Lin Hsin Hsin
Artificial Intelligence Center
Sep 13, 2026, declares Lin Hsin Hsin Gap Density Theorem & Proof








                  Theorem
                              ∀ n ≥1, ∃ n consecutive elements pi, pi+1, …, i+n−1 of 𝒽P
                              such that

                              pj+1 − pj ≤ C pjα

                              for all j = i, …, i+n−2, where C < 1 and α < 1 are absolute.             
















                  Proof
                            By the construction of 𝒽P — successive elements arising from division by an infinitesimal

                            the gaps pj+1 − pj are bounded by a constant G independent of j

                            Let α ∈ (0, 1) and C ∈ (0, 1).
                            As  j → ∞,   pjα → ∞

                            ∃ J such that for all j ≥ J:

                            G ≤ C pjα

                            For any n consecutive elements of 𝒽P all with index ≥ J.

                            Every gap in the block satisfies the required bound.      ∎