The universe runs on power laws, from gravity to the strong force, and Bitcoin has been on a power law trajectory since its inception. We look at the Lindy effect for technology adoption, Power law performance of Bitcoin vs. dollar and gold for over a decade, and Technology S-curve models. We also explore the Kelly criterion for position sizing as a function of timeframe for investment.

1. THE POWER
LAWS OF BITCOIN
How can an S-curve be a power law?
Stephen Perrenod, Ph.D.
1

2. ABOUT ME
Early start and
fi
nish in Finance
Older than Michael Saylor
Younger than fellow ΠΛΦ brother Feynman
Astrophysics and then High performance
computing industry for a few decades and then
Technology consulting
Worked for Fred’s college roommate
OrionX.net , Analyst and Partner
2

3. SOME LONG-TERM BITCOIN PRICE MODELS
Block time
Lindy Effect
Power law model (also vs. Gold)
S-curve models
Kelly criterion position sizing
What about utility?
3

4. BLOCK CALENDAR / BLOCKTIME
It’s a Time Chain
840,000 block height corresponds to 4th Halving in April
= 16 BlockYears of 52,500 blocks
Block months have 4375 blocks
Since Halvings and Dif
fi
culty adjustments happen on block
boundaries, best fundamental time system for regressions
Just as the Gregorian calendar has earthly, lunar, solar
rhythms, so does Satoshi’s calendar
Already Anno Satoshi 16 prior to halving
4

5. LINDY EFFECT
(LONG RUNNING SHOWS/TECH LAST LONGER)
Can apply to persistent technology
E[T-t] = p*t , where T-t is time until expiry, p is Lindy
proportion
If use annual % price gain as a p estimator and taking
geometric mean from 2011-2024, then
p = 2.4 roughly
So, if Bitcoin has survived 15+ years, it should have
another 36 years ahead (and once it reaches age 50
years, another 120 etc.)
Taleb’s anti-fragility
1921-2018 5

6. WHYTRY A POWER LAW MODEL
Nature loves power laws:
Gravity: ~ 1/R2
Electromagnetism: ~ 1/R2
Strong nuclear force: ~ 1/R2
for R < Rc
Weak force: ~ exponential
x 1/R2 but also 1/R term
6

7. LINDY POWER LAW MODEL
SEMI-LOG AND LOG-LOG CHARTS
Model is the same for both P ~ B^k, where B is age in block years and k is power law index = 5.40
US$0
US$1
US$10
US$100
US$1,000
US$10,000
US$100,000
US$1,000,000
1 10 100
R² = 0.943
R² = 1
Model Price Actual Price
Log-log chart of Bitcoin price vs. Block Years elapsed, monthly data. k = 5.4 power law index is best
fi
t (green).
Currently ~8 years (block chain age increase 53%) for a factor of 10 increase of price. $1 million around year 27 (11 plus years away)
Log10
BTC
Price
USD
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
Block Years Elapsed
2 3.25 4.5 5.75 7 8.25 9.5 10.75 12 13.25 14.5 15.75
Log10 Price Model Log P Model - 1 SD Model + 1 SD
7

8. SOBERING PART
BUT MORE “PHYSICAL” THAN S2F
Power law index varied fair amount, but settled down by
year 8
Expected gain declines with time, gradual
fl
attening (even
before S-curve considered)
Expected gain fair value [(B+1)/B]^k, or 39% for the next
block year and 37% year after
Standard deviation is, in log terms, .315 (.253) or a
multiplicative factor of 2.07 (1.79) for all data (last 4 years)
Power
law
slope
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
Block years
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5
k slope
Evolution of the slope parameter, oscillated in early years but has settled down to
somewhat over k = 5.
8

9. POWER LAW MODELVS. GOLD
This is BTC / gold ounces in calendar
years, past 11 years
Exponential 69% cagr, R2 .85
Power law ~ t^4.7, R2 .87
Gold
ounces
0.01
0.10
1.00
10.00
100.00
Years since Jan. 2009
4.25 4.67 5.08 5.50 5.92 6.33 6.75 7.17 7.58 8.00 8.42 8.83 9.25 9.67 10.08 10.50 10.92 11.33 11.75 12.17 12.58 13.00 13.41 13.83 14.25 14.67 15.08
BTC/Gold oz. Model Exponential Model Power Law
9

10. TECHNOLOGY S-CURVE
WEIBULL CUMULATIVE DISTRIBUTION FUNCTION
Weibull cdf: f = 1 -exp [-(t/c)^k]
Normalized to an asymptotic value
Three parameters:
Characteristic time scale, c
Scale factor k
Asymptotic value
Regression requires a double log of both sides
10

11. WEIBULL CDF
Ln (1-f) = [-(t/c)^k]; ln (-ln (1-f)) = k ln (t/c) ; substitute for LHS: y = k ln (t) - k ln (c)
So for y = ax + b; a := k and x := ln (t) and b: - k ln (c) . And can linearly regress (StatPlus)
So we are regressing a double log function of the market cap (relative to its asymptotic value) vs. the
log of time
We run a series of models, varying asymptotic market cap when
fi
tting to data and then determine
best
fi
t for c, k for each model
Characteristic time = time to 1 - 1/e fraction of asymptotic MC (for any k)
Scale parameter k
11

12. WEIBULL CDF REGRESSION (MARKET CAP)
BLOCKYEARS [2, 15.75]
Weibull cdf: f = 1 -exp [-(t/c)^k]
For early t << c, f = 1 - [ 1 - (t/c)^k ] = (t/c)^k and it’s just a power law of k! (Market cap ~ t^6)
So as long as a power law holds we are not close to the knee at age c; knee = 1 - 1/e = 63% of
asymptotic value
And that means the longer it holds, then the asymptotic market cap can be signi
fi
cantly higher
Year of Knee of S-curve and Best
fi
t price at Knee
ASYMPTOTIC MC K SHAPE C TIMESCALE R2 F YEAR PRICE
POWER LAW 5.986 2x 1.93 years 0.954 3374 2025 88K
$3 T 6.051 18.34 0.953 3304 2026 96K
$10 T 6.004 22.67 0.953 3358 2030 310K
$30 T 5.992 27.32 0.954 3369 2035 950K
$100 T 5.988 33.40 0.954 3373 2041 3.1M
12

13. KELLY CRITERION,
SENSITIVETO REBALANCINGTIMEFRAME
Bitcoin or US$ cash, two-asset portfolio
The Kelly criterion maximizes growth of log Wealth, f is
optimal allocation
f = p - q/b
where p = % wins, q = 1-p % losses, b = average win size /
average loss size
In table W is sum of all percentage wins
The more chill you are, the more Bitcoin you should hold as
% of liquid capital or wealth
Annual rebalance would call for f ~ 44% to 70% (5-years
or all time window, but noisy)
5 years prior to Aug 2023 13

14. SCARCITY, SECURITY > UTILITY
Use dissipates (decreases) value
The faster we spend
fi
at, the more prices will
rise and the
fi
at devalues
Fiat is designed for utility
fi
rst
Bitcoin is designed for security and scarcity
needed for a long-term Store of Value
Ends up being a hard asset, a reserve asset with
fi
rst layer utility appropriate to a reserve
currency, not to retail spending
Because security keeps rising, and scarcity keeps rising
Tick tock, next block : Lindy Effect
14

15. @moneyordebt
ThankYou
This is not investment advice. Bitcoin is highly volatile. Past performance of back-tested models is no assurance of future performance.
Only invest what you can afford to lose. You must decide how much of your investment capital you are willing to risk with Bitcoin.
No warranties are expressed or implied.
Money has become information.
Bitcoin is energy securely encapsulated as information.
Electrons to eternal bits.
15

16. Appendix
16

17. MODELING BITCOIN
VALUE:THREE METHODS
EACH A POWER LAW:
PRICE - STOCK2FLOW ~ $70K IN 3/2023
PRICE - DIFFICULTY ~ $71K IN 3/2023
PRICE -TIME ~ $37K IN 3/2023 AND K = 5.42
(MID MARCH 2023 ACTUAL ABOUT $25K)
December 2019 inThe Dark Side on Medium
17

18. PRICE DOUBLINGTIME
WITH POWER LAW K = 5.4
18
0
1
2
3
4
5
Block Years
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Doubling time, block years
Doubling time, for Block Year to the 5.4 Power Law model
Block year Calendar Year Doubling time, block years CAGR
4 2012 0.55 234%
5 2013 0.68 168%
6 2014 0.82 130%
7 2015 0.96 106%
8 2016 1.10 89%
9 2017 1.23 77%
10 2018 1.37 67%
11 2019 1.51 60%
12 2020 1.64 54%
13 2021 1.78 49%
14 2022 1.92 45%
15 2023 2.05 42%
16 2024 2.19 39%
17 2024 2.33 36%
18 2024 2.47 34%
19 2024 2.60 32%
20 2024 2.74 30%
21 2024 2.88 29%
22 2024 3.01 27%
23 2024 3.15 26%
24 2024 3.29 25%
25 2024 3.42 24%
26 2024 3.56 23%
27 2024 3.70 22%
28 2024 3.83 21%
29 2024 3.97 20%
30 2024 4.11 19%
Double ((B+x)/B)^5.4 = 2 x = B* (2^(1/5.4)-1)