**Magic Hourglass Anti-Closure**

Reminder. A magic hourglass of squares is

A² B² C²

E²

G² H²
I²

with

A² + B² + C² = G² + H² + I²

=
A² + E² + I²

=
B² + E² + H²

=
C² + E² + G²

A set of elements is closed under addition if the sum of any
two elements of the set always produces an element of the set.

A set of elements
is anti-closed under addition if the sum of any two elements of the set never
produces an element of the set.

An example of an addition anti-closure is the set of odd integers.
The sum of any two odd integers is never an odd integer.

The set of odd integers
is generated by the expression (2p+1) for integer p.

An example of a proof
of anti-closure is to set up a general counterexample and then find a contradiction.

Find integers p, q, r such that

(2p+1) + (2q+1) = (2r+1).

Subtract 1 and divide by 2.

p + q + 1/2 = r.

This requires r to be a non-integer.

Therefore, there is
no counterexample.

------

Another proof technique is to define the complementary set
and then show that the sum of any two elements of the set always produces an
element of the complementary set.

The set of non-odd integers is generated
by the expression (2r) for integer r.

If two odd integers generated by p
and q are added, we get

(2p+1) + (2q+1)

which is

2(p+q+1)

which has the expression form (2r) of an element in the complementary set.

==========================================

The Magic Hourglass Set is the set of positive rational numbers generated by the expression

p(1-p^2)

---------

(1+p^2)^2

for rational p, where 0 < p < 1.

Proving that this set is anti-closed under addition is equivalent
to proving that there is no magic hourglass.

Finding a counterexample to
anti-closure is equivalent to finding a magic hourglass.

Find rational numbers p, q, r such that

p(1-p^2) q(1-q^2)
r(1-r^2)

--------- + ---------
= ---------

(1+p^2)^2 (1+q^2)^2 (1+r^2)^2

There will be no duplicated entries in the 3x3 magic square as long as 0 < p,q,r < 1.

-----

How would you define the complete complementary set of the Magic Hourglass Set?

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