What's the blocksize limit after segwit and how do legacy nodes deal with segwit transactions? Normally, I heard that the bitcoin blocksize is 1MB, but with the Segwit update the blocksize can be up to 4MB, so we should say that the blocksize is 4MB, right? Moreover legacy nodes only receive the input and output data of segwit transactions. How do the legacy nodes verify the validity of segwit transactions. The legacy node can verify it during propagation from a segwit node, but how does it work for propagation between two legacy nodes, where the segwit data is missing completely? Thank you
Segregated witness therefore takes advantage of this opportunity to raise the block size limit to nearly 4 MB, and adds a new cost limit to ensure blocks remain balanced in their resource use (this effectively results in an effective limit closer to 1.6 to 2 MB).

https://bitcoincore.org/en/2016/01/26/segwit-benefits/#block-capacitysize-increase

The maximum block size is 4,000,000 bytes (4 MB). This is because the block weight calculation is base size (in MB) * 3 + total size (in MB) = block weight (see BIP 141). Since the only units are MB, the block weight's only units are also MB, thus the maximum block size is the same as the block weight.

In theory, you can get pretty close to 4M, by having transactions that consist almost entirely of witness data. In practice, that won't happen for normal financial transactions, closer to 2M.
https://bitcoin.stackexchange.com/a/54949/

SegWit simply lets you verify non-witness transaction data without downloading (then discarding) the witnesses, which solves the problem of light clients having to download things they don’t care about and can’t verify anyway.

https://blockstream.com/2017/07/31/en-segwit-myths-debunked/

When Segwit transactions are sent to Legacy nodes the witness data is stripped. The key is that these “stripped” transactions are still valid transactions on Legacy nodes, which gives us a savings over non-Segwit transactions. Thus, more transactions can fit into the block sent to Legacy nodes without going over the 1,000,000 byte limit.

Segwit nodes get Segwit transactions and blocks that include the witness data using alternate network messages. The new network messages are defined in BIP144 as part of Segwit. The Segwit blocks which include the witness data can be over 1,000,000 bytes. Legacy nodes, as mentioned, receive the same blocks and transactions, but with the witness data stripped. This is a way to make Segwit a soft fork.
The Segwit blocks are restricted by something called Block Weight. Block Weight is a new concept introduced in Segwit, and it’s calculated on a per-transaction basis.
Each transaction has a “weight” which is defined this way:
(tx size with witness data stripped) * 3 + (tx size)
Segwit transactions are transmitted to Legacy nodes without witness data, so this formula will always result in blocks communicated to Legacy Nodes that are less than or equal to 1,000,000 bytes. Again, this is why Segwit is a soft fork.
https://link.medium.com/3lvCPzo5q9

Size vs Weight: What is block weight and how is it different from block size?