Infinitude of level sets of T(n)

Demonstrate that for each positive integer t ≠ 2, there exist infinitely many n ≥ 0 such that T(n) = t, where T(n) denotes the least length t of any increasing sequence a1 < a2 < ⋯ < at beginning at n and ending at g(n) whose product a1a2⋯at is a perfect square.

Background

Building on the unboundedness and surjectivity conjecture for T(n), the authors propose a stronger statement asserting not only existence for each admissible length t ≠ 2, but the infinitude of n realizing that length. This deepens the structural understanding of the distribution of minimal corresponding sequence lengths across n.

Proving infinitude would imply that the complexity patterns in Ron Graham’s sequence are not sporadic but pervasive across the integers.