I don't think your argument about "traditional means of compression" holds. The existence of redundant data can be proven by the ability to store the same info

I don't think your argument about "traditional means of compression" holds. The existence of redundant data can be proven by the ability to store the same information with less data. Pi's information density is in fact no larger than the small function that generates Pi's digits -- even if traditional compressors fail to identify it.