None of which changes the fact that a single layer of armor is provides more resistance vs. seperate multiple layers.
Its not a fact, but repeating a the simple nonsense - otherwise you would be saying that armor designers are simply out of their minds when they use spaced, sandwiched armor etc. I don't think they are aiming at creating
less effective systems.
The italian example represents a single belt of armor which happenes to be made up of different materials in order to promote the idea of de-capping, which is not an exact science. They did not place a single thinner belt on the outer hull, followed by one or more "belts" internally to promote the shattering and breaking up of the shell. As mentioned their solution had merit but did not prove 100% effective in tests nor was the system tested in battle.
That's seems like a rubber argument to dismiss the whole idea of shattering projectile - ironically all ships vertical armor still used face hardened armor just to promote this silly idea..!
In any case, the crux of the arguement wheater the use of the upper side belt was weight efficient.
Now, you cannot really argue it added weight because it just didn't - as seen before, the tons that went into allowed the same weight shaved off from the main barbettes. It came at no extra weight and presented many benefits in armor deck, scondary and main batterie barbette protection, plus effective protection against medium calibre shells.
Now if I understand correctly, you are arguing that the use of a 145 mm KC plate on the upper side belt and the use of a 220mm KC NA on the barbette (365 mm in total) was, as opposed to Okun's calculations, less effective in ballistic protection than a single 340 mm thick barbette? (In addition there seem to be a 30 mm splinter belt too near the magazines).
It would seem to me that
in the worst case scenario, its at least as effective, given the weight speed and stability penalties imposed by the upper side belt would more than just compensate for the penalties of a multiple armor layers vs a single one.
However as you noted Italian tests showed that in two out of three cases, the 70 mm outer plate distanced 250mm from the main face hardened plate succeeded in stripping the projectile of its AP cap, which subsequently shatter on impact on the main face hardened plate. Now Bismarck's outer plating was double the thickness, and there were much more space between it and the barbette plus another 30 mm splinter belt. Given that all factors are much more beneficial for decapping the projectile, I'd say it would work even better than the Italian system and the projectile would simply shatter on impact.
The US and UK took a more conventional approach to 'decapping' with face hardened armor.
Well its a way of saying they were some 50 years behind the Germans and Italians in that regard.
The latter in particular approached the issue from a quality persective and worked to increase the effectiveness of their Cemented Armor (CA) for the KGV class. It's been estimated that this armor, vs. heavy caliber shells was 25% more effective than it's contemporaries.
I think delycrios already addressed this. In any case, neither US or UK vertical protection systems could not provide adequate protection for the vitals, so arguing about the details is a bit moot.
The slope created by the low armor turtleback will help direct shells up into upper decks and superstructure. A battleship's systems are not all located under the primary armor deck. It is true that these systems (including radar and optical fire control systems) can't be fully protected on any battleship and soft kill/disablement is a viable alternative to outright sinking via catastrophic damage. The low slung armor belt also reduced, as mentioned the actual protected bouyancy of the ship.
Let's make it clear what situation we are talking about - rounds that successfully penetrate the main belt near the waterline - otherwise they simply can't hit the turtleback slope located behind the belt, and the shell still has considerable energy to be bounced towards the superstructure before the fuse sets it off, ie. short range hits.
In this case, what do you think is the preferable end results?
a) The shell is directed up into upper decks and superstructure by the turtledeck present, and possibly ruining some of the non-vital systems of the ship
b) The shell happily continues straightly into the machinery or magazines and then it explodes there, as there's no silly turtle deck to stand in its path and deflect it upwards in a most stupid manner
You seem to be arguing that b) is preferable to a)..
Stopping shell fragments is an important consideration independent of how high or low the deck is placed. The Americans introduced this feature into their battleships on the correct acknowledgement that plunging shellfire is not a simple matter of rejection or penetration. Often armor spalling occurs even in situations where the shell itself is rejected. In other situations a partial penetration might be effected. Placing the armor deck higher, in addition to increasing the area of protected buoyancy also allows more solid protection of critical spaces from plunging fire. The lower splinter deck of SPS (Special Treatment Steel) thus enhances the protective effect provided by the primary armor deck.
Here you seem to be arguing that stopping shell fragments or armor spalling from plunging fire is more important than stopping entire shells effectively entering into the machinery or magazine spaces through the belt. I tend to disagree with your priorities.

Yes splintering and spalling can be an issue, but its minor one compared to full sized shells going off effectively inside the ship, plus there is no reason why low mounted decks could not mount splinter protection below, too, if someone is very concerned with that issue.
The ranges in which the British cruisers defeated it was apparently considered reasonable.
You seem to have been stating that the upper side belt was supposedly vulnerable to 6" cruiser fire, now you switched the arguement that it was vulnerable to 8" guns, which is supposed to
Sorry but there's no point of arguing any further - despite asked several to support the notion you presented with something, you could not deliver anything.
The upper side belt was safe from 6" cruiser fire, as per Okun and the physical realities of this world.
I did.....per Brower's text; "Nine 380mm guns in triple turrets were also considered because it would have meant an increase over the desired set displacement." This sentence does not refute Garzke's design commentary.
Of course it does. It very much reads to me that three triple turrets were considered for Bismarck vs four twin turrets, and eventually they choose the same because four twin turrets were weighting less.
You claimed they weighted more. Which has been proven incorrect.
Littorio's turrets to use one example were more heavily armored than Bismarck's and contained an additional gun barrel.
Where on Earth you get they were more heavily armored than on Bismarck. Ie.
Littorio vs Bismarck turret thicknesses:
350mm / 360 millimetres (14 in) HG turret faces
200mm / 220 millimetres turret sides
200 / 180 (forward) or 150 vs 130-150 mm (aft) roofs
Simply what you claim that they were "more heavily armored" is completely incorrect - especially considering things like the many gun openings of the closely packed guns weakening ballistic resistance.
Naturally they were heavier one for one vs. one of Bismarck's, hence your simple math exercise.
They were heavier because they had to be 30% larger to accomodate a single gun. Each turret of Littorio weight some 500 tons more, and its not due to the extra gun per turret, which weighted about 100-120 tons at best. Each barbette of Littorio had to be 30% larger on the same account, and thus was 30% heavier and longer as well.
No, not surprised because the weight savings issue is not a simple matter of comparing turret weights. A 3 turret design allows weight savings overall when all factors are considered, and allows armor to be increased.
As proven above, the notion that using three triple turrets vs four twin turrets would be more weight efficient has no merit, in effect the opposite is true. At this point You would need to support your statement with something more substantial than simply repeating it I believe.
