All the Gateway Systems are 3000 lys from the core. A Gateway System is the only one in the region (Theory). Each region is 400 lys apart. That puts each Gateway System at eight clicks (
8
) away from the core. Also, there's about 900 Gateway Systems in Euclid. 

Surface area of core sphere. A = 4π×3000²
Region circle area on sphere. R = π×200²
Number of Gateway Systems. N = A / R = 4*3000^2/200^2 = 900

115408 000ffe
See You Next Time Space Cowboy o:Evichinov
07FD:0087:07FF:0154 340 -2,8,0
119408 001fff
Gisdorns Adjunct // Gen Marie Scott 784 o:Evichinov
07FE:0087:0800:0194 404 -1,8,1
100108 000fff
Cokscohynei Mass // Gortari - 433 ly
106d08 000fff
Cokscohynei Mass // Gen Marie Scott 789 o:Katasihtr
07FE:0087:07FF:006D 109 -1,8,0
100108 000000
Cikhard Spur // [ATL] Nunasu - 375 ly
101b08 000000
Cikhard Spur // Heaven o:Digord
07FF:0087:07FF:001B 27 0,8,0
100108 001000
Ibinbile Nebula // Arzamashi - 408 ly
11f608 001000
Ibinbile Nebula // du o:Alchade-Eta IV
07FF:0087:0800:01F6 502 0,8,1
100108 002000
Cissona Anomaly // Legerdemain Haze XIV - 386 ly
101808002000
Cissona Anomaly // Pahenix
07FF:0087:0801:0018 24 0,8,2

100108 003000
100107 004000
100107 005000
100106 006000
100104 007000
100105 007000
100100 008000
100101 008000
100102 008000
100103 008000

100108 000001
100108 001001
100108 002001
100108 003001
100107 004001
100107 005001
100106 006001
100104 007001
100105 007001
100100 008001
100101 008001
100102 008001
100103 008001

100108 000002
100108 001002
100108 002002
100108 003002
100107 004002
100106 005002
100105 006002
100104 007002
100100 008002
100101 008002
100102 008002
100103 008002

100108 000003
100108 001003
100108 002003
100107 003003
100107 004003
100106 005003
100105 006003
100103 007003
100104 007003
100100 008003
100101 008003
100102 008003

100107 000004
100107 001004
100107 002004
100107 003004
100106 004004
100105 005004
100104 006004
100100 007004
100101 007004
Lafneya Nebula // Hanekoma o:Tugluq Fata XIX - 156 ly
10d801007004
Lafneya Nebula // Taco o:Eeyiij-Ure
0803:0080:0806:00D8 216 4,1,7
100102 007004
100103 007004

100107 000005
100107 001005
100106 002005
100106 003005
100105 004005
100104 005005
Sigardh Sector // Ijonag [Forgotten Colonies] o:Ijonag - 126 ly
10ff04 005005
Sigardh Sector // Taco o:Nanyov [GW#4]
0804:0083:0804:00FF 255 5,4,5
100102 006005
100103 006005
Veding Spur // Hobeytz-Amart

100100 007005
100101 007005

100106 000006
100106 001006
100105 002006
100105 003006
100104 004006
100102 005006
100103 005006
100100 006006
100101 006006

100104 000007
100105 000007
100104 001007
100105 001007
100104 002007
100103 003007
Puikeg Band // Tokori - 296 ly
107d03 003007
Puikeg Band // Don o:Nitango-Exm [GW#3]
0806:0082:0802:007D 125 7,3,3
100104 003007
100100 004007
100101 004007
100102 004007
100103 004007
100100 005007
100101 005007

100100 000008
Rugalt Anomaly // Sembouti - 596 ly
109dff 000008
Lakultu // Anskoy-Igenr [GW#1]
0807:007E:07FF:009D 157 8,-1,0
100101 000008
Gesswit // Igatar-Nas - 444 ly
10b901 000008
Gesswit // Hermenn-Edl XII
0807:0080:07FF:00B9 185 8,1,0
100102 000008
Kinush Instablilty // Takirna-Layvyy - 326 ly
109302 000008
Kinush Instablilty // Ofnuni-Lubin [GW#2]
0807:0081:07FF:0093 147 8,2,0
100103 000008
Rihaumui Band // Kikuze III - 619 ly
109302 000008
Kinush Instablilty // Ofnuni-Lubin [GW#2]
0807:0081:07FF:0093 147 8,2,0
100100 001008
Uscisco // Gepasohch - 442 ly
109200 001008
Uscisco // Tezgeb-Leyst
0807:007F:0800:0092 146 8,0,1
100101 001008
Rariha Quadrant // Tehili-Aks - 346
10b701 001008
Rariha Quadrant // Taco o:Fayenn-Onj
0807:0080:0800:00B7 183 8,1,1
100102 001008
Iferec Cluster // TACOCAT o:Gideqin - 493 lyʲ
10b701 001008
Rariha Quadrant // Taco o:Fayenn-Onj
0807:0080:0800:00B7 183 8,1,1
100103 001008
Twrigga Sector // Grand View of The End - 645+
10b701 001008
Rariha Quadrant // Taco o:Fayenn-Onj
0807:0080:0800:00B7 183 8,1,1
100100 002008
Ukyana Shallows // 3rd Stop For the Center - 337 ly
10aa00 002008
Ukyana Shallows // Last stop before the center. GL!
0807:007F:0801:00AA 170 8,0,2
100101 002008
Alsaten Terminus // Gehaixingk - 368 ly
109101 002008
Alsaten Terminus // Taco o:Zarmarga
0807:0080:0801:0091 145 8,1,2
100102 002008
Mepawe // Rekinl Cipe - 975 lyʲ
Only 662 ly to 8,1,2 Taco
107d03 003007
Puikeg Band // Don o:Nitango-Exm [GW#3]
0806:0082:0802:007D 125 7,3,3
100103 002008
Gutiap Mass // Iihamur VI - 686+123 ly
107d03 003007
Puikeg Band // Don o:Nitango-Exm [GW#3]
0806:0082:0802:007D 125 7,3,3
100100 003008
Ewyani Quadrant // Demens-Adjac V - 
10aa00 002008
Ukyana Shallows // Last stop before the center. GL!
0807:007F:0801:00AA 170 8,0,2
100101 003008
Manheirs Sector // Remyar-Yikjav VII - 520 ly
109101 002008
Alsaten Terminus // Taco o:Zarmarga
0807:0080:0801:0091 145 8,1,2
100102 003008
Nitromi Instability // Dafenho-Eding V - 693 ly
107d03 003007
Puikeg Band // Don o:Nitango-Exm [GW#3]
0806:0082:0802:007D 125 7,3,3
1001ff 000008
Lakultu // [ATL] Primo Eurus - 463 ly
109dff 000008
Lakultu // Anskoy-Igenr [GW#1]
0807:007E:07FF:009D 157 8,-1,0
1001ff 001008
Ecvari Spur // Unxianzad - 335 ly
1089ff 001008
Ecvari Spur // The Last Watering Hole o:Uayinnin
0807:007E:0800:0089 137 8,-1,1
1001ff 002008
The Arm of Gusitl // Mytili - 317 ly
1084ff 002008
The Arm of Gusitl // Sachris-Nusin
0807:007E:0801:0084 132 8,-1,2
1001ff fff008
Fluegle Sector // Mihaeju-Lajot - 716
109dff 000008
Lakultu // Anskoy-Igenr [GW#1]
0807:007E:07FF:009D 157 8,-1,0
100100 fff008
Mopkeanos Mass // Martiou4 o:Oseidiu - 408 ly
109300 fff008
Mopkeanos Mass // Tungrat
0807:007F:07FE:0093 147 8,0,-1
100101 fff008
Orfann Cloud // Galamad - 361 ly
107d01fff008
Orfann Cloud // Ogbaime
0807:0080:07FE:007D 125 8,1,-1
100102 fff008
Totraksh Cluster // Pisplan-Iae III - 438 ly
10ac02fff008
Totraksh Cluster // Taco o:Restil
0807:0081:07FE:00AC 172 8,2,-1
100100 ffe008
Vundler Adjunct // Elungo - 359 ly
108200 ffe008
Vundler Adjunct // welcome to the end
0807:007F:07FD:0082 130 8,0,-2
100100 ffd008
Ecvard // Ruvanta-Uita - 721 ly
108200 ffe008
Vundler Adjunct // welcome to the end
0807:007F:07FD:0082 130 8,0,-2

core
    
code

import math
db = set() # 108 keys * 8 = 864 total
def store():
  db.add(f'1001{x:02d} {y:03d}{z:03d}')
  db.add(f'1001{x:02d} {z:03d}{y:03d}')
  db.add(f'1001{y:02d} {x:03d}{z:03d}')
  db.add(f'1001{y:02d} {z:03d}{x:03d}')
  db.add(f'1001{z:02d} {x:03d}{y:03d}')
  db.add(f'1001{z:02d} {y:03d}{x:03d}')

for x in range(9):
  for z in range(9):
    y = 64 - x*x - z*z
    if y >= 0: 
      y = math.ceil(math.sqrt(y))
      store()

for i in sorted([i for i in db],
  key=lambda i: i[10:14]+i[7:10]+i[4:7]):
    print(i)