JCSBimp's Hexagonal Orthogonal

When one subtracts an odd positive integer m from its appropriately adjusted Ceiling Square c², with the adjustment being made by adding 1 to ceil(sqrt(m)) if necessary to make it even or odd as (m+1)/2 is even or odd, respectively, the remainder r obtained by c²-m (so that m=c²-r) is always congruent to 0 or 1 modulo 4.

To many mathematician readers, this may have followed immediately and intuitively from the concepts defined here and elsewhere, but I thought it worth spelling out separately, as it might be germane to further revelations.