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What are the tonal differences between a really stiff top across the grain and a pretty floppy one?

I've only built with stiff tops, but I have a highly figured bear claw sitka top that is really floppy. What will that top lack compared to a stiff one? Would you use it?

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"An adventure is only an inconvenience rightly considered. An inconvenience is an adventure wrongly considered." G. K. Chesterton.

The first question that comes to mind is "Why is this top so floppy?" Is it due to severe runout or is the grain well off vertical? I would have to spend a few minutes with the top to tell if I would use it.
I have returned tops which were off quarter & lacking in stiffness. The producers usually won't argue if the grain is well off vertical.
Greg Smallmann (a famous classical guitar builder in Australia) is said to prefer very loose tops for his lattice braced guitars... Much depends on how you thickness & brace the top as to how it will sound.
You could end up with a very nice fingerstyle guitar, but it may crap out if driven hard. I would expect the bass to be less defined & the overall loudness to be less than a stiffer top would produce (all else being equal) ((Which it never seems to be))
I would consider using a softer top for a small bodied guitar. I choose softer wood with a lower tap tone for my smaller instruments & (so far) it seems to be working for me.
If it is VERY floppy, I would return it for refund or replacement.

A few years ago I built two identical guitars. They were large bodied, deep guitars, similar to the Martin "Jumbo" in shape.
The back & sides (cocobolo) were from the same billet. The neck wood from the same plank. All the bracing from the same stock etc. etc....
One had a nice (not overly stiff) Engelmann top & one had a VERY stiff, high density Sitka top.
The Engelmann topped instrument attracted an immediate sale. Very nice guitar with a big warm sound, lots of rich bass, good - clearly defined mids & trebles... All the good stuff.
The Sitka topped guitar was (and is) a complete dog. It sounds "boxy" & dull. The bass E dies at the fourth or fifth fret, the mids disappear when strumming chords & the trebles are thin & tinny.
In the hands of a more experienced builder, I'm sure that Sitka top would be fine, but for me & my style of bracing / tuning the top, it didn't work at all.
All this rambling is to say that without flexing & tapping I can't advise you very well.
If the top is pretty & it shows some good silk, I would thickness it & brace it up.
If nothing else, you'd have some more impressions to ad to what you have already stored away.

A lot depends on 'how floppy'. A lot of times you can use a somewhat floppy top by altering the X angle, and the angle of the tone bars. That stuff is only 'sacred' on a production line.

What a powerful statement.

I've read this from Al before. Since it makes sense, this spring when I had the chance to use a less-well-quartered top I decided to try it. The stiffness and weight were fine, but it was obviously rubbery across the grain. What I did was to thickness it according to the long-grain stiffness and density but be careful at not weakening the crossgrain component of the bracing in any way. Classical bracing is different animal, but, sort of similar to splaying the x or reducing scallops, I made sure the soundhole cross brace is stiff (I tend to scallop or make arches in it for a perfectly quartered top) and also that the bridge is stiff (classical bridges can vary quite a lot more compared to ss belly bridges). I also made sure the end tapers on the fan braces are not too aggressive.
The guitar sounded better than expected. From the 5 I made so far (plus 2 retops) it probably sounds the second best or the best, depends on what sort of bass-med-treble balance one prefers!

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Build log

I have nothing to add.......but wanted to say I'm enjoying this topic and I'm looking forward to hearing more discussion.

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Formerly known as Adaboy.......