Physics of Bitcoin #30 Perrenod Santostasi.pdf

In this edition of the Physics of Bitcoin show we covered Metcalfe's law applied to Bitcoin and explored why Bitcoin's adoption is growing with a cube law of time. We examined institutional adoption of Bitcoin and showed it is growing as a 4th power law of time. We presented a hypothesis as to why the adoption is growing as T^3 and therefore why price grows as T^6 as a consequence of Metcalfe's Law for network emergent behavior. We also looked at recent Bitcoin price history with log-periodic wave behavior as described by Sornette.

1. PHYSICS OF BITCOIN #30 AUGUST 27, 2025
INSTITUTIONAL ADOPTION
Stephen Perrenod @moneyordebt
https://stephenperrenod.substack.com/
ROBERT METCALFE (1946- )
CO-INVENTED ETHERNET
Wednesdays 11:15 PM EDT Live : YT & X
1

2. AGENDA: INSTITUTIONS AND INSIDE
METCALFE’S LAW
Institutions are network Fat Nodes
Growth of Institutional holdings
Power law, steep
Why little to no holdings by
Largest companies?
Metcalfe’s Law: Why T
3 for
Adoption?
Revisit Hurst with GPT-5
Bitcoin bubble 2025, Log
periodic power law
2

3. PREMISE: FAT NODES TO POWER LAW TAILS
Investing in an ETF or BTC Treasury company
or having custody at an exchange is attaching to
a fat node
Fat nodes in a network support power law
behavior:
Airline hubs, Google, Bitcoin exchanges/ETFs
That is, of the form of preferential attachment
Pareto-style, with power law tails of index [2, 3]
typical
Studies of the Bitcoin graph have
found typical:
Attachment/connections index 2 to
2.5
Holdings fall off > 1.8
Holdings not same as connections
3

4. TREASURY COMPANIES PARETO CHART
BTC Held vs. Rank
as a power law
Slope ~ -0.9 for
largest companies
Slope ~ -2.4 for bulk
Many quite new, so
this is evolving
4

5. GROWTH OF INSTITUTIONAL HOLDINGS
Includes exchanges,
public and private
companies including
miners, ETFs
# BTC Increasing at ~ T
4
or 4.5
R
2 = 0.94
5
Institutional_btc_2017_2025
DATE
(MONTH
END)
BTC
PRICE
USD
CORPORATE
BTC
FUND
TRUST
BTC
ETF BTCTOTAL
INSTITUTIONAL
LOG10
AGE
PRICE
LOG
BTC
HELD
LOG
2017-1214,156140,000175,8390 315,8390.9544.1515.499
2018-123,743140,000204,2770 344,2771.0003.5735.537
2019-127,194140,000261,1920 401,1921.0413.8575.603
2020-1229,002210,470607,0390 8175091.0794.4625.912
2021-1246,306264,391644,81054,3009635011.1144.6665.984
2022-1216,548264,391644,81034,3189435191.1464.2195.975
2023-1242,2651,204,800619,00042,5001,866,3001.1764.6266.271
2024-1293,4291,508,40001,160,0002,668,4001.2044.9706.426
2025-07117,8331,865,60001,475,0003,340,6001.2205.0716.524
Except
1st
Price
vs. age
Holdings
vs. age
Except
1st
Slope vs age6.0364.4223.9804.468
Holdings vs.
price
0.6520.696
R2 (price vs age)0.8230.6760.938

6. GROWTH OF INSTITUTIONAL HOLDINGS
Includes exchanges,
public and private
companies including
miners, ETFs
Increasing at T
4.5 (or 4.0)
R
2 = 0.95
610 1613

7. GROWTH OF INSTITUTIONAL HOLDINGS
BTC Held vs.
BTC price
Increasing as
Price
0.7 (or 0.65)
R
2 = 0.83
7

8. GROWTH OF INSTITUTIONAL HOLDINGS
BTC Held vs. BTC price
Adoption (buying, Quantity) is Increasing as ~
Price
0.7 (prior slide)
But retail quantity is decreasing; ownership
diffusion
Whales selling to custodians for many
holders ?
Market Cap ~ Price, as supply nearly fixed
Inference: Price ~ Institutional Adoption
1.4
to
the extent institutions are dominating net
purchases, price trend
And that is consistent with Price ~ T
5.8 and
Institutional holdings H ~ T
4
8
✅ Contextual Percentages (2025 YTD net buying ~225k BTC total)
•Public Companies: 70%
•ETFs: 22%
•Governments: 8%
•Private Companies: ≈0% net, possibly slightly negative (due to Mt. Gox outflows).
•Retail: –247,000 BTC (large net sellers).
Slope 1.2
R
2 = 0.83
Price
Holdings

9. $TRILLION CAP CO. HOLDINGS ~ ZILCH
Institutions adopt iff Price is High
They need large capital store
Thus, Magnificent 7 are late to the
party
The price is too low for them
9
COMPANY
NVDA0
MSFT0
APPL0
GOOGL0
AMZN0
META0
TSLA12K BTC

10. EFFECT ON PRICE IF LARGEST
CORPORATIONS BUY
Even only 1% allocation relative to market cap? (Of order $20 B)
Cash holdings range between $27B and $110B for top 6 companies
Annual cash flows range from $50B to $120B
Suppose they put 10% of their cash into Bitcoin to start: $3B to $11B
For the top 6 it would be 360K BTC aggregate, $42B
If they did it within one year, it would be $106B market cap increase (~5% now) assuming
market multiple 2.5 (typical MVRV)
What they are facing is Bitcoin is rising much faster (40% p.a.) than their cash balances (13%
p.a.)
Necessity being the mother of invention, they don’t see the necessity yet
10

11. INSIDE METCALFE’S LAW
Price ~ Market cap ~ Value ~ N
m, with m ~ 2, N users
Adoption ~ Ν ~ Τ
λ , observe λ ~ 3
Price ~ T
λ*m ; λ*m ~ 6 , is general thesis
Why* is λ ~ 3?
Internet history:
Power law of index: 4.6 overall, R
2 0.97; until 2020 saturation
First half > 5, second half 2.8 index
11
Web₿
2020
*There is a ‘why’ in physics, but not in economics

12. WHY IS LAMBDA (Λ) 3?
Adoption:
Ν ~ Τ
λ , observe λ ~ 3
Three mutually reinforcing factors:
Lifetime, awareness
More on-ramps, easier to buy
Capital commitment grows
Retail, then institutions (custodians)
Actively and passively
12

13. WHY IS LAMBDA (Λ) 3? CONJECTURE
13

14. WHY IS LAMBDA (Λ) 3?
CONJECTURE
14
Institutions
10
Institutions
10
10
Awareness
Channels

15. WHY IS LAMBDA (Λ) ~3? CONJECTURE
N users ~ f (awareness, opportunity)
Awareness: know enough about BTC
Opportunity: channels for investment
(exchanges, regulatory environment,
futures, ETFs, stable coin gateways,
treasury companies…)
Commitment: level of investment
Active, stacking
Passive, simply because price rises over
time faster than stocks, real estate etc.
15
Awareness
Opportunity
Commitment
GPT5 guesstimates
Adoption =
Users * Commitment

16. HURST EXPONENT
REVISITED
H measures trend persistence, reversion
R/S = range/standard deviation ~ N
H is a
function of window size N and measure H:
H = 0.5 random walk
H < 0.5 mean-reverting
H > 0.5 trend persistence
16
“There’s no evidence” ?
15 K views,
125 likes

17. EASY HURST EXPONENT GPT-5:
MONTHLY DATA, 4 SENTENCE PROMPT
<10 MINUTE EXERCISE
17

18. EASY HURST EXPONENT W/GPT-5:
MONTHLY DATA, 4 SENTENCE PROMPT
10 MINUTE EXERCISE
18

19. EASY HURST EXPONENT GPT-5,
CORE AND BUBBLE
19
Bubble
Core

20. LATEST LOG
PERIODIC FITS 8/26
20
•Two years, daily
Long-term Daily LPPL fit:
A = 12.009
B = -0.001781
C = -0.000350
m = 0.990
w = 12.29
phi = -1.38
Tc = 885.98 days since 2023-09-07 -> 2026-02-08
R^2 = 0.963
σ = $9070 (width band $18K)

Price at Tc-3 days: $163,262
40/20/40
split
163K
99K
60K
36K

21. LATEST LOG
PERIODIC FITS
21
•One year, daily
Tc = 370.0 days since 2024-08-27 -> 2025-09-01
R^2 = 0.945
σ = $5500 (width of band $11K)
Price at Tc-3: $120,348
•4 months (4h data)
Tc = 157.1 days since 2025-04-28 -> 2025-10-3
R^2 = 0.828
σ = $2970 (width of band $6K)
Price at Tc-3: $126,677
$109K
$109K
$121K

22. ON THE OTHER SIDE?
22
Short-term 1h LPPL fit:
Tc = 74.153 days since 2025-07-28 -> 2025-10-10
R^2 = 0.7095
σ = $1430
Price at Tc-3: $107,237.88
Price at Tc-1: $106,797.31
•1 month (one hour data)
LPPL
TIMESCALE
TC R^2
PRICE K$ AT
TC - 3 D
PRICE
UNCERTAINY
K$
2 years2026-02-080.96163.39.1
1 year2025-09-010.95120.35.5
4 monthsUP: 2025-10-30.83126.73.0
1 monthDOWN: 2025-10-100.71107.21.4

23. ALL IS NOT LOST, YET: SEASONALITY
23

24. NVIDIA LPPL, ONE YEAR
24
X
Best-fit parameters
• A = 8.538
• B = -2.094
• C = 0.1015
• m = 0.100 (at the lower bound)
•ω = 9.576
•φ = 0.3886
• Tc = 417.9
(≈ 168 trading days after the last sample day)
2026-03-26
Fit quality: R^2 = 0.861
Dec. 2025 (last business day) $198
End-Jan 2026 (last business day) $232
One week before t_c (5 bdays)
2026-03-19 $422
Tc (model limit): 2026-03-26

25. PHYSICS OF BITCOIN #30 AUGUST 27, 2025
Stephen Perrenod @moneyordebt
https://stephenperrenod.substack.com/
25
Bubbles? We don’t need no stinkin’ bubbles.
We have the Power Law.
Froth, Decay vs. Clarity