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The node RPC allows you to ask for the context constants for any block via /chains/<chain>/blocks/<block>/context/constants. Is it possible for these constants to change apart from a protocol change (i.e. without the protocol hash of the <block> also changing)?

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No it is not. Constants are set in the genesis block and then changed only by amendments. They are stored in the context, which is the state passed to and returned by the protocol when it validates a block. That's why the RPC requires a block and inspects its context to read the constants.

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