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MatPlus.Net Forum Retro/Math A proof game question
 
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(21) Posted by Neal Turner [Saturday, Sep 24, 2022 23:18]

I don't know which magazines you subscribe to, but the ones I receive certainly don't contain 'numerous' helpmates with the 'exact' condition, in fact I can't actually remember any examples.

In direct mates the (very rare) ones that appear show (or should show) a special type of Black strategy where he's defending by trying to force White to mate earlier than the stipulation - a kind of fusion of direct mates and selfmates.
 
   
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(22) Posted by Kevin Begley [Sunday, Sep 25, 2022 00:19]

 QUOTE 

I don't know which magazines you subscribe to, but the ones I receive certainly don't contain 'numerous' helpmates with the 'exact' condition, in fact I can't actually remember any examples.


LOL. Such a problem (h# in exactly n moves) was published in a journal named Mat Plus (which should ring a bell to everyone reading on matplus.net).
The same journal also published an exact series mover.

You can see ~48 examples from a number of other familiar journals (StrateGems, Die Schwalbe, feenschach, Ideal-Mate Review, Problemkiste, idee & form, Pat a Mat, Deutsche Schachzeitung, harmonie, etc).
Just go to PDB, and search for the following (note: this will not filter everything out -- e.g., exact series movers will be included): stip='genau' and (stip='h#' or stip='h=') and NOT stip='BP'

If you prefer to see "diretmates in exaclty n moves", try the following filter (note: this will not filter everything out): stip='genau' and NOT (stip='h#' or stip='h=' or stip ='BP')

Seek and ye shall find.
Unfortunately, there's currently no way (that I know) to search for such problems in Win Chloe (which, itself, provides strong evidence that the lenient standard -- adopted by Win Chloe and others -- is inadequate).

While "exact" (as a stipulation term) is somewhat less uncommon in proofgames (for comparison, PDB has 56 problems stipulated as "BP in genau ..."), it is generally uncommon across all problem genres (compared to standard problems of the genre).
Popularity of usage is hardly grounds to devalue such problems (by wrongly presuming that usage can only constitute a "cop out").

 QUOTE 

In direct mates the (very rare) ones that appear show (or should show) a special type of Black strategy where he's defending by trying to force White to mate earlier than the stipulation - a kind of fusion of direct mates and selfmates.


It is dangerous (if not outright folly) to presume what strategy should be shown in problems stipulated in a manner not carefully surveyed.
Don't be too quick to disregard all things you have failed to imagine, and don't be hasty to dismiss all things you have failed to appreciate.

If you can make a "series-directmate in exactly n moves", which provides no black strategy whatsoever, you can make a "directmate in exactly n moves" which need not conform to your offhand presumption about black's strategy.
Why should a composer want to limit their possibilities by starting from a position of hasty presumptions?
Better to experiment, and welcome (most of all) those discoveries you might never have guessed would produce something of value -- that's where the magic happens!
 
   
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(23) Posted by Kostas Prentos [Sunday, Sep 25, 2022 13:01]

Neal, you make an interesting point comparing proof games with other types of problems. Although there are examples of direct mates that are intended to be solved in "exactly" the stipulated number of moves, the real comparison must be made with the close "cousins" of proof games, the helpmates. There are just a few proof games with a shorter solution/cook (with the same side on the move) that cannot be extended to satisfy the stipulation (like the PG in exactly 4 moves by Tibor Orban). In most cases of non SPGs, the shorter solution(s) (by a half move) cannot be used to solve the problem due to lack of tempo, or like in my own example in post 11, for other reasons having to do with the fairy condition.

The closest equivalent to our discussion is a helpmate with set play. How do we treat these helpmates? If there is a unique solution in the set play, it's all good. We may even use it to enrich the content of the problem. What if there are several solutions/cooks in set play? There was a discussion a few years ago, about "Parasitic set play!?" in helpmates. You probably remember it, since you initiated it. See here: https://matplus.net/start.php?px=1653959212&app=forum&act=posts&fid=gen&tid=2323 The general consensus was that those shorter "solutions" in set play were not cooks.

In any case, PGs in "exactly" n moves are not optimal. The more conditions we add to the stipulation, the worse. Furthermore, there is always the risk that some problemists may see the shorter solutions as cooks. That's why the best way out, when possible, is to add an extra half move at the end of the solution that makes the solution unique.
 
   
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(24) Posted by Kevin Begley [Sunday, Sep 25, 2022 18:00]

@Kostas,
Actually, the closer cousin to PGn is h=n (not h#n).

Like the PGn, the h=n could "theoretically" end with either player on the move (whereas the h#n must somehow convey which player is mated -- if PGn and h#n were actually cousins, the h#n would read as "help checkmate either player in n moves", and as you can see, the standards are dissimilar).
However, note that the standards for PGn and h=n are not the same -- the h=n standard insists white is stalemating black (white moves last), whereas the PGn standard is unique.

You can not argue these two standards should be synchronized one way (when convenient), but not in all other ways possible.
I maintain that argument (to maintain an illusion of consistency across a genre boundary) is specious.
 
 
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(25) Posted by Joost de Heer [Sunday, Sep 25, 2022 18:01]

>In any case, PGs in "exactly" n moves are not optimal.

Sometimes the 'exact' is the intention, see e.g. https://pdb.dieschwalbe.de/P1013079, where there are cooks in 9.0 moves, but there's only one proofgame in exactly 12.0 moves.
 
   
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(26) Posted by Joost de Heer [Sunday, Sep 25, 2022 18:30]

> However, note that the standards for PGn and h=n are not the same -- the h=n standard insists white is stalemating black (white moves last), whereas the PGn standard is unique.

The PGn standard insist that white starts, which is consistent, as PGn looks back to a fixed starter, while a h=n looks forward to a fixed ender.
 
 
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(27) Posted by Kevin Begley [Sunday, Sep 25, 2022 18:32]

I agree with Joost -- there is no reason (other than bad precedent) to suggest a "PG in exactly n moves" should seek to add an additional ply (violating the principle of "move economy").
Indeed, it makes little sense to judge problems which did not (or could not possibly) add an additional ply as inherently inferior products (as a failure to achieve that extra ply).
This policy constitutes a secondary standard (not a standard to determine correctness, but a standard for valuing problems), which is perhaps beyond our discussion (I would suggest an alternate thread).

Returning to the previous argument, where PGn is viewed as a cousin of h=n (setting aside h#n).
The h=n implies "for white" -- as does the h#n, the #n, the s#n, etc -- not only that white does the stalemating, but white plays the last move. This is not the case in the PGn.
If we are seeking a consistent standard between PGn and h=n, the PG (like the h=) must be read as a problem FOR WHITE to achieve the diagram (read: white must make the last move, unless the implied "for white" is countered by an explicit alteration in the stipulation; that is to say, "for black" would be required).

If we want proofgames to be help-games, which are stipulated in a manner consistent with all other problem genres, their default stipulation must imply it is a problem "for white" to achieve the diagram position (and must explicitly state when that principle is violated, using the consistent term: "for black").

Thus, if we insist upon consistency of standards (across the help-game and help-stalemate genres), any PGn.5 should be stipulated as a PGn, and any PGn.0 should be stipulated as a "PGn for Black."
Since nobody has argued for full compliance (complete consistency for PGn, h=n, and h#n), I would suggest a healthy skepticism that consistency can be considered the driving motivation.
The specious argument for consistency reduces to an argument for selective consistency.
Unless a fully consistent standard is advocated (which is certainly possible), it can not be argued that the standard Kostas suggests will achieve cross-genre consistency.

Once that Jenga piece is removed, I don't see that the standard Kostas has advocated confers any advantage over the standard I suggested (which confers numerous advantages, not least of which is to leave all published works sound as published, and eliminate the need for a major update of databases and software tools).
If we are not gaining a fully consistent standard (across the genre boundaries), what are we gaining for all that work?
 
   
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(28) Posted by Kevin Begley [Sunday, Sep 25, 2022 19:45]

With regard to evaluation standards for problems which use the word "exactly" in the stipulation (as previously noted, this discussion perhaps merits an independent thread, allowing this thread to focus on what standard of correctness should be codified, but my comment will be brief)...

It is worth noting that Tibor Orbán's famous "PG in exactly 4.0 moves" (referenced by Kostas, who apparently missed that I had posted the problem on the previous page of this thread) could have started his problem with 1.e3 (rather than 1.e4), thereby eliminating all four solutions in 3.5 moves (but still neither eliminating either the four solutions in 3.0 moves, nor the need for a stipulation containing the word "exact").

I would argue that 1.e4 was wisely chosen, enhancing the paradox (it is already paradoxical that a solver has difficulty in losing a full move, but even more paradoxical when the solver has a simultaneous difficulty in attempting to lose a half-move -- especially when the pawn, on e4, has the potential to lose a half-move). That is the reason this problem is so striking (it may well be the most elegant expression of the value obtained by using "exact" in the stipulation, and I can not fathom why a judge should have hoped for an additional ply -- were that possible -- to strike the word "exact" from the stipulation).

I consider that problem to be a perfectly expressed work of art, and would not want -- were it even possible -- an additional ply (which could only conceal the value of this problem, violating move economy for the sake of achieving homogeneity of stipulation).

And, for the record, it is not clear that this problem was not originally expressed as "PG4.0" (that remains to be seen -- as noted previously, it is stipulated only as "PG4.0" by Win Chloe, and marked "C+" under that stipulation; further, Jacobi renders one solution when asked to find all solutions, given that diagram, to the stipulation "PG4.0").

If it was originally stipulated as "PG4.0", then this problem, published in 1976, may have established precedent (a precedent which either bolsters or undercuts the standard suggested by Kostas).
Can anyone check the hard copy of Die Schwalbe (I believe it was April issue, 1976), to determine whether that problem stipulated "PG4.0" (or did it actually stipulate "PG in exactly 4.0 moves")?
Is anyone aware of any previous problem which stipulated (or could have stipulated) "PG in exactly n moves"?

I am hardly a fan of maintaining precedent (too often, we use bad precedent to perpetuate falsehoods -- like falsely categorizing reflex problems as something other than a fairy condition), but we should certainly be aware what is the precedent here (more importantly, we can not presume that Tibor Orbán's PG4.0 used the term "exact" in the stipulation, unless this claim can be substantiated from the hard copy).
 
   
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(29) Posted by Neal Turner [Sunday, Sep 25, 2022 20:06]

This is all very interesting, maybe my point earlier was made a bit too forcefully.
But it is something that should be resolved.
I'm thinking of the predicament a judge might be in - what does he do?
Does he discard them as cooked or does accept them and risk the ire of participants whose efforts have been bested by one of these 'flawed' examples.
I suppose there's always the cop-out (cop-out of a cop-out!) of the 'Special' categorization.

Now we have the WCCC meeting coming up quite soon, you proofgame guys should get together and come up with some proposals to set before the Codex Committee.
If they were accepted they could be implemented in just a couple of months - although it's more likely they would held over for further consideration, but at least the ball would be in play!
 
 
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(30) Posted by Frank Richter [Sunday, Sep 25, 2022 20:08]

The original stipulation was "Erspiele diese Stellung in GENAU 4 Zügen: d.h. Beweispartie in GENAU 8 Einzelzügen!"
 
   
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(31) Posted by Kevin Begley [Sunday, Sep 25, 2022 20:13]

Thank you, @Frank.
If Tibor Orbán did indeed publish the first such example (can anyone provide an earlier example which could have used "exact" in the stipulation but did not?), then we must admit that the standard suggested by Kostas has a strong claim that precedent was clearly established in 1976.

[note: though hardly a comprehensive database for proofgames, PDB shows no example predating 1976, and, as previously mentioned, I do not believe it is possible to search for such instances in Win Chloe]

@Neal, if we're going to talk about cop-outs, lol, yes, we should discuss awards containing the word "special." I could not agree more. ;-)

If ever there was a case for using "special" in an award, Tibor Orbán deserved a "Special Prize" (or "Special HM" at the very least -- not a mere Commendation) for his "PG in exactly 4.0 moves."
 
   
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(32) Posted by Joose Norri [Monday, Sep 26, 2022 03:46]

If you add an extra move at the end eliminating shorter "cooks", then the solver might not notice the tries, thereby missing what is possibly the main point of the problem. Or to put it in other words: such a modification may ruin a problem.
 
   
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(33) Posted by Kostas Prentos [Monday, Sep 26, 2022 05:40]

Joost (25): Yes, I don't see a problem with that example, and other PGs making the distinction between SPG and exact PG, with twins a and b. In fact, the goal of this PG is to emphasize this paradox. I wouldn't change anything in the stipulation. If the choice were between a single line of a PG in "exactly" n moves, and an SPG in n+0.5 moves, I would definitely go for the latter option. Of course, there may be examples in which using "exact" PG is better than the alternative - problemists are very good at finding exceptions to the rule. But in general, having the choice between an exact PG, and an SPG that avoids all controversy about shorter cooks, I would happily sacrifice the economy principle and have a PG that is slightly longer.

Kevin (28): I didn't miss the reference to Orban's exact PG 4,0 - how could I, you gave a diagram and all!. In fact I had already mentioned this problem earlier (post 11). I was just making the distinction between exact PGs based on lack of tempo (most common case) and others, like Orban's PG. I fully agree that the choice of 1.e4 in that problem was wise.

Kevin (27): I was not going for cross-genre consistency, although that would be nice. I was hoping to achieve clarity in the way a PG stipulation is presented. I also keep an open mind. If I decide (using my own criteria) that a PG is better as an exact PG rather than an SPG with a "tail" move, I will not hesitate to choose the former. But my default choice is the latter.

Neal (29): I wrote at the end of post 16 what I would do, if I had to judge an exact PG. However, since there are no hard rules, others might see it differently. As a composer of PGs, I would likely avoid pitfalls (keeping an open mind for the exception to the rule).

Joose (32): One advantage of using the term "exact PG" is that you make sure the solvers will notice the shorter solution(s). See for example the same link I gave on post 11: https://www.thbrand.de/2014/10/12/retro-der-woche-422014/#comments If there is a reason to emphasize that part of the problem, then by all means, use exact PG instead of SPG with a tail move. But when the PG shows tempo play, the solver is "forced" to find the full content, whether the problem is presented as SPG, or exact PG.
 
   
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(34) Posted by Kevin Begley [Monday, Sep 26, 2022 20:48]

After some consideration, I have changed my mind. I withdraw my advocacy for a compromised standard. I now favor the standard Kostas has suggested.

I admit I was somewhat moved by the fact that Tibor Orbán, in 1976, clearly established that standard, but I wasn't fully swayed by precedent.
What really swayed me was realizing that the vast majority of proofgames are composed with the clear intention to avoid solutions in fewer moves, that this should be reflected in the standard, and that those problems must not be burdened to change their stipulations (as originally published) when precedent clearly favors their standard.
I still believe a grace period would make this go down easier, but perhaps it would be sufficient to codify a date, such that any previous works may add "exact" to their stipulation (without the alteration being considered a correction). I believe this is necessary, because composers may have been subject to the standard of the publication editors or misled into believing the more lenient standard had been established by precedent.

The only plausible alternative standard, from my perspective, would be one seeking greater consistency across genres (read: where PGn implies white must make the last move -- implicitly insisting that white must achieve the aim, unless the stipulation explicitly says "for black" -- to maintain maximum consistency with all other established problem forms).
I would not advocate for that maximally consistent standard today, however. The considerable strain of implementation (altering previous works, retooling solving tools, and educating problemists about this new standard) is difficult to justify when even the long-term benefits are not obvious.

That said, if I were publishing the first proofgame today, I would likely adopt this maximally consistent standard (where PGn implies "for white"), whereas I had previously considered the standard Kostas has suggested to be ideal.

Finally, I would further advocate for two things:
1) codification of a single character, across all problem genres, to indicate "exactness" is required in the number of moves (that the problem can not be cooked in fewer moves than stipulated).
2) codification of a single character, across all problem genres, to indicate when a problem should be solved "for black" (not the implied default, "for white").
These suggestions will greatly reduce space in the stipulation (below the diagram), and make programming easier (more standardized) for a variety of software tools.
If others agree these are worthwhile suggestions, I leave it to their expertise to suggest what characters should be employed.
 
   
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(35) Posted by seetharaman kalyan [Thursday, Sep 29, 2022 11:04]

Can someone translate these long....... long..............
posts of Kevin, Andrew et.all for a dummy like me and

answer this simple question:

Does a solution in 8 moves cook a
PG 8.5? Or does it not?

Answer it in one line
 
   
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(36) Posted by Joost de Heer [Thursday, Sep 29, 2022 11:14]

No, unless the stipulation is 'SPG in 8.5'.
 
   
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(37) Posted by Kevin Begley [Thursday, Sep 29, 2022 11:28]

One line answer:
A "PG8.5" is nearly always composed with the intent that a shorter solution will be considered a cook, and this was the established standard in 1976; until a standard is codified, cook claims remain uncertain -- despite the fact that the "SPG8.5" standard (advocated by Joost, and previously advocated by yours truly) is used almost nowhere and is wholly inconsistent with all other problem genres (which use "exact" to stipulate exactness).

Optional notes:
1) While there are advantages to the "SPG8.5" standard, it is more likely to be read as a "series proofgame in 8.5" today ("S" is not the best indicator of non-exactness).

2) Disagreements like this are evidence that our Codex is dysfunctional (entrenchment perpetuates Codex failures, until failure becomes the default standard).

3) Until we codify a standard, I would suggest using the Charitable Standard:
If you encounter a "PG8.5" which can be cooked in fewer moves, look for a cook in as many moves.
a) if you find it in as many moves, as well, it is cooked under any standard.
b) if you find no cook in as many moves, you should charitably assume the author intended to specify "PG in exactly 8.5".

Personal note:
1) Early on, I sent a short proofgame (which I had stipulated only "SPG") in progressive chess to an online journal (note to rookies: avoid composing progressive chess proofgames, and avoid "SPG").
I thought my solution unique, but stipulated only "SPG" (wrongly assuming this more elegant stipulation was preferable, and my unique solution would be a rare bonus for "SPG").
Without my consent, the editor changed my stipulation to "PGn" (realizing my unique intent, but never consulting me). I was not happy. I demanded the problem be immediately changed, and refused to accept that an online journal had a duty to address their own error in a later issue (I expected an online journal had a responsibility to remedy their error immediately -- certain I had seen instances of such corrections, and more, taking place immediately for other authors).
Then, a cook was reported in as many moves (but not fewer). I might have argued the problem remained correct (under my original stipulation, this was technically not a cook), but I could not deny that this was never my intent (my problem was cooked). Under this strange circumstance, I could no longer accept a return to my original stipulation (which would wrongly convey no cook -- my intent was cooked), nor did I favor the stipulation erroneously imposed on me (to which I never consented). So, I asked that the problem be expunged entirely.
The truth is, I committed numerous rookie blunders (including thinking my original stipulation form was optimally elegant), but there's one blunder I would not commit: when a solution I did not intend to allow was reported, I would never deny this constitutes a cook (even when I might have argued this was technically not a cook under my intended stipulation, I would not resort to dishonesty).

Moral of this tragically LONG ... LONG story: if you report a cook to the author, the vast majority (even rookies) will tell you honestly whether they had intended to allow your solution.

2) I can no longer advocate for Joost's standard, because I can provide no good argument why Proofgames should violate both their own precedent and the standard used by all other problem genres.
 
   
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(38) Posted by seetharaman kalyan [Thursday, Sep 29, 2022 13:03]

@Joost. Thanks for perfect clarity in brevity.
@ Kevin. One-line answer please
 
   
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(39) Posted by Kevin Begley [Thursday, Sep 29, 2022 13:17]

ONE LINE: THERE IS NO AGREED STANDARD... you may be charitable, you may expect honesty from the author, but all should demand a codified standard (which formally answers your question). PERIOD.

I gave you a one line answer:
 QUOTE 

One line answer:
A "PG8.5" is nearly always composed with the intent that a shorter solution will be considered a cook, and this was the established standard in 1976; until a standard is codified, cook claims remain uncertain...


The rest of that sentence (and indeed the rest of my tragically LONG ... LONG answer) you may consider optional (it was intended for those who will appreciate that a one line answer is not possible).

If pressed for brevity, the best I can give you is five lines (if you can't read more than one line, you need only concern yourself with the first line):
1) THERE IS NO AGREED STANDARD (Joost is clearly incorrect to imply otherwise; moreover, the standard he advocates is almost never used -- SPG8.5 more often translates to "series proofgame 8.5").
2) If a solution exists, unprovided by the author, in as many moves (8.5), the problem is cooked under any standard.
3) If a solution exists, unprovided by the author, in fewer moves (8.0), but you find no solution in as many moves (8.5), you might charitably presume the author intended to convey "PG in exactly 8.5".
4) If you report your finding of a shorter solution (but no solution in as many moves) to the author, you should expect to receive an honest answer (whether this is a cook, or not).
5) We need a formal standard (codified in the Codex, universally accepted) which correctly (and concisely) conveys what the author is asking from the solver (shared by as many genres as possible).

note: item 3) is what I would call "The Charitable Standard" (and this standard -- not Joost's suggestion -- is what you will find in wide use today).
This Charitable Standard needs to formally end. The problem composer should never expect charity from the solver.

Finally, if the problem community wants to codify the standard Joost is advocating (which, I can not deny, has some tangible benefits), I would suggest two things:
1) Use a different nomenclature, so as to avoid confusion about whether "SPGx.y" means "shortest proofgame" or "series proofgame").
I would suggest "MLPG8.5" and "MLACPG8.5" (where ML = minimum length, and MLAC = minimum length for any color).
While this new standard has drawbacks (inconsistency with all other problem forms, and a failure to acknowledge established precedent) it would charitably preserve all "PGx.y" as originally stipulated.
I no longer consider this standard advisable, because it fails to avoid considerable upheaval (the vast majority of stipulated proofgames would still require modification to express their original intent).
2) Apply this new standard to all problem genres.
If you can not apply this standard to all problem genres, you need to make a case why proofgames should both enjoy a unique standard and ignore their own established precedent (this is a heavy burden, which I could not meet).
Note that this will impose even greater upheaval (good luck convincing helpmate and directmate composers that proofgame enthusiasts want all of their problems (stipulated "h#n" or "#n") to be altered (to "MLh#n" or "ML#n"), because proofgame composers refuse to accept that minimum length should be implied in the stipulation.

There are three alternatives to the standard Joost is advocating (which I had myself advocated, until I could not longer defend it):
1) Codify the Standard of Precedent (which is mostly consistent with other problem genres)
what Kostas suggests is logical, follows precedent, and is the standard to which the vast majority of proofgame composers implicitly adhere when constructing their problems (I'd venture over 99%).
It is also mostly consistent with other problem genres (much easier to learn one standard for all problems).
Even so, I would suggest changing the nomenclature (across all problem genres) so as to reduce the space needed to convey "in exactly n moves" (this can be expressed with a single character).
The downside is that we need to identify all problems which failed to meet this (more strict) standard (some of which require alteration), for as long as the Charitable Standard was dominant.

2) Codify the Standard of Maximal Consistency (which is maximally consistent with other problem genres)
Proofgames could use this codification opportunity to move to an even more consistent format, where the default assumption is that white must achieve the aim (reach the diagram, and make the last move), unless "for black" is explicitly stipulated -- as is the default rule (and nomenclature) in all other problem genres.
Even so, I would suggest changing that nomenclature (across all problem genres) so as to reduce the space needed to convey "for black" (this can be expressed with a single character).
The downside is that this requires some acclimation time, and the upheaval (identification and alteration work) is even more formidable.
The upside is maximal consistency for the sake of maximal consistency (not much more than that, I'm afraid). The long term benefits are not easily identified, except that it might simplify comprehension of stipulations (across all genres), particularly for newcomers.

3) Keep (but never codify) the Charitable Standard...
And have this LONG debate repeat every time somebody asks the simple question: "is a PG8.5 cooked if there's a solution in 8.0 moves?" (after no consistent, honest, one-line answer satisfies the query).
Upside: no renovation work is ever required, so long as we adamantly refuse to admit all past mistakes.
This is the worst alternative, of course, but in problem chess matters, the option requiring minimal work always wins the day (it's not precedent that decides, it's the principle of minimal alteration).

The problem chess motto: you will own no intelligent Codex, and you will remain blissfully content.
Asking obvious questions, and expecting a single line answer, is not in the spirit of our blissful contentment. Capeesh?

ps: don't impose a standard upon the answer, when you failed to adhere to your own standard in the posing of your grossly inefficient question. ;)
 
   
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(40) Posted by Kostas Prentos [Thursday, Sep 29, 2022 21:57]

Seetharaman (35) (influenced by Joost's answer 36): No, if the stipulation is exact PG 8.5.

As you can see, there is no universal agreement. Others would go even further and answer as follows: "Yes, unless the stipulation is exact PG 8.5". That's why I always advise to add a tail move to make the PG 8.5 a PG 9.0 without any shorter solutions. For the cases this doesn't help, using the term exact would suffice.
 
   
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