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When writing chunks and using Fletcher checksums are there any situations where the HDF5 API will do a read of a "chunk" under the covers when an application is writing a file?

Yes, the library will perform a read when partially updating a chunk that is written to. This is required when chunks are filtered because the filter (checksum in this case) is performed on the entire chunk, not just a portion of the chunk. Additionally, if the chunk cache is enabled and used (likely), the chunk will be read from the file before being partially updated in memory, and then eventually written back out.