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Just sketchily to mention couple more eval ideas.

Of course, all they will fail, but I put them here, otherwise I will simply forget about them.

Well, there are also defensive binds, aimed at stopping enemy attacks.

Of course, 2 squares where such binds will be found, are d4 and e4.

Give some small bonus, 5-8cps, for a defensive bind on d4, c3 and e3 pawns attacking d4 at the same time, or a defensive bind on e4, d3 and f3 pawns attacking e4 at the same time.

Of course, it is always good to soundly control the center, even if only in the own half of the board. That is why such binds make pretty much sense.

Actually, there are many openings that hold in specific lines entirely owing to such binds, the most notorious example being the black binds on d5 with e6 and c6 pawns in the Caro-Kann; passive, but sound.

Well, of course not all longer chains are equally important.

Considering all of them in the same way might really give lopsided results.
The most important longer chains are chains passing through the center - the 8 central c4,d4,e4,f4,c5,d5,e5 and f5 squares.

Well, Stefen tried this idea in SF in the general case, and it failed at LTC, but I think the main problem was precisely giving bonus to all long chains, and not only to chains passing through the center.

Of course, longer chains are scored with their inner pawns (what a nice term Stefan came up with!, originally I thought calling those pawns medium would be fine, but I immediately agree Stefan's name is much more elegant), a pawn that defends anoter own pawn in the chain and is defended at the same time by other own pawn of the chain.

Scoring only central c4,d4,e4,f4,c5,d5,e5 and f5 inner pawns would mean scoring only 8 of the existing 24 inner pawns on the board, so 1/3 of all inner pawns.
That would skip inner pawns on the 3rd rank, which are not important, inner pawns on the b and g files, similarly, and also very advanced b6 through g6 inner pawns on the 6th rank, which presumably should be seen fine in the search.

I think, when scoring only those 8 central inner pawns, one might give a slightly bigger bonus in terms of ranks than the patch that succeeded at STC, 1/16 of connected phalanx opposed, maybe 1/8 or even some intermediate value between 1/4 and 1/16.

In any case, what is certain is that those 8 inner pawns are by far the most important inner pawns.

[d]r4rk1/ppqn1pp1/2pbpn1p/8/2PP4/3Q1NN1/PP1B1PPP/3RR1K1 w - - 0 14

a typical Caro-Kann defensive bind with black c6 and e6 pawns controlling the center - passive, but sound in a range of lines.

[d]r2qkb1r/pp1nnppb/2p1p2p/3pP3/3P4/2P2N2/PP1NBPPP/R1BQ1RK1 w kq - 0 9

Another typical defensive bind on the d5 square with c6 and e6 pawns - passive, but sound.

White has space advantage and better development, but the central bind on the d5 square strengthens the black defensive fortress.

So, you might just give the small bonus, just for the mg, for any 2 c3 and e3, or d3 and f3 pawns, as it does not matter much if the square they attack is empty or not; in both cases a bonus is due.

Defensive binds on the e4 square with d3 and f3 pawns also happen frequently in a range of openings, not so frequently as with d4 binds, but still frequently enough.

[d]6k1/8/8/4P2P/3P2P1/2P2P2/8/6K1 w - - 0 1

Of course, scoring in this way will mean that you will score the c3-d4-e5 chain above with d4 inner pawn, but skip the f3-g4-h5 chain, with g4 inner pawn.
Indeed, the central chain passing through d4 is much more important and giving equal bonus simply distorts things very much.

[d]6k1/8/3P4/2P5/1P2P3/5P2/6P1/6K1 w - - 0 1
same here, you would give bonus to b4-c5-d6 chain with c5 central inner pawn, but skip g2-f3-e4 chain with f3 inner pawn, as this chain is not advanced and unimportant.
Having a range of test positions where non-advanced chains with inner pawns on the 3rd rank are scored might hugely distort performance, even if the bonus for non-advanced inner pawns is low, in case such non-advanced chains are the only one to occur in a game, which might not be infrequent at all.

[d]6k1/8/6P1/5P2/4P3/3P4/8/6K1 w - - 0 1

Of course, this has nothing to do with pointed chains, but, incidentally, all meaningful pointed chains pass through the 8 central squares I mentioned.

For example, this is the case with d3-e4-f5-g6 pointed chain, passing through e4 and f5.

[d]6k1/8/5P2/4P3/3P4/2P5/8/6K1 w - - 0 1

and also, with c3-d4-e5-f6 pointed chain, passing through d4 and e5.

So I really think this is a reasonable way to do longer chains.

Of course, doing it for all minor pieces will be expensive, but it might make sense to consider just the most important case of minor backward control - knight attacking the square in front of an enemy backward pawn.

[d]6k1/1p6/2p4p/2P3p1/2N3P1/6N1/8/6K1 w - - 0 1
Nc4 above is due some small bonus for attacking b6 square in front of enemy b7 backward pawn, and also Ng3 for attacking the h5 square in front of enemy h6 backward pawn.
This makes usually the backward pawns even more backward than otherwise.

the bonus might be very small, some 2-3cps, just to make a distinction.
But, on the other hand, this will also diversify your eval.

[d]6k1/8/8/3PP3/2P2P2/8/8/6K1 w - - 0 1

well, 2 defended pawns next to each other on the same rank like d5 and e5 above definitely deserve some bonus over the usual bonus points.

Reasons:

- such pawns are very strong, stronger than most connected pawns
- such pawns are flexible, they can stay for a long time where they are, but can also easily advance
- no matter what of the 2 pawns advances, this will create longer chains. For example, if e5 advances above, this will create the c4-d5-e6 longer chain, while if d5 advances above, this will create the f4-e5-d6 longer chain

So, definitely, such pawns should be bonused additionally.

Of course, in SF they could be bonused in terms of ranks, with bonus somewhere between 1/4 and 1/8 of connected phalanx opposed.

[d]6k1/8/8/3P4/2P1P3/3P4/8/6K1 w - - 0 1

well, this is one of my favourite.
you might want to try it just for the fun of it.

I call it Gandalf cross, as I have seen this pattern most often in Gandalf games (one of my favourite engines in the past)

That this pawn constellation deserves a bonus, there is absolutely no doubt.
Even the doubled pawns do not quite matter. What really matters that this pawn structure is the strongest ever you can find on the board - a real fortress.

Above, the most advanced pawn, d5, is defended twice by 2 other pawns, c4 and e4, that are in turn also defended! So this is the strongest pawn structure you can find around - no breaking through it.
But besides, this structure is also flexible.

I think the bonus should be uniform, not very large, some 10-15cps, but you can either score all such pawns found, or, which for me is the better scenario, score such pawns only when the most advanced pawn of the structure, d5 above, is on one of the following squares: c5,d5,e5,f5,c6,d6,e6,f6.
This will ensure that you skip the less important pawns of this type like pawns on b and g semiedge files, or pawns on the less advanced 4th rank.

I think also that you could skip pawns on the 7th rank, as search should be able to see them sufficiently.

Am I the only one who likes the engine Gandalf?