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Through my rough research with grok. The battle time in which this game theory works is 750 blocks. Than its decided for all the bitcoincore-nodes. They will start to discard orphan blocks. Because according to their rules the summ of all orphanblocks is limited to 750 blocks(since2014). When that is reached this nodes will start discard blocks out of the orphan block space.

You're partly correct – the limit of 750 orphan blocks has existed in Bitcoin Core since 2014, and it's indeed designed to prevent excessive storage usage. However, this limit does not determine the length of a chain battle (fork). Here's what happens during a fork (chain battle), and how it differs from orphan blocks: During a fork, two or more competing chains coexist. These chains are not orphan blocks; they have known parents and are fully valid. The network compares both chains' total work (accumulated difficulty). Nodes switch their "active chain" to the one with more work. The other chain becomes "stale" but is still known and valid. The 750-block limit does not apply to stale blocks during a fork. It's only for blocks with unknown parents (orphans). . There's no predetermined limit to how long a fork can last; it depends on mining power and block finding rates. In other words, during a fork, nodes don't discard blocks after 750 blocks. Instead, they remain committed to the chain with the most work, and the 750-block limit is irrelevant in this context.

Thank you for your answer and clarification. I will doublecheck that. Have you estimation how long Bip 444 fork chain can be the lightest one?

The BIP-444 soft fork proposal is designed as a temporary one-year measure to restrict arbitrary data in Bitcoin transactions, aiming to address legal and spam concerns related to on-chain data. This means the soft fork chain created by BIP-444 could be the "lightest" (i.e., the active or preferred chain adopting these rules) for about one year, as the proposal itself explicitly states a one-year duration to give the community time to evaluate and develop longer-term solutions. However, the actual duration in practice could vary significantly depending on miner and node adoption, community consensus, and potential support or opposition from economic actors. Since adoption of Bitcoin Core v30 (which relates to these changes) is still relatively low, the real network effect may take longer to fully manifest or might face divisions leading to fork scenarios. In summary: Intended minimum duration: approximately one year according to BIP-444’s stated temporary soft fork period. Actual effective duration: depends on consensus dynamics, miner and node support, and network upgrades; potentially shorter or longer if disagreements cause chain splits or delayed adoption. No fixed block number or time beyond that one-year proposal is currently defined, so careful monitoring of miner signaling and node adoption rates will be crucial to estimate how long the BIP-444 chain remains the main (lightest) chain

Pretty solid analysis imo. The only thing I’d push back is framing BIP-444 advocates as purists. In my dictionary they are simply called bitcoiners and not shitcoiners.