Part A: gradient, Hessian form, self-stress, test direction, target Gram, Schur complement
PASS (3) grad V = L(e) P, so law (1) is p' = -grad V (Lemma 2.1)
PASS (3) Hessian form H_P[Q] = sum 2<p_ij,q_ij>^2 + e_ij |q_ij|^2 (Lemma 2.1)
PASS Theorem 1.1(c): H_P[P] = 2 sum |p_ij|^4 + tr(P^T L(e) P)
PASS Lemma 3.2: signed cofactors give sum lambda_i = 0 and sum lambda_i p_i = 0
PASS Lemma 3.2: L(lambda_i lambda_j) = -lambda lambda^T (so L(w) P = 0: w is a self-stress)
PASS P = Wb C is centred and P^T lambda = 0
PASS (9) H_P[lambda z^T] = 2 z^T (s2 P^T P + 4 P^T D P) z - kappa s2^2 |z|^2  (= 2 z^T B^T K B z - ...)
PASS Step 1: L((lambda_i - lambda_j)^2) = diag(4 lambda_i^2 + s2) - delta 1^T - 1 delta^T + 2 lambda lambda^T
PASS Step 1: sum_{i<j} lambda_i lambda_j (lambda_i - lambda_j)^2 = -s2^2
PASS (10) X* = -(1/2) J Dbar J = P P^T + (kappa/2)(lambda lambda^T - J D J)
PASS Pi0 P = s2 P (the columns of P lie in W)
PASS (11) U^T X* U = B B^T - (kappa/2) M, in projector form Pi X* Pi = P P^T - (kappa/2) Pi D Pi   [times s2^2]
PASS (11) U^T X* lambda = -(kappa/2) b, in projector form Pi X* lambda = -(kappa/2) Pi D lambda   [times s2]
PASS (11) lambda^T X* lambda = (kappa/2)(s2^2 - s4)
PASS (12) Schur complement = B B^T - (kappa/2) N, in projector form (exact, at 25 random rational points)
Part B: Lemma 4.2
PASS Newton identities (eta1 = 0): s2, s3, s4, s5, s6, s8 in terms of sigma, eta3, eta4
PASS sigma = s2/2 and Delta = sum_{i<j} lambda_i^2 lambda_j^2 = (s2^2 - s4)/2
PASS proof of Lemma 4.2: tr(Pi D) = (3/4) s2 - s4/s2
PASS proof of Lemma 4.2: tr(Pi D Pi D) = s4/2 - 2 s6/s2 + s2^2/16 + s4^2/s2^2 + s3^2/(2 s2)
PASS proof of Lemma 4.2: lambda^T D Pi D lambda = s6 - s3^2/4 - s4^2/s2
PASS proof of Lemma 4.2: Pi D lambda = (lambda_i^3) - (s3/4) 1 - (s4/s2) lambda
PASS proof of Lemma 4.2: lambda^T D Pi D Pi D lambda = s8 + s2 s3^2/16 + s4^3/s2^2 - s3 s5/2 - 2 s4 s6/s2 + s3^2 s4/(2 s2)
PASS (5) tr M = (sigma^2 + 4 eta4)/(2 sigma)
PASS (5) tr M^2 = sigma^2/4 - 3 eta3^2/(4 sigma) + 4 eta4^2/sigma^2
PASS (5) det M = eta4 + 3 eta3^2/(8 sigma)
PASS (5) |b|^2 = 2 sigma eta4 + 3 eta3^2/4 - 8 eta4^2/sigma
PASS (5) b^T M b = 4 eta4^2 (sigma^2 - 4 eta4)/sigma^2 + eta3^2 (sigma^2 + 24 eta4)/(8 sigma)
PASS (6) tr(NK) = 2 (sigma^4 + 5 sigma^2 eta4 + 12 eta4^2 - sigma eta3^2)/Delta
PASS proof of Lemma 4.2: det K = 2 (4 sigma^3 + 16 sigma eta4 + 3 eta3^2)/sigma
PASS proof of Lemma 4.2: det N = sigma (3 sigma eta4 + eta3^2)/(2 Delta)
PASS (6) det(NK) = (3 sigma eta4 + eta3^2)(4 sigma^3 + 16 sigma eta4 + 3 eta3^2)/Delta
PASS direct 4x4 check: tr(N K) in projector form agrees with (6)
PASS direct 4x4 check: nu1 nu2 = (tr^2 - tr((NK)^2))/2 in projector form agrees with det(NK) of (6)
Part C: Proposition 4.1
PASS mu = s2^2/2 = 2 sigma^2
PASS (7) (2 mu - tr NK) Delta = 2 [(sigma^2 - 4 eta4)(sigma^2 + 3 eta4) + sigma eta3^2]   [sigma, eta3, eta4 independent]
PASS (8) (mu^2 - mu tr NK + det NK) Delta = eta3^2 (8 sigma^3 + 25 sigma eta4 + 3 eta3^2)   [sigma, eta3, eta4 independent]
PASS (7) in lambda
PASS (8) in lambda
PASS Remark 4.3: sigma^2 - 4 eta4 = (A-B)^2/4 + (l1 l3 - l2 l4)^2 + (l1 l4 - l2 l3)^2 (all real lambda, sigma = s2/2)
PASS Remark 4.3: sigma^2 + 4 eta4 = (A-B)^2/4 + (l1 l3 + l2 l4)^2 + (l1 l4 + l2 l3)^2
PASS Remark 4.3: sigma^2 + 3 eta4 = (3/4)(sigma^2 + 4 eta4) + sigma^2/4
PASS Remark 4.3: 8 sigma^2 + 25 eta4 = (25/4)(sigma^2 + 4 eta4) + (7/4) sigma^2
PASS equality case of Proposition 4.1: eta3 = 0 at lambda = (a, -a, c, -c)
   lambda = (3,-3,1,-1): eigenvalues of NK = [3672/59, 200], s2^2/2 = 200
PASS    lambda_max(NK) <= s2^2/2 at lambda = (3,-3,1,-1) (exact)
   lambda = (1,-1,0,0): eigenvalues of NK = [0, 2], s2^2/2 = 2
PASS    lambda_max(NK) <= s2^2/2 at lambda = (1,-1,0,0) (exact)
   lambda = (1,2,-4,1): eigenvalues of NK = [26, 752/7], s2^2/2 = 242
PASS    lambda_max(NK) <= s2^2/2 at lambda = (1,2,-4,1) (exact)
Part D: Proposition 6.1 (sharpness)
PASS the rectangle (0,0), (4,0), (4,sqrt 6), (0,sqrt 6) has squared distances (16,22,6,6,22,16) in the order 12,...,34
   e = [4, -4, -4, -4, -4, 4] ; affine dependency lambda = [1, 1, -1, -1] ; V = 24
PASS grad V(P) = 0, e = (4,-4,-4,-4,-4,4) = kappa lambda_i lambda_j with lambda = (1,1,-1,-1), kappa = 4, V = 24
   characteristic polynomial of Hess V(P): x**3*(x - 80)*(x - 56)*(x + 8)*(x**2 - 160*x + 1728)
PASS characteristic polynomial = x^3 (x+8)(x-56)(x-80)(x^2-160x+1728)
PASS lambda_min(Hess V(P)) = -8 = -(kappa/2)|lambda|^2 = -sqrt(8 V/3)
PASS Morse index 1, three zero eigenvalues, four positive
Part E: Examples 6.2 and 6.3
  Example 6.2 (triangle with an interior agent)
   char. poly of X* + 11^T = x*(x - 4)*(x**2 - 357*x + 6480) ; target affine dependency [5, -1, -3, -1] -> triangle with an interior agent
PASS    X* + 11^T has characteristic polynomial x*(x - 4)*(x**2 - 357*x + 6480)
PASS    target is planar (X* PSD of rank 2), affine dependency [-5, 1, 3, 1], a triangle with an interior agent
   P = [[3, -2], [2, 0], [10, 0], [-15, 2]] ; lambda = [-1, 4, -2, -1] ; kappa = 4 ; e = [-16, 8, 4, -32, -16, 8] ; V = 420
PASS    grad V(P) = 0; e = [-16, 8, 4, -32, -16, 8] = 4 lambda_i lambda_j with lambda = [-1, 4, -2, -1]
PASS    |lambda|^2 = 22
PASS    characteristic polynomial of Hess V(P) = x**3*(x**5 - 5360*x**4 + 7214832*x**3 - 2571180544*x**2 - 137562323968*x + 8413501194240)
   lambda_min = -77.6912478483582 ; -(kappa/2)|lambda|^2 = -44 ; -sqrt(8V/3) = -33.4664010613630
PASS    lambda_min rounds to -77.69 and is below -(kappa/2)|lambda|^2 = -44
PASS    exactly one negative, three zero and four positive eigenvalues
PASS    H_P[lambda z^T] = -1520 <= -968 = -(kappa/2) s2^2 |z|^2 for z = (0, 1)
  Example 6.3 (convex quadrilateral)
   char. poly of X* + 11^T = x*(x - 4)*(x**2 - 393*x + 7200) ; target affine dependency [3, -1, -3, 1] -> convex quadrilateral
PASS    X* + 11^T has characteristic polynomial x*(x - 4)*(x**2 - 393*x + 7200)
PASS    target is planar (X* PSD of rank 2), affine dependency [3, -1, -3, 1], a convex quadrilateral
   P = [[-5, 0], [15, 5], [-9, -5], [-1, 0]] ; lambda = [-2, -1, -1, 4] ; kappa = 4 ; e = [8, 8, -32, 4, -16, -16] ; V = 420
PASS    grad V(P) = 0; e = [8, 8, -32, 4, -16, -16] = 4 lambda_i lambda_j with lambda = [-2, -1, -1, 4]
PASS    |lambda|^2 = 22
PASS    characteristic polynomial of Hess V(P) = x**3*(x**5 - 5936*x**4 + 9010800*x**3 - 3736566784*x**2 - 130423352320*x + 13331005440000)
   lambda_min = -71.6871791626366 ; -(kappa/2)|lambda|^2 = -44 ; -sqrt(8V/3) = -33.4664010613630
PASS    lambda_min rounds to -71.69 and is below -(kappa/2)|lambda|^2 = -44
PASS    exactly one negative, three zero and four positive eigenvalues
PASS    H_P[lambda z^T] = -6256 <= -4840 = -(kappa/2) s2^2 |z|^2 for z = (1, -2)
Part F: Remark 5.2, lambda = (0,...,0,1,-1) in the n = l + 2 setting
PASS n = 4: lambda_max(NK)/s2^2 = 1/2 = (n-2)/4
PASS n = 5: lambda_max(NK)/s2^2 = 3/4 = (n-2)/4
PASS n = 6: lambda_max(NK)/s2^2 = 1 = (n-2)/4
PASS n = 7: lambda_max(NK)/s2^2 = 5/4 = (n-2)/4
PASS n = 8: lambda_max(NK)/s2^2 = 3/2 = (n-2)/4
ALL PASS
real 173.13
user 172.31
sys 0.70
