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BITCOIN LAYER:AUGUST 11, 2025
Stephen Perrenod @moneyordebt
https://stephenperrenod.substack.com/
Power Laws of Bitcoin:The Core and the Bubbles
Virgo Cluster, credit: DOE, NSF, Vera C. Rubin Observatory
Three power laws operating in this image 1
AGENDA
Energy + cryptography = Money 3.0
Power law, not exponential
Zettascale imparts value
Persistence of power law
Core vs. Bubble
EMH ?, Hurst exponent
Bubble decay trend
Kelly log wealth optimization
criteria
Power law vs. Global liquidity
Log Periodic Power law
14th semiannual
CryptoSuper500 Report
2
ENERGY AS DIGITAL MONEY
Energy as Money, Soddy 1921,“Wealth,Virtual Wealth, and Debt”
• Wealth originates from Energy transformation
• Debt grows exponentially whereas real world wealth has growth and decay
• Called for an energy currency
Others: Henry Ford 1920s, Buckminster Fuller 1967
Digital Money 1983, David Chaum
Bitcoin 2008/9, Nakamoto
Frederic Soddy, Nobel in Chemistry,
radioactive decay
Discoveries can take decades or a century
3
TRUE ENERGY MONEY,
FULLY SCIENTIFIC MONEY
First Precise Scienti
fi
c Monetary Standard in history
Underlying Power Law is Proof of Work
Energy = Power * Time
21 GigaWatts * 600 sec —> 1000 ExaHash/s
Aristotelian attributes
Not just
Scarce,
De
fi
ned
Finiteness,
Requiring
energy input
Intrinsic value
4
MOSTVALUABLE SOFTWARE CODE EVER?
$2 trillion revenue (3 decades)
$2 trillion revenue (2.5 decades)
$2 trillion market cap (1.6 decades)
5
ZETTASCALE = 1000 EXAHASH CRYPTOGRAPHY
Encapsulated Energy, Exponentially Hardened, onto PermanentTimeChain
Mining 2011: First GPUs were a few 100 MHash/s, Shift to
ASICs in 2013 leap of few 100 x faster
ASICs at 5 nanometers give us 300 - 800 Terahashes/s
Per system factor of 1 million x faster hashing in 14 years
And one double SHA-256 hash is about 1600 low-level bit
operations
Several million (~5?) mining rigs globally are a continual
Zettahash/s competition
The world’s largest Decentralized supercomputer with
$10s of billions invested
6
PROOF POINT
Encapsulated Energy, Exponentially Hardened, onto PermanentTimeChain
BITCOIN OR
COPY
SUPPLY
(MILLIONS)
# EXAHASH/S PRICE PRICE / HASHRATE
Bitcoin 19.9 851 $113,600 133
“Bitcoin” Cash 19.9 4.55 $554 122
7
POWER LAWS
EVERYWHERE
~ M
~ 1/γ
~ φ2, φ4
~ r4
8
EXPONENTIALS
Exponential: Price ~ c * exp (t/
tc)
Exponential = CAGR
Semilog (log-linear) plot
One axis is logarithmic
S&P 500 11.9% CAGR since
January 2009
9
Exponential: straight on Log-Linear chart
Power Law
Log - Linear
POWER LAW, TWO PARAMETERS: C *TK
Steep k = 5.69 or 5.83 depending on technique
Power law, no characteristic time scale
Log-log, straight line is a power law
Both axes logarithmic
Still > 40% year-over-year trend (1 +1/T)^k
10
GROWTH
(1 + 1/T)^K
NOW AUG. 2026
OLS $105K $146K
QR 0.5 $93.7K $132K
Geo
Mean
$99.2K $1393K
10
3 16
5
2
Age (yrs)
Straight on a Log-Log chart: Factor of 1
million x price, 10 x age
2011
2013
2017
2021
QUANTILE REGRESSION FOR GOLD
Weekly data
7 Quantile levels
Median (5.58 power law)
and three levels below
bunched up
Power law nearly as
large as against US$
(5.83)
Bubbles
Power Law Core
11
BITCOIN: EXPONENTIAL OR POWER LAW?
The data for average monthly return and monthly
return volatility is clear
Results at left are model independent
If exponential = constant CAGR both lines would
be straight
In reality, both the return and volatility fell
monotonically and considerably from Epoch 1 to
Epoch IV
Actual values are consistent with power law
model plus allowance for bubbles
12
https://stephenperrenod.substack.com/publish/posts/detail/144303111
Vol
Monthly Return
If 20%
per month
STABILITY OF POWER LAW
Measure power law looking
back from each month (label
start of year)
Power Law index settled in
very nicely
From 2016 it is in the high 5+
range
Uncertainty (band) decreasing
13
EMH REJECTED BY 20 OF 25 STUDIES
Is the Bitcoin market ef
fi
cient? A literature review, 2021, K.E. Lengyel-Almos and M. Demmler
https://doi.org/10.24275/uam/azc/dcsh/ae/2021v36n93/lengyel
“Considering the multiple and diverse tests that these 25 studies applied, there are also several reoccurring
models and tests, which help to compare the results among the diverse currencies and time periods selected.
Among the most used methods are the Ljung-Box test for autocorrelation, Bartel´s test used for
independence of returns, vector autoregression (VAR) tests and its variations such as FCVAR for random
walk analysis, Brock, Dechert and Scheinkman (or BDS) test for independence test, detrended
fl
uctuation analysis (DFA), Hurst exponent test, OLS model,Augmented Dickey-Fuller (ADF)
test, GARCH-type models, the PWY model (Phillips,Wu andYu, 2011a and 2011b), PSY model (Phillips, Shi andYu, 2015).These
latter two models, especially the PSY (2015) model, are often applied by other authors (e.g. Cheung et al., 2015;
Geuder et al., 2019), making it one of the most ubiquitous among these papers, as these authors note that it offers
the best predictive capacity.
Several recent articles favor the log-periodic power law (LPPL) model. This is more common in
studies that include data for the 2017 price increase which required the analysis of exponential growth, for
example, the study of Geuder, Kinateder & Wagner, 2019; Wheatley et al., 2018; Xiong et al., 2020.Testing for and con rming the
presence of martingale for highly explosive speculative bubble tendencies on the Bitcoin market is also
present in the article of Schilling & Uhlig (2019).As mentioned previously, most studies use multiple models, often up
to 6-8 models and tests.
With respect to the cryptocurrencies included in the 25 articles, there is also great diversity although Bitcoin
is the common denominator”
Ef
fi
cient Market Hypothesis ~> Mean-Reverting
Because they only look at
price change % and
Correlations with other assets:
They missed the most
signi
fi
cant model:
Power LawTheory
All were pre-COVID depression so missed
a factor of 25 in price growth also
14
HURST EXPONENTS
Hydrology and
environmental science
Physics and complex
systems
Neuroscience and
biology
Geophysics and
seismology
Network, Internet traf
fi
c
analysis
Image processing,
computer vision
Social sciences, linguistics
Finance, economics
British hydrologist Harold Edwin
Hurst (1880-1978) was investigating
long-term storage capacity of Nile
River:
Hurst’s rescaled range analysis and
Hurst exponent, 1951
A power law
Mandelbrot extended because
connection to fractals.
15
HURST EXPONENTS: RANDOM ORTRENDING?
Classical technique: chunk detrended residuals into windows of size N, vary N
R is the range (max-min partial sums cumulative residuals), S is standard deviation
Power Law in N: R/S ~ NH
Where N is window size and H is Hurst exponent: H = [0, 1]
H = 0.5 is random walk
H > 0.5 trending, H < 0.5 mean-reverting
> 0.5 is a Long memory “inef
fi
cient market” 16
HURST EXPONENT: POWER LAW DETREND
This study: Method R/S,
fi
rst
fi
nd Power Law (0.5
QR) then detrend the log price data
Log residuals relative to underlying power law
Full Price history: Weekly 0.89 (gold .87) ~ 750
points; Daily 0.97 (gold .96) > 5000 points
Very strong persistence, highly “inef
fi
cient market”,
means non-reverting
Validation: 10 years of SPY daily prices yields H =
0.18 strong mean-reversion (exponential detrend,
DFA)
S&P H = 0.18
Proof
of Alpha
17
HURST EXPONENT, R/S METHOD,
QUARTERLYTIMESTEP: 0.9 EXTREMELY HIGH
Hourly data R/S method
Extreme Persistence
Hurst value is Re
fi
ned;
Repeated QR for all data up
to a given time
Window 1 year, Step 1/4
year
Very high:“inef
fi
cient market”,
slowly coming down
Proof
of Alpha
18
DUAL GAUSSIAN MODELS ($, GOLD) FOR
LOG RESIDUALS FROM POWER LAW
(Vs. Gold)
19
– ChatGPT 4o
“✅ “The Bimodal Gaussian Mixture Model (GMM) is a signi cantly better
fi
t than the Gamma model.
• It captures both the normal residuals and the bubble regime separately.
• The Gamma model misses the left tail and over-smooths the right tail, meaning it does not fully
capture the bubble behavior.””
Hidden Markov
20
REMOVE SECOND GAUSSIAN (47% OF DATA REMAINS),
REDO OLS AND QR ANALYSIS
Ordinary Least Squares
• Mean, Power Law Index 5.89
• R2 0.998
Quantile Regression
• Median Power Law Index 5.92
• Tight range of slopes 5.84 to 5.96
• Pseudo R2 0.997
21
BITCOINVS. $ RESIDUALS: ONE STD. DEV.TOP 40%
SAME FOR BOTTOM 40% (LOG - LINEAR PLOT)
Peaks lessen
Transitions sharp
Red zone width 3 x green
Just reentering, > 2 x
potential
In
fl
uid
fl
ow analogy:
laminar (green) and
turbulent (red)
22
BITCOIN LAMINAR -TURBULENT SYSTEM
“Heartbeat”
23
QUANTILE REGRESSION (MEDIAN):
RESIDUALS: MEAN
3 x
2 x
1.6 x
0.7 x
“The bubble regime shows high variance and
poor Gaussian
fi
t. This suggests the bubble
zone is non-Gaussian, likely multimodal
— i.e., multiple phases like early rise, parabolic
blow-off, and collapse.”
ChatGPT:“The core market is stable and well-explained by a narrow,
unimodal Gaussian. Log residuals are closely clustered
just below the quantile regression median.”
24
BUBBLETO CORE PRICE RATIO: ~ 1/T
25
VOLATILITY: CORE AND BUBBLE
Bubble Std Dev (%) = 594.12 / (t + 2.23) + 5.40
Now: 37%
26
NEW FORECAST METHOD:
BUBBLE MEAN PLUS ITSVOLATILITY
PRICE RANGES
2026
Incorporates projected decline in bubble-to-core, volatility
~ 8 weeks of 2 year bubble ~1week
1.2% chance
8% chance
2026
27
YIN ANDYANG OF BITCOIN
(CHATGPT 4O AGREES)
28
KELLY FRACTION DERIVATION
Many papers written on this and why
maximizing Log W is the way to go
The Kelly Capital Growth and
Investment Criterion, L.C. MacLean,
E.O.Thorp, and W.T. Ziemba, editors. 2010,
World Scienti
fi
c Publishing (Chapters 3, 6,
7 by J.R. Kelly, E.O.Thorp, E.O.Thorp).
Kelly principles: embrace risk but aversion
to extreme risk, long term growth,
avoiding ruin, balancing utility of wins/
losses
Key:
p = % wins
b = payoff ratio, e.g. 2:1?
f = fraction of wealth bet
29
NATURE OFTHE CURVE
If you bet too much you lose steadily, even when
you have an edge
Here p = 0.6 you win 60% of time and the average
win size vs. loss size ratio is 1.
And yet if you bet $38 or more out of $100 you
will lose money over time (because recovering from a
20% loss requires 25% gain to recover, for example)
Example: a losing streak of 2 off the bat and you
wager 40% of purse you are down to 36% and have
to gain 178% just to get back to even!
Kelly principles: Long term growth, avoiding ruin,
balancing utility of wins/losses
30
10YEARS OF ASSET CLASS (16) RETURNS
31
10YEARS OF ASSET CLASS RETURNS
Top Asset 10 year Kelly (2015-2024)
Asset Class P W L B F
B* (r =
4%)
F* (r =
4%)
Bitcoin 0.8 292.55% -69.53% 4.21 0.752 3.92 0.750
Broad Tech 0.8 26.9% -18.11% 1.49 0.665 1.04 0.607
Small Caps 0.7 11.99% -17.42% 0.69 0.264 0.37 -0.104
Large Caps 0.7 19.37% -11.23% 1.72 0.526 1.01 0.403
Real Estate 0.6 12.51% -12.85% 0.97 0.190 0.51 -0.192
Gold 0.6 13.39% -8.05% 1.66 0.360 0.78 0.087
Commodities 0.6 12.7% -23.61% 0.54 -0.144 0.32 -0.669
International
Equities
0.7 9.61% -15.58% 0.62 0.214 0.29 -0.347
Bitcoin, 5 years
only
0.8 155.49% -68.45% 2.27 0.712 2.09 0.704
Classical Kelly
f = p - (1-p)/b
b = -<win>/<loss>
Two edges: p > .5 and b > 1
32
BITCOIN AS POWER LAW OF GLOBAL
LIQUIDITY
Michael Howell, CrossBorder Capital
Y = 9.72 X - 106 (natural log base, dashed)
Price ~ GLI ^ 9.72, nearly 10th power, R2 0.90
Over span BTC $200 to $100K
Global Liquidity is strongly Granger causal in a statistical sense and that its impact on Bitcoin is positive and with a lead-time concentrated around 11-13 weeks
33
OECD M3 GLOBAL LIQUIDITY
Power Law Bitcoin vs. OECD M3
Global Liquidity
Slope 11.51 (+/- 0.36) , R2 0.86
Longer span than GLI, from under
$0.1 (6 orders of magnitude vs. almost
3)
But is it real, since OECD interval is
just 0.4 in log
34
RANDOM FOREST: MINI-BUBBLES AND
LIQUIDITY
OECD M3YoY Growth (not
delta month-to-month)
Lags 4, 5, 2 months best
correlation
• Half of importance is in M3YoY
Growth with Lags 1, 2, 3, 4, 5 months;
weighted mean is 3.5 months
Machine learning, creates hundreds of decision trees, Bootstrap sampling,
i.e. train each on a different subset, trees are deco related by taking random subsets of features, majority voting on classi cation
35
LOG PERIODIC POWER LAW
(DIDIER SORNETTE)
Log P(t) = A + B (tc - t)m + C (tc-t)m * cos[ω log (tc-t) + φ]
• tc critical time
• m power law slope
• ω (angular) frequency
This is not the long-term core Power Law for Bitcoin, this is a model for bubbles:
Has log-periodic in time components, time relative to a future crash/critical event,
36
Intermediate
term LPPL
ChatGPT Code
Copilot
Grok agreed,
with peak on
9/15 at $130K
BITCOIN 2025 LPPL (USUAL 7 PARAMETERS)
A = $141K, t_c ≈ 989.4 days after Jan 1, 2023 → 2025-09-16
R^2 0.957 RMSE $6.0 K very good
fi
t, est. price on 9/2 $130.8K
t_c ≈ 989.4 days after Jan 1, 2023 → 2025-09-16
R^2 0.92 RMSE $5.9K estimated 9/2 $118.4K
2.6Years 1.6Years
37
“Diversi
fi
cation is Wall Street’s way of saying,‘Give us
your money, and we’ll make sure you never get rich.’”
@OrionX_net
©2025 OrionX.net 22
Six and a Half Year
Review
since Nov. 2018
Attribute Nov. 2018 June 2025 6.5 Year Growth
Coins making cut for
CryptoSuper report
Bitcoin, Ethereum,
Litecoin, Bitcoin Cash,
Monero
Bitcoin, Dogecoin Consolidation
Number of different
cryptocurrencies
2000
~ 10,000 active; millions of
meme coins
Vast Majority worth little
Bitcoin Market
Capitalization
$111 billion $2200 billion 20 x
Bitcoin Price $6,334 $110,000 17 x
Bitcoin annual
production rate and fees
$4.2 billion $18.3 billion 4.4 x
Bitcoin Hash Rate
Exahash/s 57 900 16 x
Cryptocurrency Market
Cap $220 billion $3400 billion 15 x
Top cryptos annual
mining production w/
fees
$5.6 billion $19.5 billion 3.5 x
We dropped from 6 to 2 coins,
despite the 10-fold increase in the
number of cryptos
Bitcoin price has grown over 50%
compounded annually
Bitcoin hash rate has grown over 50%
compounded
Even with Ethereum dropping out,
(and it once reached half of the value
of production), the mining
production has grown to $19 billion Table 7. Key attributes of Bitcoin and major cryptocurrencies, comparing the first CryptoSuper report in 2018
with this report 6.5 years later. Recent data is as of May 26, 2025. Bitcoin has been growing in value at a
Moore’s law-like rate (but in a power law relation with its age), and its supercomputing crypto hashing power
has been growing faster than Moore’s law.
38
–Warren Buffett
“Diversi
fi
cation is protection against ignorance. It makes little sense if you
know what you are doing.”
“Diversi
fi
cation is Wall Street’s way of saying,‘Give us
your money, and we’ll make sure you never get rich.’”
39
THANKYOU!
Be Stoic, Bitcoin is, at its core
Physics of Bitcoin live eachWed. 11:15 PM EDT onYT & X
(Thurs.AM in Asia)
With Giovanni Santostasi and Stephen Perrenod
40
EXPONENTIALS AND POWER LAWS
Exponential: Price ~ c * exp (t/tc)
Power Law: Price ~ c * tk
Power law, no characteristic time
scale Exponential: characteristic
timescale tc
Exponential = constant CAGR
Semilog (log-linear) plot, exponential
and power law
Log-log, exponential and power law
41
Exponential
Power Law
Log - Linear Log - Log
Power Law
Exponential

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Bitcoin Layer August 2025: Power Laws of Bitcoin: The Core and Bubbles

  • 1. BITCOIN LAYER:AUGUST 11, 2025 Stephen Perrenod @moneyordebt https://stephenperrenod.substack.com/ Power Laws of Bitcoin:The Core and the Bubbles Virgo Cluster, credit: DOE, NSF, Vera C. Rubin Observatory Three power laws operating in this image 1
  • 2. AGENDA Energy + cryptography = Money 3.0 Power law, not exponential Zettascale imparts value Persistence of power law Core vs. Bubble EMH ?, Hurst exponent Bubble decay trend Kelly log wealth optimization criteria Power law vs. Global liquidity Log Periodic Power law 14th semiannual CryptoSuper500 Report 2
  • 3. ENERGY AS DIGITAL MONEY Energy as Money, Soddy 1921,“Wealth,Virtual Wealth, and Debt” • Wealth originates from Energy transformation • Debt grows exponentially whereas real world wealth has growth and decay • Called for an energy currency Others: Henry Ford 1920s, Buckminster Fuller 1967 Digital Money 1983, David Chaum Bitcoin 2008/9, Nakamoto Frederic Soddy, Nobel in Chemistry, radioactive decay Discoveries can take decades or a century 3
  • 4. TRUE ENERGY MONEY, FULLY SCIENTIFIC MONEY First Precise Scienti fi c Monetary Standard in history Underlying Power Law is Proof of Work Energy = Power * Time 21 GigaWatts * 600 sec —> 1000 ExaHash/s Aristotelian attributes Not just Scarce, De fi ned Finiteness, Requiring energy input Intrinsic value 4
  • 5. MOSTVALUABLE SOFTWARE CODE EVER? $2 trillion revenue (3 decades) $2 trillion revenue (2.5 decades) $2 trillion market cap (1.6 decades) 5
  • 6. ZETTASCALE = 1000 EXAHASH CRYPTOGRAPHY Encapsulated Energy, Exponentially Hardened, onto PermanentTimeChain Mining 2011: First GPUs were a few 100 MHash/s, Shift to ASICs in 2013 leap of few 100 x faster ASICs at 5 nanometers give us 300 - 800 Terahashes/s Per system factor of 1 million x faster hashing in 14 years And one double SHA-256 hash is about 1600 low-level bit operations Several million (~5?) mining rigs globally are a continual Zettahash/s competition The world’s largest Decentralized supercomputer with $10s of billions invested 6
  • 7. PROOF POINT Encapsulated Energy, Exponentially Hardened, onto PermanentTimeChain BITCOIN OR COPY SUPPLY (MILLIONS) # EXAHASH/S PRICE PRICE / HASHRATE Bitcoin 19.9 851 $113,600 133 “Bitcoin” Cash 19.9 4.55 $554 122 7
  • 8. POWER LAWS EVERYWHERE ~ M ~ 1/γ ~ φ2, φ4 ~ r4 8
  • 9. EXPONENTIALS Exponential: Price ~ c * exp (t/ tc) Exponential = CAGR Semilog (log-linear) plot One axis is logarithmic S&P 500 11.9% CAGR since January 2009 9 Exponential: straight on Log-Linear chart Power Law Log - Linear
  • 10. POWER LAW, TWO PARAMETERS: C *TK Steep k = 5.69 or 5.83 depending on technique Power law, no characteristic time scale Log-log, straight line is a power law Both axes logarithmic Still > 40% year-over-year trend (1 +1/T)^k 10 GROWTH (1 + 1/T)^K NOW AUG. 2026 OLS $105K $146K QR 0.5 $93.7K $132K Geo Mean $99.2K $1393K 10 3 16 5 2 Age (yrs) Straight on a Log-Log chart: Factor of 1 million x price, 10 x age 2011 2013 2017 2021
  • 11. QUANTILE REGRESSION FOR GOLD Weekly data 7 Quantile levels Median (5.58 power law) and three levels below bunched up Power law nearly as large as against US$ (5.83) Bubbles Power Law Core 11
  • 12. BITCOIN: EXPONENTIAL OR POWER LAW? The data for average monthly return and monthly return volatility is clear Results at left are model independent If exponential = constant CAGR both lines would be straight In reality, both the return and volatility fell monotonically and considerably from Epoch 1 to Epoch IV Actual values are consistent with power law model plus allowance for bubbles 12 https://stephenperrenod.substack.com/publish/posts/detail/144303111 Vol Monthly Return If 20% per month
  • 13. STABILITY OF POWER LAW Measure power law looking back from each month (label start of year) Power Law index settled in very nicely From 2016 it is in the high 5+ range Uncertainty (band) decreasing 13
  • 14. EMH REJECTED BY 20 OF 25 STUDIES Is the Bitcoin market ef fi cient? A literature review, 2021, K.E. Lengyel-Almos and M. Demmler https://doi.org/10.24275/uam/azc/dcsh/ae/2021v36n93/lengyel “Considering the multiple and diverse tests that these 25 studies applied, there are also several reoccurring models and tests, which help to compare the results among the diverse currencies and time periods selected. Among the most used methods are the Ljung-Box test for autocorrelation, Bartel´s test used for independence of returns, vector autoregression (VAR) tests and its variations such as FCVAR for random walk analysis, Brock, Dechert and Scheinkman (or BDS) test for independence test, detrended fl uctuation analysis (DFA), Hurst exponent test, OLS model,Augmented Dickey-Fuller (ADF) test, GARCH-type models, the PWY model (Phillips,Wu andYu, 2011a and 2011b), PSY model (Phillips, Shi andYu, 2015).These latter two models, especially the PSY (2015) model, are often applied by other authors (e.g. Cheung et al., 2015; Geuder et al., 2019), making it one of the most ubiquitous among these papers, as these authors note that it offers the best predictive capacity. Several recent articles favor the log-periodic power law (LPPL) model. This is more common in studies that include data for the 2017 price increase which required the analysis of exponential growth, for example, the study of Geuder, Kinateder & Wagner, 2019; Wheatley et al., 2018; Xiong et al., 2020.Testing for and con rming the presence of martingale for highly explosive speculative bubble tendencies on the Bitcoin market is also present in the article of Schilling & Uhlig (2019).As mentioned previously, most studies use multiple models, often up to 6-8 models and tests. With respect to the cryptocurrencies included in the 25 articles, there is also great diversity although Bitcoin is the common denominator” Ef fi cient Market Hypothesis ~> Mean-Reverting Because they only look at price change % and Correlations with other assets: They missed the most signi fi cant model: Power LawTheory All were pre-COVID depression so missed a factor of 25 in price growth also 14
  • 15. HURST EXPONENTS Hydrology and environmental science Physics and complex systems Neuroscience and biology Geophysics and seismology Network, Internet traf fi c analysis Image processing, computer vision Social sciences, linguistics Finance, economics British hydrologist Harold Edwin Hurst (1880-1978) was investigating long-term storage capacity of Nile River: Hurst’s rescaled range analysis and Hurst exponent, 1951 A power law Mandelbrot extended because connection to fractals. 15
  • 16. HURST EXPONENTS: RANDOM ORTRENDING? Classical technique: chunk detrended residuals into windows of size N, vary N R is the range (max-min partial sums cumulative residuals), S is standard deviation Power Law in N: R/S ~ NH Where N is window size and H is Hurst exponent: H = [0, 1] H = 0.5 is random walk H > 0.5 trending, H < 0.5 mean-reverting > 0.5 is a Long memory “inef fi cient market” 16
  • 17. HURST EXPONENT: POWER LAW DETREND This study: Method R/S, fi rst fi nd Power Law (0.5 QR) then detrend the log price data Log residuals relative to underlying power law Full Price history: Weekly 0.89 (gold .87) ~ 750 points; Daily 0.97 (gold .96) > 5000 points Very strong persistence, highly “inef fi cient market”, means non-reverting Validation: 10 years of SPY daily prices yields H = 0.18 strong mean-reversion (exponential detrend, DFA) S&P H = 0.18 Proof of Alpha 17
  • 18. HURST EXPONENT, R/S METHOD, QUARTERLYTIMESTEP: 0.9 EXTREMELY HIGH Hourly data R/S method Extreme Persistence Hurst value is Re fi ned; Repeated QR for all data up to a given time Window 1 year, Step 1/4 year Very high:“inef fi cient market”, slowly coming down Proof of Alpha 18
  • 19. DUAL GAUSSIAN MODELS ($, GOLD) FOR LOG RESIDUALS FROM POWER LAW (Vs. Gold) 19
  • 20. – ChatGPT 4o “✅ “The Bimodal Gaussian Mixture Model (GMM) is a signi cantly better fi t than the Gamma model. • It captures both the normal residuals and the bubble regime separately. • The Gamma model misses the left tail and over-smooths the right tail, meaning it does not fully capture the bubble behavior.”” Hidden Markov 20
  • 21. REMOVE SECOND GAUSSIAN (47% OF DATA REMAINS), REDO OLS AND QR ANALYSIS Ordinary Least Squares • Mean, Power Law Index 5.89 • R2 0.998 Quantile Regression • Median Power Law Index 5.92 • Tight range of slopes 5.84 to 5.96 • Pseudo R2 0.997 21
  • 22. BITCOINVS. $ RESIDUALS: ONE STD. DEV.TOP 40% SAME FOR BOTTOM 40% (LOG - LINEAR PLOT) Peaks lessen Transitions sharp Red zone width 3 x green Just reentering, > 2 x potential In fl uid fl ow analogy: laminar (green) and turbulent (red) 22
  • 23. BITCOIN LAMINAR -TURBULENT SYSTEM “Heartbeat” 23
  • 24. QUANTILE REGRESSION (MEDIAN): RESIDUALS: MEAN 3 x 2 x 1.6 x 0.7 x “The bubble regime shows high variance and poor Gaussian fi t. This suggests the bubble zone is non-Gaussian, likely multimodal — i.e., multiple phases like early rise, parabolic blow-off, and collapse.” ChatGPT:“The core market is stable and well-explained by a narrow, unimodal Gaussian. Log residuals are closely clustered just below the quantile regression median.” 24
  • 25. BUBBLETO CORE PRICE RATIO: ~ 1/T 25
  • 26. VOLATILITY: CORE AND BUBBLE Bubble Std Dev (%) = 594.12 / (t + 2.23) + 5.40 Now: 37% 26
  • 27. NEW FORECAST METHOD: BUBBLE MEAN PLUS ITSVOLATILITY PRICE RANGES 2026 Incorporates projected decline in bubble-to-core, volatility ~ 8 weeks of 2 year bubble ~1week 1.2% chance 8% chance 2026 27
  • 28. YIN ANDYANG OF BITCOIN (CHATGPT 4O AGREES) 28
  • 29. KELLY FRACTION DERIVATION Many papers written on this and why maximizing Log W is the way to go The Kelly Capital Growth and Investment Criterion, L.C. MacLean, E.O.Thorp, and W.T. Ziemba, editors. 2010, World Scienti fi c Publishing (Chapters 3, 6, 7 by J.R. Kelly, E.O.Thorp, E.O.Thorp). Kelly principles: embrace risk but aversion to extreme risk, long term growth, avoiding ruin, balancing utility of wins/ losses Key: p = % wins b = payoff ratio, e.g. 2:1? f = fraction of wealth bet 29
  • 30. NATURE OFTHE CURVE If you bet too much you lose steadily, even when you have an edge Here p = 0.6 you win 60% of time and the average win size vs. loss size ratio is 1. And yet if you bet $38 or more out of $100 you will lose money over time (because recovering from a 20% loss requires 25% gain to recover, for example) Example: a losing streak of 2 off the bat and you wager 40% of purse you are down to 36% and have to gain 178% just to get back to even! Kelly principles: Long term growth, avoiding ruin, balancing utility of wins/losses 30
  • 31. 10YEARS OF ASSET CLASS (16) RETURNS 31
  • 32. 10YEARS OF ASSET CLASS RETURNS Top Asset 10 year Kelly (2015-2024) Asset Class P W L B F B* (r = 4%) F* (r = 4%) Bitcoin 0.8 292.55% -69.53% 4.21 0.752 3.92 0.750 Broad Tech 0.8 26.9% -18.11% 1.49 0.665 1.04 0.607 Small Caps 0.7 11.99% -17.42% 0.69 0.264 0.37 -0.104 Large Caps 0.7 19.37% -11.23% 1.72 0.526 1.01 0.403 Real Estate 0.6 12.51% -12.85% 0.97 0.190 0.51 -0.192 Gold 0.6 13.39% -8.05% 1.66 0.360 0.78 0.087 Commodities 0.6 12.7% -23.61% 0.54 -0.144 0.32 -0.669 International Equities 0.7 9.61% -15.58% 0.62 0.214 0.29 -0.347 Bitcoin, 5 years only 0.8 155.49% -68.45% 2.27 0.712 2.09 0.704 Classical Kelly f = p - (1-p)/b b = -<win>/<loss> Two edges: p > .5 and b > 1 32
  • 33. BITCOIN AS POWER LAW OF GLOBAL LIQUIDITY Michael Howell, CrossBorder Capital Y = 9.72 X - 106 (natural log base, dashed) Price ~ GLI ^ 9.72, nearly 10th power, R2 0.90 Over span BTC $200 to $100K Global Liquidity is strongly Granger causal in a statistical sense and that its impact on Bitcoin is positive and with a lead-time concentrated around 11-13 weeks 33
  • 34. OECD M3 GLOBAL LIQUIDITY Power Law Bitcoin vs. OECD M3 Global Liquidity Slope 11.51 (+/- 0.36) , R2 0.86 Longer span than GLI, from under $0.1 (6 orders of magnitude vs. almost 3) But is it real, since OECD interval is just 0.4 in log 34
  • 35. RANDOM FOREST: MINI-BUBBLES AND LIQUIDITY OECD M3YoY Growth (not delta month-to-month) Lags 4, 5, 2 months best correlation • Half of importance is in M3YoY Growth with Lags 1, 2, 3, 4, 5 months; weighted mean is 3.5 months Machine learning, creates hundreds of decision trees, Bootstrap sampling, i.e. train each on a different subset, trees are deco related by taking random subsets of features, majority voting on classi cation 35
  • 36. LOG PERIODIC POWER LAW (DIDIER SORNETTE) Log P(t) = A + B (tc - t)m + C (tc-t)m * cos[ω log (tc-t) + φ] • tc critical time • m power law slope • ω (angular) frequency This is not the long-term core Power Law for Bitcoin, this is a model for bubbles: Has log-periodic in time components, time relative to a future crash/critical event, 36
  • 37. Intermediate term LPPL ChatGPT Code Copilot Grok agreed, with peak on 9/15 at $130K BITCOIN 2025 LPPL (USUAL 7 PARAMETERS) A = $141K, t_c ≈ 989.4 days after Jan 1, 2023 → 2025-09-16 R^2 0.957 RMSE $6.0 K very good fi t, est. price on 9/2 $130.8K t_c ≈ 989.4 days after Jan 1, 2023 → 2025-09-16 R^2 0.92 RMSE $5.9K estimated 9/2 $118.4K 2.6Years 1.6Years 37
  • 38. “Diversi fi cation is Wall Street’s way of saying,‘Give us your money, and we’ll make sure you never get rich.’” @OrionX_net ©2025 OrionX.net 22 Six and a Half Year Review since Nov. 2018 Attribute Nov. 2018 June 2025 6.5 Year Growth Coins making cut for CryptoSuper report Bitcoin, Ethereum, Litecoin, Bitcoin Cash, Monero Bitcoin, Dogecoin Consolidation Number of different cryptocurrencies 2000 ~ 10,000 active; millions of meme coins Vast Majority worth little Bitcoin Market Capitalization $111 billion $2200 billion 20 x Bitcoin Price $6,334 $110,000 17 x Bitcoin annual production rate and fees $4.2 billion $18.3 billion 4.4 x Bitcoin Hash Rate Exahash/s 57 900 16 x Cryptocurrency Market Cap $220 billion $3400 billion 15 x Top cryptos annual mining production w/ fees $5.6 billion $19.5 billion 3.5 x We dropped from 6 to 2 coins, despite the 10-fold increase in the number of cryptos Bitcoin price has grown over 50% compounded annually Bitcoin hash rate has grown over 50% compounded Even with Ethereum dropping out, (and it once reached half of the value of production), the mining production has grown to $19 billion Table 7. Key attributes of Bitcoin and major cryptocurrencies, comparing the first CryptoSuper report in 2018 with this report 6.5 years later. Recent data is as of May 26, 2025. Bitcoin has been growing in value at a Moore’s law-like rate (but in a power law relation with its age), and its supercomputing crypto hashing power has been growing faster than Moore’s law. 38
  • 39. –Warren Buffett “Diversi fi cation is protection against ignorance. It makes little sense if you know what you are doing.” “Diversi fi cation is Wall Street’s way of saying,‘Give us your money, and we’ll make sure you never get rich.’” 39
  • 40. THANKYOU! Be Stoic, Bitcoin is, at its core Physics of Bitcoin live eachWed. 11:15 PM EDT onYT & X (Thurs.AM in Asia) With Giovanni Santostasi and Stephen Perrenod 40
  • 41. EXPONENTIALS AND POWER LAWS Exponential: Price ~ c * exp (t/tc) Power Law: Price ~ c * tk Power law, no characteristic time scale Exponential: characteristic timescale tc Exponential = constant CAGR Semilog (log-linear) plot, exponential and power law Log-log, exponential and power law 41 Exponential Power Law Log - Linear Log - Log Power Law Exponential