"Here's a little lemma"

Who said that? Bonus points for any reader who knows... While some of the people I respect greatly have reviewed the demonstration which follows and called it: "an ontological proof of America's decline" or a "didactic refutation of modern America", I make no such claim. I'm merely allowing the reader's own intuition to be my guide. Following the path of those who hold such truths as "self-evident" and steering clear of any controversial waters, here it is:

The rules to this are simple, just use your basic civic knowledge and political intuition to map these points and terms in your own mind's eye.

Now, let an x-axis be drawn and the left end of the axis place the term "Monarchy". At the right end place the term "Direct democracy". I assume this makes intuitive sense to you. It certainly made some sense to our founders, and to the Revolution they inspired. See figure 2-1.
Next, allow your intuition to place the term "Aristocracy" somewhere on this line. Would it be closer to the monarchy or direct democracy end? Most people I talk to place it a little left of center, towards monarchy. Now place the term "Republic". A little right of center? See if fig 2-2 is close. On average, this is where most people who grew up in a democratic republic tend to place these dots.
Now, let us invert this x-axis and make it a y-axis. Keep the direct democracy end at the top, and the monarchy end towards the bottom. Extend the axis through the monarchy point and draw an arrow downwards. Underneath the arrow place the proportion 1 over infinity. At the top, beside the term direct democracy place the proportion 1 over 1. Again, this should hopefully make some intuitive sense. The most direct of democracies, or any government, is that which a man has over his own self (1:1), and the most indirect of governments is that which God has over the universe (1:infinity). For the sake of a relaxed cognition, this is just figure 2-1 on its end, with some proportions written in! See figure 2-3 below.
And again, if place those dots in relatively the same position as they were before, you would have figure 2-4 below.
Now, I am going to place a dot representing the proportion 1 over 50,000. This is approximately the proportion of representatives to counted citizens (ex 2/5 slaves, Indians not taxed, etc.) according to the 1800 census numbers. In other words, in the Congress of 1800, congressional districts were about 50,000 strong. I am going to place the dot in between the "direct democracy" proportion and the dot for "republic". This is an arbitrary placement, we can put it anywhere you wish, but in general, my own feeling is that that back in those days there was a little "more democracy" and a little "less republic". Again though, you can place it anywhere you like.

Now that I have my 1800 1:50,000 ratio placed, I am going to extend an x-axis again down at the bottom of my line. I am going to call this axis "time" and represent it with an arrow at its right most extreme. I am then going to extend two dotted parallel lines across the graph from the points "republic" and "aristocracy". This merely assumes that the definitions have held more or less constant since the founding. I am then going to move out along the x-axis and place another dot, representing today's current congressional district size 1 over 660,000. This dot will be substantially below the 1:50,000 dot we placed earlier.

Finally, let's connect the two dots with a dotted curved line, representing America's history of congressional district size. You will then have figure 2-5, and you will see a rather alarming trend.
Of course, it matters entirely where you place these dots, but the trend is still there no matter how far up or how far down you start. Is "republic" a simple point on a map? Does it encompass a range of options? Of course, but how far can we move on our present course before a qualitative change takes place? How many more people can we add to congressional districts until it simply makes no sense to call us a republic any longer and we should use a term like aristocracy or some other form?

Now if we superimpose a chart of US population growth, we can begin to see an answer to that question. Not for long. As our population continues to increase, and as our Congress continues to refuse to perform its Constitutional duty, the very essence of the Republic will fade from contemporary existence. See figure 2-6.