[THOUGHT] We need to compute several quantities step by step.

First, part I: curve L is a cycloid.

(I-1) Length of curve L for t in [0, 2π]. Arc length formula: L = ∫_0^{2π} sqrt( (dx/dt)^2 + (dy/dt)^2 ) dt. Compute derivatives: dx/dt = a(1 - cos t) dy/dt = a sin t Then (dx/dt)^2 + (dy/dt)^2 = a^2[(1 - cos t)^2 + sin^2 t] = a^2[1 - 2 cos t + cos^2 t + sin^2 t] = a^2[1 - 2 cos t + 1] = a^2[2 - 2 cos t] = 2a^2 (1 - cos t). Using identity 1 - cos t = 2 sin^2(t/2), so sqrt = sqrt(2a^2 * 2 sin^2(t/2)) = sqrt(4a^2 sin^2(t/2)) = 2a |sin(t/2)|. For t in [0, 2π], t/2 in [0, π], sin(t/2) ≥ 0, so |sin(t/2)| = sin(t/2). Thus L = ∫_0^{2π} 2a sin(t/2) dt. Compute integral: ∫ sin(t/2) dt = -2 cos(t/2). So L = 2a * [-2 cos(t/2)]_0^{2π} = -4a [cos(π) - cos(0)] = -4a[-1 - 1] = -4a*(-2) = 8a. So L = 8a.

(I-2) Curvature κ_L(t) for 0 < t < 2π. Formula for curvature for parametric curve: κ = |x' y'' - y' x''| / ( (x'^2 + y'^2)^(3/2) ). We have: x' = a(1 - cos t) y' = a sin t x'' = a sin t y'' = a cos t Compute numerator: x' y'' - y' x'' = a(1 - cos t) * a cos t - a sin t * a sin t = a^2[ (1 - cos t) cos t - sin^2 t ] = a^2[ cos t - cos^2 t - sin^2 t ] = a^2[ cos t - (cos^2 t + sin^2 t) ] = a^2(cos t - 1) = -a^2 (1 - cos t). Absolute value: a^2 (1 - cos t). Denominator: (x'^2 + y'^2)^(3/2) = (2a^2 (1 - cos t))^(3/2) = (2a^2)^(3/2) (1 - cos t)^(3/2) = (2^(3/2) a^3) (1 - cos t)^(3/2). Thus κ_L(t) = a^2 (1 - cos t) / (2^(3/2) a^3 (1 - cos t)^(3/2)) = (1) / (2^(3/2) a (1 - cos t)^(1/2)). Simplify: κ_L(t) = 1 / (2√2 a √(1 - cos t)). Using identity 1 - cos t = 2 sin^2(t/2), √(1 - cos t) = √2 |sin(t/2)| = √2 sin(t/2) for 0 < t < 2π (except t=π? Actually sin(t/2) positive for t in (0, 2π) except at t=0,2π,π? At t=π, sin(π/2)=1 >0, so positive). So √(1 - cos t) = √2 sin(t/2). Then κ_L(t) = 1 / (2√2 a * √2 sin(t/2)) = 1 / (2*2 a sin(t/2)) = 1/(4a sin(t/2)). Thus κ_L(t) = 1/(4a sin(t/2)). Check at t=π: sin(π/2)=1, so κ_L(π) = 1/(4a).

Now α = L * κ_L(π) = (8a) * (1/(4a)) = 2. So α = 2.

Now part II: surface given parametrically.

(II-1) Equation without parameters. We have: x = sinh u cos v y = 2 sinh u sin v z = 3 cosh u We can eliminate u and v. Note: cosh^2 u - sinh^2 u = 1. Also from x and y: (x / cosh? Actually x = sinh u cos v, y/2 = sinh u sin v). Thus (x)^2 + (y/2)^2 = sinh^2 u (cos^2 v + sin^2 v) = sinh^2 u. So sinh^2 u = x^2 + (y/2)^2. Also z/3 = cosh u, so cosh^2 u = (z/3)^2. Then using identity cosh^2 u - sinh^2 u = 1: (z/3)^2 - (x^2 + (y/2)^2) = 1. Thus (z^2)/9 - x^2 - (y^2)/4 = 1. Or rearranged: z^2/9 - x^2 - y^2/4 = 1. This is a hyperboloid of two sheets? Actually standard form: x^2/a^2 + y^2/b^2 - z^2/c^2 = -1 for two sheets? Let's check: Our equation: z^2/9 - x^2 - y^2/4 = 1 => z^2/9 - x^2 - y^2/4 = 1. That's a hyperboloid of two sheets (since only one squared term positive on left? Actually it's of the form z^2/c^2 - x^2/a^2 - y^2/b^2 = 1, which is a hyperboloid of two sheets). Yes.

(II-2) Sketch cross-sections. At z = 5: substitute into equation: (5^2)/9 - x^2 - y^2/4 = 1 => 25/9 - 1 = x^2 + y^2/4 => (25 - 9)/9 = 16/9 = x^2 + y^2/4. So x^2 + y^2/4 = 16/9. This is an ellipse in xy-plane. x-intercept: set y=0 => x^2 = 16/9 => x = ±4/3. So A_x = 4/3 (positive). y-intercept: set x=0 => y^2/4 = 16/9 => y^2 = 64/9 => y = ±8/3. So A_y = 8/3. At y = 0: cross-section in xz-plane. Equation becomes: z^2/9 - x^2 = 1 => z^2/9 - x^2 = 1. For z-intercept: set x=0 => z^2/9 = 1 => z^2 = 9 => z = ±3. So A_z = 3.

Thus δ = A_x * A_y * A_z = (4/3)(8/3)3 = (483)/(33) = (96)/(9) = 32/3? Wait compute: (4/3)(8/3) = 32/9, times 3 = 96/9 = 32/3. Yes. ρ = A_y / A_x = (8/3)/(4/3) = 2.

(II-3) Unit normal vector n with positive z-component. We have surface r(u,v) = (sinh u cos v, 2 sinh u sin v, 3 cosh u). Compute partial derivatives: r_u = (cosh u cos v, 2 cosh u sin v, 3 sinh u) r_v = (-sinh u sin v, 2 sinh u cos v, 0) Normal vector N = r_u × r_v. Compute cross product: i component: (2 cosh u sin v)(0) - (3 sinh u)(2 sinh u cos v) = 0 - 6 sinh^2 u cos v = -6 sinh^2 u cos v. j component: (3 sinh u)(-sinh u sin v) - (cosh u cos v)(0) = -3 sinh^2 u sin v - 0 = -3 sinh^2 u sin v. k component: (cosh u cos v)(2 sinh u cos v) - (2 cosh u sin v)(-sinh u sin v) = 2 cosh u sinh u cos^2 v + 2 cosh u sinh u sin^2 v = 2 cosh u sinh u (cos^2 v + sin^2 v) = 2 cosh u sinh u. Thus N = (-6 sinh^2 u cos v, -3 sinh^2 u sin v, 2 cosh u sinh u). Simplify: factor out -3 sinh u? Actually let's see: first component: -6 sinh^2 u cos v = -6 sinh^2 u cos v. Second: -3 sinh^2 u sin v. Third: 2 cosh u sinh u. We want unit normal n = N / ||N||, with positive z-component. At u=0, sinh u=0, cosh u=1, then N = (0,0,0) degenerate? Actually at u=0, the surface point is (0,0,3). The normal there is along z-axis. But our N becomes (0,0,0) which is zero vector. That's because parametrization is not regular at u=0? Actually sinh u=0, so r_u = (cosh u cos v, 2 cosh u sin v, 0) = (cos v, 2 sin v, 0) and r_v = (0,0,0). So r_v is zero vector, so parametrization is degenerate at u=0. But we can compute normal via gradient of implicit equation. However, the problem asks for expression in terms of u and v, with z-component positive. Possibly they want the normal vector field away from u=0. But at u=0, we can define by limit. However, for (II-4) we need Gaussian curvature at u=v=0, we can compute via first and second fundamental forms, but maybe easier using formula for Gaussian curvature for parametrization: K = (LN - M^2)/(EG - F^2) where L,M,N are coefficients of second fundamental form. But we can also compute using the implicit equation.

But the problem says: "Express a unit normal vector n of the curved surface by u and v. Here, the z-component of n should be positive." So we need to compute n from N and normalize, ensuring z positive. However, at u=0, N is zero, so we need to treat separately. Possibly they expect expression valid for u>0? But we can still derive.

Compute magnitude ||N||: ||N||^2 = (-6 sinh^2 u cos v)^2 + (-3 sinh^2 u sin v)^2 + (2 cosh u sinh u)^2 = 36 sinh^4 u cos^2 v + 9 sinh^4 u sin^2 v + 4 cosh^2 u sinh^2 u = sinh^2 u [36 sinh^2 u cos^2 v + 9 sinh^2 u sin^2 v + 4 cosh^2 u] = sinh^2 u [ sinh^2 u (36 cos^2 v + 9 sin^2 v) + 4 cosh^2 u ]. Simplify inside: 36 cos^2 v + 9 sin^2 v = 9(4 cos^2 v + sin^2 v) = 9(4 cos^2 v + sin^2 v). Not simple. Maybe we can simplify using identity from implicit equation? Alternatively, we can compute unit normal via gradient of F(x,y,z) = z^2/9 - x^2 - y^2/4 - 1 = 0. Gradient ∇F = (-2x, -y/2, 2z/9). Then unit normal n = ± ∇F / ||∇F||. Choose sign so that z-component positive. Express x,y,z in terms of u,v: x = sinh u cos v, y = 2 sinh u sin v, z = 3 cosh u. Then ∇F = (-2 sinh u cos v, - (2 sinh u sin v)/2? Wait y/2 derivative: ∂F/∂y = -y/2, so -y/2 = - (2 sinh u sin v)/2 = - sinh u sin v. Actually careful: F = z^2/9 - x^2 - y^2/4 - 1. ∂F/∂x = -2x, ∂F/∂y = -y/2, ∂F/∂z = 2z/9. So ∇F = (-2 sinh u cos v, - (2 sinh u sin v)/2? Wait y = 2 sinh u sin v, so -y/2 = - (2 sinh u sin v)/2 = - sinh u sin v. Yes. ∂F/∂z = 2*(3 cosh u)/9 = 6 cosh u/9 = 2 cosh u/3. Thus ∇F = (-2 sinh u cos v, - sinh u sin v, 2 cosh u/3). Now compute magnitude ||∇F||: ||∇F||^2 = 4 sinh^2 u cos^2 v + sinh^2 u sin^2 v + (4 cosh^2 u)/9 = sinh^2 u (4 cos^2 v + sin^2 v) + (4/9) cosh^2 u. Simplify: 4 cos^2 v + sin^2 v = 4 cos^2 v + sin^2 v = 4 cos^2 v + (1 - cos^2 v) = 3 cos^2 v + 1? Actually sin^2 v = 1 - cos^2 v, so 4 cos^2 v + 1 - cos^2 v = 3 cos^2 v + 1. Thus ||∇F||^2 = sinh^2 u (3 cos^2 v + 1) + (4/9) cosh^2 u. Then unit normal n = ∇F / ||∇F|| if z-component positive? Check z-component of ∇F: 2 cosh u/3 > 0 for all u (since cosh u ≥ 1). So ∇F already has positive z-component. So n = ∇F / ||∇F||. But the problem asks expression by u and v. So we can give n = ( -2 sinh u cos v, - sinh u sin v, 2 cosh u/3 ) / sqrt( sinh^2 u (3 cos^2 v + 1) + (4/9) cosh^2 u ). That's acceptable.

(II-4) Gaussian curvature κ at u=v=0. We can compute using formula for Gaussian curvature for a surface given by implicit equation F(x,y,z)=0: K = ( ∇F · H · ∇F^T - ||∇F||^2 trace(H?) )? Actually there is formula: Gaussian curvature K = ( (F_{xx} F_{yy} F_{zz} + ... )? ) Not simple. Better: Since we have parametrization, we can compute first and second fundamental forms at u=0. But at u=0, r_v = 0, so parametrization is degenerate. However, we can use the implicit equation to compute curvature at point (0,0,3) (since u=0, v arbitrary? Actually at u=0, x=0, y=0, z=3, independent of v). So the point is (0,0,3). The surface is z^2/9 - x^2 - y^2/4 = 1. This is a hyperboloid of two sheets. At (0,0,3), the surface is like a "peak"? Actually it's a smooth point. We can compute Gaussian curvature via formula for graph: z = f(x,y) = 3 sqrt(1 + x^2 + y^2/4)? Wait solving for z: z^2 = 9(1 + x^2 + y^2/4) => z = 3 sqrt(1 + x^2 + y^2/4) (taking upper sheet since z=3>0). So surface is graph of f(x,y) = 3 sqrt(1 + x^2 + y^2/4). Then Gaussian curvature K = (f_{xx} f_{yy} - f_{xy}^2) / (1 + f_x^2 + f_y^2)^2. Compute at (0,0): f(0,0)=3. Compute partial derivatives: Let g(x,y) = sqrt(1 + x^2 + y^2/4) = (1 + x^2 + y^2/4)^(1/2). Then f = 3g. g_x = (1/2)(1 + x^2 + y^2/4)^(-1/2) * 2x = x / sqrt(1 + x^2 + y^2/4). g_y = (1/2)(1 + x^2 + y^2/4)^(-1/2) * (y/2) = y/(4 sqrt(1 + x^2 + y^2/4)). Thus f_x = 3x / sqrt(1 + x^2 + y^2/4) f_y = 3y/(4 sqrt(1 + x^2 + y^2/4)) At (0,0): f_x=0, f_y=0. Second derivatives: Compute f_xx: differentiate f_x wrt x. Let h = sqrt(1 + x^2 + y^2/4). Then f_x = 3x / h. f_xx = derivative: (3 * h - 3x * (x/h)) / h^2? Actually use quotient rule: (3h - 3x * (x/h)) / h^2 = (3h - 3x^2/h) / h^2 = (3(h^2 - x^2)/h) / h^2 = 3(h^2 - x^2) / h^3. But h^2 = 1 + x^2 + y^2/4. So h^2 - x^2 = 1 + y^2/4. Thus f_xx = 3(1 + y^2/4) / h^3. At (0,0): h=1, so f_xx = 31 / 1 = 3. Similarly f_yy: f_y = (3y)/(4h). Derivative: f_yy = (3/4) * (h - y(y/(4h)? Wait careful: f_y = (3y)/(4h). Use quotient rule: derivative = (3/4) * (h - y * (y/(4h))? Actually let's compute properly: f_y = (3y)/(4h). Then ∂/∂y f_y = (3/4) * ( (1h + y * (-1) * (1/(4h))? ) I'm mixing. Better: f_y = (3/4) * y / h. Then f_yy = (3/4) * ( (1h - y * h_y) / h^2 ), where h_y = (1/2)(1 + x^2 + y^2/4)^(-1/2) * (y/2) = y/(4h). So h_y = y/(4h). Thus f_yy = (3/4) * ( h - y * (y/(4h)) ) / h^2 = (3/4) * ( (h^2 - y^2/4) / h ) / h^2 = (3/4) * (h^2 - y^2/4) / h^3. But h^2 = 1 + x^2 + y^2/4, so h^2 - y^2/4 = 1 + x^2. At (0,0): h=1, x=0, so h^2 - y^2/4 = 1. Thus f_yy = (3/4) * 1 / 1 = 3/4. Now f_xy: differentiate f_x wrt y. f_x = 3x / h. ∂/∂y f_x = 3x * (-1) * h_y / h^2 = -3x h_y / h^2. h_y = y/(4h). So f_xy = -3x * (y/(4h)) / h^2 = -3xy/(4 h^3). At (0,0): f_xy = 0. Thus at (0,0): f_xx=3, f_yy=3/4, f_xy=0. Then numerator: f_xx f_yy - f_xy^2 = 3*(3/4) - 0 = 9/4. Denominator: (1 + f_x^2 + f_y^2)^2 = (1 + 0 + 0)^2 = 1. Thus Gaussian curvature K = (9/4) / 1 = 9/4. But careful: This is for the graph z = f(x,y). However, Gaussian curvature can be positive or negative. For hyperboloid of two sheets, at the "peak", curvature is positive. So K = 9/4. Then β = |κ| = |K| = 9/4.

Now we have: α = 2 β = 9/4 δ = 32/3 ρ = 2

Compute S = (α β δ ρ)^3. First compute product: α β δ ρ = 2 * (9/4) * (32/3) * 2. Step: 2 * (9/4) = 9/2. 9/2 * (32/3) = (932)/(23) = (288)/(6) = 48. 48 * 2 = 96. So α β δ ρ = 96. Then S = 96^3. Compute 96^3 = (100-4)^3 = 100^3 - 3100^24 + 31004^2 - 4^3 = 1,000,000 - 310,0004 + 310016 - 64 = 1,000,000 - 120,000 + 4,800 - 64 = 880,000 + 4,800 - 64 = 884,800 - 64 = 884,736. Check: 96^2 = 9216, 921696 = 9216100 - 9216*4 = 921,600 - 36,864 = 884,736. Yes. Thus S = 884,736.

Now 𝒦 = 999 S^2 + 7. Compute S^2 = (884,736)^2. That's a large number. We'll compute using Python.

Then 𝒦 = 999 * S^2 + 7.

We need integer 𝒦.

Let's compute step by step with Python. [/THOUGHT] [PYTHON] import math

Given values

alpha = 2 beta = 9/4 # 2.25 delta = 32/3 # approx 10.666... rho = 2

Compute product

product = alpha * beta * delta * rho print("product =", product)

Should be 96

S = product ** 3 print("S =", S)

Should be 884736

Compute S^2

S_sq = S ** 2 print("S^2 =", S_sq)

Compute K = 999 * S^2 + 7

K = 999 * S_sq + 7 print("K =", K)

Output as integer

print("K as integer:", int(K)) [/PYTHON]

[THOUGHT] The Python output gives 𝒦 = 781975031906311. This matches the computed value from the given definitions. Therefore, the final answer is this integer.

[/THOUGHT] \boxed{781975031906311}