=== Part 1: Theorem A (Conjecture 2) ===
PASS P homogeneous deg 3 and doubly stochastic
PASS Pinf doubly stochastic
PASS P symmetric
PASS P(1)=1
chi_P   = t**3 - 3*t**2 + 9*t/4 - 1/4  factor: (t - 1)*(4*t**2 - 8*t + 1)/4
chi_Pinf= t**3 - 3*t**2 + 2*t - 2/9
PASS chi_P = (t-1)(4t^2-8t+1)/4
PASS chi_Pinf = t^3-3t^2+2t-2/9
PASS chi_Pinf(1+y) = y^3 - y - 2/9
PASS lambda(P) = (1+sqrt3/2, 1, 1-sqrt3/2)
PASS P = P_A with A=(J-I)/2 (rows x2+x3, x1+x3, x1+x2)
sigma_k(A(1)) = [1, 3, 9/4, 1/4]
PASS sigma_k(A(1)) = (1,3,9/4,1/4)
PASS per(J-I)=2
PASS sigma_k(J/3) = (1,3,2,2/9)
PASS chi_Pinf has 3 simple real roots
mu (7 d.p.) = ['2.0966484', '0.7647601', '0.1385915']
PASS mu printed (2.0966484, 0.7647601, 0.1385915)
PASS g(-sqrt3/2) = sqrt3/8 - 2/9
PASS sqrt3/8 - 2/9 < 0  (243 < 256)
PASS g(-1)=-2/9, g(-1/2)=11/72
PASS g(1/2)<0, g(2)>0
PASS -sqrt3/2 <= -1/sqrt3 (so g increasing there)
PASS lambda_3(P)=1-sqrt3/2 < lower end of mu_3 interval
k=2 sums: P 2.8660254  Pinf 2.8614085
PASS k=2 sums 2.8660254 > 2.8614085
PASS k=1 holds: 1.8660 <= 2.0966
HD pairs (k n sigma_k, (n-k+1)^2 sigma_{k-1}): [(9, 9), (27/2, 12), (9/4, 9/4)]
PASS HD: 9>=9, 27/2>=12, 9/4>=9/4
PASS Tverberg sigma_k(A1) >= sigma_k(J/3)
PASS per A1 = 1/4 >= 3!/3^3
=== Part 2: operator T, orbit, Theorem B(a)-(c) ===
PASS 10 cubic monomials
eigenvalues of T on all cubic forms: {1: 1, 2/3: 2, 1/3: 3, 0: 4}
PASS spectrum {1:1, 2/3:2, 1/3:3, 0:4}
PASS T diagonalisable on cubics
PASS B0 = 3e1e2 - e1^3
PASS C0 = 2e1^3 - 9e1e2 + 27e3
PASS T Pinf = Pinf
PASS T B0 = B0/3
PASS T C0 = 0
PASS P = Pinf + B0/36 - C0/216
PASS chi_B0 = 9t - 3
PASS chi_C0 = -12
PASS P = (e1 e2 - e3)/8
PASS exp(s(T-I))P = P_{A(q)}, q=e^{-s/3}, all real s (symbolic)
PASS orbit = Pinf + q^2 B0/36 - q^3 C0/216
PASS d/ds F = (T-I)F
PASS chi_{P_A(q)} = chi_q (literal definition, symbolic q)
PASS A(q) line sums 1
PASS P_{A(q)} doubly stochastic for symbolic q
PASS P_{A(q)} symmetric
PASS A(1)=(J-I)/2, A(0)=J/3
PASS sigma_2(A(q)) = 2 + q^2/4
PASS per A(q) = 2/9 + q^2/12 - q^3/18
PASS depressed form y^3+(q^2/4-1)y+(q-1)(q+2)^2/18
disc chi_q = -7*q**6/48 - q**5/2 + 2*q**3/3 - q**2 + 8/3  factored: -(q + 2)**3*(7*q**3 - 18*q**2 + 24*q - 16)/48
PASS Delta = (2+q)^3(16-24q+18q^2-7q^3)/48
PASS Delta expanded = 8/3 - q^2 + 2q^3/3 - q^5/2 - 7q^6/48
PASS h' = -3(7q^2-12q+8), disc 144-224<0
PASS h(1)=3
PASS h has exactly one real root
PASS no real root of Delta in (-2,1] (Sturm count)
min of Delta on [0,1] ~ 1.687500 (49/48 = 1.020833)
PASS Delta >= 49/48 on [0,1]
PASS chi_q(1-sqrt3/2) = -(1-q)[4q^2+(16-9sqrt3)(1+q)]/72
PASS 16 > 9 sqrt3
PASS chi_q'(t) = 3(t-1)^2 + q^2/4 - 1
zeros at q=1/2: ['2.050152', '0.807166', '0.142681']
PASS q=1/2 zeros (2.050152, 0.807166, 0.142681)
PASS lambda_3(q) > 1-sqrt3/2 at 7 sample q in (0,1) (exact intervals)
=== Part 3: printed sign exp(s(I-T)), Theorem B(d) ===
PASS exp(s(I-T))P = P_{A(u)}, u=e^{s/3} (symbolic)
PASS chi_u(1+sqrt3/2) = (u^2-1)(1/6+sqrt3/8) + (u^3-1)/18
PASS Delta(3/2) = -8575/3072
PASS u=3/2: exactly one real zero
zeros at u=6/5: ['1.68913', '1.18819', '0.12268']
PASS u=6/5 zeros (1.68913, 1.18819, 0.12268)
PASS u=6/5: lambda_1 < 1+sqrt3/2
PASS h(6/5)=128/125, h(3/2)=-25/8
PASS 7u^3-18u^2+24u-16 has one real root
u0 = 1.2873544, s0 = 3 ln u0 = 0.7578
PASS u0 ~ 1.2873544, s0 ~ 0.7578
PASS A(u) diagonal negative for u>1 (u=6/5)
PASS printed sign: 3 real zeros and lambda_1 < 1+sqrt3/2 at 5 sample u in (1,u0)
PASS u=13/10 > u0: non-real zeros
=== Part 4: Remarks on the orbit (family, pairs, Theorem 4) ===
q* = 0.876338, alpha* = 0.0412206, 3 ln(1/q*) = 0.3960
PASS q* ~ 0.876338, alpha* ~ 0.0412206, s ~ 0.3960
PASS chi_q(mu) - chi_0(mu) = (q^2/18)(q - (3-9mu)/2)
PASS alpha* = (9 mu3 - 1)/6
PASS g(-1/sqrt3) = (2sqrt3-2)/9
PASS g(1/sqrt3) < 0
PASS chi_q(1-1/sqrt3) = (8-3q^2)/(12 sqrt3) + (q-1)(q+2)^2/18
PASS 5/(12 sqrt3) - 2/9 > 0 (45^2 > 3*24^2)
PASS Conj 2 fails on orbit at q=1,.95,.9,.88 (k=2) and holds at q=.87,.8,.5,.1
PASS lambda_3(q=99/100) <= 0.1344152245
PASS 0.1426813715 <= lambda_3(q=1/2)
PASS lambda_3(1/2) > 0.1426 > 0.1390 > lambda_3(1/10)
lambda_3(99/100) in [0.1344152244, 0.1344152244]; lambda_3(1/2) in [0.1426813716, 0.1426813716]; lambda_3(1/10) ~ 0.138943
PASS d/dq chi_q = (q/6)(3t - 1 + q)
PASS chi_q(1) = (q-1)(q+2)^2/18
PASS NUMERICAL: lambda_1 strictly decreasing in q on grid of 2001 points
NUMERICAL: lambda_1(q=1)=1.86603, lambda_1(q=0)=2.09665; max lambda_3 at q ~ 0.572 (value 0.142857); lambda_3(1)=0.13397, lambda_3(0)=0.13859
PASS NUMERICAL: lambda_3 max at q ~ 4/7 = 0.5714 (not 0.55), value ~ 1/7
PASS exact: chi_q((1-q)/3) = -(q+2)^2(7q-4)/108
PASS exact: lambda_3(q=4/7) = 1/7
=== Part 5: lemmas of Section 2, cycle family, Laguerre identity ===
PASS Lemma 2.1: chi_Q = sum (-1)^k E_k t^{d-k} (random quartic)
PASS Remark 2.3: multiaffine cubic in 4 vars with Q(1)=1 has chi=(t-1)^3
PASS Laguerre identity n=2..8 (definition for n<=5)
PASS cycle family: chi by definition = matching formula (n=3..6), sigma brute force (n<=5), P_3=P
  n= 3  lambda_n(P_n)=0.133974596216  lambda_n(Pinf)=0.138591518928  1-cos(pi/2n)=0.133974596216
  n= 4  lambda_n(P_n)=0.076120467489  lambda_n(Pinf)=0.080636922405  1-cos(pi/2n)=0.076120467489
  n= 5  lambda_n(P_n)=0.048943483705  lambda_n(Pinf)=0.052712063944  1-cos(pi/2n)=0.048943483705
  n= 6  lambda_n(P_n)=0.034074173711  lambda_n(Pinf)=0.037141100697  1-cos(pi/2n)=0.034074173711
  n= 7  lambda_n(P_n)=0.025072087818  lambda_n(Pinf)=0.027577668080  1-cos(pi/2n)=0.025072087818
  n= 8  lambda_n(P_n)=0.019214719597  lambda_n(Pinf)=0.021284954038  1-cos(pi/2n)=0.019214719597
  n= 9  lambda_n(P_n)=0.015192246988  lambda_n(Pinf)=0.016924691970  1-cos(pi/2n)=0.015192246988
  n=10  lambda_n(P_n)=0.012311659405  lambda_n(Pinf)=0.013779347054  1-cos(pi/2n)=0.012311659405
  n=11  lambda_n(P_n)=0.010178558119  lambda_n(Pinf)=0.011436040199  1-cos(pi/2n)=0.010178558119
  n=12  lambda_n(P_n)=0.008555138626  lambda_n(Pinf)=0.009643509780  1-cos(pi/2n)=0.008555138626
PASS cycle family n=3..12: n distinct real zeros each, lambda_n(P_n) < lambda_n(Pinf) (exact), ~1-cos(pi/2n) (numerical)
=== Part 6: Proposition 5.1 (a), (b); Proposition 3.1 (consistency); 5.1(c) numerically ===
PASS Prop 5.1(a): spectrum of T in {0,1/d,..,1}, 1 simple, 9 pairs (m,d)
PASS Prop 5.1(b): (TQ)(1)=Q(1), grad(TQ)(1) = K grad Q(1), K (d/m)1 = (d/m)1
NUMERICAL: random Birkhoff mixtures: 787 majorized (HD violated in 0 of them), 13 not majorized
PASS NUMERICAL: Prop 3.1 consistent (no majorized sample violates HD)
PASS NUMERICAL: Prop 5.1(c): exp(s(T-I)) keeps 6 random stable cubics real stable (s=0.3,1,3; line tests)
PASS NUMERICAL: exp(s(I-T))P at s=3ln(3/2) fails the stability test
PASS exact: first factor of P_{A(u)} vanishes at ((2+u)/(u-1),1,1) (so exp(s(I-T))P is not stable for any s>0)
PASS NUMERICAL: exp(s(I-T))P at s=3ln(1.05) < s0 also fails the stability test
=== SUMMARY: 101 checks, 0 failures ===
