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I can't seem to find actual documentation about the reward halving process.

This is not addressed in Bitcoin's Whitepaper, although Satoshi Nakamoto briefly mentions the decrease of the reward, but never explicitly talks about the amount of the reward.

I've also looked in the references of Bitcoin's Wikipedia page, but nothing there either. The couple of crypto books I own do not mention it either.

Is it possible that the amount is explicitly described only in the source code? (To be honest I haven't looked at the source code.)

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This is not addressed in Bitcoin's Whitepaper, although Satoshi Nakamoto briefly mentions the decrease of the reward, but never explicitly talks about the amount of the reward.

Satoshi speak about the halving process since the release notes for Bitcoin v0.1 Alpha:

Total circulation will be 21,000,000 coins.  It'll be distributed
to network nodes when they make blocks, with the amount cut in half
every 4 years.

first 4 years: 10,500,000 coins
next 4 years: 5,250,000 coins
next 4 years: 2,625,000 coins
next 4 years: 1,312,500 coins
etc...

When that runs out, the system can support transaction fees if
needed.  It's based on open market competition, and there will
probably always be nodes willing to process transactions for free.

Satoshi Nakamoto

On the Bitcoin wiki ( https://en.bitcoin.it/wiki/Controlled_supply ) you find this table that tell you when and how each of the 33 halving process will happen:

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If you need a technical explanation of how the halving work in the code here you find a dissemination about it: https://ma.ttias.be/dissecting-code-bitcoin-halving/

Keep in mind that the halving are programmed by block height, so there are only estimates on which date these blocks will be reached (and the block subsidy will cut in half), but all other info on this topic are exact and well documented.

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