Assume that

- Population is bounded from above
- The flow of immigrants is unbounded from above
- The survival probabilities of the genes of each immigrant are equal.

Then

For any given cohort of immigrants at time t, the survival probability of their genes at T > t, p(t,T) must go to zero as T goes to infinity.

Proof

Let N(t,T) be the flow from t to T.

The expected number of genes that exist at some date T is the sum of p(t’,T) N(t’) where t’ is an entering cohort and N(t’) entered at time t’ and have a survival probability p(t’,T) at T.

The sum of the N(t’) from t to T is N(t,T).

If p(t’,T) was bounded from below by epsilon, then we would have

N(t,T) epsilon

as a lower bound to the expected number of genes for the entire flow from t to T. Since N(t,T) grows without bound, so does its product with epsilon greater than zero where epsilon is fixed.

Thus the expected number of genes,

sum over t’ of the N(t’) p(t’,T) > N(t,T) epsilon

But we assumed there existed some upper bound B to population. Thus the expected number of genes will exceed the bound on them B as T grows larger.

So we have a contradiction. Thus there is no lower bound epsilon greater than zero for the survival probability of the immigrants.

So every immigrant gene that enters at time t eventually goes extinct.

QED.

Assume that for some positive k, the survival probability of those here already is bounded from above by k times the immigrant survival probability.

Then the survival probability of those here must also vanish, i.e. is not bounded below as T goes to infinity for q(t,T) where q is the survival probability for those here.

Proof

Since p(t,T) the immigrant survival probability falls below any epsilon1 for T sufficiently great, k times p(t,T) also falls below any epsilon2. Take T sufficiently great that p(t,T) falls below epsilon2/k. Then k p(t,T) is now less than epsilon. Since q(t,T) < k p(t,T), it follows that q(t,T) < epsilon. Thus q(t,T) vanishes as T grows larger.

What happens is that q(t,T) is between k p(t,T) and 0, q is squeezed between a vanishing quantity, k p(t,T), and zero, so q vanishes as well.

QED

Thus sustained immigration under these assumptions implies extinction of each year’s cohort that comes here as well as everyone here at any point in time.

Note that its only necessary to have one immigrant group whose numbers entering are unbounded and whose survival probability times some positive value is an upper bound to the rest for the theorem to apply to all those who enter and to all those here.

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Thus the Bush Kennedy Kyl McCain amnesty bill with its guest worker provision and its annual flow that is bounded from below above zero implies genetic extinction of all those who come here and all those who are here.

So does existing law.

Any law that does not require that annual immigration vanish, i.e. approach closer to zero than any positive bound, implies that the survival probabilities of those who come here and those here all go to zero, i.e. complete genetic extinction.

The causal mechanism by which the law operates is the substitution of immigrants for births. When population reaches a maximum, immigrants must substitute for births or it wouldn’t be a maximum.

This drives the fertility rate below replacement.

This can happen quite quickly.

Assume US population at 300 million was the maximum. If people live 75 years, then 4 million die per year. If 2 million enter then births = 4million deaths – 2 million entrants = 2 million.

The ratio of births to deaths is 2/4 or 1/2. The time from birth to parent is roughly 25 years. So in 50 years, one has 1/4, and in 75 years 1/8 of the starting genes.

Even if population went to 450 million, deaths per year are 6 million. With even one million entrants that gives a survival ratio of 5/6. So the number left after 25*n years is (5/6)^n which goes to zero as n goes to infinity.

It goes to zero rapidly in fact.

The above implies that any law with immigration above zero on a sustained basis is unconstitutional and a crime against humanity. Causing the extinction of a group is a violation of treaties the US has passed.

The current US law is thus void. So is the proposed law.

The drop in fertility from 1800 to 1990 in one graph shows this substitution effect pressure from immigration.

Look at the Karen Stevenson graph of fertility from 1800 to 1990 below:

http://www.elderweb.com/home/node/2919

Fertility falls except during the period of immigration restriction from the 1920’s to 1965. During part of that period fertility rose, which is called the baby boom. This was a departure from the uniform fall in fertility.

Fertility is now below replacement for many groups in accordance with the theorem.

The same applies in Europe where it also violates EU law as well as international law.

See also

1965 Immigration Act Causes U inverted U in Income Inequality and Fertility

Blogs for immigration restriction even have names like those of the theorem, e.g. Vanishing American:

http://vanishingamerican.blogspot.com/

June 14 to 16 all across America is March for America. Even if you can’t march, there are ways to participate.

http://www.lframerica.com/march2.html

See also Lawrence Auster on it:

http://www.amnation.com/vfr/archives/007957.html