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Build the fence and stop the soft slavery of illegal immigrant workers

The estimates on the numbers are that there's between 11m and 40m illegal aliens in the US, given that the population is 316m this gives us a range of 3.5% to 12.7% of the population being illegal aliens — how much would it need to be considered an invasion? Moreover, how much is the "they will overwhelm us" point? 20%? (That's 1 in 5.) 25%? (That's 1 in 4.)

Simply put, a fence is inadequate; here's my proposal from another thread:

If no additional illegals come into the country, that would stop the flow, which is the real problem.
Wikipedia on the US/Mexico border:
The border's total length is 1,954 mi, according to figures given by the International Boundary and Water Commission. It is the most frequently crossed international border in the world, with approximately 350 million legal crossings being made annually.
[…]
There are an estimated half a million illegal entries into the United States each year.

Wikipedia on the Paladin:
Crew:                     6
Effective firing range:  11 mi
Operational range:      216 mi
So to cover the border we'd need 1,954 / 22 ≈ 89 Paladins… but that's an under-count because that would entail butting them directly on the border. We would actually want them back a bit.
c² = b² + a²
11²  = 5.5² + a²
121 = 30.25 + a²
90.75 = a²
9.5 ≈ a
So, we could put them 5.5 mi away from the border, allowing them to cover 19 mi of border; this would mean that we'd need 1,954 / 19 ≈ 103 paladins [103 * 6 = 618 men].
Or, we could put them 9.5 mi away from the border, allowing them to cover 11 mi of border; this would mean that we'd need 1,954 / 11 ≈ 178 paladins [178 * 6 = 1,068 men].
(Of course we'd need logistics to support them: meals, munitions, mail, etc.)
See, securing the border is not as hard as they make it out to be.

That's the proper response to invasion.

Jeepers.
You’re talking about picking off non-combatant border crossers.

1. If it truly is an invasion, the proper response by the US military would be to go on the offensive and set-up a buffer zone, and not take up a defensive position. But even that is extreme.

2. Border crossings account for ~40% of illegal immigration. The rest comes from expired visas. Your “solution” doesn’t address that at all.