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Your #4 is the early model with the lever release.  It was introduced by Remington in approximately 1901.  The exact date is anyone's guess
FYI - Remington didn't serial number their rifles except for the REMINGTON NO 1 ROLLING BLOCK LONG-RANGE CREEDMOOR TARGET RIFLES

Bill,
Look to see if you can find the serial number located on the underside of the barrel, in front of the tip of the forearm, near the caliber stamping.
You don't need to remove the forearm to see the serial number.
When Remington came out with the later version of the #4 takedown that used the knurled screwhead with coinslot protruding out fron the left side of the frame (about 1924), they moved the serial number from the barrel, to the left side of the frame just below the "Model 4" marking.

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John Boy,
Where did you get the info on Remington serial numbers? Every #1 Rolling Block I own has a serial number, and I don't own a Creedmoor. My #1 1/2's and my #2's also have serial numbers. The #4 does indeed also have a serial number, but it is not on the frame! It is on the bottom of the barrel, just forward of the forearm wood. Right next to the caliber marking.

According to John Gyde' the first #4 was offered in the 1890 catalogs. He lists the first 10 years as solid frame, and the lever takedown being first offered in 1900. In 1924 the takedown lever was changed to a thumbscrew on the left side, and the octagon barrel was dropped, being replaced by round barrel. 

Age of your #4 can be narrowed down further by features, such as the escutcheon the stock bolt screws into, and rollstamp used on the barrel. Without more detail it will only be narrowed down to 1900-1924.

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For military contracts, I can believe it, but not commercial production.

They all have serial numbers, except for the military rifles.

Yes, a military Rolling Block is totally different, and agree that batch numbers aren't serial numbers. But I haven't seen a sporting rifle that didn't have a serial number somewhere yet, unless it was restored. My R. Evans engraved #4 was likely heavily pitted on the barrel, as the serial number and caliber are barely visible on the bottom flat.  No idea for sure what the whole number was, as parts aren't distinguishable.

John Boy,
How did your #4 come to be chambered in 32-20?
The #2 was offered in 32-20 from the factory, but I've never heard of a #4 chambered for 32-20 by Remington.
Were the #4's ever sold as part of a military contract?
If so, that would be an interesting part of Remington history to learn about.