[OK] parametrisation: u^2 = (1-q)/q
[OK] parametrisation: v^2 = q/(1-q)
[OK] parametrisation: tau^2 = q(1-q)
[OK] E chi = q u - (1-q) v = 0
[OK] E chi^2 = q u^2 + (1-q) v^2 = 1
[OK] q u = tau
[OK] (1-q) v = tau
[OK] tau v = q
[OK] tau u = 1-q
[OK] u + v = 1/tau
[OK] E|chi| = q u + (1-q) v = 2 tau
[OK] 4 kappa' tau^2 = q^2
[OK] a_B = kappa/(1-kappa) = (1-q)/(5q-1)
[OK] C_q = (q^2+(1-q)^2)/(2 tau) = (1-2tau^2)/(2tau)
[OK] C_q = 1/(2tau) - tau
[OK] Lemma 9.4 region y > a v
[OK] Lemma 9.4 region -a u <= y <= a v
[OK] Lemma 9.4 region y < -a u
[OK] A'(w) = -2 e^{-w^2/2}/(1+w^2)^2
[OK] B'(w) = -6 e^{-w^2/2}/(w^2 (w^2+3)^2)
     (A, B -> 0 as w -> oo is immediate: every term is O(e^{-w^2/2}/w).)
[OK] Stein eq. f' - w f = w - sqrt(2/pi) for w > 0
[OK] f'' = sqrt(2/pi) k, k = (1+w^2) m - w
[OK] k' = (w^3+3w) m - (w^2+2)
[OK] upper Mills bound <=> w m < (w^2+2)/(w^2+3) < 1: (w^2+2)/(w^3+3w)*w = (w^2+2)/(w^2+3)
     m(0) = sqrt(pi/2) => f(0) = 0, f'(0+) = -sqrt(2/pi), f''(0+) = sqrt(2/pi) k(0) = 1.
[OK] sqrt(2/pi)*sqrt(pi/2) = 1
[OK] q u G1 - (1-q) v G2 = tau (G1 - G2)  [coefficients]
[OK] density of U = 1/(u+v) = tau
[OK] coupling: E|chi - U| = ((1-q)^2+q^2)/(2 tau)
[OK] u + v - V/tau = (1 - V)/tau
[OK] (y-t)_+ bound: y^2/(4t) - (y-t) = (y-2t)^2/(4t)
[OK] (1-q)/(4 a v) = kappa tau / a
[OK] q/(4 a u) = kappa' tau / a
[OK] kappa - kappa' = (1-2q)/(4q(1-q))
[OK] a + kappa(1-a^2)/a = 1 + (a-1)((1-kappa)a - kappa)/a
[OK] kappa < 1 <=> q > 1/5:  1 - kappa = (5q-1)/(4q)
[OK] 4 c a tau = q^2
[OK] y > av: c y^2 + theta y - (1-q)(y - a v) = c y^2 - q y + a tau
[OK]    ... = c (y - q/(2c))^2  (zero discriminant)
[OK] y < -au: c y^2 - q(-y - a u) = c y^2 + q y + a tau
[OK]    ... = c (y + q/(2c))^2
[OK] 2 tau a + 2 c b^2 + 2 theta b tau = 2 tau psi(a)
[OK] psi(a)/b = s + kappa'/s + theta  and 1/b = sqrt(1+s^2)
[OK] s^2[(s+theta+kappa'/s)^2 - (1+s^2)] = 2 theta s^3 + (theta^2+2kappa'-1)s^2 + 2theta kappa' s + kappa'^2
[OK] 16(1-q)^2 * cubic = P(q,s)  (eq. (13))
[OK] P = c3 s^3 + c2 s^2 + c1 s + q^2
     sign facts on [3/8,1/2] (exact, by hand): 1-2q in [0,1/4], 8(1-q)^2-1 in [1, 17/8] => c2 <= -8q(1-q) < 0;
     q(1-q) increasing, 1-2q and 8(1-q)^2-1 decreasing in q => corner bound of the paper is valid.
[OK] fhat({1}) for f = 1{x1=1}: q u = tau
[OK] s(a_B)^2 = (1-q)^2 / (8 q (3q-1))

ALL IDENTITIES OK
