The Conjecture Re-stated So As to Be Most Capable of Formal Proof

Final form of the conjecture, which should not be overwhelmingly
difficult for a professional to prove:

*****
Conjecture (Third and Final Version):

Let n**2 distinct lines be rulings of n distinct planes, n distinct
hyperbolic paraboloids, and n**2-n distinct hyperbolic paraboloids such
that:

1) each of the n**2 lines rules just n+1 of the surfaces as follows:
just one plane, just one hyperbolic paraboloid, and just n-1 of the
one-sheeted hyperboloids (where these n-1 hyperboloids are distinct);

2) each of the surfaces has just n of the lines as rulings (where these
n lines are distinct).

Then if n is not a prime-power:

at least one pair of the lines will rule both a hyperbolic paraboloid
and a one-sheeted hyperboloid.

And if n is a prime-power:

a) each of the n**2 lines will rule just one plane, one hyperbolic
paraboloid and just n-1 distinct one-sheeted hyperbolic paraboloids
(for a total of n+1 surfaces per line);

b) each of the n**2 + n surfaces will be ruled by just n of the n**2
lines (where these n lines are distinct)

*******

If this conjecture were proven, it would follow immediately that:

a) if n is not a prime-power, there can be no finite affine plane of
order n whose points and lines are respectively the n**2 lines and the
n**2 + n surfaces;

b) if n is a prime-power, there will be a finite affine plane of order
n whose points and lines are respectively the n**2 lines and the n**2 +
n surfaces.

For n = 5, suppose the lines are labelled from the cells of the
n-square matrix:

11 12 13 14 15
21 22 23 24 25
31 32 33 34 35
41 42 43 44 45
51 52 53 54 55

Then:

a) the five planes are ruled by the five sets of rulings
(11,12,13,14,15), (21,22,23,24,25), (31,32,33,34,35),(41,42,43,44,45),
and (51,52,53,54,55);

b) the five hyperbolic paraboloids are rules by the five sets of
rulings (11,21,31,41,51), (12,22,32,42,52), (13,23,33,43,53),
(14,24,34,44,54), and (15,25,35,45,55);

c) the 20 one sheeted hyperboloids are ruled by the twenty sets of
rulings given by the columns of the following four 5-square matrices:

11 12 13 14 15 11 12 13 14 15 11 12 13 14 15 11 12 13 14 15
22 23 24 25 21 23 24 25 21 22 24 25 21 22 23 25 21 22 23 24
33 34 35 31 32 35 31 32 33 34 32 33 34 35 31 34 35 31 32 33
44 45 41 42 43 42 43 44 45 41 45 41 42 43 44 43 44 45 12 13
55 51 52 53 54 54 55 51 52 53 53 54 55 51 52 52 53 54 51 55

It will readily be seen that each of these matrices is obtained by
diagonalizing the columns of the previous, e.,g. the (broken) diagonal
12 25 33 41 54 of the second matrix becomes the second column of the
third matrix.

And if one tries this procedure for n = 6, the force of the conjecture
will be immediately apparent.

.