Bitcoin · power law · the prosecution
This whole site leans on the power-law floor. So here's the fair test: an honest, adversarial attempt to break the mathematics behind it, the maths Giovanni Santostasi lays out in his power-law theory. Six attacks, including the floor's own circularity, the effective sample size, and survivorship. I'll prosecute it as hard as I can, then rule on what survives.
01 / WHAT'S ON TRIAL
Santostasi's theory isn't just "price fits a line." It's a stack of claims, and to "disprove the math" we'd have to break one of the load-bearing ones. Here's the stack:
On top of the arithmetic sit three stronger claims: that the relationships are mechanistically necessary (feedback loops where "the output becomes the new input"), that "the power laws we observe in Bitcoin are what Bitcoin is" (i.e. a law, not a curve fit) and, strikingly, that scarcity plays no role at all. It's pure network effect.
To convict, we need to show one of three things: the arithmetic is wrong, the derivation doesn't follow, or the fit is an illusion. Six attacks follow.
02 / ATTACK ONE
The headline evidence is a tight straight line on a log–log chart. But log–log axes are forgiving to the point of being misleading: compress fifteen years and several orders of magnitude onto both axes and almost any monotonic, multi-order-of-magnitude growth looks linear. A high R² in log space is a low bar. Worse, regressing one trending series on another (or on time) is the textbook setup for a spurious regression (inflated R² and significance even when there's no real relationship), and Santostasi explicitly waves away the autocorrelation tests that would catch it.
This is the strongest statistical punch and it connects: the tight log–log fit is weaker evidence than it looks, and dismissing autocorrelation diagnostics is a genuine methodological hole. But it doesn't disprove the math. The relationship survives tougher tests its critics rarely run: it holds out-of-sample (a 2019 fit hind-casts the following years), the residuals are bounded in a stable channel rather than drifting, and the same exponent reappears across independent quantities (price, hash-rate, addresses). That's more than a lonely log–log line. Verdict: lands a bruise, not a knockout.
03 / ATTACK TWO
The clean integers (3, 6, 12) look like a derivation but smell like decoration. Two cracks. First, Metcalfe's law (value ∝ n²) is not settled maths: Odlyzko and others argue network value scales like n·log(n), which would change the whole composition. Second, the "derived" price exponent is 6, but the data's exponent is about 5.7, and the address exponent is admitted to be ~1.95, not a clean 2. So the elegant integers are rounded to fit the story. The real numbers are non-integer and were read off the chart first. That's curve-fitting wearing a lab coat.
This is fair, and it's the most damaging point: the specific exponents are empirical, not deduced. "It must be exactly t⁶ because Metcalfe" is overclaiming: pick n·log(n) and the tidy integers dissolve. But notice what this does and doesn't kill. It refutes the claim that the exponent is derivable from first principles. It does not refute that a power law with exponent ≈5.7 fits, since fitting a non-integer exponent is still a valid power law. Verdict: the "necessary integer" story is disproved. The power-law fit itself is untouched.
04 / ATTACK THREE
The derivation reads as a one-way street: addresses drive price, price drives hash-rate. But every arrow runs backwards too: people open addresses because price is rising, miners add hash because price is rising. Santostasi celebrates this ("the output becomes the new input"), but you can't have it both ways: if it's a feedback loop, you can't cleanly assign "price ∝ addresses²" as causation. The correlations are real. The causal direction is assigned, not proven. A reflexive loop with no exogenous anchor can wander anywhere. It doesn't have to trace a power law.
Correct that the causation is underdetermined: this is association dressed as mechanism, and a self-referential loop is not guaranteed to produce a power law. But "we can't prove the causal arrows" is not the same as "the maths is wrong." Coupled feedback systems do commonly settle into power-law scaling, and the model's job here is description, not a proof of mechanism. Verdict: it exposes the mechanism as unproven. It does not break the relationship.
05 / ATTACK FOUR
Calling it "what Bitcoin is", a law of nature, is the real overreach. A law in physics is derived, dimensionless, and exact to many decimals. This is a fitted exponent with a plausible story. And no curve fit, however tight, escapes the induction problem: fifteen years of a regularity is not a guarantee of the sixteenth. The theory even keeps escape hatches ("phase transitions," "catastrophic events," "regime changes") that can absorb any deviation, which quietly erodes its falsifiability.
This one wins outright, but against the rhetoric, not the equations. "Law of nature" is unearned, "the future must obey" is unprovable, and the escape hatches do weaken falsifiability. Strip all of that away and what's left is still intact: a strong empirical regularity with a coherent (if unproven) mechanism. Verdict: the "law" framing is disproved. The descriptive model stands.
06 / ATTACK FIVE
The four attacks above prosecute the trendline. But this site's load-bearing object is the floor, and any floor invites its own circularity charge: draw a line under fifteen years of prices after the fact, and "price held the floor" is close to tautological — the line was placed around the data. And if price ever breaks it, nothing stops a quiet refit lower, resetting the record. A support level that can be redrawn after every violation isn't support; it's hindsight with a ruler.
Three specifics blunt the attack here. First, this floor is not a hull traced under the lows — it is the fitted trendline shifted down two standard deviations of the log residuals (a −2σ quantile band). By construction, a few percent of historical days should fall below it, and some have: price spent hours beneath the band during the March 2020 crash and the model carried on. So the site's claim was never "never broken"; it is that price spends the overwhelming majority of time above the band and the excursions below have been brief. Second, the parameters are frozen and published — A = 10−16.493, exponent = 5.688, σ = 0.20 — fixed in a single source file as of 6 Jun 2026 and not refit since. Anyone can recompute tomorrow's floor from those constants and check it matches. Third, the break condition is stated in advance and survives no refit: a sustained close below the frozen −2σ line — an extended stretch well beneath it, with the log–log R² degrading and the slope forced materially lower on each refit — falsifies the model, and this site commits to saying so rather than redrawing the line. Verdict: the generic circularity charge convicts floors that are refit after the fact. A frozen, published, quantile-based floor with a pre-stated break condition is the falsifiable version — the only version this site leans on.
07 / ATTACK SIX
Two statistical objections in one. Sample size: fifteen years sounds long, but in regime terms Bitcoin has lived through roughly four halving cycles and a handful of macro-liquidity regimes. Fitting a two-parameter curve through four cycles is not a large-n result, whatever the daily data count says: the observations are heavily autocorrelated, so the effective sample is tiny. Survivorship: Bitcoin is the winner drawn from thousands of dead or dying cryptoassets. Fit a smooth growth curve to the ex-post survivor and you will always find one; the graveyard doesn't get plotted. The tight fit may say less about Bitcoin's nature than about which asset we chose to chart.
The small-effective-n point is correct and permanently caps the confidence anyone should place in any Bitcoin model, power law, S2F, or otherwise; this model earns no exemption. The survivorship point lands on "the power law was inevitable" (a claim this page already convicted in Attack Four) but not on the descriptive claim: conditioning on the asset that exists, its history still traces a stable power law, which remains informative for anyone holding that asset. What survivorship kills is extrapolating the fit as if Bitcoin were a physical system rather than a surviving one. Verdict: confidence must be discounted for both reasons, and the discount is the point of this page. The description stands, at reduced certainty.
08 / THE VERDICT
What survived did so on the merits, not on a technicality. The fit holds out-of-sample when frozen at a past date, the residuals stay in a bounded channel rather than drifting, and the same exponent recurs across independent quantities (price, hash-rate, addresses). And per Attack Five, a pre-registered frozen floor carries a stated break condition, which makes the surviving claim falsifiable, the opposite of "cannot be disproved." As a description of what has happened, the support math holds.
What the attacks did land on was everything wrapped around the equations: the claim that the exact exponent is derived rather than fitted (it's fitted, ~5.7, not a clean 6), that the causal chain is proven rather than assigned (it's assigned), that the result is a law of nature the future is obligated to obey (it isn't), that the naive never-broken floor is circular (only the frozen, pre-registered version escapes), and that confidence is capped by effective sample size and survivorship (four cycles and an ex-post winner).
So the honest bottom line is two-sided, and both sides matter:
It survives, and it's falsifiable. As a mathematical description of Bitcoin's fifteen-year history, the power law is real, tight, and robust. The version that stands is the frozen, pre-registered one, which carries its own break condition, so it can be disproved and simply hasn't been. This page attacked it hard and it held.
But it was never proof. A description that can't be disproved is not the same as a law that must continue. The support line is a powerful empirical regularity, not a theorem, and treating it as a guarantee is a category error the maths does not license.
Which is exactly how the rest of this site treats it: anchor to the floor because it is the most reliable thing we have, while never forgetting it's a regularity, not a promise. The strongest honest statement is: not disproved, and not proven. Supported.
This is an argument about evidence and inference, not financial advice. I lean on the power law myself. This page is the steel-manned case against the thing I rely on, because a thesis you won't attack isn't a thesis, it's a belief. Read Santostasi's original power-law theory and judge for yourself.