Improve upper bound for Davenport constant C_{53} to 4

[maaxgrin]

This PR updates the entry for the Davenport constant of $C_n^3$, improving the recorded upper bound

$$ C_{53}\le 20369 $$

to

$$ C_{53}\le 4. $$

The update records the stronger pointwise estimate

$$ D(C_n^3)\le 4n-P(n)-2, \qquad P(n)=\max_{p^a\parallel n}p^a. $$

Since $P(n)\ge 2$, this gives

$$ D(C_n^3)\le 4(n-1) $$

for every $n\ge 2$, and hence $C_{53}\le 4$.

The proof was discovered with assistance from GPT-5.5 Pro. The deduction from the cited zero-sum inputs was also checked in a conditional Lean 4 formalization prepared with assistance from Aristotle and GPT-5.5 Pro.

The proof PDF, references, and conditional Lean 4 formalization are available in this auxiliary repository:

https://github.com/maaxgrin/davenport-cn3-bound

[maaxgrin]

I added a conditional Lean 4 formalization of the proof:

RequestProject/Main.lean

It compiles with Lean 4.28.0 / mathlib v4.28.0. The file isolates the results from cited previous works as axioms and checks the local estimate, the induction step, and the final bound.

The formalization was prepared with the help of Aristotle and GPT-5.5 Pro.

[teorth]

I ran your submission through an AI to flag for potential issues. My main request would be to update your repository to host an informal proof as well as the formal one, as well as prominently declare (e.g., on the front page of the repository) the precise axioms used in the conditional formalization (ideally both in informal and formal mathematical language). The PR can then be updated to link to that repository. Also there is a minor unicode issue flagged by the AI to look into.

[maaxgrin]

Thank you very much for the feedback. I have updated the repository containing the proof PDF, a conditional Lean formalization, and a README that explicitly lists the axioms/results from the literature used in the formalization.