Oh I think I get the idea, really good!
Illegal Position Contest!

Do you mean in the TLDR that the h3 pawn prevents black's king from acting as a shield?
Oops. It should say that moving wPh3 allows the black king to act as the shield. The issue has been resolved.

Here's a position that's related to my previous position, but the pawns don't keep the black king from reaching the first rank, and I can even post a proof game for this position. But it is still an illegal position. Why do you suppose that is?

Without doing anything I assume 50 move role.....or rule.
Now you just need to explain why the position can't be reached in 50 moves or less after the last capture or pawn move.
Except it's not illegal to exceed 50 moves without a pawn move or capture. Whether it's illegal to exceed 75 moves depends on which rules you assume (FIDE basic rules v competition rules).
But the latter is also not illegal under the local chess variant @n9531l1 usually assumes.

Except it's not illegal to exceed 50 moves without a pawn move or capture.
But I've decided to call a composed position illegal if it requires moves after the game has ended.
Should rules for over-the-board chess be applied to a composed problem, rather than the rules found in the Codex for Chess Composition? (See what Article 17 of the Codex says about retro-problems, which include proof game problems.)

50 move rule shouldn't apply to endgame studies, long puzzles, or tablebases. It's more of a tournament regulation rule rather than a solid rule of the game, such as can't castle through check. What's funny is that games which have actually ended, such as this one:
The site doesn't declare as ended, even though it is a draw as no side can ever checkmate the other! LOL

This is a common trick puzzle, because it appears in a semi-famous book, but is it possible for this position to be reached with white to move?
Nh3 d6 Ng1 d5
Except it's not illegal to exceed 50 moves without a pawn move or capture.
But I've decided to call a composed position illegal if it requires moves after the game has ended.
Presumably meaning after any possible game has ended according to the rules stated to be in force or all generally accepted sets of rules in the absence of such a statement. Seems eminently reasonable, but I think chess.com rules or all variant GUI rules should be excluded in the latter case, because the variations from human sets are normally caused by implementation problems.
Should rules for over-the-board chess be applied to a composed problem, rather than the rules found in the Codex for Chess Composition? (See what Article 17 of the Codex says about retro-problems, which include proof game problems.)
It says
Unless expressly stipulated, the 50 moves-rule does not apply to the solution of chess compositions except for retro-problems.
Fascinating. Have you come across any retro-problems where it applies (without specifying the 50 move rule ply count as part of the puzzle)?

50 move rule shouldn't exist at all in any of that. It should only be used in OTB tournaments, preferably at least 100 moves.
Possibly so. The only problem is it does exist in many sets of rules. But as I said earlier it has very little relevance to legality. The 75 move rule under FIDE competition rules, however, does.

No move limit should exist for anything, other than OTB tournament games. Whether that should be 50, 100, or 75 I don't know.

It says
Unless expressly stipulated, the 50 moves-rule does not apply to the solution of chess compositions except for retro-problems.
Fascinating. Have you come across any retro-problems where it applies (without specifying the 50 move rule ply count as part of the puzzle)?
I interpret the Codex article to mean that the 50-move rule does apply to retro-problems. In that case, it applies to every retro-problem unless stipulated otherwise, including #7721.
Have you tried to see how many moves #7721 requires?
Have you tried to see how many moves #7721 requires?
Er, not in fact. I'll give it a go.

What's a retro-problem?
A problem whose solution requires analyzing the possible moves that could have led to the problem position.
Except it's not illegal to exceed 50 moves without a pawn move or capture.
But I've decided to call a composed position illegal if it requires moves after the game has ended.
Presumably meaning after any possible game has ended according to the rules stated to be in force or all generally accepted sets of rules in the absence of such a statement. Seems eminently reasonable, but I think chess.com rules or all variant GUI rules should be excluded in the latter case, because the variations from human sets are normally caused by implementation problems.
Should rules for over-the-board chess be applied to a composed problem, rather than the rules found in the Codex for Chess Composition? (See what Article 17 of the Codex says about retro-problems, which include proof game problems.)
It says
Unless expressly stipulated, the 50 moves-rule does not apply to the solution of chess compositions except for retro-problems.
Fascinating. Have you come across any retro-problems where it applies (without specifying the 50 move rule ply count as part of the puzzle)?
Nobody ever specifies a 50-move rule move count but there are quite a lot of compositions (e.g. by N.Plaksin) where the diagram position is provably near the 50-move limit which therefore affects the solution. Btw, the 50-move convention is not completely clear and appears to accept that a diagram position is legally past that count but at that point immediately draws. Which is weird since "how did you get there?"
Everything on this site is named "puzzle" and nobody knows what that means in terms of the rules and conventions. What we know for sure is that "the game rules are not enough" to solve just about any puzzle. For instance, nobody can prove that a diagram is not the 3rd occurrence of the same and therefore a (claimable) draw. So you need "extras" and the field of composition provides them. More important: nobody else provides them so the composition conventions are about the only place to go. And it does not only affect the 50-M rule but also repetitions, castling, e.p. moves and "on-move" decisions. Plus some other things following from the interaction of some conventions with with the "dead positions" rule.
The priority order of applicable rule sets is (first 3 for compositions):
- The composition author can change any Codex- or FIDE-rule
- The Codex overrides some FIDE-game-system-rules - like removing the 50M restriction for common problems
- The Codex appends choice rules to fill in the "unknowns" - like castling right.
- Chess.com rules override some FIDE-game-system-rules but have no defined impact on compositions.
- FIDE-rules define how to play standard chess games and provide a basis for compositions.
I was going to post a shorter explanation for confirmation, but I'll just get started on my longer one right now. Am I correct?
TLDR; no possible shields exist for bBg1 to be set up. wPh3 allows for black's king to act as this shield.
Edit: improved readability
Do you mean in the TLDR that the h3 pawn prevents black's king from acting as a shield?