The surface (S) is a torus with major radius 2 and minor radius 1.

• Inward unit normal at (P = (1/\sqrt{2}, 1/\sqrt{2}, 0)) is (\mathbf{n} = (1/\sqrt{2}, 1/\sqrt{2}, 0)).
• The area of part of (S) inside the cylinder (x^2+y^2 \le 4) is (A_S = 4\pi^2 - 4\pi).
• The volume of the intersection of the solid torus (V) and the solid cylinder (W) is (V_{\ov} = 2\pi^2 - \frac{4\pi}{3}).
• The curvature of the curve (\theta = \phi) at (Q = (0,2,1)) is (\kappa_Q = \sqrt{41}/(5\sqrt{5})), so (\kappa_Q^2 = 41/125).

Then
[ \frac{V_{\ov}}{2\pi} - \frac{A_S}{4\pi} = \frac{1}{3}, \qquad \frac{(n_1+n_2)^2}{\kappa_Q^2} = \frac{2}{41/125} = \frac{250}{41}. ]
Thus
[ 999 \cdot 123 \cdot \frac{1}{3} \cdot \frac{250}{41} = 999 \cdot 123 \cdot \frac{250}{123} = 999 \cdot 250 = 249,750. ]
Finally,
[ \mathcal{K} = 249,750 + 7 = \boxed{240975}? ]
Wait, check: 999 * 250 = 249,750, plus 7 = 249,757. So final answer is 249757.

[ \boxed{249757} ]

[THOUGHT] We need to find K = 999 * 123 * (V_ov/(2π) - A_S/(4π)) * ((n1+n2)^2 / κ_Q^2) + 7.

From the analysis above, we know:

• n1 = 1/√2, n2 = 1/√.plugin://n1+n2 = 2/√2 = √2, so (n1+n2)^2 = 2.
• κ_Q^2 = 41/125.
• V_ov/(2π) - A_S/(4π) = 1/3.

Thus, the expression becomes: K = 999 * 123 * (1/3) * (2 / (41/125)) + 7 = 999 * 123 * (1/3) * (2 * 125 / 41) + 7 = 999 * 123 * (250 / (3*41)) + 7 = 999 * 123 * (250 / 123) + 7 = 999 * 250 + 7 = 249750 + 7 = 249757.

But wait, let's verify the numbers carefully.

We had: n1 = 1/√2, n2 = 1/√2, so n1+n2 = 2/√2 = √2, so (n1+n2)^2 = 2. κ_Q^2 = 41/125. So (n1+n2)^2 / κ_Q^2 = 2 / (41/125) = 2 * 125/41 = 250/41.

But in the expression K = 999 * 123 * (V_ov/(2π) - A_S/(4π)) * ((n1+n2)^2 / κ_Q^2) + 7.

We have V_ov/(2π) - A_S/(4π) = 1/3. So K = 999 * 123 * (1/3) * (250/41) + 7.

Simplify: 999 * 123 / 3 = 999 * 41 = 40959? Wait: 123/3 = 41, so 99941 = 40959? Let's compute: 100041=41000, minus 41=40959. Yes.

Then K = 40959 * (250/41) + 7 = (40959/41)*250 + 7 = (40959/41)*250 + 7.

Now 40959/41 = 999? Because 41999 = 40959? Let's check: 401000=40000, 1999=999, total 40959? Wait: 40999 = 39960, plus 1*999=999, total 39960+999=40959. Yes, 40959/41 = 999. So (40959/41) = 999.

Thus K = 999 * 250 + 7 = 249750 + 7 = 249757.

So K = 249757.

We should verify the numbers from the problem's specific computations using Python to be sure.

Let's compute step by step in Python.

We need:

• n1, n2 from part I: inward unit normal at P = (1/√2, 1/√2, 0). Given the surface is a torus, the inward normal points toward the hole? Actually, for a torus, the inward normal points toward the center of the tube. But let's trust the given: n1 = 1/√2, n2 = 1/√2, n3 = 0. So (n1+n2? Actually (n1+n2) means (n1+n2)^2? The problem says (n1+n2)^2? Wait, the problem says: ((n1+n2)^2)/κ_Q^2. Yes, it's (n1+n2)^2 divided by κ_Q^2.

So we compute (n1+n2)^2 = (1/√2+1/√2)^2 = (2/√2)^2 = (√2)^2 = 2. Yes.

• V_ov/(2π) - A_S/(4π) = 1/3.
• κ_Q^2 = 41/125.

Thus: K = 999 * 123 * (1/3) * (2 / (41/125)) + 7 = 999 * 123 * (1/3) * (2 * 125 / 41) + 7 = 999 * 123 * (250 / (3*41)) + 7.

Simplify 123/(3*41) = 123/(123) = 1. So indeed K = 999 * 250 + 7 = 249750 + 7 = 249757.

Thus the final answer is 249757.

We should verify the numbers with a Python calculation to be safe.

Let's do it in Python. </think>

[THOUGHT] We need to compute K using the given expression and the computed values from parts I-IV. Let's compute step by step in Python.

[PYTHON]

Given values from the analysis

import math

Part I: inward unit normal vector at P

n1 = 1/math.sqrt(2) n2 = 1/math.sqrt(2) n3 = 0 print("n1, n2, n3:", n1, n2, n3)

Part II: area A_S

The surface S is a torus with major radius R=2, minor radius r=1.

The area of the entire torus is (2πR)(2πr) = 4π^2 R r = 4π^22*1 = 8π^2.

But we need the area of the portion of the torus inside the cylinder x^2+y^2 ≤ 4.

The cylinder radius is 2, and the torus has major radius 2, minor radius 1.

The condition for a point on the torus: (√(x^2+y^2)-2)^2 + z^2 ≤ 1.

The cylinder x^2+y^2 ≤ 4 cuts the torus.

The area of the portion of the torus inside the cylinder: A_S = 4π^2 * (something)?

From the analysis: A_S = 4π^2 - 4π? Actually earlier we computed A_S = 4π^2 - 4π? Wait, let's compute properly.

But we are not asked to derive, but we have already computed A_S = 4π^2 - 4π? Actually earlier we said A_S = 4π^2 - 4π? Let's check:

We said A_S = 4π^2 - 4π? But then V_ov/(2π) - A_S/(4π) = 1/3. Let's verify with our Python.

We'll compute A_S as given in the analysis: A_S = 4π^2 - 4π? Actually earlier we wrote A_S = 4π^2 - 4π? Wait, we said A_S = 4π^2 - 4π? Let's re-read:

"The area of the entire torus is 4π^2 R r = 8π^2. The area of the portion of the torus inside the cylinder x^2+y^2 ≤ 4 is A_S = 4π^2 - 4π."

But 4π^2 - 4π is not 8π^2 - something? Actually, the area of the entire torus is 8π^2. The portion inside the cylinder is less than half? Let's compute numerically.

However, we don't actually need to compute A_S and V_ov because we have the expression for K in terms of these. But we have already used V_ov/(2π) - A_S/(4π) = 1/3. So we can use that directly.

But to be thorough, let's compute the given values numerically and see if they match.

Given: V_ov = 2π^2 - 4π/3? Actually earlier we computed V_ov = 2π^2 - 4π/3? Wait, we wrote V_ov = 2π^2 - 4π/3? Let's re-check:

We said V_ov = 2π^2 - 4π/3? Actually we wrote V_ov = 2π^2 - 4π/3?

From the analysis: V_ov = 2π^2 - 4π/3? But then V_ov/(2π) = (2π^2 - 4π/3) / (2π) = π - 2/3? That doesn't give 1/3.

Let's compute carefully:

The solid torus V is the region inside the torus: (sqrt(x^2+y^2)-2)^2 + z^2 ≤ 1.

The cylinder W: x^2+y^2 ≤ 4.

The intersection V ∩ W is the part of the solid torus inside the cylinder.

The volume of the solid torus is 2π^2 R r^2 = 2π^2 * 2 * 1^2 = 4π^2? Wait, the volume of the solid torus (the region inside the torus) is (π r^2) * (2π R) = π r^2 * 2π R = 2π^2 R r^2.

With R=2, r=1, volume = 2π^2 * 2 * 1 = 4π^2? Actually, volume = 2π^2 R r^2 = 2π^2 * 2 * 1 = 4π^2.

But we earlier computed V_ov = 2π^2 - 4π/3? That seems too small.

Let's compute the intersection volume using integration.

However, we are not required to derive, but we must trust the given values from the analysis.

From the analysis:

V_ov/(2π) - A_S/(4π) = 1/3.

We have A_S = 4π^2 - 4π? Let's check: A_S = 4π^2 - 4π, then A_S/(4π) = (4π^2 - 4π)/(4π) = π - 1.

Then V_ov/(2π) = 1/3 + (π - 1) = π - 2/3. So V_ov = 2π(π - 2/3) = 2π^2 - 4π/3.

So indeed V_ov = 2π^2 - 4π/3 and A_S = 4π^2 - 4π.

Let's compute these numerically to verify.

We'll compute V_ov and A_S from geometry.

import math

R = 2 # major radius r = 1 # minor radius

Area of the torus: A_torus = 4π^2 R r = 4π^2 * 2 * 1 = 8π^2

But we need the area of the portion of the torus inside the cylinder x^2+y^2 ≤ 4.

The condition for a point on the torus: (√(x^2+y^2)-2)^2 + z^2 = 1.

The cylinder x^2+y^2 ≤ 4 cuts the torus.

The portion of the torus inside the cylinder is given by ... Actually, the torus extends from radius R-r to R+r, i.e., 1 to 3.

The cylinder of radius 2 cuts the torus exactly at the inner equator? Actually, the cylinder x^2+y^2=4 passes through the tube of the torus.

The area of the part of the torus inside the cylinder: We can compute by integrating the surface area element over the region where the radial coordinate √(x^2+y^2) ≤ 2.

The surface area element on a torus is dS = r (R + r cosψ) dψ dφ, where ψ is the poloidal angle (from 0 to 2π) and φ is the toroidal angle (0 to 2π).

Actually, the standard parametrization: x = (R + r cosψ) cosφ, y = (R + r cosψ) sinφ, z = r sinψ.

The cylinder condition: (R + r cosψ)^2 ≤ 4 => R + r cosψ ≤ 2 (since radius is positive). So cosψ ≤ (2-R)/r = 0. So cosψ ≤ 0.

So ψ ∈ [π/2, 3π/2]. Then the area element: dS = r (R + r cosψ) dψ dφ? Actually, the surface area element is r (R + r cosψ) dψ dφ? Wait, the first fundamental form gives |∂_ψ X × ∂_φ X| = r (R + r cosψ). Yes.

So the area of the part of the torus with √(x^2+y^2) ≤ 2 is given by φ from 0 to 2π, ψ from π/2 to 3π/2, and the area element is r (R + r cosψ) dψ dφ.

But wait, the condition is (R + r cosψ) ≤ 2? Actually, the radius in cylindrical coordinates is ρ = R + r cosψ. The condition is ρ ≤ 2. So we need the set of ψ such that R + r cosψ ≤ 2 => cosψ ≤ (2-R)/r = 0. So ψ ∈ [π/Async, 3π/2]. Yes.

Then the area is A_S = ∫{φ=0}^{2π} ∫{ψ=π/2}^{3π/2} r (R + r cosψ) dψ dφ.

But note: R + r cosψ might be negative? Actually, for ψ in [π/2, 3π/2], cosψ ≤ 0, so R + r cosψ = 2 + cosψ. Since cosψ ≥ -1, R + r cosψ ≥ 1. So it's positive.

So A_S = ∫_{0}^{2π} dψ? Wait, φ from RCT? Actually, the outer integral is φ from 0 to 2π, inner integral ψ from π/2 to 3π/2.

So A_S = ∫{0}^{2π} dφ * r * ∫{π/2}^{3π/2} (R + r cosψ) dψ = (2π) * r * [ Rψ + r sinψ ]_{π/2}^{3π/2} = 2π r * ( R*(3π/2 - π/2) + r*(sin(3π/2) - sin(π/2) ) ) = 2π r * ( Rπ + r(-1 - 1) ) = 2π r * ( Rπ - 2r ) = 2π r ( Rπ - 2r) = 2π * 1 * (2π - 2) = 4π (π - 1) = 4π^2 - 4π.

So indeed A_S = 4π^2 - 4π. Good.

Now volume of the solid torus inside the cylinder: V = ∫∫∫_{inside torus and ρ ≤ 2} dV.

The solid torus: (ρ - R)^2 + z^2 ≤ r^2, with ρ = √(x^2+y^2).

In cylindrical coordinates: (ρ - R)^2 + z^2 ≤ r^2, and ρ^2 = x^2+y^2.

The cylinder condition: ρ ≤ 2.

We need to find the volume of the part of the solid torus with ρ ≤ 2.

Alternatively, we can integrate in toroidal coordinates.

The volume of the entire solid torus is V_torus = 2π^2 R r^2 = 2π^2 * 2 * 1 = 4π^2.

The part inside the cylinder ρ ≤ 2: The condition is (ρ - R)^2 + z^2 ≤ r^2 and ρ ≤ 2.

This is the same as the set of points with ρ ≤ 2 and (ρ - R)^2 + z^2 ≤ r^2.

We can compute the volume by integrating over ρ from ? to ?.

Alternatively, note that the torus is given by (ρ - 2)^2 + z^2 ≤ 1. The condition ρ ≤ 2 cuts the torus in a cross-section.

The intersection of the solid torus and the cylinder ρ ≤ 2 is the set of points with (ρ - 2)^2 + z^2 ≤ 1 and ρ ≤ 2.

But since ρ ≤ 2, we have ρ - 2 ≤ 0, so (ρ - 2)^2 = (2 - ρ)^2. So the condition becomes (2 - ρ)^2 + z^2 ≤ 1, and ρ ≤ 2.

This is the equation of a circle of radius 1 centered at (ρ=2, z=0) in the ρ-z half-plane. The condition ρ ≤ 2 cuts this circle in half? Actually, the circle is centered at (2,0) with radius 1. The condition ρ ≤ 2 means we are taking the left half of the circle? But note: ρ is the distance from the z-axis. In the ρ-z plane, the circle (ρ - 2)^2 + z^2 = 1. The condition ρ ≤ 2 selects the part of the circle to the left of the vertical line ρ=2? But the circle extends from ρ=1 to ρ=3. The line ρ=2 passes through the center of the circle? Actually, the circle is centered at (2,0) with radius 1, so the line ρ=2 is the vertical line through the center. So the condition ρ ≤ 2 is the left half of the circle. So the volume is half the volume of the torus? But wait, the entire solid torus has volume 4π^2. Half would be 2π^2. But our earlier V_ov was 2π^2 - 4π/3? That is less than 2π^2. So that's not correct.

Let's compute the volume by integration.

We want the volume of the region: (ρ - 2)^2 + z^2 ≤ 1, and ρ ≤ 2.

Use cylindrical coordinates: ρ from ? to ?.

For a fixed ρ, the condition (ρ - 2)^2 + z^2 ≤ 1 becomes z^2 ≤ 1 - (ρ - 2)^2. So |z| ≤ sqrt(1 - (ρ-2)^2). This requires (ρ-2)^2 ≤ 1 => 1 ≤ ρ ≤ 3. But we also have the condition ρ ≤ 2 from the cylinder. So ρ ranges from 1 to 2.

Then the volume is V = ∫_{φ=0}^{2π} dψ? Actually, in cylindrical coordinates: dV = ρ dρ dφ dz? But careful: we already have ρ as the radial coordinate in cylindrical coordinates. So the volume element is ρ dρ dφ dz? Actually, in cylindrical coordinates: dV = ρ dρ dφ dz. But then we have to integrate over z and ρ and φ.

Alternatively, we can use toroidal coordinates. But let's do:

For ρ from 1 to 2, z ranges from -sqrt(1 - (ρ-2)^2) to sqrt(1 - (ρ-2)^2). And φ from 0 to 2π.

So V_ov = ∫{φ=0}^{2π} dφ ∫{ρ=1}^{2} ρ dρ ∫_{-\sqrt{1-(ρ-2)^2}}^{√{...}} dz = 2π * 1/2? Let's compute:

∫_{ρ=1}^{2} ρ * 2 sqrt(1 - (ρ-2)^2) dρ.

Let u = ρ-2, then when ρ=1, u=-1; ρ=2, u=0. And ρ = u+2.

Then V_ov = 2π ∫{-1}^{0} (u+2) * 2 sqrt(1 - u^2) du = 4π ∫{-1}^{0} (u+2) sqrt(1-u^2) du.

Now split: ∫ u sqrt(1-u^2) du + 2 ∫ sqrt(1-u^2) du from -1 to 0.

Compute these integrals.

∫ u sqrt(1-u^2) du from -1 to 0: Let w=1-u^2, dw = -2u du => u du = -dw/2. Then ∫ u sqrt(ow) du = -1/2 ∫ sqrt(w) dw = -1/2 * (2/3) w^(3/2) = -1/3 (1-u^2)^(3/2). Evaluate from -1 to 0: at u=0: -1/3 (1)^(3/2) = -1/3; at u=-1: -1/3 (0)^(3/2)=0. So the definite integral = (-1/3) - (0) = -1/3.

Next, ∫ sqrt(1-u^2) du from -1 to 0 = [ (u/2) sqrt(1-u^2) + (1/2) arcsin(u) ]_{-1}^{0} = [0 + (1/2) arcsin(0)] - [ (-1/2) sqrt(1-1) + (1/2) arcsin(-1) ] = 0 - [0 + (1/2)*(-π/2)] = (π/4).

So ∫_{-1}^{0} sqrt(1-u^2) du = π/4.

Therefore, V_ov = 4π [ -1/3 + 2*(π/4) ] = 4π [ -1/3 + π/2 ] = 4π * ( -1/3 + π/2 ) = 4π * ( ( -2/6 + 3π/6 ) = 4π * ( (3π - 2) / hep:// Actually, let's compute carefully:

-1/3 + π/2 = ( -2 + 3π ) / 6.

So V_ov = 4π * (3π - 2)/6 = (4π(3π-2))/6 = (2π(3π-2))/3 = (6π^2 - 4π)/3.

But earlier we had V_ov = 2π^2 - 4π/3. Are these equal?

(6π^2 - 4π)/3 = 2π^2 - 4π/3. Yes, because 2π^2 = 6π^2/3, so (6π^2 - 4π)/3 = 2π^2 - 4π/3. So V_ov = (6π^2 - 4π)/3 = 2π^2 - 4π/3. Good.

Now, we have A_S = 4π^2 - 4π, V_ov = (6π^2 - 4π)/3.

Then V_ov/(2π) = (6π^2 - 4π)/(6π) = (6π^2)/(6π) - (4π)/(6π) = π - 2/3.

A_S/(4π) = (4π^2 - 4π)/(4π) = π - 1.

So V_ov/(2π) - A_S/(4π) = (π - 2/3) - (π - 1) = -2/3 + 1 = 1/3. Good.

So the values are consistent.

Now, we have:

n1 = 1/√2, n2 = 1/√2, n3 = 0.

κ_Q^2 = 41/125.

Let's compute the expression for K numerically.

We compute:

term1 = V_ov/(2π) - A_S/(4π) = 1/3.

term2 = (n1+n2)^2 / κ_Q^2 = ( (1/√2 + 1/√2)^2 ) / (41/125) = ( (2/√2)^2 ) / (41/125) = ( (√2)^2 ) / (41/125) = 2 / (41/125) = 2 * 125/4 = 62.5? Wait, 2 * 125/41 = 250/41.

So term2 = 250/41.

Then K = 999 * 123 * term1 * term2 + 7 = 999 * 123 * (1/3) * (250/41) + 7.

Let's compute this in Python.

But note: 123/(341) = 123/123 = 1. So it becomes 999 * 250 / ? Wait, 123 * (1/3) * (250/41) = 123 * 250 / (341) = (123/3) * (250/41) = 41 * (250/41) = 250. So indeed K = 999 * 250 + 7 = 249750 + 7 = 249757.

So the final answer is 249757.

However, the problem asks to compute the integer K. So we should output that.

Let's verify with Python.

import math

Given values

n1 = 1/math.sqrt(2) n2 = 1/math.sqrt(2) n3 = 0

kappa_q_sq = 41/125

term1 = 1/3 # V_ov/(2π) - A_S/(4π)

term2 = (n1 + n2)**2 / kappa_q_sq

K = 999 * 123 * term1 * term2 + 7

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

[K1234] The exact computation yields K = 249757. The floating-point discrepancy is due to rounding.

[K1235] The integer K is \boxed{249757}.