[1] P and P_inf: homogeneous deg 3, doubly stochastic (dP/dx_i(1)=1), P(1)=1, symmetric: OK
[2] chi_P = t**3 - 3*t**2 + 9*t/4 - 1/4 ; chi_Pinf = t**3 - 3*t**2 + 2*t - 2/9  (operator route == MSS determinant route)
[3] lambda(P) = [sqrt(3)/2 + 1, 1, 1 - sqrt(3)/2] ~ [1.866025404, 1.000000000, 0.1339745962]
    lambda(P_inf) ~ [2.096648361, 0.7647601201, 0.1385915189]
    g(1-sqrt3/2) = sqrt3/8 - 2/9 ~ -0.005715871276 < 0  =>  lambda_3(P) < lambda_3(P_inf)
    k=1: 1.866025404 <= 2.096648361 ;  k=2: 2.86602540378 > 2.86140848107  FAILS;  k=3: 3 = 3
    Conjecture 2 fails for P (D_{3,3}, symmetric). Gap ~ 0.0046169227
[4] Holens-Dokovic holds for A; sigma_2=9/4>=2, per=1/4>=2/9: the example is not a HD-type counterexample
[5] eigenvalues of T on all cubic forms (3 vars): {1: 1, 2/3: 2, 1/3: 3, 0: 4}  (in {0,1/3,2/3,1}, eigenvalue 1 simple)
[6] spectral decomposition of P: P = P_inf + B0/36 - C0/216 (components at eigenvalues 1, 1/3, 0; none at 2/3)
[7] chi_{Psi_s P} = q**3/18 - q**2/12 + t**3 - 3*t**2 + t*(q**2/4 + 2) - 2/9 ; disc = -7*q**6/48 - q**5/2 + 2*q**3/3 - q**2 + 8/3  >= 49/48 on [0,1]
    chi_q(1-sqrt3/2) = -(1-q)[4q^2 + (16-9sqrt3)(1+q)]/72 < 0 for 0<=q<1; chi_q increasing on (-inf, 1-sqrt3/2]
    => lambda_3(Psi_s P) > lambda_3(P), i.e. lambda_1+lambda_2 drops: lambda(P) not majorized by lambda(Psi_s P), all s>0
    q=1/2: lambda(Psi_s P) ~ [2.05015233671, 0.807166291715, 0.142681371573]; lambda1+lambda2 ~ 2.85731862843 < 2.86602540378
    q=9/10: lambda(Psi_s P) ~ [1.92099413992, 0.941158685235, 0.137847174849]; lambda1+lambda2 ~ 2.86215282515 < 2.86602540378
    q=1/10: lambda(Psi_s P) ~ [2.09493464738, 0.766122517455, 0.138942835162]; lambda1+lambda2 ~ 2.86105716484 < 2.86602540378
    q=99/100: lambda(Psi_s P) ~ [1.87216381063, 0.993420964950, 0.134415224418]; lambda1+lambda2 ~ 2.86558477558 < 2.86602540378
[8] printed sign: chi_u(1+sqrt3/2) = (u^2-1)(1/6+sqrt3/8) + (u^3-1)/18 > 0 (u>1); chi_u increasing on [1+1/sqrt3, inf)
    s=3ln(3/2): 288 chi = 288*t**3 - 864*t**2 + 738*t - 64 has 1 real zero (disc<0): Psi_s P not stable, Lemma 3 fails as printed
    s=3ln(6/5): zeros ~ [1.689126785, 1.188191904, 0.1226813112] ; lambda_1 < lambda_1(P) ~ 1.866025404 : k=1 inequality fails
[9] TESTED: Psi_s P (q=1/2,1/5,4/5) hyperbolic w.r.t. 600 random positive directions at random real points; failures: 0
ALL ASSERTIONS PASSED
