Interesting, FascinatedWanderer.
Certainly check your stopper position, as others have indicated. (Note where it is set first, in case it had been very cleverly set, and you want to return to it! Or it had been badly wrongly set, and you want to avoid it!) Normally, a stopper distance issue shows up as an issue in the upper 2nd and third octaves, whereas I’m interpreting what you are reporting is a systemic flatness of the low octave compared to the second octave. It would be helpful to have some figures on how flat, how sharp and which notes!
But, I’m wondering if you are encountering a variant of an issue that plagued 19th century English players and subsequently plagued Irish players who picked up their old flutes in the 20th century. For a number of reasons, flutes from that period suffered what I termed “flat foot syndrome” - the notes in the second octave were generally well in tune, but particularly the low notes in the octave below were really painfully flat. Sometimes just a bit, but sometimes dramatically flat, like say 60 cents! The players in both centuries worked out the same solution - force the energy in the lower octave notes into the higher (in-tune) harmonics, by using an offset jet. We call that blowing down towards the bottom of the embouchure hole, rather than blowing across at the top of the embouchure hole.
Woah, wait a minute, I hear you holler. If I force the energy into the higher harmonics say on low D, won’t it sound middle d? Surprisingly, no. It still sounds low D, but but a nice edgy version of it. The reason is that you are still exciting the harmonic structure of low D (1, 2, 3, 4, 5) not the harmonic structure of middle d (2, 4, etc). Excepting “1” doesn’t sound much or at all, so you end up with (2, 3, 4, 5), not (2, 4, etc). Our very clever brains interpret that as an edgy low D, not a middle D. Turns out we are smarter than we think (thank goodness!)
Now, in the classic “flat foot syndrome” , it’s only the foot notes that require this manipulation (lipipulation?), as it’s only they that are dramatically flat. But you seem to be suggesting that your whole lower octave at large is flatter than your middle octave? If so, I would imagine the same approach would work - play the whole bottom octave using the offset jet approach - the whole low octave will sound edgy but in tune. Because you are relying on the second octave tuning not the first. It would require you to tune the second octave A to the tuner or the accordion or other fixed instrument, or use the offset jet lower octave A, not the naive lower octave A.
Does that fit your experience, or have I wandered off into the wilderness?
It’s nice here in the wilderness…