Convolving two arrays can be done perfectly accurately in O(n log n), despite every element being combined with every other element.
Or consider the even more basic sum of products a[i] * b[j] for all possible i, j:
total = 0
for i in range(len(a)):
for j in range(len(b)):
total += a[i] * b[j]
Your logic that 'the result contains terms of all pairs, therefore the algorithm must be quadratic' simply doesn't hold.
\iiint f(x, y, z) dx dy dz = \int [\int g(x, y) dx]*[\int h(y, z) dz] dy
which greatly accelerated numerical integration (O(n^2) rather than O(n^3)).
My advisor was not particularly impressed and objectively I could have skipped it and let the simulations take a bit longer (quite a bit longer--this integration was done millions of times for different function parameters in an inner loop). But it was clever and all mine and I was proud of it.
Attention is a global operation.
Fundamentally, multiplication need to look at every pair of integer from the two input numbers. It must be O(n^2); N digits looking at N other digits is quadratic. Any sub-quadratic multiplication must hence necessarily lose some information.
Attention also has some specific properties.
And sometimes results are just unexpected. Did you know that anything a Turing machine can do in t tome steps, a different Turing machine can do in O(sqrt(t log t)) memory cells? https://news.ycombinator.com/item?id=44055347