Imagine the CDC yelling: "Don't throw that away! I need more details! Please! If you won't finish the job, please let me do that work!"

That's my translation of the document I found when looking into this issue. More details below...

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"CDC is suggesting CT value of 28 for detecting breakthrough infections after vaccinations... So that means, a breakthrough infection needs to have 128 times viral load in someone who has been vaccinated to be considered as positive, than it is required to have considered as positive in a non-vaccinated person."

This sounded strange to me, so I searched for more information about it. It looks like you just misunderstood the CDC's policy.

This document regarding breakthrough case investigations was the second result in a Google search for "cdc ct value 28". The relevant text can be found on page 5 of that document.

https://stacks.cdc.gov/view/cdc/105217/cdc_105217_DS1.pdf

"If SARS-CoV-2 sequencing will not be performed locally and a specimen is available, the state public health laboratory should request the residual clinical respiratory specimen for subsequent shipping to CDC. For cases with a known RT-PCR cycle threshold (Ct) value, submit only specimens with Ct value <=28 to CDC for sequencing."

In other words... Imagine some lab just found a breakthrough case. And this breakthrough case had an especially high viral load (Ct 28). And the lab was just going to report a positive and throw away the sample...

Imagine the CDC yelling: "Don't throw that away! I need more details! Please! If you won't finish the job, please let me do that work!"

That's what the document is saying. There's a rare event that needs extra analysis. CDC is just letting the labs know in advance that if they ever see this event, and didn't have the resources to fully analyze it, please send that sample to the CDC so it gets the attention it deserves.

Nothing to do with whether the test is considered positive or a breakthrough - just about whether to put extra effort into gathering more details on that particular case. CDC is volunteering to do this extra effort only for higher Ct values. Whether or not they do this extra work, it's still a positive result either way.