check_note.py -- OWR-4531-016 (Dyakonov's Fermat problem for model spaces, n = 3)

Part A. Exact identities used in the proofs
  [PASS] A0 omega^3 = 1 and 1 + omega + omega^2 = 0
  [PASS] A1 V is tangent to x^3 + y^3 = 1
  [PASS] A2 chart at infinity: S' = -C^2 (exactly)
  [PASS] A3 chart at infinity: C' = -S^2 modulo x^3 + y^3 = 1
  [PASS] A4 the field -C^2 d/dS - S^2 d/dC is tangent to 1 + C^3 = S^3
  [PASS] A5 points at infinity (S, C) = (0, -omega^k) lie on the curve; the field there is (-omega^(2k), 0) != 0
  [PASS] A6 rotation: d/du[cm(omega u)] = -omega sm(omega u)^2 = -(omega^-1 sm(omega u))^2
  [PASS] A7 sm = -i gives cm^3 = 1 - i; sm = +i gives cm^3 = 1 + i (both nonzero)
  [PASS] A8 q + i q_H = 1 (hence F + iH = B)
  [PASS] A9 q^3 + q_G^3 - q_H^3 = (sm^3 + cm^3 - 1)/(sm + i)^3 (Fermat identity)
  [PASS] A10 (cm/sm)^3 = (1 - sm^3)/sm^3 = sm^-3 - 1 on the curve (q_G is holomorphic at the poles of sm)
  [PASS] A11 for real sm, cm: conj(q) = q~, conj(q_G) = q~_G, conj(q_H) = q~_H
  [PASS] A12 Weierstrass form satisfies sm' = cm^2 (mod P'^2 = 4P^3 - 1/27)
  [PASS] A13 Weierstrass form satisfies cm' = -sm^2
  [PASS] A14 Weierstrass form satisfies sm^3 + cm^3 = 1 (mod P'^2 = 4P^3 - 1/27)
  [PASS] A15 P = p0*omega = -p0/2 + i h with h = (sqrt 3/2) p0
  [PASS] A16 pi p0/h = 2 pi/sqrt 3 (shift of the strip variable under the deck map)
  [PASS] A17 2ih = p0 (1 + 2 omega), a period (monodromy of psi around +-1)
  [PASS] A18 coefficients of sm^3 - (a sm + c)^3/b^3 - 1: [1 - a^3/b^3, -3a^2c/b^3, -3ac^2/b^3, -(c^3/b^3 + 1)]   constant term: c^3/b^3 = -1
  [PASS] A19 that system forces b = 0 (so the case b != 0 is impossible)   solutions: [{a: 0, b: 0, c: 0}]
  [PASS] A20 (1 + s^2)^3 - (1 - s^3)^2 = s^2 (3 s^2 + 2 s + 3), discriminant -32 < 0 (norm remark)
  [PASS] A21 (1 + e^{it})/(1 - e^{it}) = i cot(t/2) (6 values of t, 50 digits)
  [PASS] A22 1 - |tanh(s/2)|^2 = 4 e^sigma cos(tau)/|e^s + 1|^2 (203 points)
  [PASS] A23 Blaschke estimate 1 - |z|^2 <= C0 e^{-|sigma|}, C0 = 4/(1 - 1/e)^2 = 10.010601 >= e   max ratio 0.39954
  [PASS] A24 deck map gamma = psi^-1(psi + p0) is z -> (z + t)/(1 + t z), t = tanh(pi/sqrt 3) = 0.948216534125
  [PASS] A25 pseudo-hyperbolic distance is invariant under disc automorphisms (Lemma 2.2; spot check)

Part B. Dixon's functions to 40 digits (Weierstrass function, g2 = 0, g3 = 1/27)
  p0 = 2 int_{e1}^inf dt/sqrt(4t^3 - 1/27) = 5.29991625085634987194106849895    h = 4.58986211117157862473958251693
  [PASS] B1 p0 = B(1/3,1/3) = Gamma(1/3)^2/Gamma(2/3) = 3 int_0^1 (1-t^3)^(-2/3) dt (= pi_3)   |p0 - B(1/3,1/3)| = 0.0, |3 int - B| = 0.0
  [PASS] B2 Laurent series satisfies P'^2 = 4P^3 - 1/27 (4 points, |v| <= 3.05)   max rel. residual 1.59e-38
  [PASS] B3 lattice test: series at v and v - lambda agree for lambda = p0, p0 w, p0 w^2, p0(1+w)   max diff 3.76e-41; control p0/2: 0.4766
  [PASS] B4 sm' = cm^2, cm' = -sm^2, sm^3 + cm^3 = 1 (6 points across the plane); sm(u) = u + O(u^4), cm(0) = 1   ODE 2.46e-40, first integral 2.94e-40
  [PASS] B5 Abel inversion: sm(int_0^x (1-t^3)^(-2/3) dt) = x (8 values of x, no Weierstrass input)   max 2.31e-41
  [PASS] B6 periods p0, P = p0 omega, 2ih; sm(omega u) = omega sm(u), cm(omega u) = cm(u); conjugation symmetry   max 4.62e-40
  [PASS] B7 sm, cm real on Im u = 0 and Im u = h (18 points)   max relative |Im| 0.0
  [PASS] B8 sm(p0/3) = 1, cm(p0/3) = 0; real pole of sm at 2p0/3 (1 - 3P' = 0 there)
  zeros of sm + i in [0, p0) x (0, h):
     (0.7497704548828558397945 + 3.646997258195469354795j)   cm^3 = (1.0 - 1.0j)
     (1.766638750285449957314 + 1.88572970595221853989j)   cm^3 = (1.0 - 1.0j)
     (2.783507045688044074833 + 3.646997258195469354795j)   cm^3 = (1.0 - 1.0j)
  zeros of sm - i in [0, p0) x (0, h):
     (0.1335489202598691388624 + 0.9428648529761092699448j)   cm^3 = (1.0 + 1.0j)
     (3.399728580311030775765 + 0.9428648529761092699448j)   cm^3 = (1.0 + 1.0j)
     (4.416596875713624893284 + 2.70413240521936008485j)   cm^3 = (1.0 + 1.0j)
  [PASS] B9 exactly three zeros of sm + i and three of sm - i per period; cm^3 = 1 -/+ i there (simple zeros)
  [PASS] B10 the zeros agree with the 25-digit values recorded by the first computation (ODE stepper)   max diff 6.96e-25
  distance of the zeros to the lines Im u = 0, h: 0.942864852976109 ; mutual distances in Z_S: ['2.033736591', '2.033736591', '2.033736591']
  centroid of the three listed points of Z_S: (1.76663875028545 + 3.05990807411439j) ; p0/3 + 2ih/3 = (1.76663875028545 + 3.05990807411439j)
  [PASS] B11 Z_S is an equilateral triangle with centroid p0/3 + 2ih/3; Z~_S = conj(Z_S) + P (mod p0); distance to the lines > 0.94   reflection mismatch 2.31e-41
  poles of sm in [0, p0) x [0, h): ['(0.883319375142725 + 1.52995403705719j)', '(3.5332775005709 + 0.0j)', '(3.5332775005709 + 3.05990807411439j)']
  [PASS] B12 sm has exactly three poles in [0, p0) x [0, h), one of them the real pole 2p0/3

Part C. The counterexample on the disc (double precision, strip coordinates)
  Blaschke sums sum_k (1 - |a_k|), |k| <= 14, over the six orbits (Z: first three, Z~: last three):
    0.5188 (ratio 2.657993e-02); 0.6210 (ratio 2.657993e-02); 0.3634 (ratio 2.657993e-02); 0.5641 (ratio 2.657993e-02); 0.3926 (ratio 2.657993e-02); 0.7949 (ratio 2.657993e-02)
  [PASS] C1 Blaschke condition: geometric decay with ratio e^(-2 pi/sqrt 3) = 2.657993e-02
  [PASS] C2 quadrature: total mass 1; |z| = |B| = |B~| = 1 on the nodes   mass - 1 = 0.0e+00
  [PASS] C3 boundary identity theta_0 conj(z) conj(f) = f~ for f = F, G, H on 15000 nodes   max 1.6e-14
  [PASS] C3b F, G, H, F~, G~, H~ have no Fourier coefficients of index -1..-8 (no poles left in D); wrong input W4 (B~ in place of B) is rejected   max 2.1e-16; W4: 1.195e-01
  [PASS] C4 membership from the definition: |<f, theta_0 z^k>| for f = F, G, H, k = 0..8   max 2.2e-16
  [PASS] C5 wrong input W1, theta = B B~ (factor z omitted), is rejected   max 1.016e-01
  [PASS] C6 wrong input W2, theta = z B (B~ omitted), is rejected   max 2.819e-01
  [PASS] C7 wrong input W3, strip of height 0.97 h, is rejected   max 7.862e-03
  boundary spectrum {phi^-1(-1), phi^-1(1)} = {-0.1354+0.9908i, 0.9959-0.0903i}, angle 1.7971 (pi would be antipodal)
  [PASS] C8 repaired corollary: F o phi, G o phi, H o phi orthogonal to z (B o phi)(B~ o phi) z^k, k = 0..8   max 2.6e-16, mass - 1 = -2.2e-16
  [PASS] C9 naive transport theta_0 o phi is rejected: |<F o phi, theta_0 o phi>| = |phi(0) F~(phi(0))|   5.728581e-02 vs predicted 5.728581e-02; k >= 1: 1.2e-16
  [PASS] C10 F^3 + G^3 = H^3 and F + iH = B at 4000 random points of D   max 6.4e-16, 2.5e-16
  [PASS] C11 |F|, |G|, |H| are invariant under the deck map gamma (orbit of one point, |k| <= 8)   |F| = 0.096578, |B| = 0.139335
  [PASS] C12 sup of |F|, |G|, |H|, |F~|, |G~|, |H~| over the closed period rectangle (= over D): at most 1, close to 1   0.999998 0.999998 0.999978 0.999998 0.999998 0.999978
  [PASS] C13 F, G, H are linearly independent (singular values of 40 x 3 sample matrix)   1.5014 1.2623 0.8395

TOTAL failures: 0
