Replace a chain of image blurs with one blur

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In this question I asked how to implement a chain of blurs in one single step.

Then I found out from the gaussian blur page of Wikipedia that:

Applying multiple, successive gaussian blurs to an image has the same effect as applying a single, larger gaussian blur, whose radius is the square root of the sum of the squares of the blur radii that were actually applied. For example, applying successive gaussian blurs with radii of 6 and 8 gives the same results as applying a single gaussian blur of radius 10, since sqrt {6^{2}+8^{2}}=10.

So I thought that blur and singleBlur were the same in the following code:

cv::Mat firstLevel;
float sigma1, sigma2;
//intialize firstLevel, sigma1 and sigma2
cv::Mat blur = gaussianBlur(firstLevel, sigma1);
        blur = gaussianBlur(blur, sigma2);
float singleSigma = std::sqrt(std::pow(sigma1,2)+std::pow(sigma2,2));
cv::Mat singleBlur = gaussianBlur(firstLevel, singleSigma);
cv::Mat diff = blur != singleBLur;
// Equal if no elements disagree
assert( cv::countNonZero(diff) == 0);

But this assert fails (and actually, for example, the first row of blur is different from the first one of singleBlur).

Why?

UPDATE:

After different comments asking for more information, I'll update the answer.

What I'm trying to do is to parallelize this code. In particular, I'm focusing now on computing all the blurs at all levels in advance. The serial code (which works correctly) is the following:

   vector<Mat> blurs ((par.numberOfScales+3)*levels, Mat());
   cv::Mat octaveLayer = firstLevel;
   int scaleCycles = par.numberOfScales+2;

   //compute blurs at all layers (not parallelizable)
   for(int i=0; i<levels; i++){
       blurs[i*scaleCycles+1] = octaveLayer.clone();
       for (int j = 1; j < scaleCycles; j++){
           float sigma = par.sigmas[j]* sqrt(sigmaStep * sigmaStep - 1.0f);
           blurs[j+1+i*scaleCycles] = gaussianBlur(blurs[j+i*scaleCycles], sigma);
           if(j == par.numberOfScales)
               octaveLayer = halfImage(blurs[j+1+i*scaleCycles]);
       }
   }

I'm sorry for the horrible indexes above, but I tried to respect the original code system (which is horrible, like starting counting from 1 instead of 0). The code above has scaleCycles=5 and levels=6, so 30 blurs are generated in total.

This is the "single blur" version, where first I compute the sigmas for each blur that has to be computed (following Wikipedia's formula) and then I apply the blur (notice that this is still serial and not parallelizable):

   vector<Mat> singleBlurs ((par.numberOfScales+3)*levels, Mat());
   vector<float> singleSigmas(scaleCycles);
   float acc = 0;
   for (int j = 1; j < scaleCycles; j++){
       float sigma = par.sigmas[j]* sqrt(sigmaStep * sigmaStep - 1.0f);
       acc += pow(sigma, 2);
       singleSigmas[j] = sqrt(acc);
   }

   octaveLayer = firstLevel;
   for(int i=0; i<levels; i++){
       singleBlurs[i*scaleCycles+1] = octaveLayer.clone();
       for (int j = 1; j < scaleCycles; j++){
           float sigma = singleSigmas[j];
           std::cout<<"j="<<j<<" sigma="<<sigma<<std::endl;
           singleBlurs[j+1+i*scaleCycles] = gaussianBlur(singleBlurs[j+i*scaleCycles], sigma);
           if(j == par.numberOfScales)
               octaveLayer = halfImage(singleBlurs[j+1+i*scaleCycles]);
       }
   }

Of course the code above generates 30 ... (more)