[Dennis Sweeney <[email protected]>]
> Here's my attempt at some heuristic motivation:

Thanks, Dennis!  It helps.  One gloss:

> ....
> The key insight though is that the worst strings are still
> "periodic enough", and if we have two different patterns going on,
> then we can intentionally split them apart.

The amazing (to me) thing is that splitting into JUST two parts is
always enough to guarantee linearity.  What if there are a million
different patterns going on ?  Doesn't matter!  I assume this
remarkable outcome is a consequence of the Critical Factorization
Theorem.
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