The calcium will leach away leaving carbolic acid behind, that re-acidifies the soil so you have to reapply the calcium carbonate to reduce the acidity of the soil.
When you read about lime, you are reading papers prepared by folks that are totally vested in "Modern Agriculture Methods" these folks are in every segment of horticulture so that even flower and vegetable farmers are listening to the same information.
What that does is get most of the growers onboard with applying lime or sulfur (depending on the base pH starting point) to adjust the soil pH.
The interesting (to me) part of this is that while it does work, it is only a temporary fix that will necessitate repurchases of the materials, meaning the seller gets to sell you the stuff over and over and over.
When you bring up rainforest, or any forest for that matter, you are talking about undisturbed land, growing the same plant matter year after year, most of which is perennial.
Farmers are more about annual plants, so they are going to till and plant and fertilize and adjust every year.
CaCo3 prices vary all around the planet and countries can have variant prices too, so it would be best to locate suppliers and get quotes from them.
For me, there is no getting around adding lime, or a lime alternative. We are rather low in PH in the Northeast anyway, but with the mid-west powerplants pumping out acid rain, and the jet-stream running this way, our water bodies and soil is just teeming with acidic moisture. It is no big deal, our pastures and hay fields can thrive on low PH, along with potatoes and other crops, but the trouble begins when we want to grow other crops like corn or alfalfa that like a much higher PH.
If we want to grow that stuff, we got to play the lime game.
But there are other ways around buying lime. We can use wood ash here, and seaweed, both of which are free, or very cheap to buy. It takes a fair amount of it, but it is available. To me it is just the price we pay for not having to irrigate crops. We get free water, but it comes at a price. Putting lime on the fields is just a part of farming here, and I would rather do that once every 6-7 years then have to put water on my crops to get them to grow.
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Permaculture Technology Jamboree: June 29th-July 10th, 2020, Wheaton Labs