Arbitrary Local-Cohomology Lengths between Consecutive Ideal Powers

For every field k and every integer i >= 0, we construct a fixed complete regular local ring (R,m) of dimension 2i+2 and a fixed prime complete-intersection ideal I with the following property. Every sequence (a_n) of nonnegative integers is realized by ideals J_n satisfying I^(n+1) contained in J_n contained in I^n and length H^i_m(R/J_n) = a_n. All lower local cohomology vanishes, and all transition maps below degree i+1 are zero. Thus even adjacent-power bounds impose no asymptotic restriction on these lengths for arbitrary decreasing cofinal families. The construction inserts a one-column module presentation into each successive homogeneous layer; its length is computed directly in the top Cech module of a power-series ring. The families need not be multiplicative. The result concerns the unrestricted-family formulation and makes no assertion about ordinary, symbolic, or Frobenius powers. This is an unrefereed preprint prepared with AI assistance; no independent review, formal verification, or absolute priority certification is claimed.