gap^2' is confusing: from the context it should mean alpha' M_gap^2 = 1/2, i.e., the string's second massive state at the cutoff. Please rephrase.","section":"Sec. 4.2, paragraph after Fig. 2"},{"comment":"The definition 'Y = 3a2,0 - 2a2,1 / a0,0' should be parenthesized as 'Y = (3a2,0 - 2a2,1)/a0,0' to avoid ambiguity.","section":"Eq. (3.18)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good fit for JHEP and the derivation of the nonlinear constraints is careful and well benchmarked. The main issue is that the abstract and Section 4.4 overstate the uniqueness claim relative to the caveats in Appendix C; a major revision that qualifies the claim and clearly labels the finite-kmax extrapolation should be sufficient. No concerns about attribution or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, this paper does something real: it takes the hidden zero/split properties known from earlier work and shows that imposing the 5-point splits on a general EFT ansatz fixes nearly all 5-point contact terms and forces nonlinear relations among the 4-point Wilson coefficients (Table 1). That is a clean analytic result, and it is new. The subsequent bootstrap implementation—linearizing those constraints via scans over X and Y, then running SDPB—is careful, and the non-convex allowed region with the sharp corner near the beta function is a nice, visible consequence. The shrinking island around the string in d=10 and d=4 is genuine numerical evidence, and the comparison to the monodromy-based island is useful.\n\nThe soft spot is the one flagged in the paper itself. The uniqueness conjecture relies on the improved Regge behavior (1.10), which is stronger than the Froissart-Martin bound. Appendix C shows that under the standard twice-subtracted dispersion relations the bifurcation does not cleanly occur up to kmax=8; the island may escape between grid points. So the conclusion is not ‘unitarity + splits pick out the string’ in full generality; it is that combination plus a specific non-generic Regge assumption. The authors are upfront about this. That is a real limitation on the central claim, not a hidden flaw.\n\nOther minor concerns: no code or SDPB input files are released, so exact reproduction of the islands is harder than it should be. The finite kmax extrapolation is taken a bit quickly—sizes shrink impressively, but a 10^-5 range at kmax=10 is still finite truncation. The analytic part is the strongest piece; the numerics are honestly described.\n\nWho is this for? Anyone working on S-matrix bootstrap, positivity bounds, or string amplitude uniqueness. The analytic constraints in Table 1 and the factorization/exponentiation remark in Section 5 are worth having even if the uniqueness conjecture stays conditional. It deserves a serious referee—I would send it to review, asking the authors to release code and to soften the abstract's uniqueness wording to match the conditional evidence.","headline":"Solid advance: 5-point splits impose genuinely new nonlinear constraints on 4-point Wilson coefficients, but the uniqueness claim is conditional on the improved Regge assumption, which the paper itself shows is crucial.","tokens_in":37829,"tokens_out":1542,"would_cite":true,"duration_ms":17122,"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":"This paper argues that 5-point split constraints, unitarity, and the absence of infinite spin towers leave the string beta function as the unique 4-point amplitude.","keywords":["S-matrix bootstrap","hidden zeros","split factorization","string beta function","effective field theory","Wilson coefficients","non-convex bootstrap","unitarity"],"falsifier":"Exhibit one unitary, weakly coupled scalar EFT whose 4-point amplitude has the hidden zero and 5-point split compatibility, has only finitely many states at the mass gap, and whose $(X,Y,Z)$ lies outside the $k_{\\rm max}=10$ island of Table 2; for $d=10$ and cutoff $2$, that means outside roughly $0.730760 < X < 0.730764$, $1.644933 < Y < 1.644937$, and $0.630367 < Z < 0.630384$. Alternatively, show numerically or analytically that the island stops shrinking at nonzero size as $k_{\\rm max}$ increases.","tokens_in":36725,"feed_emoji":"🎯","tokens_out":8797,"duration_ms":81784,"temperature":0.7,"pith_summary":"This paper enters the perturbative S-matrix bootstrap with information from 5-point amplitudes. The authors start from the observation that certain scalar effective field theories, and the string beta-function amplitude, have hidden zeros, special kinematic loci where the amplitude vanishes, and that near those loci the 5-point amplitude splits into a product of 4-point amplitudes. Imposing these split conditions on a general 5-point EFT ansatz fixes almost all its contact terms and, more importantly, imposes nonlinear relations among the 4-point Wilson coefficients. Bootstrapping the 4-point amplitude with these relations turns the usual convex allowed region into a non-convex one with a sharp corner at the string beta function. Supposing there is no infinite tower of spinning states at the mass gap, the allowed region bifurcates and the island containing the string shrinks as more constraints are added, indicating that the string beta function is the unique unitary 4-point amplitude compatible with hidden zeros and 5-point splits.","feed_headline":"5-point splits shrink the allowed region onto the string","feed_subtitle":"Unitarity and split factorization leave the Veneziano beta function as the only 4-point amplitude.","key_machinery":"The machinery is the hidden-zero/split structure of tree amplitudes, imported into the numerical bootstrap. A hidden zero is a kinematic locus, like $t=0$ in $A_4$, where the amplitude vanishes without a pole; a split is the factorization of a higher-point amplitude into lower-point amplitudes on that locus, for example $A_5|_{s_{13}=0}=A_4(s_{12},s_{15})A_4(s_{23},s_{34})$. The paper constructs a general 5-point EFT amplitude from pole residues and cyclically invariant contact terms, imposes the splits order by order, and finds nonlinear constraints among the 4-point Wilson coefficients. To feed these into the semidefinite bootstrap, the nonlinear relations are linearized by scanning over the variables $X=a_{1,0}/a_{0,0}$ and $Y=(3a_{2,0}-2a_{2,1})/a_{0,0}$, either singly or in a double scan. The remaining input is a set of once-subtracted dispersion relations for the $c_{k,q}$ coefficients, plus null constraints from crossing; these depend on the assumed improved Regge behavior $A_4(s,u)/s \\to 0$ as $|s|\\to\\infty$. The final result is a non-convex allowed region that, under the finite-spectrum-at-the-gap assumption, bifurcates into a scaled trivial region and a shrinking island around the $\\beta$ function.","core_discovery":"On the paper's own terms, the discovery is that higher-point split factorization is a powerful constraint on $2 \\to 2$ amplitudes, not a kinematic curiosity. Working at weak coupling with amplitudes of the $\\mathrm{Tr}(\\Phi^3)$ hidden-zero/split type, the paper derives nonlinear constraints on the 4-point Wilson coefficients from the 5-point split equations; for example, at order $k=2$, $a_{2,1} = \\frac{3}{2} a_{2,0} - \\frac{1}{2g^2} a_{0,0}^2$, and in general every $a_{k,q}$ with $q\\ge 1$ is fixed in terms of $a_{k,0}$ and lower-order coefficients. The bootstrapped region is non-convex, and with a finite number of states at the mass gap the parameter space separates into a trivial high-energy region and an island containing the string $\\beta$ function amplitude $A_4 = \\Gamma(-\\alpha' s)\\Gamma(-\\alpha' u)/\\Gamma(-\\alpha'(s+u))$. The island shrinks with increasing derivative order; in $d=10$ with cutoff $2$, it constrains three Wilson-coefficient ratios to within about $4\\cdot 10^{-6}$ of the string values at $k_{\\rm max}=10$. The paper therefore claims that, absent single-mass infinite spin towers, the string $\\beta$ function is the unique unitary 4-point amplitude compatible with the hidden zero and the 5-point splitting constraints.","pith_inferences":["If the uniqueness conjecture survives at higher $k_{\\rm max}$, it gives a purely field-theoretic characterization of the open-string tree amplitude: no worldsheet input is needed, since the splits already hold for $\\mathrm{Tr}(\\Phi^3)$.","The linearization-by-scan trick used here could be adapted to other non-convex higher-point constraints, such as those from soft limits or supersymmetry, where the same nonlinearity would otherwise block semidefinite methods.","The apparent absence of new constraints from 6-point splits suggests the decisive higher-point information is already contained at 5 points; a testable extension is to check whether 7-point splits fix additional 5-point contact terms corresponding to multiple zeta values.","One could replace the finite-spectrum assumption with a slope-one Regge trajectory input to see whether the island persists; if it does, the uniqueness statement would be robust to this spectral detail."],"forward_implications":["Any weakly coupled unitary 4-point amplitude with hidden zero and 5-point split compatibility must, if it has no infinite spin tower at the mass gap, have Wilson coefficients that approach the string values as the derivative order $k_{\\rm max}$ is increased.","The allowed region of the 4-point EFT is no longer convex; sums of individually valid amplitudes are in general invalid once 5-point splits are imposed.","At each derivative order only one 4-field Wilson coefficient, $a_{k,0}$, remains free; the splits fix all other $a_{k,q}$ in terms of it and lower-order data.","The split conditions saturate the multipositivity bounds of [77] at low orders, so those bounds add no further constraints in this setup.","The numerical bounds on $X=a_{1,0}/a_{0,0}$, $Y$, and $Z$ reach within about $4\\cdot 10^{-6}$ of the string values in $d=10$ with cutoff $2$ at $k_{\\rm max}=10$, with comparable though weaker bounds in $d=4$ with cutoff $1.2$."],"supporting_citations":[{"why":"Establishes the hidden zeros and split factorizations for colored scalar, pion, and gluon amplitudes, the property imposed on the EFT amplitudes.","marker":"[79]"},{"why":"Extends the splits to all orders, justifying the order-by-order imposition of the split equations.","marker":"[80]"},{"why":"Introduces smoothly splitting amplitudes and semi-locality, the earlier formulation of split behavior used as background.","marker":"[78]"},{"why":"Supplies the contour-deformation derivation of the once-subtracted dispersion relations used for the Wilson coefficients.","marker":"[47]"},{"why":"Gives the fixed-$u$ and fixed-$t$ null constraints used to impose crossing in the bootstrap.","marker":"[34]"},{"why":"The semidefinite program solver used for the numerical bootstrap bounds.","marker":"[83]"},{"why":"Proves that at most spin 1 can be exchanged at the mass gap under the finite-spectrum assumption, justifying the two-state gap input.","marker":"[66]"},{"why":"Provides the analytic bound on spins at the mass gap used to replace the no-infinite-tower assumption by scalar-plus-vector exchange at the gap.","marker":"[84]"},{"why":"Provides the comparison bootstrap of the open superstring whose island size and spectrum are benchmarked against the present results.","marker":"[60]"}],"fun_headline_variants":["5-point splits isolate string beta function in bootstrap","Splitting conditions shrink bootstrap island to string","Higher-point constraints pin down string amplitude uniquely","String beta function emerges as sole 4-point amplitude","Nonconvex bootstrap island collapses onto string beta"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The uniqueness conclusion rests on assuming that the 4-point amplitude falls off faster than $1/s$ at large $s$ with fixed momentum transfer; the authors say this is not generic and is stronger than the standard Froissart-Martin bound, and with the standard Regge behavior the bifurcation and shrinking island cannot be cleanly established.","fun_headline_variants_meta":{"raw":{"variants":["5-point splits isolate string beta function in bootstrap","Splitting conditions shrink bootstrap island to string","Higher-point constraints pin down string amplitude uniquely","String beta function emerges as sole 4-point amplitude","Nonconvex bootstrap island collapses onto string beta"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000177,"raw_usage":{"total_tokens":1372,"prompt_tokens":1103,"completion_tokens":269,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":719,"completion_tokens_details":{"reasoning_tokens":198}},"tokens_in":719,"tokens_out":269,"duration_ms":4145,"temperature":1.0,"reasoning_tokens":198,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:04:11.718907+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit one unitary, weakly coupled scalar EFT whose 4-point amplitude has the hidden zero and 5-point split compatibility, has only finitely many states at the mass gap, and whose $(X,Y,Z)$ lies outside the $k_{\\rm max}=10$ island of Table 2; for $d=10$ and cutoff $2$, that means outside roughly $0.730760 < X < 0.730764$, $1.644933 < Y < 1.644937$, and $0.630367 < Z < 0.630384$. Alternatively, show numerically or analytically that the island stops shrinking at nonzero size as $k_{\\rm max}$ increases.","supporting_citations":[],"review_version":1}