araver

images are clearer.Reusing past established "facts":

• Fact: All p-balanced scenarios have a complement of (n-p)-balanced scenarios
• Corollary: All 5=balanced scenarios ARE complements of 7-balanced scenarios (and viceversa).
• Fact: All p-balanced scenarios where p is not prime and gcd(p,n)=1 can (must) be decomposed into balanced non-overlapping groups.
• Fact: All p-balanced scenarios where p is prime are composed of equally distant samples, are atomic, and gcd(p,n)=p.
• Corollary: All p-balanced scenarios where p is not prime and gcd(p,n)=1 can (must) be decomposed into balanced non-overlapping atomic groups.

TSLF's solutions for n=12 and p=5 / 7 were:
*5 is  balanced on (12,3,4,8,9)
*7 is balanced  on (12,3,4,8,9,1,7) or (12,3,4,8,9,11,5),

All the above balanced scenarios can be decomposed into atomic groups (5=2+3),(7=2+2+3). And all have a complement scenario which is also balanced and can be split into atomic groups.
TSLF's solutions and complements below:
 Centrifuge12.png   19.51K   15 downloads

I dare conjecture based on the above that for any n which is a multiple of 6 (or equivalently have 2 and 3 as divisors), all cases except for 1 and n-1 samples can be decomposed into atomic non-overlapping groups of 2 or 3 samples. This seems intuitive because any number between 2 and n-2 can be written as a sum of 2s and/or 3s. So F(n)=n-2 for gcd(6,n)=6.

For a more general n, it is still tricky and I can't pinpoint to a way of ensuring non-overlapping. F(N)<=n-2 for sure.

Writing on the phone, I didn't recheck the sentences I wrote.
 
Yes, my first impression on composition/decomposition was using (3D) layers of a cake on top of each other:
If all layers/groups of samples are balanced AND one could shift them and project them without overlapping, maybe one would get another balanced set.
That is true for all SAME-groups regardless of overlapping (because of symmetry), but sometime during writing I became convinced that one cannot get both balance and avoid overlapping if groups are different. 
 
Logic miss:
*Composition of balanced SAME groups preserves balance (all n-phi(n) above have easy to construct balanced atomic or SAME groups)
*Composition of balanced DIFFERENT groups doesn't ALWAYS preserve balance even if they don't overlapped(EDITED)
 
The missing link is the fact that if groups are different, there MAY be a shift such that composition is balanced. EDIT: Or not
 
For 12, there is such a shift for both 5 and 7 according to TSLF. 
 
For 1 and 11 there is not (in general 1 and n-1 are never balanced). EDIT for n>=2
 
In general if p samples can be balanced the negative image of n-p samples are also balanced. EDIT: Less sure after I wrote this. Anyway, that only needs to be true for gcd(n,p)=1, the missing case above. In which case gcd(n,n-p)=1.

And I don't think it's easy to construct these special cases using decomposition ...

n-phi(n) for n geq 2 if having it spin empty is not considered a valid way. Plus 1 if no samples is considered a case of succesful processing rather than a waste of energy.
For n=12 that is 12-4=8. Unfortunately that is just fitting the data with a formula, not really an answer.

For 12 exactly, only 1, 5,7 and 11 samples can't be balanced.

Assume that q samples are balanced then the sample distribution could be decomposed into groups, each with at least 2 samples such that each group could be processed balancefully on its own. Decomposition may not be unique.
An atomic /not decomposable further) such group would have the distance among the samples constant. So the cardinal of the group would be a divisor of n to ensure the equal distance.
Being an atomic group kinda makes that cardinal prime othwrwise it can be decomposed into equal balanced groups

That being said The integration step is tricky or my algebra is very rusty. The solution - the number of q's between 1 and n exist such that there exists a balance position of q samples on n equally distanced placeholders. The number is clearly smaller or equal to the formula above. However, constructing a sample distribution for each q not relatively prime to n is not easy to describe.

Induction should work for n prime(trivial) and for q being a divisor of n (equally space them. Done)
plus the fact that if gcd(q,n)=p then p is balanced as a divisor of n. If q=p*r with r relatively prime with n by gcd definition then one needs to acknowledge that r groups each of p samples can be placed in r balanced layers each shifted with 1 position clockwise then the layer below and projected without overlapping since the length of consecutive samples is less than n/p the distance between the sanples in each group so shfting by 1 the distance can't be covered unless q=n in which case they still wouldn't manage to overlap/collide, just fit neatly.

All in all a great image is better than the thousand characters above.