Has this $100,000 by the end of 2021 sacrosanct?
Hell no!
The $ values seem to be off though...
Log price usd is a linear function of the square root of time.
The vertex of the parabola in the first chart of the OP is located at September 8, 2010 on the horizontal axis and -1.64982665 on the vertical axis (in dollars, that's 10^-1.64883 = $0.0224). The subsequent charts use the same data and the same quadratic equation, except that the arc length graph uses a single set of weekly weighted values instead of the double chart with weekly highs and weekly lows.
The dollar scale of the arc length chart doesn't give you the same spacing as a simple log chart. On the vertical dollar axis, you're looking at the logarithm of a square root.
So the dollar values are "off" your expectation of what a log chart should look like. But they're mathematically correct if the spreadsheet and my computer are making correct calculations.
Another interesting implication of this model -- if bitcoin continues to color between the lines like the talented kindergartner it is (not saying it will, but for argument's sake), then future volatility will decline. It's a result of the dollar values being "off" because of the square root function. That's how this model works. If you don't think that bitcoin's volatility will decline, then you should reject the model. I find it reassuring.
I would welcome some mathematical proofreading. This is the equation for arc length (p is price):
(228.868256*(LOG(p)+1.64982665)*SQRT(52380.6786*(LOG(p)+1.64982665)^2+1)+LN(228.868256*(LOG(p)+1.64982665)+SQRT(52380.6786*(LOG(p)+1.64982665)^2+1)))/457.736512
We good?
