Enough

 

Too Marvelous For Words

You're just too marvelous, too marvelous for words
Like glorious, glamorous and that old standby amorous
It's all too wonderful, I'll never find the words
That say enough, tell enough, I mean they just aren't swell enough

How much is enough?

If there is a single absolute, then it is everywhere and is without error. A mathematician would say there is nowhere, 0, that the absolute is not, and the standard error of that absolute is 0. Taken together this would be 0±0. While this seems to be similar to the statement μ±σ, this does NOT mean that the mean, μ, of the absolute is 0 and the variance, σ² of the absolute is 0.

Instead, if the absolute is taken to be infinite series of repeating triangular waves, completely filling the range in which the absolute is defined, then the mean can be considered to be the phase of each wave, and the fact that opposing phases interfere with each other, 0=μ-μ, then this is true for every value of μ and not just μ=0. Similarly the standard error at a point n on the wave has a limit as n approaches infinity of zero, according to n→∞, σ/√n = 0. This is true for every value of σ, not just σ=0.

A group of n individuals can also operate as a single wave. To be consistent with the absolute, that single wave, the derivative of that wave, and the integral of that wave should have an amplitude related to the absolute. The amplitude, A, of  any wave is related to the variance of that wave according to the formula ½A²=σ². If the individuals are viewing the absolute from a hyperbolic surface, this is consistent with the individuals following a logistics, hyperbolic secant squared, distribution. From that hyperbolic surface, the absolute, an infinite series of triangle waves, would appear to have a variance of π/6, its derivative would have a variance of half of that, π/12, and its integral would have a variance of twice of that, π/3. While the period would vary accordingly, the mean/median/phase will repeat and can considered to be π/2 for the wave, its  derivative, and its integral.

However when viewed from a hyperbolic surface, the variance would appear to be 5/6 of the absolute. This makes the range variable, s, of the logistics distribution appear to be ½. From a hyperbolic surface the logistics distribution repeats but only on other imaginary surfaces. This means from the perspective of a real surface where the coefficient of the imaginary dimension is 0, the distribution will not appear to repeat. Also the logistics distribution is only consistent with values on the one of the sheets of the hyperbola ( e.g. only the positive portion of the wave.), the sheet which is being observed.

The limitation of 5/6 of the variance of the absolute is also referred to a Nash Equilibrium. When viewed from the perspective of the individual, it should be limited to 5/6 of the value of the absolute. Thus an individual’s contributions to the group should be on average 1/6, 16.7%, of the total, while each individual should try for no more than 83.3% of the total. If any individual tries to achieve 100% of the absolute, other individuals can also try to achieve 100% of the absolute and they will block each other. If a user also assumes that the median is also zero, this is a User Optimal and is not a stable equilibrium for a group. If an individual assumes that that the median is π/2 but the variance is that of the absolute then this is a System Optimal and it is also not a stable equilibrium for the group. Only the Nash Equilibrium, a mean/median of half of the absolute and a variance of 5/6 of the absolute, is stable for individuals forming a group. So if individuals set a goal of 5/6 of the variance of the absolute, what happens to the other 1/6?

It is suggested that 5/6, 83.3%, is why many building and engineering codes are set as between 80% and 85% of the maximum, and that taxation of individuals for the group should on average not exceed 16.7% of the Gross Domestic Product. (NOTE this does NOT mean that the effective rate for all individuals can not be higher than 16.7%, just than on average across all individuals in a group it should not exceed this value).  The current tax code appears to confuse marginal with effective rates, but has effective rates on average from the group close to this amount, although its median seems to be $0, not 50% of the absolute total.  A previous blog post proposed a Nash Equilibrium of tax brackets, https://dbeagan.blogspot.com/2024/08/laffer-curve.html, although the brackets cited in that post should obviously increase as the absolute total of the economy also increases. So in answer to what is enough, 5/6 of the total is enough.