A hard upper edge
An exact proof reports that every Gegenbauer partial-wave coefficient of the bosonic Veneziano amplitude is nonnegative for every mass level and allowed spin in dimensions above 3 and up to 26. The upper endpoint is sharp: once the dimension is greater than 26, the scalar coefficient is negative.
This is an all-level statement, not a result from checking a short list of examples. The analysis covers an infinite family of inequalities, with one inequality for every mass level and allowed spin. The coefficients are examined symbolically, so the claim concerns the full indexed family described by the amplitude rather than a finite sample of observations.
The paper asks whether those coefficients stay at or above zero across a range of dimensions. Its dimension-descent step extends the conclusion to every real dimension above 3 and no greater than 26, rather than leaving it only at D = 26.
The machinery behind the result
The calculation starts from the beta function and uses a D-dimensional Gegenbauer partial-wave projection implemented through a plane-wave expansion. In ordinary terms, the expansion sorts the residue into angular components labeled by allowed spin and assigns a coefficient to each component for the sign test.
An exact generating-function transform reduces each allowed partial wave to one Taylor coefficient. That gives the proof a single sign target for each level and spin, instead of requiring a separate sign argument for a large collection of expressions.
The reduced coefficients are then governed by a triangular recurrence. In the bulk of that recurrence, all 20 coefficients are positive, and the proof establishes h_p > 0 for p from 0 through ell minus 6. This handles the interior range, while a finite boundary remains to be certified.
That boundary is handled by a finite certificate. It proves the exit weight E_R is positive for every j at least 0 and every R from 6 through ell. A standalone exact-arithmetic program verifies the certificate, and the verification does not scan values of j, R or ell.
The exceptions are tightly confined
The proof also checks the low trajectories exactly. For ell from 0 through 5, it finds h_ell at least zero, with zeros only at the index pairs singled out in the analysis. At the critical dimension D = 26, those additional zeros are (n, j) = (1, 0) and (2, 1).
The residue has definite parity, which imposes a parity selection rule on the partial-wave expansion. In practical terms, the expansion includes only the parity-compatible spins allowed by that rule. This is why the headline result is stated for every allowed spin.
Together, the recurrence, boundary certificate and dimension-descent argument cover the infinite family rather than a finite cutoff in level. The conclusion is nonnegativity of every partial-wave coefficient at every mass level and spin throughout dimensions above 3 and up to 26.
A result with a defined scope
Because this is a symbolic proof of an amplitude-level property, its conclusion is about the analyzed coefficients themselves. The paper does not report participants or a sampled dataset; it works through an infinite family of inequalities, one for every mass level and allowed spin.
That scope matters. The result establishes a sign pattern within the specified partial-wave calculation, but it does not by itself extend the claim beyond the coefficients and dimensions that were analyzed.
The document is a preprint: its header identifies it as arXiv version 1, dated 25 Aug 2026.
Paper data and sources
Original title: Exact All-Level Positivity of the Bosonic Veneziano Amplitude
Authors: Qi Chen, Yuan Yin
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text