sqrt(2/pi) < 0.7979  [EXACT, from pi > 3.14159]
general-q identities (Lemma K1 constants, Case B/C majorants, cubic reduction, zero-bias constant)  [EXACT]
  also E|chi| = 2 tau, 4 kappa' tau^2 = q^2, Lemma 9.4 at rational points, a_B - 1 = 2(1-3q)/(5q-1) (a_B <= 1 iff q >= 1/3)  [EXACT]

=== q = 3/8:  tau^2 = 15/64,  2 tau = 0.9682458,  theta = 1/4,  kappa = 5/12,  kappa' = 3/20
Case A: a <= 3/10: zero-bias constant (q^2+(1-q)^2)/(2tau) = 17*sqrt(15)/120 = 0.548673; E|L| < 0.7979 + 0.164602 = 0.962502 < 2 tau  [EXACT]
        constant = sqrt(289/960) < 0.5487, times a_A < 0.1647, E|L| < 0.9626 < 0.9682 < 2 tau = sqrt(15/16); margin 2 tau - 0.7979 - a_A * constant = 0.005744 >= 0.0049  [EXACT]
Case B: a >= a_B = kappa/(1-kappa) = 5/7  (kappa >= kappa', kappa < 1)  [EXACT]
Case C: 3/10 < a < 5/7: 400 * cubic = 200*s**3 - 255*s**2 + 30*s + 9
        values at -1, 0, 31/100, 21/20, 2: -476, 9, -2473/10000, -729/80, 649
        => one root in each of (-1,0), (0,31/100), (21/20,2); cubic < 0 on [31/100,21/20]  [EXACT]
        s(a_A) = 0.31449 >= 31/100,  s(a_B) = 1.02062 <= 21/20  [EXACT]
        s(a_A) = sqrt(9/91), s(a_B) = sqrt(25/24)  [EXACT]
        (float) max psi on [a_A, a_B] = 0.99348
        max psi on [a_A, a_B] < c = 0.9935: the quartic (s^2 + theta s + kappa')^2 - c^2 s^2 (1 + s^2) has no root on [0.31448545, 1.02062073] (rational endpoints) and is negative there  [EXACT]

=== q = 2/5:  tau^2 = 6/25,  2 tau = 0.9797959,  theta = 1/5,  kappa = 3/8,  kappa' = 1/6
Case A: a <= 1/3: zero-bias constant (q^2+(1-q)^2)/(2tau) = 13*sqrt(6)/60 = 0.530723; E|L| < 0.7979 + 0.176908 = 0.974808 < 2 tau  [EXACT]
        constant = sqrt(169/600) < 0.5308, times a_A < 0.1770, E|L| < 0.9749 < 0.9797 < 2 tau = sqrt(24/25); margin 2 tau - 0.7979 - a_A * constant = 0.004988 >= 0.0049  [EXACT]
Case B: a >= a_B = kappa/(1-kappa) = 3/5  (kappa >= kappa', kappa < 1)  [EXACT]
Case C: 1/3 < a < 3/5: 900 * cubic = 360*s**3 - 564*s**2 + 60*s + 25
        values at -1, 0, 7/20, 1, 2: -959, 25, -1531/200, -119, 769
        => one root in each of (-1,0), (0,7/20), (1,2); cubic < 0 on [7/20,1]  [EXACT]
        s(a_A) = 0.35355 >= 7/20,  s(a_B) = 0.75000 <= 1  [EXACT]
        s(a_A) = sqrt(1/8), s(a_B) = sqrt(9/16)  [EXACT]
        (float) max psi on [a_A, a_B] = 0.96634
        max psi on [a_A, a_B] < c = 0.9664: the quartic (s^2 + theta s + kappa')^2 - c^2 s^2 (1 + s^2) has no root on [0.35355339, 0.75000000] (rational endpoints) and is negative there  [EXACT]

=== q = 5/12:  tau^2 = 35/144,  2 tau = 0.9860133,  theta = 1/6,  kappa = 7/20,  kappa' = 5/28
Case A: a <= 7/20: zero-bias constant (q^2+(1-q)^2)/(2tau) = 37*sqrt(35)/420 = 0.521178; E|L| < 0.7979 + 0.182412 = 0.980312 < 2 tau  [EXACT]
        constant = sqrt(1369/5040) < 0.5212, times a_A < 0.1825, E|L| < 0.9804 < 0.9860 < 2 tau = sqrt(35/36); margin 2 tau - 0.7979 - a_A * constant = 0.005701 >= 0.0049  [EXACT]
Case B: a >= a_B = kappa/(1-kappa) = 7/13  (kappa >= kappa', kappa < 1)  [EXACT]
Case C: 7/20 < a < 7/13: 7056 * cubic = 2352*s**3 - 4340*s**2 + 420*s + 225
        values at -1, 0, 9/25, 1, 2: -6887, 225, -1195767/15625, -1343, 2521
        => one root in each of (-1,0), (0,9/25), (1,2); cubic < 0 on [9/25,1]  [EXACT]
        s(a_A) = 0.37363 >= 9/25,  s(a_B) = 0.63901 <= 1  [EXACT]
        s(a_A) = sqrt(49/351), s(a_B) = sqrt(49/120)  [EXACT]
        (float) max psi on [a_A, a_B] = 0.95383
        max psi on [a_A, a_B] < c = 0.9539: the quartic (s^2 + theta s + kappa')^2 - c^2 s^2 (1 + s^2) has no root on [0.37363235, 0.63900966] (rational endpoints) and is negative there  [EXACT]

(float) scan: q, a_A(q) (exact Case A threshold), a_B(q), max psi_q on [a_A, min(a_B, 1)]
  q=0.300  a_A=0.1875  a_B=1.4000  max psi=1.13183  FAILS
  q=0.305  a_A=0.1965  a_B=1.3238  max psi=1.11566  FAILS
  q=0.310  a_A=0.2055  a_B=1.2545  max psi=1.10094  FAILS
  q=0.315  a_A=0.2143  a_B=1.1913  max psi=1.08747  FAILS
  q=0.320  a_A=0.2231  a_B=1.1333  max psi=1.08176  FAILS
  q=0.325  a_A=0.2318  a_B=1.0800  max psi=1.07798  FAILS
  q=0.330  a_A=0.2403  a_B=1.0308  max psi=1.07426  FAILS
  q=0.335  a_A=0.2487  a_B=0.9852  max psi=1.07061  FAILS
  q=0.340  a_A=0.2570  a_B=0.9429  max psi=1.06702  FAILS
  q=0.345  a_A=0.2652  a_B=0.9034  max psi=1.06313  FAILS
  q=0.350  a_A=0.2731  a_B=0.8667  max psi=1.05499  FAILS
  q=0.355  a_A=0.2810  a_B=0.8323  max psi=1.04384  FAILS
  q=0.360  a_A=0.2886  a_B=0.8000  max psi=1.03128  FAILS
  q=0.365  a_A=0.2961  a_B=0.7697  max psi=1.01816  FAILS
  q=0.370  a_A=0.3034  a_B=0.7412  max psi=1.00499  FAILS
  q=0.375  a_A=0.3105  a_B=0.7143  max psi=0.99211  ok
  q=0.380  a_A=0.3174  a_B=0.6889  max psi=0.97973  ok
  q=0.385  a_A=0.3241  a_B=0.6649  max psi=0.97387  ok
  q=0.390  a_A=0.3305  a_B=0.6421  max psi=0.96891  ok
  q=0.395  a_A=0.3368  a_B=0.6205  max psi=0.96421  ok
  q=0.400  a_A=0.3428  a_B=0.6000  max psi=0.95977  ok
  q=0.410  a_A=0.3540  a_B=0.5619  max psi=0.95159  ok
  q=0.420  a_A=0.3643  a_B=0.5273  max psi=0.94431  ok
  q=0.430  a_A=0.3734  a_B=0.4957  max psi=0.93791  ok
  q=0.440  a_A=0.3815  a_B=0.4667  max psi=0.93239  ok
  q=0.450  a_A=0.3883  a_B=0.4400  max psi=0.92777  ok
  q=0.460  a_A=0.3940  a_B=0.4154  max psi=0.92412  ok
  q=0.470  a_A=0.3985  a_B=0.3926  max psi=(empty)  ok
  q=0.480  a_A=0.4017  a_B=0.3714  max psi=(empty)  ok
  q=0.490  a_A=0.4036  a_B=0.3517  max psi=(empty)  ok
  q=0.500  a_A=0.4042  a_B=0.3333  max psi=(empty)  ok
(float) the three-case argument closes for all grid points q >= 0.375
(float) asserted on the grid: closes at every q >= 0.375, Case C fails at every q <= 0.370, Cases A and B overlap (a_A >= a_B) exactly for q >= 0.470
(float) q=3/8: max E|L| over 540 random unit vectors, n<=10: 0.968246  vs 2tau = 0.968246
(float) q=2/5: max E|L| over 540 random unit vectors, n<=10: 0.979796  vs 2tau = 0.979796
(float) q=5/12: max E|L| over 540 random unit vectors, n<=10: 0.986013  vs 2tau = 0.986013

(float, adversarial sanity check) local search for E|L| > 2 tau started away from dictators: for each n = 2..10, 40 random unit starts
  with max|alpha_i| <= 0.8; Nelder-Mead (adaptive) on alpha = x/|x|, E|L| by exact enumeration of the 2^n sign patterns; (i) unconstrained,
  (ii) with max|alpha_i| <= 0.8 enforced by the exact penalty 10 (max|alpha_i| - 0.8)_+ (E|L| is 1-Lipschitz in alpha)
  q=3/8: 2 tau = 0.9682458366; (i) best E|L| = 0.9682458366 (n = 4, max|alpha_i| = 1.000000: a dictator), best local maximum with max|alpha_i| < 0.99: 0.862609; (ii) best E|L| with max|alpha_i| <= 0.8: 0.858228
  q=2/5: 2 tau = 0.9797958971; (i) best E|L| = 0.9797958971 (n = 8, max|alpha_i| = 1.000000: a dictator), best local maximum with max|alpha_i| < 0.99: 0.860437; (ii) best E|L| with max|alpha_i| <= 0.8: 0.860437
  q=5/12: 2 tau = 0.9860132972; (i) best E|L| = 0.9860132972 (n = 3, max|alpha_i| = 1.000000: a dictator), best local maximum with max|alpha_i| < 0.99: 0.862109; (ii) best E|L| with max|alpha_i| <= 0.8: 0.862109
(float) no unit vector with E|L| > 2 tau + 1e-9 found; away from dictators E|L| stays well below 2 tau
