It occurs to me my new factorization method is likely not to be any faster than the old, because of a simple observation: I am still iterating values to add to the Ceiling Square of m and testing.
There will not be an improvement, at least not the one I seek, until I can simply, from looking at the remainder when m is subtracted from c2, or perhaps even the first several remainders after “raising the ceiling,” determine formulaically the Ascent to which I should raise it in order to find the greater perfect square I need.
The ability to do so is, after all, what I was hoping to tease out of the numbers when I was developing these characteristic and row polynomials.
Back to work studying. If the answer is in this, it won’t be in finding some new way to do the old tricks. If the answer is in this, I need to find it by developing/discovering a Whole New Trick.