There is an infinite descending chain at (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,1). Here is to show it is standard. The result is here. It is actually the last example on the calculator

  • (0,0,0,0,0)(1,1,1,1,1)
  • = (0,0,0,0)(1,1,1,1)(2,2,2,2)(3,3,3,3)...
  • > (0,0,0,0)(1,1,1,1)(2,2,2,2)
  • = (0,0,0,0)(1,1,1,1)(2,2,2,1)(3,3,3,1)...
  • > (0,0,0,0)(1,1,1,1)(2,2,2,1)...
  • = (0,0,0,0)(1,1,1,1)(2,2,2,0)(3,3,3,1)...
  • > (0,0,0,0)(1,1,1,1)(2,2,2,0)
  • = (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,4,1,1)(5,5,1,1)...
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,4,1,1)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,4,1,0)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,4,0,0)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,3,1,1)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,3,1,0)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,3,0,0)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,1,1)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,1,0)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,3,1,1)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,3,1,0)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,3,0,0)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,2,0,0)
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,1)

Here I show it doesn't terminate

  • = (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,1)
  • = (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,0)(8,4,2,1)(9,5,2,1)(10,6,2,1)(11,5,0,0)(12,4,1,1)(13,5,1,1)...
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,0)(8,4,2,1)(9,5,2,1)(10,6,2,1)(11,5,0,0)(12,4,1,1)(13,5,1,1)
  • = (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,0)(8,4,2,1)(9,5,2,1)(10,6,2,1)(11,5,0,0)(12,4,1,1)(13,5,1,0)(14,6,2,1)(15,7,2,1)(16,8,2,1)(17,7,0,0)(18,6,1,1)(19,7,1,1)(20,8,1,0)(21,9,2,1)(22,10,2,1)(23,11,2,1)(24,10,0,0)(25,9,1,1)...
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,0)(8,4,2,1)(9,5,2,1)(10,6,2,1)(11,5,0,0)(12,4,1,1)(13,5,1,0)(14,6,2,1)(15,7,2,1)(16,8,2,1)(17,7,0,0)(18,6,1,1)(19,7,1,1)...
  • = (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,0)(8,4,2,1)(9,5,2,1)(10,6,2,1)(11,5,0,0)(12,4,1,1)(13,5,1,0)(14,6,2,1)(15,7,2,1)(16,8,2,1)(17,7,0,0)(18,6,1,1)(19,7,1,0)(20,8,2,1)(21,9,2,1)(22,10,2,1)(23,9,0,0)(24,8,1,1)(25,9,1,1)(26,10,1,0)(27,11,2,1)(28,12,2,1)(29,13,2,1)(30,12,0,0)(31,11,1,1)(32,12,1,0)(33,13,2,1)(34,14,2,1)(35,15,2,1)(36,14,0,0)(37,13,1,1)...
  • > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,0)(8,4,2,1)(9,5,2,1)(10,6,2,1)(11,5,0,0)(12,4,1,1)(13,5,1,0)(14,6,2,1)(15,7,2,1)(16,8,2,1)(17,7,0,0)(18,6,1,1)(19,7,1,0)(20,8,2,1)(21,9,2,1)(22,10,2,1)(23,9,0,0)(24,8,1,1)(25,9,1,1)
  • ...

Extra: There is a problem with BM3.1/3.2 at (0,0,0,0)(1,1,1,1)(2,2,0,0)(3,1,1,1).

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