The original questions were:

"Has anyone been able to verify the capacity of these? AA are specified at 4000mah. "

No.

"What is the advantage of these new batteries?"

They last a longer time than some other AA batteries.

"They last 10 years?"

The shelf life of alkaline batteries can reach 7 to10 years. If a load is put on the battery, yes, they can drive a small load for 7 to 10 years.

http://www.alkalinebatteries.us/Shelf_life_of_Alkaline_Batteries.html

For example, a NAND gate in the following link's Figure 41 uses 1 femtojoule to switch with a fanout of two.

https://smartech.gatech.edu/bitstream/handle/1853/31652/wunderlich_richard_b_200912_masters.pdf;jsessionid=A8FC50BF5BD90D3F800BA9A55B742329.smart2?sequence=1

Running at 1 Ghz, it uses one microjoule per second. So the AA cell with 21000 joules can last for 21 billion seconds, unless that is more than the shelf life of 7 years.

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Home electricity cost from wall socket example calculation:

11 cents per kilowatt hour (kwh) in New England

or

43 cents per kwh in Hawaii

one kilowatt hour is 1000 watts * 3600 seconds = 3.6 megajoules

in New England, USA

(11 cents/kwh) * (1 kwh/3.6Mjoules) * (1 billion joules) = $30.55

in Hawaii, USA

(43 cents/kwh) * (1 kwh/3.6Mjoules) * (1 billion joules) = $120

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

AA cell battery cost to run a household that spends $200 per month on electricity

Compare a battery cost to a wall socket's electrical cost :

1 billion joules costs $30.55 from my wall socket.

How many batteries will give that much energy?

1 billion Joules / 21 kiloJoules per AA battery = 47000 AA batteries

So find the value of the batteries to compare to a wall socket :

$30.55/47000 batteries = $0.00065 per battery.

But batteres cost $1.50 for an AA cell, not .065 cents.

According to this calculation, batteries are 2307 times more expensive than a wall socket's electricity in New England.

So it would cost 2307 * $200 to power a house using AA cells for a month.

That is $460,000 of AA cells to supply electrical energy to a house for one month.

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

The AA cell battery has 21000 joules. If that energy could be used in one second it would have a horsepower of:

21000 J * (1 horsepower/745 watts) * (1 watt*second/joule) = 28 horsepower second