Each axis splits into an ordered pair and a singleton. Choosing one block on each axis gives a sector. Packing places its coordinates together, so the simultaneous pair layer becomes four independent calls.
Fig 1Sector packingSelect a sectorOriginal order · 0 of 9 moved
The ordered pair is (2, 0), with singleton (1), on both axes. Nonconsecutive coordinates can therefore belong to the same sector.
Original array: coordinates are indexed by I = 3a + b.
The selected sector
Two pairs
For each axis, p counts coordinates in earlier blocks, q is the chosen block’s width, and t is its local position. The first axis has suffix width 3.
Follow all nine coordinate addresses
Original I
(a, b)
Packed S + o
Sector
0
(0, 0)
3
1
1
(0, 1)
5
2
2
(0, 2)
2
1
3
(1, 0)
7
3
4
(1, 1)
8
4
5
(1, 2)
6
3
6
(2, 0)
1
1
7
(2, 1)
4
2
8
(2, 2)
0
1
Indices identify the entries. Packing permutes their positions; the values are unchanged.
Follow the argument
This illustrates the packing step in Lemma 4.1 and the pair-slot decomposition in Proposition 4.2. The full Fourier algorithm also needs the other stages in the manuscript’s roadmap.