{"id":"ece8865b-aebb-4e84-9801-b23712ed5159","arxiv_id":"2506.15299","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Scaffolding from the tachyon to higher-level string states preserves and extends the hidden zeros of string amplitudes, with extra supersymmetric zeros from the superstring.","lead":"The paper shows that scattering amplitudes for heavier string states inherit the special momentum choices where simpler tachyon amplitudes vanish, and that supersymmetric string amplitudes have extra vanishing regions absent in the bosonic string. This gives a systematic map of when string and superstring amplitudes vanish, which matters for understanding the hidden organization of scattering amplitudes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (3.27), the F-polynomial identity that carries the zero-inheritance theorem, is verified in one triangulation only, and derivative terms from scaffolding residues are not shown to preserve the scaleless-integral criterion; the general n, N claim is therefore not established.","rationale":"The reader's weakest-assumption analysis identifies the same combinatorial identity, eq. (3.27), as the pivot on which the central zero-inheritance claim turns, and also flags the interaction of the identity with derivatives from the scaffolding residues. I agree: the paper checks the identity in one five-point example and then uses it as a general bridge to all higher levels and to the superstring. The present evidence is strong enough for conditional acceptance, but not for a categorical 'all zeros' claim. The explicit four- and six-point gluino results, the OPE/BRST derivation of higher-level vertex operators, and the low-point tests are genuine independent support; they make rejection inappropriate. The missing piece is a proof or an exhaustive verification of the F-polynomial lemma and a controlled treatment of the derivative terms in the scaffolded integrand. The verdict should remain conditional: accept the scaffolding program as a plausible and well-supported proposal, but require the missing combinatorial proof before the full generality of the theorem is asserted.","tokens_in":27327,"tokens_out":11529,"duration_ms":126108,"concrete_test":"Run a symbolic computer-algebra check that (i) enumerates all triangulations of the 8-, 10-, and 12-point polygons containing the diagonals (2s-1,2s+1), (ii) builds every F-polynomial via eqs. (2.5)-(2.6), and (iii) verifies eq. (3.27) in each case. Then apply the derivative operators in eq. (3.26) or eq. (4.18) symbolically, impose the zero kinematics of eq. (3.30), and test whether each resulting term is polynomial in the target y_r so that the y_r-integral is scaleless. Any triangulation where eq. (3.27) fails, or any derivative term with non-integer y_r-dependence, refutes the zero-inheritance theorem; if none appears, the missing lemma should be stated and proved as a theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.3 uses eq. (3.27) as the bridge between parent and amputated F-polynomials: after setting ys = 0, F_{2i-1,2j-1}, F_{2i,2j-1}, F_{2i-1,2j}, and F_{2i,2j} are all asserted to equal the amputated F^{amp}_{i,j}. The only verification is the single 10-point triangulation in Figure 4, shown in eqs. (3.28)-(3.29); no proof is given for arbitrary n, for all triangulations containing the scaffolding diagonals, or for the iterated N = 2 case. This identity is load-bearing: it converts the n-point tachyon zero criterion into a zero criterion for the scaffolded amplitude and is applied again in Section 3.4 and Section 4.2.\n\nThe second gap is the scaffold residue itself. Eq. (3.26) replaces each residue by a derivative with respect to y_{2s-1,2s+1}, evaluated at ys = 0. The actual integrand is a sum over which F factors are differentiated, so it contains terms with derivatives such as -alpha' c_{ij} F^{-alpha' c_{ij}-1} partial_{ys} F, evaluated at ys = 0. Eq. (3.27) controls only the undifferentiated product. The text states that 'only the latter is relevant,' but it does not demonstrate that the differentiated terms preserve the scaleless-integral criterion for the remaining integration variable y_r. If a derivative-generated term has non-polynomial dependence on y_r, the amplitude need not vanish under the conditions in eq. (3.30). The same issue recurs in the level-2 double scaffolding and in the superstring analysis, where exponents receive additional half-integer shifts from Pfaffian prefactors.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies zeros of open-string tree amplitudes using the curve-integral (positive-geometry) representation, building on prior scaffolding constructions. For the bosonic string, the tachyon amplitude is shown to be identical, in curve-integral form, to a kinematically shifted stringy Tr(phi^3) amplitude, and scaffolding is interpreted as OPE residues that produce higher-level string amplitudes. The main claim is that the n-point level-N scattering amplitude inherits the zeros of both the n-point tachyon amplitude and its (2^N)n-point prescaffold parent, via an identity (Eq. 3.27) relating parent F-polynomials evaluated on the scaffolding locus to the F-polynomials of an amputated graph. For the superstring, the super-tachyon amplitude is argued to share the tachyon zeros plus new Pfaffian zeros, which become the eps_i·eps_j = 0 zeros of super Yang-Mills after scaffolding; the gluino seed is then used to derive four- and six-point gluino amplitudes, their zeros, and a field-theory limit.","tokens_in":27712,"tokens_out":2930,"duration_ms":29896,"significance":"If the central zero-inheritance claim holds, the paper provides a uniform, representation-theoretic explanation of hidden zeros across bosonic strings, superstrings, and their field-theory limits, connecting OPE/scaffolding to the existence of zeros. The interpretation of scaffolding as vertex-operator OPE residues is conceptually clean, and the explicit BRST checks for the level-2 vertex and the correlator-level scaffolding in Appendix A are valuable. The paper also presents explicit curve-integral forms for four- and six-point gluino amplitudes, which are new and potentially useful. The main results are presented without fitted parameters and the logic is largely derivational, but the central combinatorial identity and the treatment of derivative terms in the residue procedure are not fully proven.","major_comments":[{"comment":"The identity in Eq. (3.27) — that after setting the scaffolding variables y_s = 0, all four parent F-polynomials F_{2i-1,2j-1}, F_{2i,2j-1}, F_{2i-1,2j}, F_{2i,2j} reduce to the amputated F_{i,j}^{amp} — is load-bearing for the zero-inheritance theorem, but it is only verified for a single 10-point triangulation in Eqs. (3.28)-(3.29). No proof is given for arbitrary n, for arbitrary triangulations that contain the scaffolding diagonals, or for the iterated N = 2 case. Since this identity is reused in Section 3.4 and Section 4.2, the paper needs a general combinatorial argument or an explicit counterexample-free proof; without it, the central claim is not established beyond the example shown.","section":"§3.3, Eq. (3.27)"},{"comment":"The scaffolding residue is implemented as derivatives with respect to y_{2s-1,2s+1}, and the text states that 'only the latter is relevant' for scalelessness. However, the differentiated terms contain factors such as -α' c_{ij} F^{-α' c_{ij}-1} ∂_{y_s} F evaluated at y_s = 0. Eq. (3.27) controls only the undifferentiated product, not these derivative terms. The paper does not demonstrate that the derivative-generated terms preserve the scaleless-integral criterion for the remaining integration variable y_r; if a derivative term has non-polynomial dependence on y_r, the amplitude need not vanish under the conditions in Eq. (3.30). This gap also affects the level-2 discussion in Section 3.4 and the superstring analysis in Section 4.2.","section":"§3.3, after Eq. (3.26)"},{"comment":"The level-2 argument says that one simply applies Eq. (3.27) twice, with the amputated graph obtained by chopping four consecutive legs into one effective node. But after the first scaffolding, the integrand contains derivative terms and modified F-polynomial exponents; it is not shown that the second residue acts on an integrand of the same structural form that Eq. (3.27) addresses. The paper needs an explicit demonstration that the intermediate integrand, including the derivative contributions, still satisfies the claimed F_{i,j}|_{y_s=0} = F_{i',j'}^{amp} reduction for the relevant set of variables.","section":"§3.4, level-2 zeros"},{"comment":"In the analysis of the super Yang-Mills zeros, the paper claims that under α' c_{ij} ∈ -N_0 for (i,j) ∈ N_YM, every Pfaffian partition term vanishes or becomes scaleless. This conclusion depends on the exponents of the F-polynomials after the scaffolding residue. However, the Pfaffian sum in Eq. (4.8) contains factors 1/F_{a,b} and half-integer powers y^{n^πα}, which can shift the effective exponents of the remaining variables. The paper only checks that the picture choice keeps (a,b) outside N_YM, but does not verify that the half-integer shifts and the derivative terms from the residue do not create non-scaleless non-polynomial dependence. A more systematic accounting of all exponent shifts is needed.","section":"§4.2, after Eq. (4.18)"}],"minor_comments":[{"comment":"The abstract contains a typo: 'from it's prescaffold image' should be 'from its prescaffold image', and the phrase 'Finally we consider the field theory limit... Finally we consider...' repeats 'Finally' twice.","section":"Abstract"},{"comment":"Eq. (2.3) is referred to as the u-equations, but the numbering is implicit; please number displayed equations consistently and refer to them by number.","section":"§2, Eq. (2.3)"},{"comment":"The F-polynomials in Eq. (3.28) are written in terms of y_{1,7}, y_{1,5}, y_{7,9}, y_{1,3}, but the correspondence with the triangulation T in the text would benefit from a clear label of which edge each variable corresponds to, to aid the reader in verifying Eq. (3.29).","section":"§3.3, Eq. (3.28)"},{"comment":"The cone-by-cone expansion for the four-point gluino limit in Eqs. (5.18)-(5.20) asserts that only certain terms contribute at O(α'^{-1}); a short justification of why the other terms are subleading would make the derivation more transparent.","section":"§5.1, field-theory limit"},{"comment":"The identity in Eq. (5.10), namely that sum_i C_i^{π=1} = 0 and sum_i D_i^{π=1} = 0, is stated without proof or reference; please provide a derivation or an explicit pointer to the literature.","section":"§5.1, Eq. (5.10)"},{"comment":"The proposed super-u variables in Eq. (6.2) are interesting, but the notation '|z_{i,j} - θ_i θ_j|' is ambiguous in a real-worldsheet context; please clarify the meaning of the absolute value for grassmann-odd variables.","section":"§6, Eq. (6.2)"}],"recommendation":"major_revision","confidential_remarks":"The paper presents an attractive and potentially important unification of hidden zeros in string and superstring amplitudes, but the central combinatorial identity and the handling of derivative terms in the scaffolding residue are currently verified only in an example and asserted for the general case. With a proof or a systematic argument for Eq. (3.27) and the scalelessness of derivative terms, the paper would be publishable; without it, the main theorem is not yet established. I see no grounds for concern about novelty or attribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a useful extension, not a breakthrough. It takes the scaffolding mechanism from Arkani-Hamed et al. and Cao et al. and shows that the zero structure of higher-level open-string amplitudes is inherited from tachyon/super-tachyon seeds. The new pieces are the level-2 analysis, the super-tachyon pfaffian zeros, and the 4- and 6-point gluino curve integrals. The paper is clearly written and gives credit where it's due.\n\nThe good parts: the observation that scaffolding is an OPE limit is nicely explained, and the curve-integral representation makes the zero-inheritance argument almost transparent. The identity eq. (3.27) linking parent F-polynomials to amputated ones is exactly the right tool, and the five-point example is convincing. For the superstring, the new zeros from the pfaffian are a nice avatar of supersymmetry, and they correctly project out non-susy operators like F^3.\n\nThe soft spots: eq. (3.27) is only verified for one triangulation. The paper treats it as a general lemma, but no proof is given for arbitrary n and arbitrary triangulations containing the scaffolding diagonals. If it fails, the whole zero-inheritance theorem for higher levels collapses. This feels fixable—the structure of F-polynomials under amputation is combinatorial, and a proof might be straightforward—but as written it's an unproven assumption.\n\nSecond, and more concerning, the paper dismisses the derivative terms that arise when scaffolding residues are taken. Eq. (3.26) defines the residue as a derivative with respect to the scaffolding variable, and that derivative can act on F-polynomials, changing their exponents by -alpha' c - 1. The paper says only the undifferentiated part is relevant and that's it. The stress-test note is right: if a derivative term produces a non-polynomial dependence on the remaining integration variable, the scaleless-integral argument may not apply. The paper does not show this can't happen. This is a genuine gap in the argument, not a stylistic quibble.\n\nThere's also a scope mismatch: the title says 'all zeros' but the paper actually proves zeros for specific amplitudes and specific conditions. The gluino section is explicitly limited to 4 and 6 points; the field-theory limit is worked out in detail only at 4 points. That's fine for a first paper, but it should be labeled as partial.\n\nWho this is for: people working on surfaceology and the scaffolding program. They'll find the level-2 and superstring extensions useful. I'd send it to a serious referee, but I'd ask the referee to demand a proof or a much more extensive check of eq. (3.27) and an explicit treatment of the derivative terms. With those fixed, this would be a solid contribution.","headline":"A useful extension of the scaffolding/zero program to higher levels and superstrings, but the central zero-inheritance lemma is unproven beyond one example and derivative terms from scaffolding are waved away.","tokens_in":28213,"tokens_out":2550,"would_cite":true,"duration_ms":25125,"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":"The paper establishes that every higher-level open string amplitude inherits its zeros from the tachyon amplitude through scaffolding.","keywords":["string amplitudes","hidden zeros","scaffolding","curve-integral representation","tachyon amplitude","super Yang-Mills","gluino amplitudes","F-polynomials"],"falsifier":"Take a non-ray-like triangulation for a scaffolded level-2 amplitude, set the kinematic invariants in the predicted zero set (for instance $\\alpha' c_{i,j} = -N_0$ in the index range from Section 3.4), and numerically evaluate the curve integral; a single non-vanishing result would refute the general lemma eq. (3.27) and with it the claimed zero inheritance.","tokens_in":27122,"feed_emoji":"🧵","tokens_out":6748,"duration_ms":62255,"temperature":0.7,"pith_summary":"The paper claims that the zeros of essentially all tree-level open string amplitudes are governed by a single seed: the tachyon amplitude in its curve-integral representation. Scaffolding—taking successive two-particle factorization limits, equivalent to OPE residues of vertex operators—promotes the tachyon to gluons, then to level-2 massive states, and iteratively to arbitrary level-$N$ excitations. The central technical point is that after scaffolding, the $F$-polynomials of the parent $2n$- or $(2^N)n$-point surface reduce to the $F$-polynomials of an amputated $n$-point surface, so any kinematic configuration that makes the tachyon integral scaleless also makes the descendant amplitude vanish. In the superstring, the same logic applies starting from the super-tachyon, with additional pfaffian zeros that become the $\\epsilon_i\\cdot\\epsilon_j = 0$ zeros of super Yang-Mills. A sympathetic reader would care because this gives a uniform, combinatorial explanation of hidden zeros across mass levels and sectors.","feed_headline":"All string zeros descend from the tachyon seed","feed_subtitle":"Higher-level string amplitudes vanish exactly where the tachyon does, in bosonic and superstring alike.","key_machinery":"The central object is the curve-integral representation of the string amplitude, in which the Koba-Nielsen factor is written in terms of the $n-3$ variables $y_I$ associated with a triangulation of an $n$-gon, with $F$-polynomials $F_{i,j}(y)$ built from left/right turn matrices along the dual fatgraph. The operation that carries the argument is scaffolding: taking residues on the factorization poles $X_{2i-1,2i}$, which is equivalent to the OPE residue of two vertex operators and produces the next higher-level vertex operator. The load-bearing identity is eq. (3.27), which says that after setting the scaffolding variables $y_s = 0$, all four parent $F$-polynomials $F_{2i-1,2j-1}$, $F_{2i,2j-1}$, $F_{2i-1,2j}$, and $F_{2i,2j}$ collapse to the single amputated $F^{\\mathrm{amp}}_{i,j}$ of the $n$-point fatgraph; this is what transfers the scaleless-integral zero mechanism from parent to descendant. For the superstring the extra ingredient is the reduced pfaffian $\\mathrm{Pf}(D^{a,b})$ of the super-tachyon correlator, whose rank-deficient zeros become the $\\epsilon_i\\cdot\\epsilon_j = 0$ zeros after scaffolding.","core_discovery":"On the paper's own terms, the discovery is that the $n$-point level-$N$ bosonic string amplitude, obtained by $N$-fold scaffolding from the $(2^N)n$-point tachyon amplitude, has exactly the zeros of the $n$-point tachyon amplitude plus the zeros inherited from its prescaffold parent. The argument runs through the curve-integral representation: the tachyon integrand is the kinematic-shifted Tr $\\phi^3$ integrand, and scaffolding is a residue operation on the curve variables $y_s$ that corresponds to the OPE limit. The key identity $F_{2i-1,2j-1}|_{y_s=0}=F_{2i,2j-1}|_{y_s=0}=F_{2i-1,2j}|_{y_s=0}=F_{2i,2j}|_{y_s=0}=F^{\\mathrm{amp}}_{i,j}$ implies the descendant $F$-polynomials are those of the amputated $n$-point graph, so the scaleless-integral mechanism that produces tachyon zeros operates unchanged. The paper verifies this for a five-point example and uses it to conclude the zero-inheritance theorem for all levels, and extends the same scaffolding logic to the super-tachyon, where pfaffian zeros of the reduced pfaffian become the $\\epsilon_i\\cdot\\epsilon_j = 0$ zeros of super Yang-Mills. It also presents four- and six-point gluino amplitudes in curve-integral form and identifies their tachyon-type zeros.","pith_inferences":["If eq. (3.27) holds for all triangulations and all levels, the zero structure of massive higher-spin amplitudes is fixed by amputated fatgraph combinatorics alone, which would give a purely combinatorial classification of hidden zeros across mass levels.","The same mechanism should apply to mixed-level amplitudes beyond the single level-2 with three tachyons example worked out in the paper; one can test this by scaffolding a level-2 and a level-1 vertex from the same parent and checking the predicted zero locus.","A super-$u$-variable formulation suggested in the conclusions would turn the pfaffian zeros into positivity constraints on a supersymmetric positive geometry, potentially linking supersymmetry to the geometry of quartic fatgraphs.","The gluino zero conditions displayed in the paper are demonstrated only at four and six points; extending the six-point analysis to $2n$ points would give a concrete prediction that gluino amplitudes vanish on all ray-like triangulation zero sets with strictly negative integer $\\alpha' c_{i,j}$."],"forward_implications":["Every higher-level bosonic string amplitude at $n$ points vanishes on the same kinematic locus as the $n$-point tachyon amplitude, so the tachyon zero set is universal across the whole Regge tower.","The zero set of any scaffolded amplitude includes the zero set of its $(2^N)n$-point tachyon parent, so increasing mass level adds zeros rather than destroying the existing ones.","For the open superstring, the pfaffian zeros imply that super Yang-Mills amplitudes vanish when all polarization inner products $\\epsilon_i\\cdot\\epsilon_j$ vanish, projecting out non-supersymmetric corrections such as the $F^3$ term.","Scaffolding from the gluino seed yields the same super Yang-Mills vertex operators, so the R-sector provides an independent route to super Yang-Mills and its zeros.","The curve-integral form of gluino amplitudes admits a field-theory limit via tropicalization, reproducing colored fermion amplitudes in a form where the zeros remain visible."],"supporting_citations":[{"why":"Supplies the curve-integral representation, the scaleless-integral mechanism for hidden zeros, and the scaffolding idea connecting Tr $\\phi^3$ to Yang-Mills and NLSM.","marker":"[4]"},{"why":"Establishes the scalar-scaffolded gluon construction that the paper reinterprets as tachyon OPE limits.","marker":"[5]"},{"why":"Provides the $F$-polynomial/fatgraph parameterization and the open-string curve-integral representation used throughout.","marker":"[10]"},{"why":"Gives the super-tachyon curve-integral formula and the pfaffian exponent bounds the paper relies on for superstring zeros.","marker":"[19]"},{"why":"Provides the original observation of tachyon amplitude zeros that the paper generalizes to all levels.","marker":"[24]"},{"why":"Supplies the positive parameterization and tropicalization tools used for the field-theory limit.","marker":"[11]"},{"why":"Provides the six-fermion superstring correlator used to build the six-point gluino amplitude.","marker":"[26]"},{"why":"Gives the four-point bosonic Yang-Mills amplitude with $F^3$ corrections that the $\\epsilon_i\\cdot\\epsilon_j=0$ zeros project out.","marker":"[25]"}],"fun_headline_variants":["String zeros all descend from tachyon seed","Tachyon zeros rule all string amplitudes","Scaffolding shows string zeros are tachyon's","Superstring zeros born from tachyon's zeros","All string zeros trace back to tachyon"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the identity that after setting the scaffolding variables to zero, the four parent $F$-polynomials $F_{2i-1,2j-1}$, $F_{2i,2j-1}$, $F_{2i-1,2j}$, and $F_{2i,2j}$ all equal the amputated $n$-point $F^{\\mathrm{amp}}_{i,j}$; this is checked for one five-point triangulation and then assumed to hold for all triangulations and after the derivatives introduced by scaffolding residues.","fun_headline_variants_meta":{"raw":{"variants":["String zeros all descend from tachyon seed","Tachyon zeros rule all string amplitudes","Scaffolding shows string zeros are tachyon's","Superstring zeros born from tachyon's zeros","All string zeros trace back to tachyon"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000177,"raw_usage":{"total_tokens":1347,"prompt_tokens":1051,"completion_tokens":296,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":223}},"tokens_in":667,"tokens_out":296,"duration_ms":3306,"temperature":1.0,"reasoning_tokens":223,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:37:57.912743+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a non-ray-like triangulation for a scaffolded level-2 amplitude, set the kinematic invariants in the predicted zero set (for instance $\\alpha' c_{i,j} = -N_0$ in the index range from Section 3.4), and numerically evaluate the curve integral; a single non-vanishing result would refute the general lemma eq. (3.27) and with it the claimed zero inheritance.","supporting_citations":[{"cited_title":"D’Adda, S","cited_arxiv_id":null,"evidence_quote":"Provides the original observation of tachyon amplitude zeros that the paper generalizes to all levels."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the six-fermion superstring correlator used to build the six-point gluino amplitude."}],"review_version":2}