If there are different pollinators in area from trees with varying sized nuts than you should go with the big nuts, I think. Pollen parent affects dna of seed but not fruit and nuts are seeds. I assume a pollinator with large nuts would increase the size of nuts over a tree that produces smaller ones.

You'll probably get an answer from someone much better versed in genetics, but if only as a mental exercise, that is my answer.

THink of it this way. Back when plants werent propagated/farmed or bred, our progenitors basically found a tree they liked, and planted the seeds, and kept the ones with the desierable features (back then Im assuming it would be sweetness and fruit size, or nut size etc).

Its also safe to assume they would eventually try to cross those seedlings with trees with the same general features, and pollinated those, whether that be with new generation or back crossing (breeding with the species or previous generations).

Simply (and very generally speaking) yes, basic breeding of wanted features (big nuts in this case), should pass on large fruits/nuts to SOME of the offspring. Just be prepared to be planting dozens upon dozens of trees, and possible years of crossing before you find out!

I'm not sure I have this right, but are you saying you have seedlings from trees that make different sized nuts on the same tree? If this is what you have, keep the seedlings from the smaller nuts, as they will grow into good trees. The larger nuts off the same tree will make a bigger first year seedling, but the small ones catch up and it doesn't seem to affect probablity of a given seedling producing larger/smaller nuts. Many nuts on a branch vs a few, nuts in the centre of the tree where the leaves get less light, a branch over a root that has thin soil.....all these things affect the size of a nut on that branch. I have even planted a half eaten nut that a squirrel dropped, and although the first year seedling is tiny, it catches up in later years. As long as the embryo is not damaged and the nut is not mouldy, the litle tree from it will do well. When the nut grows up, the little trees will ALL be different, especially if there are many possible pollinators.

have fun

I am talking about nuts from the same tree, with I would believe the same pollen parent as in the immediate area the other chestnuts are limited to my trees. The differing sizes due to a reason as you are stating, many nuts on that branch, lighting, and some were maybe the smallest in a husk with 3 nuts. I understand genetic variability and trees producing superior nuts are more likely to pass on those traits. What you stated about the nut size effecting the size and growth the first year makes sense as well with more stored resources in the larger nut giving it an initial edge in growth, but that the eventual nut potential will not be setback because it was the runt nut in a husk of 3 nuts, or one that did not develop as well for other environmental reasons. They were the smaller rejects I did want to process for eating. Thanks for the inputs