Update C88a chronology and reference for H₁ ≤ 240

constants/88a.md

@@ -20,7 +20,8 @@ More generally one writes $H\_m := \liminf\_{n\to\infty}(p\_{n+m} - p\_n)$ for $
 | $4680$ | [Pol14a] | Polymath8a, by optimizing Zhang's equidistribution estimates. |
 | $600$ | [May15] | Maynard, via a multidimensional generalization of the Selberg sieve, using only Bombieri–Vinogradov. Obtained independently by Tao. |
 | $246$ | [Pol14b] | Polymath8b, generalizing the Maynard sieve further together with extensive numerical work. Stood as the record for twelve years. |
-| $240$ | [Sta26] | Stadlmann: "the Bombieri–Vinogradov theorem can be combined with newer equidistribution estimates for smooth moduli", the latter being the exponent of distribution $\tfrac12 + \tfrac1{40}$ of [Sta25]. Preprint of 31 August 2026. |
+| $240$  | [Zha26]  | Zhang: proof discovered with GPT-5.6 Sol. Preprint publicly deposited on Zenodo on 28 August 2026, with an exact-arithmetic verification code released separately. |
+| $240$  | [Sta26]  | Stadlmann: independent proof using the Bombieri–Vinogradov theorem combined with newer equidistribution estimates for smooth moduli. Preprint of 31 August 2026. |
 | $186$ | [OAI26] | Establishes $\mathrm{DHL}[40,2]$: every admissible $40$-tuple has infinitely many translates containing at least two primes.  Combines the equidistribution estimates of [Pol14a] and [Sta25] with factorization conditions making suitable least common multiples of divisor products *triply* densely divisible, which widens the support of the multidimensional Selberg sieve, together with an improved numerical optimization.  Applied to the admissible $40$-tuple $\\{0, 2, 6, \dots, 182, 186\\}$ of diameter $186$.  Preprint of 30 August 2026, one day before [Sta26], which it describes as independent concurrent work; the proof is attributed to the model GPT 6 Astra.  See the note on its formalization below. |
 
 ## Known lower bounds
@@ -49,6 +50,7 @@ More generally one writes $H\_m := \liminf\_{n\to\infty}(p\_{n+m} - p\_n)$ for $
 - [Pol14b] D. H. J. Polymath. *Variants of the Selberg sieve, and bounded intervals containing many primes.* Research in the Mathematical Sciences **1** (2014), Art. 12. DOI: [10.1186/s40687-014-0012-7](https://doi.org/10.1186/s40687-014-0012-7). Erratum: *ibid.* **2** (2015), Art. 15, DOI: [10.1186/s40687-015-0033-x](https://doi.org/10.1186/s40687-015-0033-x). [arXiv:1407.4897](https://arxiv.org/abs/1407.4897).
 - [May15] Maynard, James. *Small gaps between primes.* Annals of Mathematics **181** (2015), no. 1, 383–413. DOI: [10.4007/annals.2015.181.1.7](https://doi.org/10.4007/annals.2015.181.1.7).
 - [Sta25] Stadlmann, Julia. *On primes in arithmetic progressions and bounded gaps between many primes.* Advances in Mathematics **468** (2025), Paper No. 110190. DOI: [10.1016/j.aim.2025.110190](https://doi.org/10.1016/j.aim.2025.110190). [arXiv:2309.00425](https://arxiv.org/abs/2309.00425).
+- [Zha26] Zhang, Hanxin. A bound of 240 for gaps between primes. Preprint, 28 August 2026. Zenodo. DOI: 10.5281/zenodo.22135842. Computational certificate and verification code: https://github.com/hanxinzhang/prime-bound.
 - [Sta26] Stadlmann, Julia. *Bounded gaps between primes.* Preprint, 31 August 2026. [arXiv:2608.31126](https://arxiv.org/abs/2608.31126).
 - [OAI26] OpenAI. *Improved short gaps between primes.* Preprint, 30 August 2026. [PDF](https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf).  Conditional Lean 4 formalization and Python numerical certificate: [openai/PrimeGaps186](https://github.com/openai/PrimeGaps186) (Apache-2.0).  A companion document, *Numerical certificate for prime gaps at most 186*, carries the coefficient tables and verification records.