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Main discovery. Hidden zeros of tree-level string amplitudes appear on kinematic loci where certain dimensionless momentum contractions take integer values arranged in rectangular blocks on the kinematic mesh. These “integer rectangles” organize zeros first seen for colored scalars and extend to open-string amplitudes with gluons, fermions and higher-spin states at finite mass levels. For closed strings, including graviton amplitudes, persistent cubic zeros in both chiral sectors can remove disk poles so that the Kawai–Lewellen–Tye (KLT) momentum-kernel zero implies amplitude zeros, provided the stated chiral conditions hold.
Method and concrete result. Using disk monodromy relations, the authors derive a zero-locus factorization. At regular points where the monodromy (sine-insertion) system is invertible, the restricted amplitude on an integer rectangle equals a weighted sum of physical three-point amplitudes sewn to complete on-shell (n−1)-point amplitudes. The surviving fusion contributions are organized into an m!×m! sine-insertion matrix S_sin. Where det S_sin ≠ 0 on the physical family, the restricted amplitudes satisfy A_q = π S_sin^{-1} T_q, with the right-hand vector T_q given by oriented sums of physical residues. Nonspectral cross pairs contribute zero residue, and the integer condition selects the physical fusion mass for the fused leg.
Low-energy limit and field-theory match. On the massless zero-integer branch as α′→0, the sine matrix expands as S_sin = 2πα′ S_lin + O(α′^3). The factorization then reduces to the known on-shell rectangular factorization of Yang–Mills and super‑Yang–Mills amplitudes, S_lin A_FT = (1/2) r_FT (the 1/2 follows from the kinematic conventions used).
Concrete example. For four massless gluons with s13 fixed to an integer q, the factorization identifies the selected fusion channels. If q≥2 the fusion space is empty and the amplitude vanishes: A4|_{s13=q}=0 for q≥2. At q=1 and q=0 the selected tachyon or massless-vector couplings are annihilated by appropriate external-polarization conditions, again giving zero in those cases.
Worldsheet origin. Cross operator product expansions (OPEs) provide a common worldsheet origin for the fusion conditions across spins and mass levels. Under stronger cross‑OPE regularity the remaining cross dependence decomposes into finitely many homogeneous relative‑scale sectors, each leading to formally scaleless integrals.
Significance. The work unifies previously separate hidden-zero phenomena. It generalizes scalar integer-rectangle zeros to the full open-string spectrum and, under persistent chiral cubic-zero conditions, to closed strings via KLT. It also explains how the string zeros reduce to familiar field-theory rectangular factorizations in the low-energy limit.
Supported caveats and limitations. The factorization and the on‑shell 3×(n−1) reconstruction require admissibility and regularity hypotheses. These include: 1) working in a smooth on‑shell neighborhood where the cross deviations δ_{ij} are independent local transverse coordinates; 2) using holomorphic physical‑state representatives; and 3) excluding other physical poles in all orderings that enter the monodromy relations. Where inversion of the sine-insertion system is used, an additional requirement is 4) det S_sin ≠ 0 on the chosen physical kinematic family; formal nonvanishing of the determinant does not guarantee invertibility on every physical Gram component, which requires a realized on‑shell momentum assignment compatible with the spacetime Gram rank. The fusion‑zero criterion (that all selected three‑point couplings vanish) is a sufficient but not necessary condition for an amplitude zero; cancellations among nonzero fusion contributions can also produce zeros. For closed‑string zeros via KLT, the chiral cubic identities must hold persistently along each fusion divisor with the other parameters free; vanishing only at the rectangular intersection can leave compensating poles.
Notes. Explicit four‑ and six‑point formulas and examples (including massive bosonic and Ramond external states), derivations of the external-state conditions that annihilate three-point couplings, and local OPE and relative-scale analyses are provided in the supplemental material of the source.