The
Determinism
Problem
A Forensic Examination of Bitcoin Power Law Modelling:
From Retail Monte Carlo to Academic Physics
From Retail Monte Carlo to Academic Physics
Most Bitcoin power law models cannot fail.
That is the problem.
Bitcoin’s power law is presented as a structural floor, a predictive framework, and — at the extreme — a physical law.
This paper demonstrates something much simpler and much more dangerous: the model is consistent with the past because it is constructed from the past.
It does not constrain the future. It does not generate falsifiable predictions. In several cases, it is structurally incapable of being wrong.
Before you rely on a model that cannot fail.
Structural Fault Summary
9
Structural
faults identified
faults identified
2
Analyses
examined
examined
5/7
Retail: checklist
failures
failures
6/7
Academic: checklist
failures
failures
0.96
R² — cannot distinguish
law from description
law from description
25yr
Falsification
horizon
horizon
Central conclusion: Both analyses demonstrate that Bitcoin’s price history is well-described by a power law. Neither demonstrates that it will continue to be.
The Determinism Problem
Description ≠ Prediction
R² = 0.96 is not a physical law
Circular boundary condition
9 structural faults
Composition identity is algebraic tautology
Bayesian posterior: 5× too tight
No out-of-sample validation
Free white paper — April 2026
Granger causality not tested
Volatility decay is a coordinate artifact
The S&P passes the same ADF test
Paul Faulkner // The Rogue Protocol
The Determinism Problem
Description ≠ Prediction
R² = 0.96 is not a physical law
Circular boundary condition
9 structural faults
Composition identity is algebraic tautology
Bayesian posterior: 5× too tight
No out-of-sample validation
Free white paper — April 2026
Granger causality not tested
Volatility decay is a coordinate artifact
The S&P passes the same ADF test
Paul Faulkner // The Rogue Protocol
The Structural Case
Nine faults.
Two analyses.
One conclusion.
The retail Monte Carlo and the Santostasi–Perrenod academic derivation are examined on separate grounds. The retail variant fails on five counts. The academic variant fails on four additional counts. Any single fault would be material. Together they are structural.
09
Structural faults
documented
documented
I
Retail Analysis — Five Faults
Monte Carlo // Power Law Floor // Volatility Regimes
Part I
01
The Floor and What Stationarity Does Not Prove
Passing the test does not prove the model — everything trending passes the test.
ADF residual stationarity (p ≈ 0.0075) confirms historical consistency. The S&P 500 exponential trend passes the identical test with near-identical p-values. The test does not discriminate between a structural law and a good trend description. Passing it is necessary but not sufficient.
Retail
02
The Circular Boundary Condition
The model cannot break because breakage is excluded by construction.
The Monte Carlo floor cannot be breached because the simulation was built assuming the floor holds. Paths that breach it are eliminated or reflected. The non-violation of the floor is not evidence — it is a construction choice. No out-of-sample test distinguishes a structural constraint from a coincidental trend.
Retail
03
The Log-Scale Volatility Illusion
Volatility appears to fall because of the chart, not because of the market.
The claimed decay of volatility toward zero by 2050 is a log-scale coordinate artifact. The same percentage volatility occupies shrinking visual space as price grows. The 2022 drawdown was proportionally consistent with prior cycles. True decay toward zero would not permit that.
Retail
04
Regime Dredging and In-Sample Validation
The model is tuned and validated on the same data.
Four halving-cycle volatility regimes are fitted and tested on the same dataset. This is entirely in-sample. No out-of-sample validation demonstrates superiority over a simpler model. The result reflects trend capture, not structural insight.
Retail
05
The Missing Mechanism
Without a mechanism, persistence is assumed, not demonstrated.
No generative mechanism explains why the power law should persist. “Adoption drives price” is not a mechanism. A mechanism specifies structure, stability, and failure conditions. None are defined. Without that, the power law is description — not explanation.
Retail
II
Academic Variant — Four Additional Faults
Santostasi & Perrenod (2026) // Mechanistic Derivation Claimed
Part II
06
Composition Identity is Algebra, Not Evidence
The “three confirmations” are mathematically guaranteed to agree.
The paper presents three independent estimates converging on β = 5.69. They are derived from the same dataset. Their agreement is a direct consequence of algebraic substitution — not independent validation. Figure 2’s three arrows illustrate algebra, not evidence.
Academic
07
Endogeneity and the Missing Causal Test
Price may be driving adoption — not the other way around.
The model assumes addresses drive price. The reverse causal path is equally plausible and untested. No Granger causality analysis is performed. If causation runs from price to addresses, β_M is a reflexivity exponent, not a network value exponent — and the compositional framework collapses.
Academic
08
The Bayesian Precision is Artificial
The confidence interval is ~5× tighter than it should be.
Rolling regressions reuse nearly identical data, creating extreme autocorrelation. The effective sample size collapses from 1,899 to 40–100. Correcting for this widens the 95% CI from [5.703, 5.754] to approximately ±0.29 — spanning the full observed range of rolling estimates. The “physical constant” precision is a statistical artifact.
Academic
09
The Mechanism Predicts Its Own Failure
Even if correct, the model implies it will stop working.
The adoption mechanism requires slowing growth as saturation occurs — reducing the exponent, and with it, the price projection. At the same time, ETF custody structurally breaks the address–user relationship: BlackRock IBIT alone holds ~570,000 BTC for 500,000+ investors behind a handful of addresses. The key input variable is degrading in the period treated as confirmatory.
Both
A curve fit is not a physical law. An R-squared of 0.96 on a non-stationary regression is not science. Description is not obligation.
Paul Faulkner — The Determinism Problem, April 2026
The Central Conclusion
What the evidence
does and doesn’t establish.
✓ What Both Analyses Do Show
Legitimate empirical findings
Bitcoin’s price history from 2010–2026 is well-described by a power law with exponent ~5.69
Residuals are stationary over the historical observation period (ADF p ≈ 0.0075)
Address growth follows a cubic trend and price scales superlinearly with address count
These three observations are mutually consistent — as algebra requires them to be
Bitcoin exhibits stronger scale invariance than NASDAQ or gold over the historical record
✗ What Neither Analysis Can Show
The unsupported claims
That beta = 5.69 will remain the correct exponent in cycles 5, 6, or beyond
That the adoption network has the structural properties the epidemic analogy requires
That the floor is a genuine causal constraint rather than a regression line through trending data
That the Bayesian stability result is informative about future stability rather than historical consistency
That price at any given future date will respect the power law’s confidence interval
Diagnostic Tool
The advocacy
checklist.
The paper includes a seven-question diagnostic that any practitioner can apply to identify when a quantitative model has crossed from analysis into advocacy. A model that fails any two tests should be treated as a narrative. A model that fails four or more should be treated as advocacy dressed as analysis.
Checklist Failures — Both Variants
5/7
Retail
Monte Carlo
Monte Carlo
6/7
Academic
Santostasi
Santostasi
Diagnostic Question
Retail Variant
Academic Variant
Does the model derive a future price path from historical data alone, without reference to macro conditions, regulatory environment, or market structure?
✗ Fails
✗ Fails
Are the uncertainty bands wide enough that the model cannot be directionally wrong over a 12-month period?
✗ Fails
✗ Fails
Is the floor defined relative to the model itself — repricing upward so it cannot be permanently breached?
✗ Fails
✗ Fails
Are the falsification criteria set at horizons practically untestable for a retail investor?
✗ Fails
✗ Fails
When challenged, does the model’s advocate claim critics haven’t understood the model rather than engaging with specific methodological objections?
✗ Fails
✗ Fails
Does the model produce a dramatically higher price target — sufficiently distant to be unverifiable in the short term?
✗ Fails
✗ Fails
Has the model been validated out-of-sample, on withheld data, with pre-specified criteria for success and failure?
✗ Fails
~ Partial
Historical Parallel
The Stock-to-Flow
pattern repeats.
Stock-to-Flow // Plan B // 2019–2022
The Predecessor
OLS regression of I(1) price series against a deterministic trend variable, without cointegration testing
High R-squared interpreted as causal relationship rather than statistical artifact
Mechanism asserted rather than derived: supply scarcity drives value
Uncertainty bands wide enough to accommodate a very large range of outcomes
Falsification criteria that retreated as each projected price level was missed
Power Law // Santostasi & Perrenod // 2026
The Successor
OLS regression in log-log space with residual stationarity claimed but causality untested
R-squared of 0.961 cannot be compared to standard distributional benchmarks
Mechanism post-hoc rationalised: epidemic spreading analogy fitted to observed exponent
±0.30 dex bands (factor of 2 on each side) cannot be directionally wrong
Falsification conditions structured to be untriggerable before 2050
Stock-to-Flow Outcome
Projected $288,000 by December 2021 // Bitcoin reached $69,000 // Fell to $15,500 // Model declared its own bands had been “misread”
The sophistication of the apparatus is not evidence of the validity of the model. It is evidence of the sophistication of the people who built it. These are not the same thing.
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Read the
Read the
full analysis.
22 pages. Nine structural faults. A diagnostic checklist for any quantitative model. The Stock-to-Flow parallel documented. Free. No email required. No subscription. Just the work.
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Paul Faulkner
25+ years across JPMorgan Chase, PwC, SG Kleinwort Hambros, Bradford & Bingley. The same rigour applied to subprime risk documentation in 2007 is applied to digital asset market structure today. Founder of The Rogue Protocol — forensic intelligence for practitioners who need the analysis behind the headline.
