So, I don't have the theory background a PhD candidate might, so forgive my ignorance, but why would a theory be expected to build on the already proven partial results? I was under the impression their use was as a quality heuristic, not something a proof of a lower bound of the general case could make much use of.
Generally because that’s how mathematics tends to work - people publish a partial result & then they or others build on it to improve that result in various ways[1]. The authors are also usually embedded in a wider community of mathematicians who act as a pre-publication bullshit filter by asking pointed questions at seminars & that kind of thing.
It’s quite rare for a genuinely novel solution to a major problem in mathematics to spring up out of nowhere in the literature.
[1] Think of, eg, the Twin Prime conjecture (that there are infinitely many pairs of primes only 2 integers apart). For decades there was no progress on this problem, before Yitang Zhang published a paper last year proving that there were infinitely many primes less than a finite bound apart (70 million in his case). Then a horde of matheaticians attacked the problem using his & other new approaches to progressively reduce that large bound down to the point where it’s now been proven that there are infinitely many primes < 246 apart (according to the polymath group - I’m not sure whether this is a published result yet). The twin prime conjecture is in sight!
Yea, but to my understanding the partial work done thus far does not apply to an understanding of the general case solution. Let's say there is a linear solution for some form--how does that help establish a lower bound on the general form?
Recognition of past work I can understand. But it seems like occam would argue with me against the use of some of the partial results without understanding why they would be useful.
