k=  7 m= 3: deg phi_(m+1) = 2, (C1) True, (C2) True, (C3) True  -> OK
k=  9 m= 4: deg phi_(m+1) = 2, (C1) True, (C2) True, (C3) True  -> OK
k= 11 m= 5: deg phi_(m+1) = 3, (C1) True, (C2) True, (C3) True  -> OK
k= 13 m= 6: deg phi_(m+1) = 3, (C1) True, (C2) True, (C3) True  -> OK
k= 15 m= 7: deg phi_(m+1) = 4, (C1) True, (C2) True, (C3) True  -> OK
k= 17 m= 8: deg phi_(m+1) = 4, (C1) True, (C2) True, (C3) True  -> OK
k= 19 m= 9: deg phi_(m+1) = 5, (C1) True, (C2) True, (C3) True  -> OK
k= 21 m=10: deg phi_(m+1) = 5, (C1) True, (C2) True, (C3) True  -> OK
k= 23 m=11: deg phi_(m+1) = 6, (C1) True, (C2) True, (C3) True  -> OK
k= 25 m=12: deg phi_(m+1) = 6, (C1) True, (C2) True, (C3) True  -> OK
k= 27 m=13: deg phi_(m+1) = 7, (C1) True, (C2) True, (C3) True  -> OK
k= 29 m=14: deg phi_(m+1) = 7, (C1) True, (C2) True, (C3) True  -> OK
k= 31 m=15: deg phi_(m+1) = 8, (C1) True, (C2) True, (C3) True  -> OK
k= 33 m=16: deg phi_(m+1) = 8, (C1) True, (C2) True, (C3) True  -> OK
k= 35 m=17: deg phi_(m+1) = 9, (C1) True, (C2) True, (C3) True  -> OK
k= 37 m=18: deg phi_(m+1) = 9, (C1) True, (C2) True, (C3) True  -> OK
k= 39 m=19: deg phi_(m+1) = 10, (C1) True, (C2) True, (C3) True  -> OK
k= 41 m=20: deg phi_(m+1) = 10, (C1) True, (C2) True, (C3) True  -> OK
k= 43 m=21: deg phi_(m+1) = 11, (C1) True, (C2) True, (C3) True  -> OK
k= 45 m=22: deg phi_(m+1) = 11, (C1) True, (C2) True, (C3) True  -> OK
k= 47 m=23: deg phi_(m+1) = 12, (C1) True, (C2) True, (C3) True  -> OK
k= 49 m=24: deg phi_(m+1) = 12, (C1) True, (C2) True, (C3) True  -> OK
k= 51 m=25: deg phi_(m+1) = 13, (C1) True, (C2) True, (C3) True  -> OK
k= 53 m=26: deg phi_(m+1) = 13, (C1) True, (C2) True, (C3) True  -> OK
k= 55 m=27: deg phi_(m+1) = 14, (C1) True, (C2) True, (C3) True  -> OK
k= 57 m=28: deg phi_(m+1) = 14, (C1) True, (C2) True, (C3) True  -> OK
k= 59 m=29: deg phi_(m+1) = 15, (C1) True, (C2) True, (C3) True  -> OK
k= 61 m=30: deg phi_(m+1) = 15, (C1) True, (C2) True, (C3) True  -> OK
k= 63 m=31: deg phi_(m+1) = 16, (C1) True, (C2) True, (C3) True  -> OK
k= 65 m=32: deg phi_(m+1) = 16, (C1) True, (C2) True, (C3) True  -> OK
k= 67 m=33: deg phi_(m+1) = 17, (C1) True, (C2) True, (C3) True  -> OK
k= 69 m=34: deg phi_(m+1) = 17, (C1) True, (C2) True, (C3) True  -> OK
k= 71 m=35: deg phi_(m+1) = 18, (C1) True, (C2) True, (C3) True  -> OK
k= 73 m=36: deg phi_(m+1) = 18, (C1) True, (C2) True, (C3) True  -> OK
k= 75 m=37: deg phi_(m+1) = 19, (C1) True, (C2) True, (C3) True  -> OK
k= 77 m=38: deg phi_(m+1) = 19, (C1) True, (C2) True, (C3) True  -> OK
k= 79 m=39: deg phi_(m+1) = 20, (C1) True, (C2) True, (C3) True  -> OK
k= 81 m=40: deg phi_(m+1) = 20, (C1) True, (C2) True, (C3) True  -> OK
k= 83 m=41: deg phi_(m+1) = 21, (C1) True, (C2) True, (C3) True  -> OK
k= 85 m=42: deg phi_(m+1) = 21, (C1) True, (C2) True, (C3) True  -> OK
k= 87 m=43: deg phi_(m+1) = 22, (C1) True, (C2) True, (C3) True  -> OK
k= 89 m=44: deg phi_(m+1) = 22, (C1) True, (C2) True, (C3) True  -> OK
k= 91 m=45: deg phi_(m+1) = 23, (C1) True, (C2) True, (C3) True  -> OK
k= 93 m=46: deg phi_(m+1) = 23, (C1) True, (C2) True, (C3) True  -> OK
k= 95 m=47: deg phi_(m+1) = 24, (C1) True, (C2) True, (C3) True  -> OK
k= 97 m=48: deg phi_(m+1) = 24, (C1) True, (C2) True, (C3) True  -> OK
k= 99 m=49: deg phi_(m+1) = 25, (C1) True, (C2) True, (C3) True  -> OK
k=101 m=50: deg phi_(m+1) = 25, (C1) True, (C2) True, (C3) True  -> OK
PART 1: all conditions hold for odd k in [7, 101]
PART 2 (exact expansion for k = 7, 9):
  k=7: coefficient of t^1 in p q^6 P (minus k beta p (q^6 - 1)) reduces to 0
  k=7: coefficient of t^4 in p q^6 P  reduces to 0
  k=7: coefficient of t^7 in p q^6 P  reduces to 0
  k=7: coefficient of t^10 in p q^6 P  reduces to 0
  k=7: coefficient of t^13 in p q^6 P  reduces to 0
  k=7: numeric check of t = (1/(m sigma)) sum lam^-j P(lam^j t) at three points: max error 1.59e-56; |sigma| = 13.0499
  k=7: EXACT IDENTITY VERIFIED
  k=9: coefficient of t^1 in p q^8 P (minus k beta p (q^8 - 1)) reduces to 0
  k=9: coefficient of t^5 in p q^8 P  reduces to 0
  k=9: coefficient of t^9 in p q^8 P  reduces to 0
  k=9: coefficient of t^13 in p q^8 P  reduces to 0
  k=9: coefficient of t^17 in p q^8 P  reduces to 0
  k=9: numeric check of t = (1/(m sigma)) sum lam^-j P(lam^j t) at three points: max error 2.26e-57; |sigma| = 1318.16
  k=9: EXACT IDENTITY VERIFIED
RESULT: PASS
