"""test_624 -- MB open problem 7: non-zonal conformal factors with (df (x) df)^TT = 0 exist. Construction (Gemini, 12 November 2026). Let v = 1 + b x1 with |b| < 1 and F(x) = v psi(x0/v) on R^4, so F is homogeneous of degree one in (x0, v): its Hessian has rank <= 1 wherever v > 0, which holds on the convex neighbourhood {|x| < R} n {v > 0} of S^3 for R slightly above 1. For H = dF (x) dF the linearised Riemann tensor about the flat metric is -(F_ik F_jl - F_il F_jk) (up to the factor convention), so it vanishes; on a convex (hence contractible) set Saint-Venant then gives H = L_xi delta. Pulled back to the unit S^3, iota* L_xi delta = L_{xi^T} gamma + 2 (xi.x) gamma, so df (x) df = iota* H lies in the Lie and conformal parts and has no TT part. f = F|_{S^3} is not zonal for b != 0. P1 [exact] rank D^2 F <= 1: det of the (x0, x1) Hessian block vanishes, for psi = s^2, s^3, sin s (symbolic). P2 [exact] the linearised Riemann tensor of H = dF (x) dF on R^4, dR_ijkl = (1/2)(d_j d_k H_il + d_i d_l H_jk - d_i d_k H_jl - d_j d_l H_ik), vanishes identically for psi = s^2, s^3, sin s. P3 [numeric, MB Corollary 5.11's route] J(e^{2tf} gamma) = 2 pi^2 + t^4 Q + O(t^5) with Q <= 0, and Q = 0 exactly when (df (x) df)^TT = 0. Q is int c4 dV with test_353's c4 = q(det H + H(p,p))/2 - (sigma_2(H) + |p|^2)^2/8 - |Hp|^2/2, p = grad f, H = Hess f on S^3 (for non-homogeneous F: p = P grad F, H = P D^2F P - (x . grad F) P). By 2-d Gauss-Legendre quadrature (the integrand depends on x0, x1 only): the controls x0 x1 -> -1/24 and x0^2 -> 0 of test_353 P2 are reproduced, and f = x0^2/(1 + b x1) for b = 0.3, 0.6 and x0^3/(1 + 0.5 x1)^2 give |Q|/(2 pi^2) below 1e-12, while int c1, c2, c3 vanish to the same accuracy. P4 [numeric] f = x0^2/(1 + 0.3 x1) is not zonal: for every axis a = (cos beta, sin beta, 0, 0) (180 values) there are points of S^3 with the same a.x and values of f differing by more than 1e-3. (A zonal f about an axis outside the (x0, x1) plane would not be invariant under the rotations of (x2, x3), which f is.) P5 [numeric] Polynomial examples. By Hartman and Nirenberg a C^2 function on all of R^4 with rank D^2 F <= 1 is g(a.x) plus an affine function, and the affine part need not be along a. So f = x0^2 + x1 and x0^3 + x1 (rank-one Hessian on R^4) are non-zonal polynomial solutions: Q = 0 to 1e-12, and the non-zonality gap of P4 exceeds 1e-3. P6 [numeric] The Möbius images psi(x0/v), v = 1 + 0.3 x1, of zonal functions (level sets non-concentric round 2-spheres) are not solutions: Q/(2 pi^2) < -1e-4 for psi = s, s^2, s^3, against the quadrature floor of P3, while the weighted v (x0/v)^2 of the second family gives |Q| < 1e-12. The kernel of DC_gamma is invariant under h -> Omega^-2 phi^* h, not under h -> phi^* h. History: Gemini's handoff (12 November 2026) gave the v psi(x0/v) family and added that a polynomial F with rank-one Hessian is psi(a.x), so that polynomial searches could only find zonal solutions. That omits the affine part allowed by Hartman-Nirenberg; P5 records the polynomial non-zonal solutions it allows. """ import sys import numpy as np import sympy as sp FAIL = [] def check(name, ok, detail=""): ok = bool(ok) if not ok: FAIL.append(name) print(f"[{'PASS' if ok else 'FAIL'}] {name}" + (f" ({detail})" if detail else ""), flush=True) x = sp.symbols('x0:4', real=True) b = sp.Symbol('b', real=True) s_ = sp.Symbol('s') psis = {"s^2": s_**2, "s^3": s_**3, "sin s": sp.sin(s_)} # P1, P2 ok1 = ok2 = True for name, ps in psis.items(): v = 1 + b*x[1] F = v*ps.subs(s_, x[0]/v) Hb = sp.Matrix(2, 2, lambda i, j: sp.diff(F, x[i], x[j])) ok1 &= sp.simplify(Hb.det()) == 0 dF = [sp.diff(F, xi) for xi in x] H = [[dF[i]*dF[j] for j in range(4)] for i in range(4)] for i in range(4): for j in range(4): for k in range(4): for l in range(4): if not (i < j and k < l): continue e = (sp.diff(H[i][l], x[j], x[k]) + sp.diff(H[j][k], x[i], x[l]) - sp.diff(H[j][l], x[i], x[k]) - sp.diff(H[i][k], x[j], x[l]))/2 if sp.simplify(e) != 0: ok2 = False check("P1 F = v psi(x0/v), v = 1 + b x1: rank D^2 F <= 1 (psi = s^2, s^3, sin s)", ok1) check("P2 the linearised Riemann tensor of dF (x) dF vanishes identically (psi = s^2, s^3, sin s)", ok2) # P3 def Q_of(Fexpr, n=160): grad = [sp.diff(Fexpr, xi) for xi in x] hess = [[sp.diff(Fexpr, xi, xj) for xj in x] for xi in x] fg = sp.lambdify(x, grad, 'numpy'); fh = sp.lambdify(x, hess, 'numpy') gx, gw = np.polynomial.legendre.leggauss(n) chis = (gx + 1)*np.pi/2; wch = gw*np.pi/2 ths = chis; wth = wch tot = np.zeros(4) for ci, wc in zip(chis, wch): for ti, wt in zip(ths, wth): X = np.array([np.cos(ci), np.sin(ci)*np.cos(ti), np.sin(ci)*np.sin(ti), 0.0]) g = np.array(fg(*X), dtype=float); D2 = np.array(fh(*X), dtype=float) P = np.eye(4) - np.outer(X, X) p = P @ g Hm = P @ D2 @ P - (X @ g)*P # orthonormal tangent basis Bq, _ = np.linalg.qr(np.column_stack([P[:, 1], P[:, 2], P[:, 3], P[:, 0]])) T = [] for k in range(4): c = Bq[:, k] - (Bq[:, k] @ X)*X if np.linalg.norm(c) > 1e-8 and len(T) < 3: for t_ in T: c = c - (c @ t_)*t_ if np.linalg.norm(c) > 1e-8: T.append(c/np.linalg.norm(c)) T = np.array(T) H3 = T @ Hm @ T.T; p3 = T @ p q = np.trace(H3); s2 = (q**2 - np.trace(H3 @ H3))/2; dH = np.linalg.det(H3) Hpp = p3 @ H3 @ p3; pp = p3 @ p3; Hp = H3 @ p3 c1 = -q; c2 = (s2 - pp)/2; c3 = (dH + Hpp)/2 c4 = q*(dH + Hpp)/2 - (s2 + pp)**2/8 - (Hp @ Hp)/2 w = wc*wt*np.sin(ci)**2*np.sin(ti)*2*np.pi tot += w*np.array([c1, c2, c3, c4]) return tot/(2*np.pi**2) ctrl1 = Q_of(x[0]*x[1]); ctrl2 = Q_of(x[0]**2) cases = {"x0^2/(1 + 0.3 x1)": x[0]**2/(1 + sp.Rational(3, 10)*x[1]), "x0^2/(1 + 0.6 x1)": x[0]**2/(1 + sp.Rational(3, 5)*x[1]), "x0^3/(1 + 0.5 x1)^2": x[0]**3/(1 + sp.Rational(1, 2)*x[1])**2} res = {k: Q_of(v) for k, v in cases.items()} ok3 = abs(ctrl1[3] + 1/24) < 1e-12 and abs(ctrl2[3]) < 1e-12 and all(np.max(np.abs(r)) < 1e-12 for r in res.values()) check("P3 test_353's quartic coefficient: controls x0 x1 -> -1/24, x0^2 -> 0; the three non-zonal f give Q = 0", ok3, "; ".join(f"{k}: Q/2pi^2 = {r[3]:.1e}" for k, r in res.items()) + f"; control {ctrl1[3]:.12f}") # P4 f = lambda X: X[0]**2/(1 + 0.3*X[1]) worst = np.inf for beta in np.linspace(0, np.pi, 180, endpoint=False): a = np.array([np.cos(beta), np.sin(beta), 0, 0]); e = np.array([-np.sin(beta), np.cos(beta), 0, 0]); e2 = np.array([0, 0, 1.0, 0]) gap = 0.0 for u in np.linspace(-0.95, 0.95, 39): y1 = u*a + np.sqrt(1 - u*u)*e; y2 = u*a + np.sqrt(1 - u*u)*e2 gap = max(gap, abs(f(y1) - f(y2))) worst = min(worst, gap) check("P4 f = x0^2/(1 + 0.3 x1) is not zonal about any axis in the (x0, x1) plane", worst > 1e-3, f"smallest gap {worst:.3f}") # P5 poly = {"x0^2 + x1": x[0]**2 + x[1], "x0^3 + x1": x[0]**3 + x[1]} resp = {k: Q_of(v, n=120) for k, v in poly.items()} fz = lambda X: X[0]**2 + X[1] worst5 = np.inf for beta in np.linspace(0, np.pi, 180, endpoint=False): a = np.array([np.cos(beta), np.sin(beta), 0, 0]); e = np.array([-np.sin(beta), np.cos(beta), 0, 0]); e2 = np.array([0, 0, 1.0, 0]) gap = max(abs(fz(u*a + np.sqrt(1 - u*u)*e) - fz(u*a + np.sqrt(1 - u*u)*e2)) for u in np.linspace(-0.95, 0.95, 39)) worst5 = min(worst5, gap) check("P5 polynomial non-zonal solutions x0^2 + x1, x0^3 + x1: Q = 0, not zonal", all(np.max(np.abs(r)) < 1e-12 for r in resp.values()) and worst5 > 1e-3, "; ".join(f"{k}: Q/2pi^2 = {r[3]:.1e}" for k, r in resp.items()) + f"; gap {worst5:.3f}") # P6 vv = 1 + sp.Rational(3, 10)*x[1] mob = {"x0/v": x[0]/vv, "(x0/v)^2": (x[0]/vv)**2, "(x0/v)^3": (x[0]/vv)**3} resm = {k: Q_of(F_, n=120)[3] for k, F_ in mob.items()} wq = Q_of(vv*(x[0]/vv)**2, n=120)[3] check("P6 the Moebius images psi(x0/v) are not solutions (Q < 0); the weighted v (x0/v)^2 is", all(r < -1e-4 for r in resm.values()) and abs(wq) < 1e-12, "; ".join(f"{k}: {r:.2e}" for k, r in resm.items()) + f"; v (x0/v)^2: {wq:.1e}") print("\nALL PASS" if not FAIL else f"\nFAILURES: {FAIL}") sys.exit(1 if FAIL else 0)