Part 1 -- the map of Theorem 1.2 (Lemma 5.1, (3), Section 5)
[PASS] h_{1+2i} built from the affine definition = [x0^3 x1^2 : x1^3 x2^2 : x0^4 x2] -- [x0**3*x1**2, x1**3*x2**2, x0**4*x2]
[PASS] g o h equals the three forms (3) printed in the note
[PASS] the forms are homogeneous of degree 10
[PASS] gcd(f0, f1, f2) over Q is a constant (no common factor) -- gcd = 1
[PASS] gcd(f0, f1, f2) modulo 3 is a constant -- gcd = 1
[PASS] gcd(f0, f1, f2) modulo 5 is a constant -- gcd = 1
[PASS] gcd(f0, f1, f2) modulo 7 is a constant -- gcd = 1
[PASS] gcd(f0, f1, f2) modulo 11 is a constant -- gcd = 1
[PASS] gcd(f0, f1, f2) modulo 13 is a constant -- gcd = 1
[PASS] gcd(f0, f1, f2) modulo 101 is a constant -- gcd = 1
[PASS] control: modulo 2 the forms have the common factor u0+u1+u2 of degree 5 (g = id in characteristic 2) -- gcd mod 2 = x0**4*x2 + x0**3*x1**2 + x1**3*x2**2
[PASS] f_j = u_j P_j and P_0 + P_1 = 2 u_2, P_0 + P_2 = 2 u_1, P_1 + P_2 = 2 u_0
[PASS] x0 does not divide f1, x1 does not divide f2, x2 does not divide f0
[PASS] g o g = [m x0 : m x1 : m x2] with m = (x1+x2-x0)(x2+x0-x1)(x0+x1-x2)
[PASS] the swap sigma of x1, x2 commutes with g
[PASS] BDJ's affine formula for g (introduction) homogenizes to the forms (2) of the note -- [x0**2 - x0*x1 - x0*x2, -x0*x1 + x1**2 - x1*x2, -x0*x2 - x1*x2 + x2**2]
[PASS] Jacobian determinant of (f1/f0, f2/f0) non-zero at (2/3, -7/5) and (5/9, 11/4): f dominant in characteristic 0
[PASS] f o f: raw degree 100, common factor x0^16 x1^16 x2^2 (degree 34), so deg f^2 = 66 -- gcd = x0**16*x1**16*x2**2 (0.5 s)

Part 2 -- monomial maps (Proposition 1.4, Section 6)
[PASS] Hasselblatt-Propp matrix gives the monomials [YW, ZW, X^2, XW] -- [(0, 1, 0, 1), (0, 0, 1, 1), (2, 0, 0, 0), (1, 0, 0, 1)]
[PASS] exponent matrix of the HP map re-derived with sympy = [[-1,1,0],[-1,0,-1],[-1,0,0]] -- [[-1, 1, 0], [-1, 0, -1], [-1, 0, 0]]
[PASS] HP map: direct composition with cancellation = formula (6), n = 1..60
[PASS] HP degrees for n = 1..10 are 2,3,4,6,9,12,17,25,33,45 -- [2, 3, 4, 6, 9, 12, 17, 25, 33, 45, 65, 85]
[PASS] characteristic polynomial of the HP matrix is T^3 + T^2 + T - 1 -- T**3 + T**2 + T - 1
[PASS] all 256 sequences L^(gamma) satisfy L_{n+3} = -L_{n+2} - L_{n+1} + L_n
[PASS] deg(h^n) = max over the 256 sequences, n = 1..60
[PASS] formula (6) = direct composition for 30 random matrices in each dimension 2, 3, 4 (entries in [-4,4]), n <= 12 -- 1080 comparisons
[PASS] N = 2: formula (6) for both conventions of h_zeta equals BDJ's psi(zeta) for |a|, |b| <= 80
[PASS] zeta = 1+2i: d_j = deg(h^j) = psi(zeta^j) and d_j <= sqrt(5)|zeta|^j for j <= 400; d_1..d_8 = 5,8,22,48,117,205,585,1054 -- [5, 8, 22, 48, 117, 205, 585, 1054]
[PASS] zeta = 1+2i: direct composition of h gives d_j for j <= 15

Part 3 -- sanity: the dynamical degree (BDJ Proposition 2.8) and the recursion (BDJ Proposition 2.7)
[PASS] zeta = 1+2i: the root lies in (6.857557409279008598, 6.857557409279008599) (exact certification, J = 220) -- tail bound < 1e-93
[PASS] zeta = -3+4i: the root lies in (13.449607681785413449, 13.449607681785413450) (exact certification, J = 220)
[PASS] zeta = 1+2i: mpmath root 6.857557409279008598501347 starts with BDJ's printed digits 6.8575574092
[PASS] zeta = -3+4i: mpmath root 13.44960768178541344984423 starts with BDJ's printed digits 13.4496076817
[PASS] BDJ recursion (zeta = 1+2i): e_1..e_6 = 10, 66, 454, 3114, 21368, 146488 -- [10, 66, 454, 3114, 21368, 146488, 1004626, 6889208]
[PASS] e_400/e_399 agrees with the certified root to 15 digits -- 6.857557409279008809
[PASS] BDJ recursion (zeta = -3+4i): deg f = 16 and e_1..e_5 = 16, 224, 2970, 39900, 537482 -- [16, 224, 2970, 39900, 537482]

Part 4 -- Lemma 5.2 and the hypothesis of Nguyen's Theorem D for the matrix of h_{1+2i}
[PASS] rho = (1+2i)/(1-2i) = (-3+4i)/5 -- -3/5 + 4*I/5
[PASS] minimal polynomial of rho over Q is (up to a constant) T^2 + (6/5) T + 1, not in Z[T]: rho is not an algebraic integer, hence not a root of unity -- 5*T**2 + 6*T + 5
[PASS] Im((1+2i)^n) != 0 for n = 1..3000 (exact)
[PASS] eigenvalues of [[1,2],[-2,1]] are 1+2i and 1-2i (complex conjugates with quotient rho, not a root of unity): hypothesis of [Ngu26, Theorem D(1)]
[PASS] det [[1,2],[-2,1]] = 5 != 0 (h dominant)

Part 5 -- the growth rate of the Hasselblatt-Propp degrees (Section 6, last sentence)
[PASS] roots of T^3 + T^2 + T - 1: one real root 0.5436890127..., a complex pair of modulus 1.3562030656... -- moduli [0.5436890127, 1.3562030656, 1.3562030656]
[PASS] log(deg h^n)/n for the HP map at n = 1000 and 2000 is within 2e-3 of log 1.3562030656 (the modulus of the complex pair, the only root of modulus >= 1); the consecutive ratios oscillate -- deg^(1/n) = 1.356786, 1.356617

Part 6 -- the example for the switched reading (Section 1.1)
[PASS] rho = sqrt2 - 1: a_n / a_{n-1} in {2, 3} for n = 1..100000
[PASS] log(a_n)/n at n = 10^7 is within 1e-6 of log(2 (3/2)^rho)
[PASS] Gelfond-Schneider applies: 3/2 is algebraic, not 0 or 1, and rho is algebraic irrational (minimal polynomial T^2 + 2T - 1)

Part 7 -- numerical sanity for Lemma 3.4 (|u_n| >= M^n e^{-eps n} for large n) on integer recurrences
[PASS] 2Re((1+2i)^n), M = sqrt5: no zeros for 1000 <= n <= 4000 and n log M - log|u_n| <= 0.01 n there -- max deficit 0.00355
[PASS] 2Re((1+2i)^n) + 2Re((3+i)^n), M = sqrt10: no zeros for 1000 <= n <= 4000 and n log M - log|u_n| <= 0.01 n there -- max deficit 0.00314

48 checks, 0 failed: ALL CHECKS PASSED
(running time 0.9 s)
