Does SIFT work the same on different images sizes?

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Thanks.
Yes SIFT is scale invariant by definition for a given image. But between two images, let say one 400px width and its 3200px width original the keypoints founds are absolutely not the same.

For example, on the big image, >90% of the retained keypoints for the matches between this image and another large one, are on a wall in the background, all located in the top left corner. But on the small images, all matches are spread a little everywhere on the image, there is no more keypoints matches on the wall.
Thus, kp and matches are strongly dependent of the original image size.