Your "proof" contains only d, e, f, g and m, n, k naturalness requirement and nothing about a, b, c
I brought a counter-example where d, e, f, g are natural and natural m, n, k also exist, so your conclusion that d*e*f*g can never be natural is wrong.
You hadn't explain in a proper way why a product of (a^2+b^2)(a^2+c^2)(b^2+c^2)(a^2+b^2+c^2) can never be a full square.
And also I didn't catch your point, why the upper product has to be represented exactly as a quadratic polynomial (ua^2+vb^2+wc^2)^2.
The upper product is a polynomial of the sixth degree. Why did you decide to simplify it to quadratic polynomial — this is an enigma for me.