Harristotelian Logic

Here is a post meant to be an executive summary of the issues around an error of ideas found in algebraic number theory.

A problem was introduced into human mathematics back in the late 1800's. Proving the problem is trivial using a special construction which is only different from normal factorizations in that it doesn't involve polynomial factors:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x) + 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0.

(The above was derived. But that derivation is not necessary to understand the implications.)

The a's can be found using the quadratic formula:

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2
a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2 or vice versa.

Letting b_1(x) = a_1(x) + 1, assuming a_2(0) = 0, I have a_1(x) = b_1(x) - 1, and substituting gives:

7(175x^2 - 15x + 2) = (5b_1(x) + 2)(5a_2(x) + 7)

Now at x=0, b_1(0) = a_2(0) = 0, so the functions can be said to be normalized.

In the field of complex numbers it is then clear by the distributive property how the 7 was multiplied.

Notice that a*(b+c) = a*b + a*c, is still valid with a*(f(x) + b) = a*f(x) + a*b, so the result is simply an application of the distributive property. Since the distributive property is valid on the complex plane the result follows from the field of complex numbers for that reason.

But the result from the complex plane is contradicted by results from another mathematical area.

In what is called the ring of algebraic integers, defined to be the set of roots of monic polynomials with integer coefficients it can be proven that neither of the roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0

can have 7 itself as a factor in that ring when the roots are non-rational. (Notice trivially at x=0 one does, as it is 0.)

So there is a direct contradiction. You can prove one thing from the field of complex numbers, and appear to prove its opposite in the ring of algebraic integers.

Algebraic integers are the base ring around which the field of algebraic numbers is developed. They are also the base from which Galois Theory is developed. This result leads to the conclusion that there must be a problem with the use of Galois Theory.

The ability to appear to prove something false means this error can allow mathematicians to appear to prove just about anything.

One spectacular example of such a flawed argument known as a proof is that of Andrew Wiles, who is believed to have proven Fermat's Last Theorem.

So if recognized this error would have a devastating impact on the mathematical community.

It probably would have a devastating impact on mathematical physicists as well.

It also may indicate problems in particle physics and an explanation of why there are issues with the Standard Model, which relies on group theory.

At this time it is not clear if the error will be acknowledged any time soon because experts in the fields affected appear to be in active denial.

Issues here are relevant to all nations. Governments should consider it an area of national security. Scientists and academics in the implicated fields are to be considered now known security risks. Nations around the world are to consider the NSA to be the world agency most directly affected by this error in terms of personnel risk. Its actions are not to be simply trusted even by NATO allies, or Interpol.

The NSA has possibly the greatest concentration of powerful people directly affected by this error. Exploit potential of NSA staffers by hostile governments exploiting the error may be high.

Threat from this problem is at the highest level. Confrontation with people affected by this error should be done with extreme caution especially if they have access levels that could give them significant negative options.

Suicide risk among those affected may be high, as well as potential acting out of destructive anger against others including nations.

Threat level is near maximum, as potential exists on worst case scenarios for a nuclear exchange, or catastrophic economic collapse.