Bitcoin Power Law — Empirical Test Suite

80 tests and 20 simulations behind Bitcoin’s Price Power Law Decomposed — published as run, including the wrong turns. Full index and reproduction guide: README.

β = βA × (1 + 1/α)  ≈  3.0 × 1.83  ≈  5.5   |   measured 5.62 ± 0.21

The monetary program (the detour)

M2 deflation figure
Test 1 detour
M2 deflation
Does deflating by US M2 lower the exponent as predicted?
verdict: supported · results
M2 vs CPI figure
Test 2 detour
M2 vs CPI
Money creation, not price inflation: ratio 3.14 vs 2.40 mechanical.
verdict: supported · results
Deflator hierarchy figure
Test 4-5 detour
Deflator hierarchy
M2 and S&P are valid deflators; gold and oil are not.
verdict: supported · results
Global M2 figure
Test 6 detour
Global M2
A global M2 index removes 22-24% more exponent than US M2.
verdict: supported · results
China weight sweep figure
Test 8 detour
China weight sweep
The global-M2 result holds for any China weight 5-42.5%.
verdict: mixed · results
MSCI World deflator figure
Test 9 detour
MSCI World deflator
Global equities are not a better deflator than the S&P 500.
verdict: refuted · results
Rolling M2 ratio figure
Test 10 detour
Rolling M2 ratio
The global/US ratio is unstable in rolling windows.
verdict: inconclusive · results
Forex-adjusted M2 figure
Test 11 detour
Forex-adjusted M2
The global-M2 advantage survives USD conversion (1.08-1.17).
verdict: supported · results
DXY figure
Test 17 detour
DXY
Dollar strength compresses BTC/USD (exponent -6.4).
verdict: supported · results
Alt-CPI Cantillon figure
Test 20 detour
Alt-CPI Cantillon
The ~2/3 asset channel survives 6 CPI methodologies (67% +/- 5%).
verdict: supported · results
Global M2 pipeline figure
Test 40-43 detour
Global M2 pipeline
17-35 countries: epsilon(Colgate) improves to 2.7%.
verdict: supported · results
Deep residual figure
Test 44 detour
Deep residual
No variant (DXY, credit, velocity) beats raw global M2.
verdict: refuted · results
documentary — no figure
Test 46 detour
Two-channel M2
Non-US M2 acts through adoption, not the denominator.
verdict: supported · results
documentary — no figure
Test 47 detour
Deflator comparison
M2 x Loans is the optimal deflator (epsilon = 0.07%).
verdict: supported · results
documentary — no figure
Test 48 detour
Global Cantillon
Globally the split is 49/51, not 68/32 - the US result was local.
verdict: refuted · results
Credit US/non-US figure
Test 51 detour
Credit US/non-US
Multicollinearity prevents separating the credit channels.
verdict: mixed · results
Forex residual figure
Test 52 detour
Forex residual
DXY adjustment does not reduce the residual (1.4% to 1.3%).
verdict: refuted · results

Measurement and proxies

ETF correction figure
Test 3 corroborative
ETF correction
The post-2024 address stall is an ETF aggregation artefact.
verdict: supported · results
Proxy deep dive figure
Test 12 corroborative
Proxy deep dive
The balance proxy fits better (R² 0.947) and kills B_x.
verdict: mixed · results
Bivariate N vs t figure
Test 13 corroborative
Bivariate N vs t
N and t are too collinear (R² 0.976) for causal attribution.
verdict: mixed · results
Block fullness figure
Test 15 corroborative
Block fullness
Full blocks (81% in 2024+) suppress address growth (p=0.015).
verdict: supported · results
Ordinals figure
Test 16 corroborative
Ordinals
Inscriptions did not inflate address counts - they fell.
verdict: refuted · results
Questioning t³ figure
Test 36 ★ load-bearing
Questioning t³
t³ is not rejected (p = 0.68); proxy degradation undecidable.
verdict: mixed · results
documentary — no figure
Test 49 corroborative
UTXO gamma
UTXO balances measure wealth concentration, not network topology.
verdict: inconclusive · results
ETF structural break figure
Test 53 ★ load-bearing
ETF structural break
betaA collapses 3.19 to 0.55 post-ETF; beta stays at 5.65.
verdict: supported · results

Model mechanics

Multifactor model figure
Test 14 corroborative
Multifactor model
P ~ N^1.26 x SP^2.35 fits best (R² 0.969) - but see Test 18.
verdict: supported · results
Coefficient stability figure
Test 18 ★ load-bearing
Coefficient stability
Multifactor coefficients are unstable (CV 258%); the temporal law wins.
verdict: refuted · results
Factorisation stability figure
Test 21 ★ load-bearing
Factorisation stability
beta = betaA x betaM: product stable (4%), components drift.
verdict: supported · results
w(t) decomposition figure
Test 28 corroborative
w(t) decomposition
Capital per adopter grows as t^3.19; regime change found.
verdict: mixed · results
Compensation (flows) figure
Test 29 superseded
Compensation (flows)
Exchange flows cannot measure the compensation.
verdict: inconclusive · results
Realized cap figure
Test 30 corroborative
Realized cap
The N/w compensation is real in cost-basis terms (r = -0.864).
verdict: supported · results
w_realized anatomy figure
Test 31 corroborative
w_realized anatomy
w_realized is smoothed price - Test 30 is an accounting identity.
verdict: refuted · results
Convolution model figure
Test 32 ★ load-bearing
Convolution model
k ~ 0 with alpha = 1.82; the 2010-20 law predicts 2021-25 at -0.002 dex.
verdict: mixed · results
Why 1.82? figure
Test 33 superseded
Why 1.82?
Richest-first Pareto cannot do it - resolved by inverse ordering.
verdict: mixed · results
Inverse Pareto figure
Test 34 corroborative
Inverse Pareto
Later adopters bring MORE capital (marginal ~ N^1.10).
verdict: supported · results
Beta as attractor figure
Test 35 ★ load-bearing
Beta as attractor
betaA and betaM compensate (r = -0.79); beta is the stable object.
verdict: mixed · results
Zipf vs Metcalfe figure
Test 37 superseded
Zipf vs Metcalfe
Bitcoin sits between Zipf and Metcalfe scaling (gamma ~ 2.72).
verdict: supported · results
Topology formula figure
Test 38 superseded
Topology formula
Three independent measures recombine within 0.8%.
verdict: mixed · results
Residual analysis figure
Test 39 superseded
Residual analysis
The 0.8% residual is statistically insignificant (0.36 sigma).
verdict: supported · results
Gamma expanding figure
Test 50 superseded
Gamma expanding
Implied gamma converges to 2.67 with no trend (p = 0.50).
verdict: supported · results

The adoption epidemic (betaA ~ 3)

Halving attenuation figure
Test 7 corroborative
Halving attenuation
Post-halving bubbles shrink monotonically: 1.45 to 0.42 dex.
verdict: supported · results
Bass saturation figure
Test 19 corroborative
Bass saturation
Bitcoin's share of global money is still accelerating (1.52%).
verdict: supported · results
Exchange flows figure
Test 22 corroborative
Exchange flows
Contra-cyclical accumulation; outflow grows as t^3.55.
verdict: supported · results
DCA deepening figure
Test 23 corroborative
DCA deepening
Per-holder deepening exists but is too noisy to isolate.
verdict: inconclusive · results
Google Trends figure
Test 24 ★ load-bearing
Google Trends
Cumulative narrative exposure grows as t^3.0-3.75 (R² > 0.94).
verdict: supported · results
documentary — no figure
Test 25 corroborative
Adoption gradient
Every funnel stage (curiosity to custody) follows ~t³.
verdict: supported · results
documentary — no figure
Test 26 corroborative
ETF vs exchange flows
No paper-Bitcoin signal; ETFs are 38% of exchange flow.
verdict: refuted · results
documentary — no figure
Test 27 corroborative
Institutional holdings
3.82M BTC institutional (19.3% of supply); MSTR is 61% of corporate.
verdict: supported · results
Adoption cloud figure
Test 45 corroborative
Adoption cloud
Conceptual visualization of adoption time x wealth.
verdict: illustration · results
Cognitive thresholds figure
Test 54 corroborative
Cognitive thresholds
One psychological mechanism gives t^1.15, not t³ - three needed.
verdict: refuted · results
Adoption cloud v1 figure
Test 56 superseded
Adoption cloud v1
First combined cloud - superseded by the Test 57 series.
verdict: supported · results
Global adoption clouds figure
Test 57 corroborative
Global adoption clouds
Eight entity classes: individuals ~5%, companies 0.13%, central banks 0%.
verdict: supported · results
Wealth & Rogers figure
Test 58 corroborative
Wealth & Rogers
Background: global wealth pyramid and Rogers lifecycle.
verdict: figures
GDP vs GII figure
Test 59 ★ load-bearing
GDP vs GII
Institutional adopters cluster at the innovation frontier (median GII rank 13/133).
verdict: supported · results
Size & ownership vs GII figure
Test 60 ★ load-bearing
Size & ownership vs GII
Two adoption drivers: innovation and monetary necessity.
verdict: supported · results
HODL waves figure
Test 63 corroborative
HODL waves
49% of supply held >1y; long-term holdings grow as t^0.63.
verdict: supported · results

The wealth Pareto (alpha ~ 1.2)

Pareto predicts gamma figure
Test 55 ★ load-bearing
Pareto predicts gamma
alpha = 1.22 from the wealth literature predicts the exponent, no free parameter.
verdict: supported · results
Pareto by stratum figure
Test 61 corroborative
Pareto by stratum
Sampling bias dominates; individuals converge to alpha ~ 1.25.
verdict: inconclusive · results
On-chain Pareto figure
Test 62 ★ load-bearing
On-chain Pareto
BTC wealth is a clean Pareto, alpha ~ 1.07 - a different quantity than fiat alpha.
verdict: mixed · results
Merged-strata alpha figure
Test 81 corroborative
Merged-strata alpha
The merged entity-size distribution is not a single Pareto; the naive 1+1/alpha identity fails.
verdict: inconclusive · results

On-chain cross-checks

Daily narrative figure
Test 64 ★ load-bearing
Daily narrative
Wikipedia cumulative pageviews ~ t^3.09 - third independent t³.
verdict: supported · results
Hot/cold supply figure
Test 65 corroborative
Hot/cold supply
The actively-moving fraction declines as t^-0.63.
verdict: supported · results
ETF decomposition figure
Test 66 corroborative
ETF decomposition
IBIT is 57% of ETF BTC (alpha = 0.42, most concentrated stratum).
verdict: supported · results
M2 cross-validation figure
Test 67 corroborative
M2 cross-validation
Independent global-M2 agrees (r = 0.90); the '10-week lead' vanishes in returns.
verdict: mixed · results
Sentiment figure
Test 68 corroborative
Sentiment
Sentiment lags the power-law residual - procyclical, not predictive.
verdict: refuted · results
Metric exponents figure
Test 69 corroborative
Metric exponents
Hashrate ~ t^11.6 = 2 beta; MVRV exactly trend-neutral.
verdict: supported · results
Feedback loop figure
Test 70 corroborative
Feedback loop
Narrative leads price; price leads adoption and hashrate.
verdict: supported · results
Realized price + SOPR figure
Test 71 corroborative
Realized price + SOPR
Aggregate cost basis grows at the same rate as price (t^5.54).
verdict: supported · results
Power law zoo figure
Test 72 corroborative
Power law zoo
16 exponents, 7 algebraic relationships, all within 3.5%.
verdict: supported · results

The adversarial round & external controls

alpha_implied mechanics figure
Test 73 ★ load-bearing
alpha_implied mechanics
The 'convergence' plot is mechanical, not physical.
verdict: refuted · results
Honest error bars figure
Test 74 ★ load-bearing
Honest error bars
Real uncertainty is x14 naive: beta = 5.62 +/- 0.21.
verdict: mixed · results
t0 sensitivity figure
Test 75 ★ load-bearing
t0 sensitivity
beta depends on the clock convention; betaM = 1.83 is the invariant.
verdict: mixed · results
Negative controls figure
Test 76 ★ load-bearing
Negative controls
Only Bitcoin has the surplus: +0.84 vs ~0 for ETH and LTC.
verdict: supported · results
Penetration schedule figure
Test 77 ★ load-bearing
Penetration schedule
Capital/holder ~ N^1.0 where flat orderings predict N^0.
verdict: supported · results
Domain names figure
Test 78 ★ load-bearing
Domain names
A non-crypto cognitive epidemic with scarcity: betaA = 3.01.
verdict: supported · results
Mobile phones figure
Test 79 ★ load-bearing
Mobile phones
A weakly-gated contagion grows exponentially, not as t³ - registered prediction.
verdict: supported · results
documentary — no figure
Test 80 corroborative
Cross-sectional Metcalfe
A 4-point cross-chain fit cannot constrain the exponent.
verdict: refuted · results
Ordering noise (simulation) figure
sim22 ★ load-bearing
Ordering noise (simulation)
The surplus requires geometric tail penetration; rank ordering produces none.
verdict: supported · results
Facebook beta (simulation) figure
sim23 corroborative
Facebook beta (simulation)
On a flat ARPU population the social-network value exponent needs no wealth mechanism.
verdict: supported · results
Empirical weights (simulation) figure
sim24 corroborative
Empirical weights (simulation)
Injecting measured per-stratum Pareto shapes leaves gamma unchanged: the hand-set weights did no hidden work.
verdict: supported · results

Agent-based simulations — the exploration record

Composition delta = alpha x gamma figure
sim01 corroborative
Composition delta = alpha x gamma
Is the temporal exponent the product of adoption and value exponents? Exactly — ratio 1.000 across all five topologies.
verdict: supported · results
Dynamic vs static gamma figure
sim02 corroborative
Dynamic vs static gamma
Measuring value on the growing adopter set vs the final network changes gamma massively — the dynamic exponent is the right object.
verdict: supported · results
Reflexivity figure
sim03 corroborative
Reflexivity
Value-to-adoption feedback mostly changes speed, not exponents: composability survives.
verdict: supported · results
Pareto weights: hubs = rich figure
sim04a corroborative
Pareto weights: hubs = rich
Wealth placed on network hubs: unstable and not very informative.
verdict: mixed · results
Pareto weights: late = rich figure
sim04b corroborative
Pareto weights: late = rich
Wealth arriving late raises gamma in a stable, significant way.
verdict: supported · results
Sequential strata figure
sim04c corroborative
Sequential strata
Stratified activation is the most powerful mechanism tested: gamma_weighted = 2.33.
verdict: supported · results
Where is alpha = 3? figure
sim05 corroborative
Where is alpha = 3?
Only 8.1% of the topology grid yields alpha in [2.8, 3.2] — a moderate basin, not a universal attractor.
verdict: mixed · results
Delayed activation figure
sim06 corroborative
Delayed activation
Adoption is not value creation: activation delay significantly raises the measured gamma.
verdict: supported · results
Churn figure
sim07 corroborative
Churn
De-adoption barely moves the exponents — the law is robust to churn.
verdict: supported · results
Facebook/Tencent control figure
sim08 corroborative
Facebook/Tencent control
Real networks with known data: Facebook validates the framework exactly (gamma 1.20); Tencent deviates.
verdict: supported · results
Real Bitcoin weights figure
sim09 corroborative
Real Bitcoin weights
Naive V = sum(w*w) with real weights gives gamma = 4.44 — far too high; motivated the sublinear value function (sim17).
verdict: refuted · results
Weight ratio sweep figure
sim11 corroborative
Weight ratio sweep
gamma = 1.9 corresponds to inter-stratum weight ratios r = 3-4.
verdict: supported · results
Delay + strata figure
sim13 corroborative
Delay + strata
The graded delay profile gives gamma(adopted) = 1.86 — close to Bitcoin's 1.9.
verdict: supported · results
Bitcoin on-chain validation figure
sim14 corroborative
Bitcoin on-chain validation
The framework reproduces Bitcoin's own on-chain exponents, especially with total addresses.
verdict: supported · results
Sublinear value f(w) = w^beta figure
sim17 corroborative
Sublinear value f(w) = w^beta
beta = 0.4 brings gamma back to 1.93-1.98 with real Bitcoin weights.
verdict: supported · results
Full calibrated model figure
sim18 corroborative
Full calibrated model
All mechanisms assembled: alpha = 3.03 and gamma = 2.01 simultaneously (product ratio 1.005).
verdict: supported · results
Thermostat figure
sim19 corroborative
Thermostat
An adoption shock is compensated by value adjustment — the attractor mechanism in silico.
verdict: supported · results
Pareto sweep figure
sim20 superseded
Pareto sweep
Once read as verifying gamma = 1 + 1/alpha; downgraded — sensitivity ~0.3 instead of 1.
verdict: downgraded · results
Triple process figure
sim21 corroborative
Triple process
Three coupled ~t processes compose to alpha = 3 (variant B: 3.14).
verdict: supported · results