# Part 3: Lemmas 2.1, 2.2, 2.3, 2.5 on short words
PASS Lemma 2.1 recurrence W_(m+1) = S W_m + (a-1) T W_(m-1), m <= 7, at (a,x,y)=(3/7,2/3,5/4)
PASS   discriminant >= 8 and 0 <= W_m <= (m+1) Lambda^m (m <= 8) at a=3/7
PASS Lemma 2.1 recurrence W_(m+1) = S W_m + (a-1) T W_(m-1), m <= 7, at (a,x,y)=(0,3/2,1/3)
PASS   discriminant >= 8 and 0 <= W_m <= (m+1) Lambda^m (m <= 8) at a=0
PASS Lemma 2.1 recurrence W_(m+1) = S W_m + (a-1) T W_(m-1), m <= 7, at (a,x,y)=(5/2,1,1)
PASS   discriminant >= 8 and 0 <= W_m <= (m+1) Lambda^m (m <= 8) at a=5/2
PASS Lemma 2.1 recurrence W_(m+1) = S W_m + (a-1) T W_(m-1), m <= 7, at (a,x,y)=(-3/4,2,1/2)
PASS Lemma 2.1 recurrence W_(m+1) = S W_m + (a-1) T W_(m-1), m <= 7, at (a,x,y)=(7,1/5,4)
PASS   discriminant >= 8 and 0 <= W_m <= (m+1) Lambda^m (m <= 8) at a=7
PASS W_m(1,1) >= c_a lambda^m for m <= 60 at a=0
PASS W_m(1,1) >= c_a lambda^m for m <= 60 at a=1/2
PASS W_m(1,1) >= c_a lambda^m for m <= 60 at a=1
PASS W_m(1,1) >= c_a lambda^m for m <= 60 at a=2
PASS W_m(1,1) >= c_a lambda^m for m <= 60 at a=7 (a=7: lambda=6.0, r=0.33333333, c_a=0.5)
PASS W_m(1,1) >= c_a lambda^m for m <= 60 at a=30
  G(0) = (2+sqrt2)^2 = 11.65685425 ; 6+4sqrt2 = 11.65685425
PASS Lemma 2.2: R involution, c(R w) = c(w), endpoint negated (all words, m <= 7)
PASS Lemma 2.3 at a=0: (m+1)^2 endpoints, N_m(P)=N_m(-P), L_(m+1) >= sum N^2 >= W^2/(m+1)^2 (m <= 7)
PASS Lemma 2.3 at a=1/2: (m+1)^2 endpoints, N_m(P)=N_m(-P), L_(m+1) >= sum N^2 >= W^2/(m+1)^2 (m <= 7)
PASS Lemma 2.3 at a=1: (m+1)^2 endpoints, N_m(P)=N_m(-P), L_(m+1) >= sum N^2 >= W^2/(m+1)^2 (m <= 7)
PASS Lemma 2.3 at a=3: (m+1)^2 endpoints, N_m(P)=N_m(-P), L_(m+1) >= sum N^2 >= W^2/(m+1)^2 (m <= 7)
PASS Lemma 2.5: P w F is a quadrant loop of length 2n+4k+4 with c(PwF)=c(w) (k <= 2, n <= 4)
# Part 4: Lemma 2.4 exactly
PASS Lemma 2.4: B^x_n(k) = B^y_n(k) <= max(1,a)(2n-1)(n+1)^2 e^(-2d(k+1)) Lambda_d^(2n) (a in {0,1/2,1,3}, k <= 3, n <= 12, seven d)
  largest ratio B/bound per (a,k): {'0,0': '0.2247', '0,1': '0.02993', '0,2': '0.009079', '0,3': '0.003731', '1/2,0': '0.2148', '1/2,1': '0.02733', '1/2,2': '0.007924', '1/2,3': '0.003112', '1,0': '0.2058', '1,1': '0.02511', '1,2': '0.006977', '1,3': '0.002626', '3,0': '0.05922', '3,1': '0.006236', '3,2': '0.001496', '3,3': '0.0004859'}
  sample: a=0, k=2, n=9: B^x = 26522576 ; bound at delta=1: 1.4946149e+12
# Part 5: the constants of the final step and of Proposition 3.1
PASS a=0: K = 2 max|f'| = 1.83919 > 1, f' > 1/2, (eq:K) and (eq:decay) on grids
PASS a=1: K = 2 max|f'| = 2.0 > 1, f' > 1/2, (eq:K) and (eq:decay) on grids
PASS a=0.5: K = 2 max|f'| = 1.9207 > 1, f' > 1/2, (eq:K) and (eq:decay) on grids
PASS a=2: K = 2 max|f'| = 2.22474 > 1, f' > 1/2, (eq:K) and (eq:decay) on grids
PASS a=7: K = 2 max|f'| = 3.0 > 1, f' > 1/2, (eq:K) and (eq:decay) on grids
PASS a=100: K = 2 max|f'| = 8.10634 > 1, f' > 1/2, (eq:K) and (eq:decay) on grids
PASS Prop 3.1 at a=0: C^2 lambda/(2K) = 6, exponent <= -6 log n where delta <= 1, k(n+1)-k(n) in {0,1} for 10^4 <= n < 2*10^4 (C = 2.5425)
PASS Prop 3.1 at a=1: C^2 lambda/(2K) = 6, exponent <= -6 log n where delta <= 1, k(n+1)-k(n) in {0,1} for 10^4 <= n < 2*10^4 (C = 2.4495)
PASS Prop 3.1 at a=7: C^2 lambda/(2K) = 6, exponent <= -6 log n where delta <= 1, k(n+1)-k(n) in {0,1} for 10^4 <= n < 2*10^4 (C = 2.4495)
# Part 6: six-step identity, Section 5 identity, shuffle-class sums
PASS Remark 4.1(a): Q_n(a) = sum over six-step quadrant excursions of (a-1)^(#diagonal) (n <= 10, five a)
PASS marked corner: intermediate point in the quadrant when both ends are
PASS Section 5: 2(1-a)^2/(-a) = (2+s)^2(2-s)^2/(2-s^2) = lambda^2(1+2(1-s)^2/(2-s^2)), a = s^2/2-1
  conjectured growth rates: {'-1': '8.0', '-3/4': '8.16666667', '-1/2': '9.0', '-1/4': '10.3989795', '0': '11.6568542', '1': '16.0', '2': '19.797959'}
PASS shuffle sum S_(-13/20)(V1,H1) = -124660942032543/256000000000 = -124660942032543/256000000000
PASS shuffle sum S_(-2/5)(V2,H2) = -63526557090248/1220703125 = -63526557090248/1220703125
PASS at a=-1: S = binom(i+j,i): 5005 and 92378
  V1 Dyck: True ; H1 Dyck: True ; V2 Dyck: False (bilateral: True , min height -3 ) ; H2 Dyck: True
PASS shuffle DP = brute force on (NNSS, EWEW) at a=-2/5
TOTAL failures: 0
