My simple guide to Euler conditions

Let us design an oval race track. But we change the construction of our race cars.

First we design a race car that can take fractional jumps around the track, 1/5,2/6,...  So, I stretch the race track so it takes the five jumps and finish equals start.

Second, let us say we have our race track, and it has no rational solution.  So, my cars take the best one fifth jump they can, and land within a range of points around the track.  The pir hold the error term and the car can re-use it for the next jump. It maintains bounded error.

Third, I deign a car that can execute a recursive multiply algorithm, and this algorithm will always converge to smaller error faster than a car can jump.  Now I am Euler. I can assume my agents all have access to the recursive algorithm, they never use it to infinity, but we always know there is a finite number of executions that computes toward a solution much faster than cars jumping.

In the last case, I can guarantee that all my Taylor series converge consistently.  In our basket brigade model, the Euler conditions means we always have sufficient rank, and reduced bit error, our generators to get the exact match everywhere.    We are dense, we can use Newton's grammar.  This condition is never met, basket brigade is model number two.

The crypto folks use model number one, but can also use model two if backed up by counterfeit efforts.  In other words, a smart cash card does not have to indicate its exact  spending history, just something close, and not necessarily close enough. It is still verified because the counterfeit cost of defeating close enough is high. Including built in cash limits and timeouts, along with counterfeit efforts, means the load on developing exact crypto histories is greatly reduced. Most pits need only verify an absolute measure of risk. 

For example, use a few duplicate keys among the cards, consumer cards with a $200 limit and one year renewal? Why not, risk is low. We are still backed by honest accounting, as well as any other card.

Euler and Quantization

he recursive multiply means we can get the exact ratio of a can of beans, purchased when we need it.  This is the economic version of multiply, subdivision, and we are limited to a container algebra and price with round off  There are a couple of views, all equivalent. Euler is like the checkout counters have zero transaction interval, and the number needed expands or shrink with unlimited precision.conservation.  The pits assume the one, negligible transaction intervals, transaction costs zero; really we let that fall into bit error. But we assume a mostly fixed number of checkout counters.