In order to see the dough formulation you used with all the numbers properly lined up, and to recall the particulars of the dough formulation, I went back to the dough formulation as set forth in Reply 9 at http://www.pizzamaking.com/forum/index.php/topic,10578.msg93963.html#msg93963
. From that post, I can see that you are using a natural preferment at a rate of 20% of the total formula flour. At that rate, it is probable that you have been getting high levels of organic acids that may be tightening up the protein and creating a gluten structure that is elastic rather than extensible. In Reply 23 in the same thread, at http://www.pizzamaking.com/forum/index.php/topic,10578.msg95262.html#msg95262
, I suggested a couple of things that might have addressed that problem. Did you ever try either of those proposed solutions?
I can sympathize with your plight trying to make a cold fermented dough using a natural preferment. Some time ago, I attempted to do something similar when I made a naturally leavened Papa John's clone dough. As you will see from Reply 38 at http://www.pizzamaking.com/forum/index.php/topic,6758.msg60892.html#msg60892
, I made and cold fermented a dough that was based on an Ischia preferment. My Ischia starter was not particularly virile at the time so I used it at a rate of 25% of the total formula flour. Normally, I would use about 15-20%. The dough fermented in the refrigerator for about 4 1/2 days. As you experienced, when I wanted to use the dough, I saw that it had risen very little. In my case, I needed another 20 hours or so at room temperature for the dough to rise to the desired volume. When I ended up using the dough (after about another 4 1/2 hours), I easily formed the dough into a skin without any elasticity problems. However, in my case, I was using over 7% oil, which no doubt helped the extensibility of the dough.
While a naturally leavened dough is cold fermenting, everything gets slowed down. However, the enzymes and bacteria are still on the job and you can end up with a dough with decent levels of organic acids and related byproducts of fermentation. However, the yeast, whether natural or commercial, can remain lethargic. Consequently, the dough may require a long period at room temperature, or in a temperature-controlled environment, to get the desired volume expansion of the dough after removing the dough from the refrigerator. That is just the nature of naturally leavened cold fermented doughs, especially if little or no fermentation takes place before refrigerating the dough or the starter culture is not quite ready.
I know from your other thread on this subject that you have read Jeff Varasano's material at his website (or elsewhere) and that he made naturally leavened cold fermented doughs that went into the refrigerator for about 1-6 days. However, I don't know whether he was using IDY in addition to the natural preferment (9%). I do recall that his dough was subjected to about 55-60 minutes of preparation time (with no autolyse) during which the dough was given time to ferment before being refrigerated. That no doubt helped jump start the fermentation process. However, I don't know if that would have been enough to allow the dough to be used shortly after removing it from the refrigerator. In your case, you could supplement the natural preferment with commercial yeast, but in a hybrid situation the natural yeast and the commercial yeast compete for nutrients. In such a case, the commercial yeast is likely to win the battle, resulting in reduced organic acid production and less crust flavor and aroma. But a natural side effect--and a possibly beneficial side effect in your case--is that the reduced acid levels may have less tightening effect on the gluten structure, resulting in reduced elasticity. Of course, you could use only commercial yeast, as was earlier proposed by scott123, I believe, but then you would be foregoing the benefits of using the natural preferment.
You can read a bit further on some of the above matters at Reply 125 at http://www.pizzamaking.com/forum/index.php/topic,1298.msg13410/topicseen.html#msg13410
Another possibility for you to consider is to use a regimen as suggested by Matthew in an earlier post in this thread. In my opinion, that might produce a more "balanced" fermentation protocol (biochemically speaking). You could also increase the total formula hydration as was suggested by scott123, and you could add some oil, which is common for a NY street style pizza. Usually, when oil is used, it is used at a rate of up to 3% of the total formula flour. If you try any of these methods and your dough still ends up with excessive elasticity, then the next logical step in my opinion would be to use less natural preferment or alter its composition, as I noted above. As you consider the possibilities, you might also want to read the following articles on preferments by Didier Rosada, at http://web.archive.org/web/20040814193817/cafemeetingplace.com/archives/food3_apr2004.htm
and at http://web.archive.org/web/20050829015510/www.cafemeetingplace.com/archives/food4_dec2004.htm
. The Rosada writings are directed to commercial-yeast preferments although there are some references to starters/levain
Please keep us posted on your progress.