matrix of T on (m3, m21, m111): [[1/3, 2/9, 0], [1/3, 2/3, 1/9], [0, 4/3, 1/3]]
eigenvectors: T(P_inf)=P_inf, T(B0)=B0/3, T(C0)=0 with
  B0 = -x1**3 + 3*x1*x2*x3 - x2**3 - x3**3 
  C0 = 2*x1**3 - 3*x1**2*x2 - 3*x1**2*x3 - 3*x1*x2**2 + 12*x1*x2*x3 - 3*x1*x3**2 + 2*x2**3 - 3*x2**2*x3 - 3*x2*x3**2 + 2*x3**3
P = P_inf + B0/36 - C0/216
chi_{Psi_s P}(t) = q**3/18 - q**2/12 + t**3 - 3*t**2 + t*(q**2/4 + 2) - 2/9    (q = e^{-s/3})
disc_t(chi) = -7*q**6/48 - q**5/2 + 2*q**3/3 - q**2 + 8/3
discriminant has no zero in [0,1] and is positive at 0: three distinct real zeros for all s >= 0
chi_q(1 - sqrt3/2) = (q - 1)*(4*q**2 - 9*sqrt(3)*q + 16*q - 9*sqrt(3) + 16)/72
=> chi_q < 0 on (-inf, 1 - sqrt3/2], so lambda_3(Psi_s P) > lambda_3(P) for every s > 0, and
   lambda_1+lambda_2 (Psi_s P) = 3 - lambda_3(Psi_s P) < 3 - lambda_3(P) = lambda_1+lambda_2 (P):
   lambda(Psi_0 P) is NOT majorized by lambda(Psi_s P) for any s > 0. Conjecture 5 fails (r = 0).
s = 3 ln 2: 144*chi = 144*t**3 - 432*t**2 + 297*t - 34
  lambda_1 in [1976312/963983, 2865215/1397562] ~ 2.05015233671
  lambda_2 in [2949585/3654247, 2777368/3440887] ~ 0.807166291715
  lambda_3 in [168379/1180105, 184266/1291451] ~ 0.142681371573
  lambda_1+lambda_2 <= 2.85731862842713  <  lambda_1(P)+lambda_2(P) = 2 + sqrt3/2 ~ 2.86602540378444
Theorem 4 check: lambda_1(Psi_s P) increases as q = e^{-s/3} goes 1 -> 0: [1.86603, 1.98297, 2.05015, 2.08566, 2.09665]
printed sign: chi_u(1 + sqrt3/2) > 0 for u > 1 and chi_u increasing on [1+1/sqrt3, inf) (u < 2):
   every real zero of chi_{Psi_s P} is < lambda_1(P) for s > 0, so the k = 1 inequality fails
   s = 3 ln(6/5): real zeros = 3 , approx [0.12268131, 1.1881919, 1.6891268]
   s = 3 ln(3/2): 288*chi = 288*t**3 - 864*t**2 + 738*t - 64 has only 1 real zero -> Psi_s P is NOT stable;
   so with the printed sign Lemma 3 fails for this P, and lambda(Psi_s P) is undefined there.
