Bitcoin has a significantly positive convexity from its long-run structural trajectory, enhanced by its volatility. Its log-periodicity changes the timing and intensity but not the underlying structure. Once the structural scale invariance is removed the residual risk appears closer to Gaussian than raw returns suggest.

Because Bitcoin has a long-term power law trajectory, its highly positive convexity can be considered as having four components. We can explain that with two main buckets: the structural scale invariance bucket, and the residual risk bucket. Table 1 shows how the two structural components (first two rows) and the two residual components (last two rows) evolve with Bitcoin’s age.

The scale invariant convexity is curvature of the fitted price trajectory through time; residual convexity is a Jensen adjustment arising from a return distribution. For this article, both are treated as approximate one-year log price contributions.

Scale Invariance

Bitcoin’s scale invariance has two components, one is the the continuous scale invariance (CSI) of the power law, and the second is the discrete scale invariance (DSI) of its observed log periodicity. A large portion of the bubble and trough variations can be explained with log periodicity with as few as one or two modes. See:

Models with either one or two log periodic modes and smooth amplitude decay outperform a pure power law in standard out-of-sample testing on 6 to 30 month horizons. Think of the DSI component as being the cyclical modulation of structural convexity.

Expected log-price = Power law + DSI + residual convexity (volatility, skew)

P(t) = A t^β (β >> 1) , expresses a power law of price vs. age of Bitcoin t.

dP²/dt² = A β (β-1) t ^β-2 expresses the curvature of the trajectory via its second derivative in time.

The normalized convexity of the power law is the ratio β (β-1)/t².

Currently the percentage contribution from CSI equals:

0.5 β (β-1)/t² , or 4.5% for an age of 17.66 and power law exponent of 5.85 (β, based on QR median as used in the above referenced article).

It is identifiable as the second order term in the Taylor series expansion of total power law annual return:

(1 + 1/t)^β - 1, the first order term is just β/t, which is the mean instantaneous (and uncompounded) return.

DSI

The DSI component has been observed to fall in log price amplitude as approximately a reciprocal with offset. Thus while the CSI component is declining as the inverse square of age, the DSI component falls approximately as the inverse cube. However, it varies dramatically over its cycle, which unfolds in log time, the phase matters greatly.

I am not showing all the details here—which account for both the long-term decay in DSI amplitude and the current phase—but at age 17.66, DSI with two modes (ω = 8.57 and 23.58) contributes about 1.1 times the current CSI convexity of 4.5%. The two contributions of CSI and DSI are additive in log-price curvature. That puts the DSI portion at about 5.0% and the total at 9.5%.

Both the DSI and total SI peaks are projected to occur around May 2027. That is a maximum boost to the power law as the DSI signal is currently in decline due to the second mode, and is expected to trough and turn upward. At that point the second derivative will be maximally positive. However, the total SI contribution remains positive until well into 2029 (past age 20).

Again, the DSI component is sensitive to the phase in the cycle, which however is now long, around 13 years for the fundamental, and 5 years for the higher order mode.

Current SI convexity (CSI + DSI) corresponds locally to about a 9.5% annualized log-return enhancement relative to a linear extrapolation of the current trajectory. Around the May 2027 peak the DSI will be more favorable in phase terms and the total SI estimate increases to 11.7% convexity. A year after that it should decrease to about 7.6%.

Over the 10-year period from Bitcoin age 15 to age 25, the total SI convexity is positive 80% of the time; the DSI component alone is negative (concave) 47% of the time.

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Residual Convexity

The remaining price volatility after removing the power law and log periodicity, provides additional convexity, as does skewness if it is positive (otherwise it subtracts from the convexity). Figure 2 shows the residuals after removing both the power law and two-mode log periodicity (top panel), and the volatility and skewness using monthly residual returns normalized to annual values. Both volatility and skewness dropped considerably from 2016 on, volatility remains large, but skewness has been quite small in magnitude during the current decade.

Combining SI and Residual Convexity

Figure 4 combines both categories of convexity including projection out to age 25 (2034). Residual convexity dominates across the full 10-year period. Both the SI bucket and the residual bucket are declining on average. The convexity drops below 15 from age 20 based on this projection.

Conclusion: Bitcoin Has a Changing Convexity Structure

The central realization is that Bitcoin’s convexity has two distinct sources: structural convexity from its scale-invariant price trajectory and distributional convexity from movements around that trajectory. Separating them provides a more complete picture than either power-law modeling or conventional volatility analysis alone.

The structural component combines a persistent power-law contribution with an oscillating log-periodic contribution. Continuous scale invariance supplies a positive baseline that declines smoothly with age. Discrete scale invariance modulates that baseline, alternately reinforcing or opposing it. DSI can therefore produce strongly convex or locally concave phases without invalidating the long-run power-law structure.

Removing the power law and two-mode DSI structure does not eliminate volatility. Monthly residual returns retain substantial dispersion, which supplies the largest estimated source of convexity over most of the modeled interval. DSI captures considerable variation in log-price levels but comparatively little month-to-month return variance under this global specification. Structural predictability and residual risk coexist.

The residual skewness contribution is generally small relative to the volatility contribution and was mildly negative through much of the latest five-year period. Residual convexity is therefore primarily a volatility effect rather than the result of persistent positive skewness.

Across the projected age-15-to-25 interval, residual convexity dominates, while SI supplies a smaller but economically meaningful cyclical component. At age 17.66, the model projects approximately 9.5 log-percentage points of SI convexity and 14.2 points of residual convexity, producing a grand total near 23.7. These are modeled convexity contributions rather than guaranteed returns. A log contribution (C) has the simple-return equivalent (e^C-1).

Both sources are projected to decline as Bitcoin matures, but at different rates. Residual convexity falls relatively smoothly with volatility. CSI declines with the inverse square of age, while DSI decays more rapidly on average but continues to oscillate in log time. Temporary DSI peaks can therefore interrupt the longer-term decline in aggregate convexity.

The new perspective is not simply that Bitcoin is convex, but that it has a changing convexity structure. Early in its history, structural acceleration and residual volatility were both large. As Bitcoin matures, residual volatility is projected to decline and power-law curvature weakens, yet log-periodic phase changes can still generate intervals of renewed structural convexity. Bitcoin may become less convex without becoming conventionally linear.

Finally, convexity measures curvature in the modeled price path and the effect of residual return dispersion; it is not guaranteed excess return. Its economic realization depends on investment horizon, path, exposure, rebalancing policy, and model validity. Subject to these qualifications, the framework provides a unified interpretation of Bitcoin’s long-run growth, recurrent cycles, declining volatility, and evolving return distribution.

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This is not investment advice. Stephen Perrenod engages in econophysics and network science research as Vice-Director at the Scientific Bitcoin Institute.

Appendix: Summary of Modeling Steps

Monthly Bitcoin–U.S. dollar prices were obtained from the Coin Metrics public dataset, beginning in July 2010. Bitcoin age, (T), was measured in years from the genesis date of January 3, 2009. Prices and returns were expressed in natural logarithms.

The structural price model comprised a power-law backbone and two discrete-scale-invariant oscillatory modes. The power-law exponent was fixed at the quantile-regression estimate (β=5.85), while the intercept was estimated as the median log-price offset. The DSI specification used the global parameters:

(A₁,ω₁,φ₁) = (0.2597,8.5713,1.3630) and (A₂,ω₂,φ₂) = (0.1626,23.5847,2.4479).

Because the published DSI amplitudes were expressed in log10 price, they were converted to natural-log units by multiplying by ln 10. Both modes were proportionally rescaled using the long-term, two-sided amplitude envelope (4/(T+8)). Interpreting this as a peak-to-peak envelope gives a combined one-sided amplitude of (2/(T+8)).

Monthly log-price residuals were then calculated as e_t = ln P_t - ln P_model_t

Rolling volatility and skewness were calculated from the residual returns using a trailing window of fixed log-age width, Δ ln T=0.25.

Because a fixed log-age window expands in calendar time as Bitcoin matures, the latest window contained 47 monthly residual returns. Volatility was calculated as the sample standard deviation of monthly residual returns and annualized by multiplying by sqrt{12}. Skewness was calculated using the bias-corrected sample estimator.

Residual volatility and skewness were converted into approximate one-year log-price convexity contributions using a cumulant expansion.

Total residual convexity was defined as C_{vol} + C_{skew}.

To project residual convexity from Bitcoin ages 15–25, the latest five years of rolling estimates were fitted as functions of log age. Annualized volatility was fitted in logarithms to ensure positive projected values, while skewness was fitted linearly against log age. Values beyond the latest observation therefore represent trend extrapolations rather than direct observations.

Total scale-invariant convexity was calculated from the power-law-plus-DSI structural price curve.

The approximate one-year SI convexity contribution incorporates both the permanently positive power-law contribution and the signed, phase-dependent DSI contribution. DSI can therefore increase or reduce total structural convexity at different ages.

Finally, Grand Total convexity was calculated pointwise as

C_SI +C_residual.

All plotted convexity values were multiplied by 100 and reported as approximate one-year log-price convexity contributions. They should not be interpreted as guaranteed simple returns. A plotted value (100 C) corresponds to an exact simple-return equivalent of (e^C-1), if the modeled contribution were realized.

Bitcoin’s price and network adoption appear to interact through an asymmetric feedback process. Strong price increases attract new participants and produce persistent growth in non-zero-balance addresses, while comparable price declines do not reverse adoption to the same degree. I call this mechanism Bitcoin’s Metcalfe Ratchet: price is volatile and reversible, but part of the adoption generated during rising markets becomes embedded in the network.

The declustered results show that address growth following top-decile price weeks exceeds that following bottom-decile weeks by approximately 3.0% after four weeks and 6.0% after eight weeks. Importantly, the address count still continues to grow even during and following bottom-decile weeks. These two observations are supportive of a ratchet interpretation.

The corresponding HAC test statistics correspond to Z scores of 3.14 and 2.85, and both results survive Holm correction across the tested horizons. Placebo tests find no corresponding top-minus-bottom divergence before the extreme price weeks, making it less likely that the result merely reflects address trends already underway.

Monthly data provide lower-powered but consistent cross-frequency confirmation. After declustering and applying the same general controls, the address-growth asymmetry reaches approximately 18.5% at two months, a 2.79 σ result that also survives multiple-testing correction. The larger three-month estimate is less precise because relatively few independent monthly episodes remain.

The ratchet has weakened and slowed since 2018, consistent with Bitcoin’s maturation and the declining completeness of address count as an adoption proxy. Exchanges, ETFs, custodians, and treasury companies can represent many beneficial owners through relatively few visible addresses. Nevertheless, the core asymmetry survives across weekly and monthly sampling and after controls for clustering, serial dependence, and pre-event trends.

These findings do not prove that address creation independently causes Bitcoin appreciation. They support a more specific interpretation: price gains stimulate network expansion, adoption is comparatively sticky when prices fall, and price and adoption therefore evolve as a coupled feedback system. This asymmetric accumulation provides a plausible mechanism through which volatile market cycles can produce persistent long-run network growth and reinforce Bitcoin’s approximately Metcalfe-like price–adoption relationship.

Long Run Relationship to Proposed Mechanism

In my prior article to this one I presented five new tests that support Giovanni Santostasi’s (see his book Physics of Bitcoin) and my view, that Metcalfe’s law for network value is explanatory for Bitcoin’s price adhering to a power law of age. Addresses grow as the third power of Bitcoin age and the price grows as approximately the 1.8 to 1.9th power of addresses, a generalized Metcalfe’s law formulation with exponent slightly under the classical value of 2.0.

The Metcalfe Ratchet Hypothesis

Bitcoin’s Metcalfe ratchet works like this:

• Rising prices stimulate new address formation.

• When price exceeds its address-implied equilibrium, addresses catch up.

• When price subsequently declines, the enlarged address base persists rather than falling proportionately.

• The expanded network therefore establishes a higher base from which the next price expansion begins.

Data and Methodology

The primary analysis herein uses 828 Coin Metrics Sunday close-to-Sunday close observations from July 2010 through May 2026. Bitcoin’s best and worst deciles (highest 10% and lowest 10%) of weekly returns initially contain approximately 83 events each. Non-zero-balance addresses are from Coin Metrics AdrBalCnt column.

To prevent extended rallies or crashes from being counted repeatedly, consecutive same-tail weeks are combined into distinct episodes, leaving 54 positive and 72 negative episodes. Note the clustering is stronger in bubbles above the power law trend than in the troughs below. Controlled local projections account for prior price and address behavior, volatility, Bitcoin age, and the prevailing price–address equilibrium gap.

Address Response and Evidence for Asymmetry

Figure 1 shows the unconditional cumulative address response over thirteen weeks following the strongest and weakest deciles of weekly price performance. Address count rises after both groups of events, reflecting its powerful secular trend, but it rises appreciably faster following top-decile weeks. This is the descriptive pattern implied by the ratchet hypothesis; the controlled projections, declustering analysis, and placebo test that follow provide the more demanding evidence.

Address counts rise more during price outperformance but they do not generally fall during price declines.

Figure 2 shows the results normalized to the address growth during the middle 80% of weeks considering price moves. Here the top decile produces progressively greater address growth than the middle 80%, while the bottom decile increasingly underperforms it. The error bars represent 95% confidence intervals.

Declustering and Pre-trend Tests

Figure 3 tests whether the result survives event declustering. If two adjacent weeks (or three or four consecutive weeks) are all in the top decile, they are treated as a single episode. The rise in ratchet asymmetry (top minus bottom decile difference) with declustering applied is somewhat lower, but overall quite similar.

Figure 4 illustrates a test for prior events causing the observed behavior. There is no significant top-minus-bottom difference of significance that can be ascribed to trends during the prior 13 weeks before a top decile event.

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Monthly Data Analysis

The same qualitative ratchet appears in monthly data, but with reduced statistical significance. After declustering and controlling for prior conditions, the top-minus-bottom address response reaches approximately 18.5% at two months (2.79 σ, Holm-adjusted p=0.024). The larger three-month estimate is less precisely measured because relatively few independent monthly episodes remain. By six months the gap is being reduced in relative terms but continues to grow in absolute terms.

Conclusion: The Metcalfe Ratchet

The earlier evidence established a persistent, nearly quadratic relationship between Bitcoin price and non-zero-balance addresses, together with bidirectional short-run feedback and asymmetric adjustment toward that long-run equilibrium. The present event study supplies a plausible dynamic mechanism behind those findings.

Across 828 weekly observations, address growth following top-decile price weeks exceeds that following bottom-decile weeks by approximately 3.4% after four weeks and 7.7% after eight weeks. After consecutive extreme weeks are consolidated into distinct episodes, the corresponding differences remain approximately 3.0% and 6.0%. Both survive dependence-robust inference and correction for testing multiple horizons, while the pre-event placebo analysis finds no comparable divergence before the price events. Monthly data provide lower-powered but directionally consistent confirmation.

I call this asymmetric accumulation Bitcoin’s Metcalfe ratchet. Rising prices stimulate address formation, but severe price declines do not produce an equal and opposite contraction in the aggregate address base. The aggregate address base continues to grow on average even during the worst decile. Market valuation may therefore be highly reversible while associated network expansion remains embedded in Bitcoin’s subsequent state.

The ratchet does not independently prove Metcalfe’s law, nor does this study establish that it quantitatively generates Bitcoin’s approximately (t³) address trajectory. It does, however, strengthen the Metcalfe interpretation by showing how a price-led, bidirectional feedback system can transform volatile market cycles into persistent network growth. Cointegration and DOLS identify the long-run relationship; the VECM identifies feedback and equilibrium correction; and the present event study identifies the asymmetry that allows adoption to accumulate across cycles.

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This is not investment advice. Stephen Perrenod is engaged in econophysics and network science research as the Vice-Director of the Scientific Bitcoin Institute.

Santostasi and Perrenod [1,2] have argued that Bitcoin exhibits a long-run power-law structure linking price, network adoption, and network value through approximately scale-invariant relationships. A recent Amphora Analytics Substack article [3] challenged several aspects of this framework, focusing on uncertainty in the fitted price–age exponent, the econometric and causal basis of the price–adoption relationship, and the decline in the address-growth exponent since 2018.

I agree that uncertainty in the price–age exponent should be stated more carefully. Consistent with the Amphora analysis, I estimate a one-standard-error Newey–West HAC uncertainty of 0.188 around the current exponent of 5.66—approximately 3.3% of the exponent itself. Daily, weekly, and monthly estimates are nearly identical, while an updated backward-looking analysis confirms the result reported in our paper: after substantial early variation, the fitted exponent has converged steadily toward its present value over the past decade.

The central issue is whether Bitcoin’s price–adoption relationship is consistent with Metcalfe scaling or merely reflects two variables trending with time. Dynamic Ordinary Least Squares, which is designed for long-run relationships among persistent variables, raises the estimated price–address exponent to approximately 1.92. Its dependence-robust confidence interval includes the quadratic value of 2 predicted by Metcalfe’s law. Engle–Granger tests also support cointegration between price and non-zero-balance addresses at weekly and monthly frequencies, although mixed integration-order and split-sample results counsel against an unqualified causal interpretation.

New vector error-correction tests strengthen the case that the relationship contains genuine feedback rather than only a common trend. Lagged address changes add significant information to the price equation, while lagged price changes predict subsequent address formation even more strongly. Adjustment is asymmetric: when price exceeds its address-implied equilibrium, addresses tend to catch up; when price falls below equilibrium, price tends to recover. These findings are consistent with a coupled price–adoption system organized around a persistent, approximately Metcalfe long-run relationship, although they do not prove that adoption is the sole or independently exogenous cause of price appreciation.

Finally, Amphora Analytics correctly observes that the address–age exponent declined from approximately 3.51 in January 2018 to 3.06 in January 2026. Non-zero-balance addresses, however, are an evolving proxy rather than a direct count of users or economic adoption. The increasing concentration of beneficial ownership within exchanges, ETFs, corporate treasuries, governments, custodians, and other institutional “fat nodes” allows substantially more capital and ownership to be represented by comparatively few visible addresses. A bounding calculation indicates that the rise in institutional ownership from roughly 2% to more than 20% could account for a substantial portion of the observed decline, although that estimate depends on assumptions about custody concentration.

Overall, the new evidence moves the interpretation beyond a generic common trend. Bitcoin displays both a stable price–age exponent, and a persistent and nearly quadratic price–adoption relationship. It also displays bidirectional short-run predictive feedback and asymmetric correction toward long-run equilibrium. An approximately Metcalfe feedback relation is therefore the leading interpretation among those tested, while the precise causal channels and the changing measurement of adoption remain open research questions.

Error Bars for the Price-Time Power Law Exponent

I have reanalyzed the power-law exponent on daily (n = 5789), weekly (n = 828), and monthly (n= 191) time scales using Coin Metrics data until May 24, 2026. Five different measures of the 95% uncertainty range are graphed in Figure 1, for each of the time scales.

The most appropriate measure, in agreement with Amphora Analytics, is the Newey-West HAC range (orange error bar) that corrects for autocorrelation and heteroskedasticity. Newey–West HAC errors adjust statistical uncertainty for both changing residual volatility and serial correlation over time. A moving-block bootstrap (red bars) provides a similar 95% uncertainty interval.

Figure 2 shows how the power law exponent for price-age has evolved since 2013. It stabilized to below 6.0 from early 2016 and for OLS has apparently locked in close to 5.7 with a Newey-West uncertainty (one standard deviation) of less than 0.2. The one sigma uncertainty is only about 3% of the exponent value. Accounting for serial correlation does not materially weaken the conclusion that Bitcoin price follows a remarkably well-defined secular power-law trajectory.

Metcalfe’s Law and Bitcoin’s Price-Address Relationship

Directionality

I examined the Bitcoin price and address lead-lag relationship using weekly detrended residuals (price vs. age against addresses vs. age). The x-axis shows lag in weeks with negative lag indicating price leading addresses. The most prominent feature is a short-term 3 to 12-week lead of prices vs. addresses. This is not surprising; increased prices tend to lead to increased adoption by momentum followers.

A smaller positive correlation appears near an 80-week lag, suggesting the possibility that address expansion sometimes precedes price over intermediate horizons. However, this feature does not survive the more conservative block-bootstrap and distributed-lag tests and should therefore be regarded as exploratory rather than established. A ~1.5-year lag is actually reasonable for adoption —> price, because exchanges, custody, funds/ETFs, treasury infrastructure, lending, payments, and institutional compliance channels take months to years to develop. Address growth would plausibly be functioning more as an intermediate term leading indicator of an expanding economic network that induces businesses to invest.

Is it Metcalfe-Like?

Table 1 shows the results of both OLS and DOLS regressions of price vs. address over the full history. The exponents are 1.84 for each of daily, weekly, and monthly regressions, rising to approximately 1.92 in each case for the DOLS method that reduces bias from short-run dynamics, simultaneity, and serial correlation when estimating the long-run coefficient β_M.

It is suggestive that the DOLS values are higher than the OLS values and close roughly half the gap between 1.84 and the classical value of 2.0 for Metcalfe’s law. The 95% interval using HAC is also shown for the DOLS values, and in each case the classical value of 2.0 falls well within the range. Observed ‘generalized’ Metcalfe’s law style communication networks allow for exponents well below 2.

Also shown are the Engle-Granger p-values. For weekly and monthly sampling, these fall under 0.05, indicating likelihood of cointegration, complementing the directionality results. The daily sampling p-value is just at the margin with a value of 0.059.

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Vector Error-Correction Model

I estimated a two-equation vector error-correction specification around the DOLS long-run Metcalfe relation. Define the equilibrium deviation as e_t = ln P_t - α - β_M ln A_t , with α and β_M obtained from Dynamic OLS. Separate equations modeled weekly changes in log price and log non-zero-balance addresses as functions of the previous week’s equilibrium deviation and 13 weeks of lagged changes in both variables, using Newey–West HAC inference with one-year lags.

We estimated both a symmetric specification, with one adjustment coefficient for all deviations, and an asymmetric specification separating price above equilibrium from price below equilibrium (e_t < 0). Joint tests of the cross-variable lags measured short-run bidirectional predictive feedback, while the error-correction coefficients identified whether disequilibrium was subsequently reduced through price adjustment, address adjustment, or both.

The results are provided in Figure 5. The bar chart in Panel A uses the negative log base 10 of p-value on the y-axis. All four bars support bidirectional feedback with substantially stronger influence running from price to addresses, strengthening the Metcalfe’s law interpretation. Panel B shows the response of price and address when price is above or below equilibrium. The lowest three relationships are all p < 0.05 and thus the most informative. For example, there is a negative response of price when it is below equilibrium, in other words, it tends to return to equilibrium.

Price changes strongly predict subsequent address growth. The asymmetric error-correction estimates further suggest that addresses are comparatively sticky; when price falls below the Metcalfe relation, adjustment occurs primarily through price recovery rather than proportional address contraction.

Decline of the Address-Age Exponent

Amphora Analytics notes that the address power law of age exponent decreased steadily from approximately 3.51 when it peaked in 2018, and falling to approximately 3.06 recently.

There are a couple of things to consider about this. First of all Metcalfe’s Law is about value going as the square of adoption. The number of addresses is just a proxy for adoption. It’s a reasonable one, but known to be flawed for various reasons. One is that typically there are multiple addresses per user. Satoshi’s coins are held across tens of thousands of addresses.

But another effect works in the opposite direction. That is, with the substantial growth in institutional Bitcoin holdings, the number of addresses per user may be much less than one as exchanges, ETFs, and treasury companies consolidate thousands, tens of thousands, and perhaps hundreds of thousands of Bitcoin potentially into a single cold storage address or a modest number of addresses.

The apparent slowing of address growth partly reflects a change in ownership architecture. Publicly tracked funds, companies, governments, and other concentrated vehicles increased from perhaps 2% of Bitcoin supply in early 2018 to approximately 20% in 2026. If one subtracts 3-4 million estimated lost coins (Satoshi’s included) from the current supply, that 4 million Bitcoin in institutional and custodial hands could represent as much as 25% of available Bitcoin.

Let’s estimate the growth in institutional percentage as rising from 2% to 22% over the interval. That suggests an upper bound on the magnitude of decrease in the exponent, assuming many fewer addresses per user, running as:

ln(.78/.98)/ln(17/9) = -0.36

where ln is natural log, and the non-institutional portion fell from 0.98 to 0.78 as the age increased from 9 to 17. This yields an estimated decline in the exponent of -0.36, compared to the -0.46 observed decrease.

Under this extreme assumption that institutionally held bitcoin contributes proportionally negligible incremental addresses, the ownership shift could mechanically reduce the fitted address–age exponent by roughly 0.36— about four-fifths of the observed 0.46 decline. This is an upper bound, but it demonstrates that changing custody architecture could be quantitatively important.

The declining address exponent therefore likely overstates the slowdown in underlying economic adoption considerably, because an increasing number of beneficial owners are represented by comparatively few custodial wallets.

Conclusions

Amphora Analytics raises legitimate questions about uncertainty, serial dependence, causality, and the evolving meaning of address count. Dependence-robust methods substantially widen the uncertainty around Bitcoin’s price–age exponent, but its central estimate remains stable near 5.7 across daily, weekly, and monthly sampling. Dynamic OLS similarly places the long-run price–address exponent near 1.92, statistically consistent with quadratic Metcalfe scaling.

The additional vector error-correction analysis strengthens the case that this is more than a generic common trend. Short-run predictive influence runs in both directions: lagged address changes add information to the price equation, while lagged price changes predict subsequent address growth even more strongly. Adjustment is also asymmetric. When price is above its address-implied equilibrium, address growth tends to catch up; when price is below equilibrium, price tends to recover. These results are consistent with a coupled feedback system organized around a persistent price–adoption relationship.

They do not, however, establish that address growth independently or unidirectionally causes price appreciation. Price-to-address feedback is particularly strong, the longer-horizon address-to-price evidence remains less secure, and mixed integration-order and split-sample results warrant caution. Common drivers—liquidity, technological development, institutional participation, and broader adoption—may affect both variables.

The post-2018 slowdown in measured address growth must also be interpreted in light of Bitcoin’s changing ownership architecture. A growing share of beneficial owners and capital is represented through exchanges, ETFs, corporate treasuries, governments, and custodians that generate comparatively few visible on-chain addresses. Address count may therefore increasingly understate economic adoption, weakening its comparability across eras without invalidating the underlying network relationship.

The defensible conclusion is therefore stronger than simple correlation but narrower than proof of causality: Bitcoin exhibits a robust, persistent, and nearly quadratic relationship between market price and measured network adoption, together with bidirectional short-run feedback and asymmetric correction toward their long-run equilibrium. Among the explanations tested, an approximately Metcalfe feedback relation is now the leading interpretation, although its observable proxies, causal channels, and internal composition have evolved substantially over time.

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This is not investment advice. Stephen Perrenod is engaged in econophysics research of Bitcoin in his role as the Vice-Director of the Scientific Bitcoin Institute.

References

1. Giovanni Santostasi, Stephen Perrenod 2026. “A Mechanistic Derivation of the Bitcoin Price Power Law: Network Adoption Dynamics and Generalised Metcalfe Scaling”

DOI: 10.5281/zenodo.19387099

2. The Physics of Bitcoin. Giovanni Santostasi 2026.

3. Amphora Analytics 2026.

Amphora Analytics

The Regularity Is Real. The Derivation Is Not Established.

In April 2026, Giovanni Santostasi and Stephen Perrenod published a paper making an unusually strong clai…

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18 days ago · Amphora Analytics

• Bitcoin’s global median power law remains robust, β = 5.848 and descriptive R² = 0.958 for a QR regression.

• Two global log-periodic modes raise the overall descriptive R² to 0.980.

• Re-estimating modal amplitudes and phases over the latest five years explains approximately 87% of residual variance in that window.

• The five- and eight-year scenarios suggest broadly flat behavior over the next year, followed by recovery during 2027–28; these are conditional scenarios, not precise price targets.

Background

Bitcoin resembles an astrophysical system in one important respect: we can observe it with increasingly sophisticated instruments, but we cannot run controlled experiments on its development. Short-term prices can be influenced by exchanges, derivatives, and market structure, but the longer-term phenomenon is the evolution of a self-organizing monetary network. Consensus, security, adoption, liquidity, and perceived value develop jointly through reinforcing feedback loops that have no close analogue among conventional currencies, commodities, or equities. Major currencies and commodities have been falling in power law terms relative to Bitcoin over its lifetime.

It has been well established that Bitcoin follows a long-term power law with a very robust R² of 0.96 (A Mechanistic Derivation of the Bitcoin Price Power Law: Network Adoption Dynamics and Generalised Metcalfe Scaling, Giovanni Santostasi and Stephen Perrenod DOI: 10.5281/zenodo.19387099).

Since a year ago as I saw that no significant bubble above the power law trend was appearing during 2025, I have been developing log periodic models of the price residuals that remain after subtracting out the power law. Structural log periodicity in Bitcoin is well-supported by peak and trough spacing, by FFT, Lomb-Scargle, and wavelet spectral analysis, and by fits to price history.

We have suggested that the power law and the log periodicity are both expressions of the underlying network. The relationship C = 2π β/ω = β ln λ ~ 4 describes a candidate coupling invariant — a hidden variable between the power law with exponent β and a fundamental angular frequency of log periodicity ω ~ 9, (spacing λ ~ 2 in log age, related via ln λ = 2π /ω). The hypothesis is that β and ω are respectively the real and imaginary components for complex scaling, s = β + i* ω, such that taking the real part of t^s = t^β * e^{{i* ω ln t}} produces a power law modulated periodically in log time.

In Bitcoin’s Not so Hidden Structure https://stephenperrenod.substack.com/p/bitcoins-not-so-hidden-structure I showed that adding one or two log periodic modes improved out-of-sample forecasting over the range 6 to 24 months forward, relative to the power law, which itself performs better than all other models for horizons of 12 months and longer.

In Forecasting Bitcoin beyond the Power Law https://stephenperrenod.substack.com/p/forecasting-bitcoin-beyond-the-power I showed that medium-term localization substantially improves the model’s explanatory power. Holding the globally identified log-periodic frequencies fixed while re-estimating their amplitudes and phases over a trailing five-year window increased the proportion of recent power-law residual variance explained by two modes to as much as 82%. This suggests a useful separation: the frequencies may reflect persistent global properties of Bitcoin’s scaling structure, while their amplitudes and phases evolve (drift) with the network’s local dynamical state.

The declining percentage size of successive bubbles suggests that constant-amplitude log-periodic coefficients are inadequate over the full history. This motivates allowing amplitudes and fitted phases to vary over intermediate-term windows. Phase drift is also expected in dynamical systems of this nature.

Procedure

The procedure followed here was to fit the fundamental parameters β and ω globally, but to use intermediate-term local fits for the amplitudes and phases of two log periodic modes. The fundamental and second LP mode frequencies were determined from a Lomb-Scargle fit (similar to an FFT analysis for dominant frequencies, but more suited for log spacing analysis). In the earlier work we had good results with a 5-year window, here we looked at 3, 4, 5, 6, and 8-year windows and found that the two shorter ones were insufficiently short to capture intermediate term parameters.

Table 1 reports the globally estimated structural parameters. The global power law explains 95.8% of the variance in log price, while adding two globally calibrated log-periodic modes raises the overall descriptive R² to 0.980 and explains 49.3% of the remaining residual variance. When the two frequencies are held fixed but their amplitudes, phases and intercept are re-estimated over the latest five-year window, the two-mode model explains approximately 87% of the residual variance within that interval. This large local improvement motivates using five-year coefficients for the conditional forecast scenarios below.

Robustness

As a leakage-free robustness check, I repeated the exercise from earlier forecast origins using five- and eight-year local windows and six- and twelve-month horizons. Overall, the power-law baseline outperformed the localized two-mode forecasts, although the five-year model modestly improved one six-month prediction. These limited tests do not establish out-of-sample superiority and indicate that locally estimated amplitudes and phases can be unstable. The current projection should therefore be interpreted as a conditional endogenous scenario rather than a validated point forecast. These results do not negate the evidence for globally calibrated log periodicity; rather, they show that repeatedly localizing amplitudes and phases introduces additional estimation risk.

Power Law and Overall 2-year Projection

Figure 1 shows the median quantile-regression power law fit and also a two year projection with a 5-year training window for the two LP modes, on a log-log-scale. Note the three largest bubbles are approximately equally spaced in log time. The more recent bubble structure has influence from the higher mode, whose amplitude of 0.16 in log₁₀ price is of the same order as the 0.26 amplitude fundamental. And for the most recent 5-year window the higher mode grew to an amplitude exceeding that of the fundamental (shown below). Growing power in non-fundamental modes is another typical aspect of this class of dynamic systems.

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Giovanni Santostasi and others, including myself, have demonstrated with a wide variety of methods — around one dozen techniques — and with robust statistical validation — that Bitcoin adheres to a power law when measured in dollar terms.

In his book The Physics of Bitcoin, in Chapter 7, Dr. Santostasi also presents the power laws seen for Bitcoin priced in other currencies, including the Euro, Yen, and Australian dollar, and also when measured relative to two commodities: gold and oil. All of these denominators yield strong regression results, with high R² values well above 0.9.

Here I extend the analysis with several months of additional data and include here the six major currencies and also four commodities for comparison and reinforcement. Ten numeraires in total were examined.

Bitcoin’s long-run price trend remains a consistent steep power law with only modest changes in slope; the underlying relationship between price and age remains. Volatility is similar for all numeraires, about 0.3 dex or a linear factor of two, as one standard deviation.

Monetary inflation does not explain these results.

Fiat currencies strongly declining relative to Bitcoin

Across the six major currencies: the Dollar, the Euro, the Yen, the Yuan, the Pound and the SDR (a weighted basket of the prior 5 currencies), all have strong fits with R² around 0.96 and with power law indices ranging from 5.54 for the British pound (strongest) to 5.96 for the Japanese yen (weakest). But the variation is small, less than 8% difference across the exponents.

One sees that the six lines on the log-log chart in Figure 1 are nearly parallel to one another. The dominant fact is power law of Bitcoin’s age with an index 5.4 or greater across all of the major fiat currencies.

It becomes even more striking when one inverts the relationship and measures the currencies with Bitcoin in the denominator. This is shown in Figure 2. The power law exponents are the same for each currency, but now with a minus sign in front. Now the appearance is of fiat in decline when measured against the world’s first money that is defined to scientific precision and anchored by a fixed ultimate supply.

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Table 1 lists the residual standard deviations in log10 price terms for each of the Bitcoin / currency pairs, and in all cases the one standard deviation value is close to 0.3 dex, which is a multiplicative factor of two in linear terms. The values for the various currencies are surprisingly close, there is only a 5% range in the linear multiplicative factor.

Thus for example if the fair value on the power law trend was $120K at some point in time, then the Gaussian approximation is that 68% of the data points would lie in the range of $60K to $240K, corresponding to Z = -1 and +1 scores, respectively. A factor of two is not unusual, and does not undermine the power law statistical relationship. One has to keep in mind that the price moves by a factor of about 1 million in Figures 1 and 2 for each of the six currencies.

Inflation?

A related question is, does Bitcoin’s apparent power-law appreciation merely reflect monetary inflation? The evidence suggests otherwise.

Bitcoin’s annualized supply inflation generally exceeded U.S. M2 growth until the 2016 halving, yet Bitcoin appreciated rapidly against every fiat and commodity numeraire examined here, and at a remarkably consistent power-law rate. Monetary expansion may influence the level and cyclical behavior of nominal prices, but it neither explains an exponent near 5.7, nor does it explain the persistence of similar scaling when Bitcoin is measured against gold, silver, oil, and corn.

Commodities strongly declining relative to Bitcoin

Figure 3 shows the power law of age (Bitcoin’s age) decline of prices of four commodities when measured in Bitcoin terms. The power law exponents are similar to those found for currencies, and range from 5.39 for gold to 5.84 for oil (West Texas Intermediate). Again the R² are very high, in the range of 0.95 to 0.97.

Summary for 6 Currencies, 4 Commodities

Figure 4 summarizes all the power law exponent values for the regressions displayed in Figures 1, 2, and 3. They are all within 6% of the USD exponent of 5.68, ranging from 5.39 for gold to 5.96 for the Japanese Yen. This is an exceptionally tight range, and combined with the R² values for all numeraires ranging from 0.95 to 0.97, we have strong evidence that Bitcoin’s power-law relationship is robust to the choice of numeraire.

Summary

• Bitcoin’s long-run price follows a remarkably consistent power law. Across six major currencies, the fitted exponent remains tightly concentrated between 5.54 and 5.96, with R² values near 0.96.

• The scaling law is largely independent of the unit of account. Changing the numeraire shifts the estimated slope modestly, but does not obviate the underlying relationship between Bitcoin’s price and its age.

• The result extends beyond fiat currencies. Bitcoin exhibits similar power-law appreciation against gold, silver, oil, and corn, with commodity exponents ranging from 5.39 to 5.84.

• Shorter-term volatility occurs around a stable long-run trend. Across the six fiat numeraires, the typical one-standard-deviation deviation from the fitted trajectory is approximately a factor of two in either direction, large in Tradfi terms, but small relative to the many orders of magnitude of Bitcoin’s price appreciation.

• Monetary inflation alone cannot explain the result. Bitcoin appreciated rapidly even when its own supply inflation exceeded U.S. M2 growth, while comparable scaling against scarce commodities further indicates that the observed power law is not simply a consequence of fiat debasement.

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This is not investment advice. Stephen Perrenod is engaged in econophysics research at the Scientific Bitcoin Institute.

In Beyond the Power Law articles I-III, I examined Bitcoin’s dynamic geometry in uniform log time, considering both the log price residuals after removing the power law, and the second-order residuals after removing the fundamental log periodic (DSI) mode as well.

In Part I, I demonstrated a six-dimensional embedding attractor structure of lower dimensionality, and in Part II, I examined stretching, curvature, and deformation as possible indicators of enhanced performance over intermediate timescales. In Part III the vector field and flow were extracted, revealing an elongated transverse membrane of effective dimension near 2.

In this Part IV article I present several methods to test whether the geometry is robust and persistent.

Both the first-order and second-order residuals have transverse effective dimension near 2, and the dominant plane of r(u) aligns closely with the fitted DSI plane. The second-order residual rank-2 geometry remains significant against three surrogate test methods. Out-of-sample testing supports persistence of a transverse dimension near 2, but not stable prediction from a fixed transverse basis.

Methodology

The work presented in this article, as in articles I, II, III, is primarily based on a D = 6 embedding space in log price after mapping some 5,839 daily prices onto a uniform 3,000-point grid in log time u = ln t, where t is the age of Bitcoin. After testing alternative tine spacings, I settled on τ = 50 for the delay-coordinate comb construction, with all six tines being in the past or at the latest data point. Thus a tine spacing is 0.0406 in log age, or about 4.1% between adjacent coordinates; with this choice we are exploring intermediate structure.

I also define both first order and second order residuals. The first order residuals r(u) in log price remove the continuous scale invariance (CSI) of the power law trend. The second order residuals q(u) remove both the CSI and DSI (discrete scale invariance of the log periodicity fundamental mode, of measured frequency ω = 8.80 and that corresponds to a spacing parameter λ = 2.04).

Global Dimensionality

The choice of D = 6 was made as a balance between allowing room for complexity to manifest and also trying to keep things moderately interpretable, after first investigating how the K-Y effective dimension of the reconstructed attractor changes as a function of D, the number of embedding dimensions. The effective attractor dimension does not converge as D is raised to 10, but it does grow more slowly than linearly in D.

The transverse result was checked across embedding dimensions D= 4–8 and across alternative delay and cross-sectional choices. Although the available normal space increases from three to seven dimensions, the absolute transverse effective dimension remains close to 2. This indicates that the rank-2 result is not created by selecting D=6 or by one particular cross-sectional construction.

For diagnostic purposes I explored two-component compression and participation dimension across embedding dimensions D= 4-8 also. The compression is how much of the variance for either r(u) or q(u) is explained by the two principal components, and is plotted in Figure 1A below. At D = 6, 80% of the variance is found in just the two components, and it remains above 70% even when D = 8. The first order residuals are much more compressed than the second order residuals q(u). This is attributed to the orderly nature of the DSI fundamental.

In Figure 1B we see the participation ratio, which is about 2.9 (only half of the available six) for the effective dimension from principal component analysis of r(u), and it is 4.2 for q(u), at D = 6. Only a lower dimensional volume of the full phase space is being occupied, the attractor has notable non-random structure.

These values do not conflict with the later discussion in this article of a transverse dimension that is very near to 2.0. The two statistics answer different questions:

• Global PCA dimension: dimensionality of the complete reconstructed trajectory.

• Transverse effective dimension: dimensionality of the cross-section normal to the trajectory.

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Comparison with DSI Fundamental Mode

For the r(u) residuals — but not for the q(u) residuals — the overlap between the fitted DSI and the observed reconstructed attractor structure is very strong. Table 1 shows the leading principal component squared overlap to be at the 0.989 level indicating nearly complete directional alignment. The same measure is only 30% for the second order residuals. And the rank-2 (planar) overlap is at the 0.961 level.

About 39% of the variance is in the fundamental mode for r(u), but with the second order residuals that mode is removed the variance explained drops to less than 2%. The 2ω harmonic explains only 0.5% of variance and is not a prominent component. The leading eigenvalue for the transverse (to the overall trajectory in six dimensions) falls by 2/3 as we move from the first-order to second-order residuals. This is also supportive of the significance of the DSI fundamental mode.

Transverse Effective Dimension Surrogate Testing

In Part III of this series, I asserted an effective transverse dimension of ~2 out of the 5 dimensions perpendicular to the principal trajectory in the embedding space. For this article surrogate testing has been performed on both the r(u) and q(u) residuals using three different techniques.

For the q(u) residuals (lower panels in Figure 2), all three reject null models at the 5% level, they are all below 0.025. The observed 1.99 transverse effective dimension for the second-order residuals is strongly supported.

For the r(u) residuals the observed effective dimension is 1.91 and the variance matched p-value is much less than 1%, but the phase randomized test is about double the 0.05 level and the IAAFT test result is well above. This is not surprising when one considers that for both the IAAFT and phase randomized tests, the DSI fundamental frequency ω is retained by the surrogate distribution logic.

Overall the results unambiguously indicate that the transverse organization remaining in q(u) is not readily explained by its spectrum, amplitude distribution, or variance profile alone.

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Out of Sample Testing

Out of sample testing examines the persistence and predictability of the results. The OOS results for q(u) are presented in Figure 3. The training window increased age by a factor of e (2.718 x) and four subsequent test windows each increased age by e^0.25 (1.284 x) without overlapping. Each test window corresponds to a 28.4% increase in Bitcoin’s age. The future transverse dimension stayed close to 2 for each test window (panel A). Future variance capture was variable and did not exceed the random benchmark except in one window (panel B). And tests (C) and (D) both indicated substantial change in orientation (rotation) of the transverse subspace. The subspace persists near rank-2 but the ability to forecast is confounded by its substantial rotation.

Interpretation

Bitcoin’s residual dynamics occupy an anisotropic transverse membrane whose effective dimension is close to 2 for both r(u) and q(u). In r(u), the dominant transverse direction is closely aligned with the fitted DSI mode. The membrane retains approximately the same dimensional form through time, but its orientation rotates within the higher-dimensional reconstructed state space.

The evidence is supportive for DSI, that could explain the two principal components of cosine and sin of the log periodic form. One observes that:

• The first-order residual r(u), which retains DSI, is much more globally compressed than q(u).

• Removing DSI (log periodicity) substantially reduces the dominance of the leading global PCs.

• The first-order residual r(u)’s transverse dimension is not unusual under phase and IAAFT surrogates, both of which preserve its spectral peak.

• The second-order q(u)’s transverse dimension remains significant after that spectral structure has been removed.

Conclusions

The results support a hierarchical picture of Bitcoin’s dynamics in which endogenous organization and external perturbations coexist. The primary trajectory is described by the power law, representing Bitcoin’s long-run scale-invariant growth. Superimposed on that trajectory is the fitted DSI mode: a coherent log-periodic oscillation whose frequency, phase evolution and alignment with the dominant transverse plane of r(u) support its interpretation as an additional endogenous organizing component.

The geometric contribution of DSI is substantial. The leading transverse direction of r(u) aligns almost perfectly with the leading direction of the fitted DSI-only embedding, while the complete rank-2 planes have an overlap of 0.961. Removing DSI reduces the leading transverse eigenvalue by approximately 66% and substantially weakens the conspicuous global compression and rotational loop.

Yet DSI does not account for all of the observed organization. After its removal, q(u) retains an effective transverse dimension close to 2, and this low-dimensional structure is statistically unusual under phase-randomized, IAAFT and variance-matched surrogate models. The observed low dimensionality is not readily explained by the fitted DSI spectrum, the marginal distribution of the residuals, or their overall variance.

The transverse dynamics should not be considered exclusively exogenous. They will contain external influences—macroeconomic conditions, liquidity shocks, regulation, technological events and changing investor behavior—but may also contain endogenous Bitcoin mechanisms not represented by the fitted fundamental DSI mode. These could include protocol-driven effects such as halvings, interactions between adoption and price, changing holder composition, or other internally generated modes. The present analysis finds little remaining power specifically at the fitted (2ω) harmonic so does not establish DSI harmonics as the source of the residual rank-2 geometry.

Out-of-sample results reinforce the geometric conclusion while limiting any predictive interpretation. Future windows continue to show an effective transverse dimension near 2, however the orientation of the plane changes substantially. The persistent object has the dimensional form of the membrane, but not a fixed pair of directions needed for a stable forecasting basis.

Taken together, the evidence supports an endogenous CSI–DSI backbone surrounded by a low-dimensional transverse geometry containing a mixture of exogenous perturbations and additional, not yet identified endogenous dynamics. Bitcoin’s reconstructed residual state space is organized, but its internal orientation evolves. An open question remains: what economic, protocol, and behavioral forces rotate and deform the transverse geometry through time.

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This is not investment advice. Stephen Perrenod is engaged in econophysics research as Vice-Director of the Scientific Bitcoin Institute (.org), a non-profit dedicated to education and research of the Bitcoin ecosystem.

Executive Summary

In the first article of this series, I showed that Bitcoin’s departures from long-term power-law growth are not purely stochastic. Delay-coordinate reconstruction revealed organized, low-dimensional structure within the residual dynamics. The second article characterized the local geometry of this reconstructed system through measures of stretching and curvature. This article asks the next question: what is the global dynamical structure of the reconstructed state space?

Because Bitcoin’s long-term evolution is approximately scale invariant, the appropriate developmental clock is logarithmic age, (u= log t), rather than calendar time. I therefore revisit the original analysis using a uniformly sampled log-time grid. Selected comparisons with the earlier linear-time construction show why this distinction matters: equal intervals in log time represent equal proportional increases in Bitcoin’s age and avoid mixing fundamentally different developmental scales.

A uniform-log-time reconstruction strengthens the central geometric result. In a six-dimensional delay embedding, the residual dynamics are globally compressed and display coherent rotational organization. Their five-dimensional cross-sections are highly anisotropic: approximately 91% of the transverse variance is concentrated in two principal directions, with an effective transverse dimension close to 2.0 rather than the approximately 2.5 obtained from the linear-time construction. The reconstructed system is therefore better described as a membrane-like geometry than as a five-dimensional volumetric cloud.

The revised analysis does not, however, support the earlier conclusion that this transverse tube progressively narrows with Bitcoin’s age. On a uniform log-time grid, late transverse widths are comparable to, or modestly greater than, early widths. The robust conclusion is consequently low-dimensional transverse organization—not progressive contraction.

The comparison between two residual definitions further clarifies the source of this organization. The power law residual, r(u), retains the fitted discrete-scale-invariant component and exhibits a conspicuous loop-like global geometry. After the fitted DSI (log periodic) component is removed, the resulting second-order residual, q(u), is less globally compressed but retains an approximately two-dimensional transverse structure. The dominant rank-2 transverse plane of (r(u)) aligns closely with the sine–cosine plane generated by the fitted DSI mode, indicating that much of its most visible rotational organization is the geometric expression of discrete scale invariance.

Together, these results suggest a hierarchical description of Bitcoin’s long-term dynamics. The power law defines the primary scale-invariant growth trajectory; discrete scale invariance generates a coherent log-periodic rotational mode around that trajectory; and the remaining residual dynamics occupy a compressed, membrane-like reconstructed geometry. This is evidence for organized dynamical structure, but not by itself proof of a contracting attractor or a stable forecasting mechanism. Robustness, surrogate testing, and out-of-sample persistence are taken up in the next article.

The Vector Field

In Beyond the Power Law I, I reconstructed Bitcoin’s residual dynamics using a six-dimensional delay-coordinate embedding. In Beyond the Power Law II, I examined the local geometry of that reconstruction through Jacobian stretching and curvature. Here I turn to the vector field: the average direction in which nearby reconstructed states subsequently evolve.

The time coordinate is important. Bitcoin’s long-run growth and its log-periodic oscillations are naturally expressed in logarithmic age, u = ln t, where t is Bitcoin’s age. Equal increments of u correspond to equal proportional increases in age. By contrast, equal calendar-time intervals represent quite different fractions of Bitcoin’s lifetime near the beginning and end of the historical record.

The log-time vector field is constructed from 3,000 equally spaced observations in u, rather than from daily observations in t. The six-dimensional delay embedding uses a separation of 50 log-time grid steps, corresponding to δu ~ 0.0406. Local motion is measured over 10 grid steps, or δu ~ 0.0081. All velocities and neighborhood comparisons are consequently defined on the same logarithmic developmental clock.

I examine two related residuals. The first, r(u), removes the continuous-scale-invariant power-law trend but retains the fitted discrete-scale-invariant oscillation. The second,

also removes the fitted DSI component. Comparing them allows us to distinguish geometry associated with the known log-periodic mode from structure that remains after that mode has been removed.

The CSI residual: DSI retained

The reconstructed r(u) states form a conspicuous loop-like geometry when projected onto their first two principal components. These two components capture approximately 80.5% of the total six-dimensional variance: 50.5% in PC1 and 30.1% in PC2.

The locally averaged velocity vectors follow this organization rather than pointing independently in arbitrary directions. This indicates that much of the visible global geometry of r(u) is dynamically coherent. The flow has a substantial tangential component, with a median absolute tangential-to-radial ratio of approximately 1.3.

The fraction of locally estimated flow directed inward is approximately 0.53. This is close to an even division between inward and outward motion, however, and does not establish a globally attracting spiral sink. The defensible conclusion is coherent rotational organization, not persistent contraction toward a fixed center.

The CSI+DSI residual: DSI removed

After removing the fitted DSI component, q(u) becomes substantially less compressed in its first two global principal components. PC1 and PC2 capture approximately 34.0% and 27.8% of the variance respectively, for a combined share of approximately 61.9%.

The conspicuous loop in r(u) is correspondingly weakened, but the q(u) reconstruction does not become a random cloud. Local velocity estimates here also display organized directional patterns, and tangential flow remains prominent. The median absolute tangential-to-radial ratio rises to approximately 2.2, indicating that local motion is more tangential than radial in this projection.

Once again, there is no strong net attraction toward the estimated center: the inward fraction is approximately 0.50. The post-DSI residual therefore retains rotationally organized local motion but does not support the interpretation of a contracting spiral sink.

The apparent clockwise or counterclockwise direction in a PCA projection do not carry physical significance. The sign of each principal axis is arbitrary, so reversing one axis reverses the apparent rotational direction without changing the underlying dynamics. The meaningful quantities are the coherence and relative rotational strength of the projected flow, not its handedness.

What changes in logarithmic time?

The earlier calendar-time construction also suggested coherent rotational motion, but it combined a fixed number of days across very different stages of Bitcoin’s development. The uniform-(u) reconstruction places early and late observations on the same proportional-age scale.

This revised analysis sharpens the interpretation. The strong global loop in r(u) is closely associated with the retained DSI oscillation. Removing that oscillation produces a less globally compressed q(u) geometry, while leaving significant local organization. The reconstructed flow therefore appears to contain at least two levels of structure: a dominant log-periodic rotational mode and a less visually conspicuous residual geometry that remains after the fitted mode is removed.

The vector-field analysis alone does not demonstrate that the residual states converge toward an attracting center, nor does it establish predictability. It shows that their movement through the reconstructed state space is organized and strongly non-radial, and that much of the most prominent global organization is connected to discrete scale invariance.

Flow Contours

Unlike local measures such as Jacobian stretch or curvature studied in article II, the reconstructed vector field estimates the average direction of motion throughout the attractor and therefore provides a global description of the system’s dynamics. The locally estimated vector field can also be represented as a set of flow contours. These streamlines provide a smoothed view of the average direction of motion in the first two principal-component coordinates.

For r(u) the flow contours display conspicuous rotational organization around the central region. The median absolute ratio of tangential to radial flow is approximately 1.3, indicating that motion around the center is generally stronger than motion directly toward or away from it. This organization is consistent with the loop-like geometry visible in the reconstructed states and with the presence of the fitted log-periodic mode.

The inward-flow fraction is 0.53, an approximately even division between inward and outward motion. The field should therefore not be described as a clearly attracting spiral or spiral sink. It demonstrates rotational organization, but not persistent convergence toward the estimated center.

For q(u), after removal of the fitted DSI component, the global contours are less regular but remain strongly non-radial. Its median absolute tangential-to-radial ratio is approximately 2.2, while its inward-flow fraction is 0.50. Thus, removing DSI weakens the visually dominant global loop without eliminating local rotational organization. Indeed, relative to radial motion, the remaining flow is even more tangential.

The displayed direction of rotation—clockwise in one panel or counterclockwise in another—is not physically meaningful by itself. Principal-component axes have arbitrary signs, and reversing either axis reverses the apparent handedness of the plot. The invariant result is the strength and coherence of rotational motion, not its displayed chirality.

Taken together, the two flow fields separate two aspects of the reconstructed dynamics. The fitted DSI mode accounts for much of the smooth, conspicuous global rotation in r(u), while q(u) retains a less globally ordered but still strongly tangential local flow. Neither field provides evidence of sustained net contraction toward a central point.

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An Approximately 2-D Transverse Membrane

A six-dimensional reconstructed trajectory has five directions transverse to its local direction of motion. To determine how many of these directions are meaningfully occupied, I divide each uniform-log-time reconstruction into 18 equal-(u) cross-sections and calculate the five eigenvalues of the local transverse covariance matrix.

If variance were distributed evenly across all five available directions, the effective transverse dimension would approach 5. Instead, both residual constructions have an effective dimension close to 2. The median is 1.92 for r(u) and 2.00 for q(u), after DSI removal.

The low effective dimension results from a strong concentration of variance in the leading transverse directions. For r(u), the largest transverse eigenvalue accounts for a median 66.7% of the variance, while the two largest together account for 90.7%. For q(u), the corresponding shares are 65.9% and 90.9%. The remaining three directions collectively contribute only about 9% of the typical transverse variance.

The two leading directions are not equally strong. The dominant direction usually carries roughly two-thirds of the variance, producing an elongated, anisotropic ellipsoidal cross-section rather than that of a circular disk. The second direction supplies most of the remaining organized width. The resulting geometry is therefore best pictured as a thin, locally two-dimensional membrane embedded within the five-dimensional transverse space.

The similarity between first and second order residual results, using r(u) and q(u), is important. Removing the fitted DSI component substantially changes the global appearance and overall compression of the reconstructed trajectory, but it does not eliminate its approximately rank-2 transverse geometry. DSI therefore accounts for much of the conspicuous global loop in r(u), while the local concentration of transverse variance into two directions is a broader property shared by both residuals.

This conclusion is sharper than in the earlier calendar-time construction, which produced a median effective transverse dimension near 2.5 and placed approximately 82% of the transverse variance in the leading two directions. On the uniform-(u) grid, the corresponding dimension is near 2 and the leading-two share rises to approximately 91%. Fixed calendar-time sampling made the transverse geometry appear thicker and more diffuse.

The uniform-log-time analysis does not show that the transverse tube becomes progressively narrower with age. Late cross-sectional widths are comparable to, or modestly greater than, early widths. The result supported here is persistent low-dimensional anisotropy: the reconstructed states remain concentrated mainly within two of the five available transverse directions, but there is no evidence of systematic contraction in their overall transverse width.

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The Dynamics Interpreted

The results of this article and the preceding work suggest a hierarchical picture of Bitcoin’s dynamics. The power law provides the primary scale-invariant growth trajectory and accounts for approximately 96% of the variance in log price. Discrete scale invariance contributes a damped log-periodic mode that explains approximately half of the remaining residual variance. Other endogenous and exogenous innovations account for the residual movement not captured by these fitted components.

These innovations displace the observed system from its CSI–DSI reference trajectory. In a six-dimensional delay reconstruction, their evolution is organized within a strongly anisotropic transverse geometry: approximately 91% of the transverse variance lies in two principal directions, producing an effective transverse dimension close to 2 out of the 5 available directions.

Removing the fitted DSI component substantially weakens the dominant global loop in r(u), but leaves this normalized cross-sectional geometry almost unchanged. Both r(u) and q(u) occupy an elongated, approximately two-dimensional transverse membrane. DSI therefore accounts for much of the prominent global rotational geometry, but not all of the lower-dimensional organization remaining in Bitcoin’s residual dynamics.

Conclusions

Across this three-article series, Bitcoin’s dynamical geometry has been examined through a six-dimensional delay-coordinate reconstruction. Its long-term evolution appears consistent with three interacting components: deterministic power-law growth, a damped log-periodic mode associated with discrete scale invariance, and residual innovations of endogenous and exogenous origin.

On a uniform logarithmic-time grid, the reconstructed residual dynamics display coherent rotational organization and a membrane-like transverse geometry with an effective dimension close to 2. DSI accounts for much of the dominant global loop, while approximately rank-2 transverse organization remains after the fitted DSI component is removed.

The resulting picture is not that of a stationary equilibrium point or a demonstrated spiral sink. Rather, Bitcoin evolves around a scale-invariant reference trajectory, with structured residual motion occurring within a compressed reconstructed state space. The analysis supports low-dimensional organization but provides no evidence of systematic transverse narrowing with age.

The next article examines robustness across embedding choices, surrogate-process significance and out-of-sample persistence.

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This is not investment advice. Stephen Perrenod is engaged in econophysics and nonlinear science research as Associate Director of the Scientific Bitcoin Institute.

In an article posted a few days ago I showed that Bitcoin’s departures from its long-term power-law growth are not consistent with random fluctuations alone. Instead, delay-coordinate embedding in logarithmic time reconstructs a low-dimensional dissipative attractor with positive Lyapunov exponents, indicating that the residual dynamics possess deterministic nonlinear structure. This article asks the natural next question: how does that attractor evolve?

To answer this, I examine the local geometry of the reconstructed six-dimensional state space. Two complementary geometric quantities are measured throughout the attractor: Jacobian stretch, which quantifies how rapidly nearby trajectories diverge, and curvature, which measures how sharply the average trajectory changes direction. Together these quantify local stretching and bending of trajectories, the geometric ingredients of the stretch-and-fold dynamics characteristic of dissipative chaotic systems.

The reconstructed attractor exhibits a highly organized spatial distribution of stretch and curvature rather than random variation. A geometric phase diagram shows that different regions of this six-dimensional hyperspace are associated with systematically different subsequent residual evolution. This suggests that Bitcoin’s future behavior depends not only on its position within the log-periodic cycle for discrete scale invariance, but also on its geometric state within the reconstructed attractor.

To investigate this relationship more rigorously, I identify episodes of unusually high local deformation and then perform an event study around these episodes. These events are interpreted conservatively as periods of elevated geometric deformation rather than deterministic turning points. On average, they are followed by sustained positive residual drift over the subsequent year, with a median improvement of approximately 0.14 to 0.18 log units (about 38% to 51%) relative to the power-law baseline, albeit with substantial dispersion across events. Bitcoin is not currently in such a high-deformation regime; present-day stretch, curvature, and coherence remain well below the thresholds used to define these episodes.

Finally, I show that stretch, curvature, and the probabilities of extreme deformation vary systematically with log-periodic phase. Rather than occurring randomly, the stretching and folding of the reconstructed hyperspace are synchronized with Bitcoin’s log-periodic oscillation, suggesting that the familiar boom-and-bust cycles emerge from an underlying cyclic deformation of the attractor itself. While these findings do not establish a complete physical model of Bitcoin’s dynamics, they provide new evidence that the market’s long-term residual fluctuations are governed by an organized geometric process rather than by unstructured noise. This moves the discussion beyond fitting curves toward understanding the dynamical mechanisms that shape Bitcoin’s evolution.

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For more than a decade, Bitcoin’s price has been known to follow an extraordinary long-term power law with an exponent near 5.7 and an R^2 now exceeding 0.96. The long-term power law describes Bitcoin’s secular growth, while previous work has shown that the residuals exhibit log-periodic structure with a decaying amplitude. The natural next question is whether these oscillations are simply statistical regularities or the observable signature of an underlying dynamical system.

To investigate this possibility, we first remove Bitcoin’s long-term power-law growth, leaving only the residual dynamics. Rather than analyzing these residuals in ordinary calendar time, we reconstruct their evolution in logarithmic time using delay-coordinate embedding, a standard technique from nonlinear dynamical systems theory. This allows us to examine the geometry of the system that governs Bitcoin’s departures from its long-term adoption trajectory.

The reconstructed state space reveals a coherent attractor rather than an unstructured cloud of points. Quantitative analysis yields two positive Lyapunov exponents, consistent with sensitive dependence on initial conditions, together with an overall contraction of phase-space volume, demonstrating that the reconstructed dynamics are dissipative rather than explosive. Taken together, these findings provide strong evidence that Bitcoin’s departures from its long-term power-law trajectory arise from a deterministic low-dimensional dynamical system rather than unconstrained random fluctuations.

Bitcoin power-law residuals are inconsistent with being purely random market noise.

These findings naturally complement earlier evidence that Bitcoin exhibits discrete scale invariance and log-periodic oscillations around its long-term power-law growth. Rather than viewing Bitcoin as a stochastic process fluctuating around a deterministic trend, it may be more accurate to regard the power law as the equilibrium manifold of a dissipative dynamical system whose chaotic evolution gives rise to the familiar boom-and-bust cycles. In this picture, the power law describes Bitcoin’s long-term adoption, the log-periodicity describes its discrete scaling behavior, and the reconstructed attractor describes the underlying state-space dynamics that connect them.

Workflow

A key step is delay-coordinate embedding, which transforms a one-dimensional record of Bitcoin prices into a reconstructed state space, allowing the dynamics themselves—not merely the price—to be studied.

Why do we do it? Because dynamics require state. A single time series does not define a dynamical system. A trajectory through state space does.

The output of this workflow is not another regression or statistical fit. It is a reconstruction of the system’s state space—a geometric object known as an attractor. If Bitcoin’s residual fluctuations are generated by an underlying deterministic process, that process should leave a recognizable geometric signature in the reconstructed state space.

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Log-time delay embedding

We start out with nearly 5800 daily observations. Then we interpolated those into 3000 equally spaced points in the log of Bitcoin age. Bitcoin is better represented in log space as revealed in its power law scale invariance, so that is the natural basis to work in, rather than linear time.

Imagine dragging a comb along the log-time residual series. Each tine of the comb samples the residual at a different delay. The number of tines is the embedding dimension m; the spacing between them is the (log time) delay τ. Every position of the comb produces one point in reconstructed state space.

Reconstructed Attractor Views

What we present here is a reconstructed attractor, an approximation to the true attractor. What the reconstructed attractor is:

* The power law tells us where Bitcoin tends to be: Destination

* The attractor tells us how Bitcoin moves: Path

Figures 2a and 2b are two views of the same Bitcoin state space revealed by delay-coordinate embedding. One sees immediately that this reconstructed attractor is confined to certain regions of the volume, and the core is a pronounced smaller volume. Indeed the initial (blue) and final (orange) states both lie within the same compact region of the reconstructed state space despite spanning more than sixteen years of market history, illustrating the bounded nature of the reconstructed dynamics and the fidelity to the power law.

The reconstructed attractor is a three-dimensional projection of a six-dimensional delay-coordinate embedding constructed from the residual series r(u), where u= log10 (Age). Each point represents the state vector [r(u), r(u-τ), … r(u-5τ)], with only the first three delayed coordinates shown for visualization.

Identifying a reconstructed attractor is only the first step. In nonlinear dynamics, the next question is not whether an attractor exists, but where the system currently resides on it. Local properties such as stretching, curvature, and phase determine whether nearby trajectories are stable, diverging, or approaching critical transitions. In a future article I will examine the geometry of the Bitcoin attractor itself and ask whether these local dynamical quantities contain information about future market behavior.

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Lyapunov Spectrum

The Lyapunov exponents (usually denoted Λ_i) quantify the average exponential rate at which nearby trajectories in a dynamical system either separate or converge. Taken together, the ordered set

Λ = (Λ₁, Λ₂ … Λ_m) is called the Lyapunov spectrum.

Lyapunov exponents describe the reconstructed state-space dynamics, not the original scalar price series directly. Under the assumptions of Taken’s embedding theorem, the reconstructed attractor preserves the essential dynamical invariants of the underlying system.

One chooses an embedding dimension m, a neighborhood size K, and a delay τ. The delay is chosen as a stable compromise and in this case amounts to about 4% of age. The neighborhood where stability began to be apparent is around K = 200, or 1/6 of the age at any given time. The smooth nature of the curve vs. K is encouraging. dimension is varied to explore the natural dimensionality of the state space.

I explored possibilities ranging from m of 3 up to 10 and found the Kaplan-Yorke dimensionality rose with m, but more slowly. We show the Lyapunov spectrum of m = 6 which suggests a Kaplan-Yorke dimension for the system of 4.1, but this must be viewed as a lower bound. Nevertheless given the complexity of the network and the thousands of observations, a relatively finite dimensionality for the state space is promising for further investigation.

For our reconstructed Bitcoin residual dynamics (embedding m=6), the key qualitative results are:

• Two positive exponents (Λ₁ >0, Λ₂ >0)

  • Nearby trajectories diverge exponentially in two independent directions.

  • This is strong evidence of deterministic chaos rather than simple periodic motion.

• Remaining exponents negative

  • Perturbations contract along the remaining directions.

  • Trajectories are continually pulled back toward a lower-dimensional attracting set.

• Negative sum of all exponents

  • Sum over all Λ_i < 0

  • State-space volumes shrink with time.

  • The system is dissipative, not conservative.

• Kaplan–Yorke dimension D_KY ~ 4.1

  • When embedded in six dimensions, the long-term dynamics occupy only about four effective dimensions. Again, the true dimension may be higher and requires more exploration.

Robustness

After testing a variety of values, I settled on τ = 50 as a practical optimum, corresponding to about a 4% age delay, α range of one month to 1/2 a year or so more recently. It provided:

• enough temporal separation to unfold the dynamics,

• deterministic evolution,

• a stable Lyapunov spectrum,

• good neighbor statistics, and

• still leaves approximately 2,750 embedded states from the available data.

For the neighborhood size, K = 200 was used as a reference, spanning about 1/6 of Bitcoin’s age at any given time if τ is 50. As shown in Figure 4, the Lyapunov spectrum behaves in a monotonic fashion with K, and is beginning to appear stable with two positive exponents for m = 6 and flattening profiles for the Λ values.

Conclusions

1. The reconstructed state space is highly constrained. The delay-coordinate reconstruction produces a compact, structured trajectory rather than a diffuse cloud, indicating deterministic structure in the residual dynamics.

2. The reconstructed dynamics are consistent with an attractor. The trajectory repeatedly visits the same region of state space, suggesting an underlying attractor rather than independent stochastic fluctuations.

3. The underlying dynamical system requires more than four dimensions. An embedding dimension of six is needed for the reconstruction, while the estimated attractor dimension remains substantially lower, implying the true system has at least several effective degrees of freedom.

4. The Lyapunov spectrum contains two positive exponents (for the m = 6 reconstruction). This is evidence of local exponential divergence along two directions in the reconstructed dynamics, consistent with deterministic chaos.

5. The dynamics are dissipative. Persistent volume contraction (negative sum of Lyapunov exponents) together with a finite Kaplan–Yorke dimension indicate a dissipative system rather than a conservative one.

1. Bitcoin power-law residuals do not behave as random noise, rather they exhibit deterministic low-dimensional structure that is amenable to nonlinear dynamical systems analysis.

The coexistence of positive Lyapunov exponents, persistent volume contraction, and a lower than m Kaplan–Yorke dimension is consistent with Bitcoin’s power-law residual dynamics evolving on a reconstructed dissipative attractor in log-time state space. The reconstructed dynamics are not adequately described by three or four degrees of freedom. The effective dimension based on exploration at larger m, not detailed here, is likely to be at least six.

At present, we can conclude only that the reconstructed state space possesses a highly structured geometry, including a distinctive stretched and curved frontier. The dynamical significance of this feature remains an open question and will be the focus of a subsequent article.

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Stephen Perrenod is engaged in econophysics and nonlinear science research as Associate Director of the Scientific Bitcoin Institute.

Bitcoin investors often ask whether price has hit the “floor”. There is no floor. A better question is: What percentile of Bitcoin’s historical age-adjusted valuation distribution does today’s price occupy? This article shows how to calculate that percentile in seven simple steps. The power law ingredients have been prepared for you in advance and we share those.

For this article we use a two-sided residual model with reciprocal decay to estimate the expected magnitude of positive and negative deviations from Bitcoin’s long-term power-law trend. These estimates are then converted into local volatility and age-adjusted Z-scores, allowing every point in Bitcoin’s history to be expressed as a comparable percentile.

But you say, why not just look up recent one-year volatility and take it from there? It’s probably fine in many instances, but an age-adjusted residual method takes into account the full Bitcoin history and long-term scaling. It evolves the volatility value smoothly as Bitcoin matures and allows deviations from different eras to be compared on the same statistical scale.

Quantile regression is one valid method of looking at percentile levels; the reciprocal of Bitcoin age residual method used here is a different approach, and the table below outlines the differences between the two approaches.

In a previous article, Bitcoin’s Two Leashes, I showed that the residuals on either side of the power law trend can be reasonably modeled by two log-normal distributions, at least for purposes of ordering and normalized Z-scores. The residuals overall skew positive and are narrower than normal (possess negative kurtosis). However dual log-normals, one for each side of the power law trend, provide a reasonable approximation, allowing us to remove most of the age dependence of the volatility, and estimate percentage levels.

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7 Steps to a Percentile

We now have the background and ingredients for a 7-step procedure to calculate a Z-score for a given day and price level.

Step 1

Calculate Bitcoin age and log age. Example: If today is July 3, 2026, Bitcoin is 17.5 years old.

Log age = log10 (17.5) = 1.243. All logs are log10 herein.

Step 2

Calculate the power-law price.

log P_{PL} = -1.928 + 5.690*log (Age) = -1.928 + 5.69*1.243 = 5.145.

Price = 10^5.145 = $139,637, on power law trend at age 17.5.

Step 3

Calculate the residual for the price level. Example $58K; log P = log10(58000) = 4.763

r = log P - log P_{PL}= 4.763 - 5.145 = -0.382

Step 4

Determine which reciprocal equation to use.

If residual > 0 use:

E(r|r>0) = 2.63 / (Age+2.5)

otherwise, as in our present case:

E(r|r<0) = -3.44 / (Age+8.78) = 3.44/(17.5+8.78) = 0.131

Step 5

Convert the conditional mean residual to an equivalent σ (sigma). Assuming the positive and negative residuals are approximately half-normal, the standard deviation σ is estimated from the mean by multiplying by a factor equal to the square root of π/2 = 1.253. Thus the resulting

σ = 0.131*1.253 = 0.164 .

Step 6

Calculate:

Z = r / σ = -0.382 / 0.164= -2.329

Step 7

Convert Z to a percentile. One can use a table (or ask AI), it turns out for Z = -2.33 we are at the 1% percentile exactly. A Z-score vs. percentile level table is below.

In other words, at $58,000 and if one is at the beginning of July 2026, only about 1% of the history has been as deeply below the power-law trend, after accounting for Bitcoin’s age to normalize Z-values.

Bitcoin is at the 1st percentile of its age-adjusted residual distribution. That is a much more informative statement than saying Bitcoin has “broken the floor.” There is no single floor; there is a probability distribution.

Summary

A power law is the center of an age-dependent probability distribution. The seven-step procedure above converts any Bitcoin price into an age-adjusted Z-score and Gaussian-equivalent percentile, making valuations directly comparable across Bitcoin’s entire history.

Z-score Percentile Levels

Z-score: Normal Equivalent Percentile

+3.0 99.87%

+2.5 99.38%

+2.0 97.72%

+1.5 93.32%

+1.0 84.13%

+0.5. 69.15%

0.0. 50.00%

-0.5. 30.85%

-1.0 15.87%

-1.5 6.68%

-2.0 2.28%

-2.5. 0.62%

-3.0. 0.13%

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Disclaimer: This article is for educational and informational purposes only and should not be construed as investment, financial, legal, or tax advice. The methodology described is a statistical framework for interpreting Bitcoin’s historical deviations from a long-term power-law trend and does not predict future prices or guarantee investment outcomes.

About the Author: Stephen Perrenod is Associate Director of the Scientific Bitcoin Institute (SBI), a not-for-profit organization dedicated to research and education on Bitcoin, economics, and complex systems. The views expressed here are his own.

The median Bitcoin power-law trend explains nearly 96% of long-term price variation yet the remaining residuals contain substantial structure rather than random noise.

Earlier work demonstrated that these residuals exhibit discrete scale invariance (DSI), producing recurring oscillations with a preferred scaling ratio near λ ≈ 2.04. In this article we show that financial conditions provide an additional independent source of predictability but also confirm that the endogenous log-periodic structure dominates.

A fixed-frequency log-periodic model explains approximately 23% of residual variance using only the fundamental mode. Adding two freely fitted non-integer harmonics increases this to roughly 34%. When lagged Chicago Fed ANFCI financial conditions are included, explanatory power rises to approximately 47%, representing one of the largest single improvements obtained from any macroeconomic variable studied to date.

Even more striking are the most recent five years. During the institutional Bitcoin era, only two log-periodic modes explain more than 82% of residual variance. Adding lagged ANFCI raises total explanatory power to almost 90%, indicating that most of Bitcoin’s medium-term deviation from its long-run power-law trajectory is endogenous structure, as well as being influenced by financial conditions.

Because ANFCI is already known several weeks beyond the latest Bitcoin observation and the optimal lag is approximately nine weeks, the model naturally permits a genuine out-of-sample forecast that can be evaluated prospectively.

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Bitcoin has been shown in numerous studies to approximately follow a power-law relationship between price and network age. In this article we employ a third logarithmic transformation by constructing candlesticks that occupy equal intervals of logarithmic time. These Log-Log-Log Candles represent equal multiplicative expansions of Bitcoin’s lifetime rather than equal calendar intervals, producing a scale-invariant representation of market history.

The resulting candles reveal a remarkably consistent geometric structure across Bitcoin’s evolution. Bull and bear markets become directly comparable despite spanning vastly different calendar durations. Ordinary least squares, quantile regression, and two candle-based regression methods all recover nearly identical long-term trends, demonstrating the robustness of Bitcoin’s power-law growth.

The candle construction also provides a natural framework for estimating growth-optimal portfolio allocation. Classical Kelly, empirical Kelly optimization, and continuous-time Kelly estimates were computed across multiple candle resolutions. The optimal classical Kelly fraction declines from approximately 0.81 to 0.44 as candle width decreases, and the continuous estimates are roughly consistent with the classical method. The empirical optimization remains at the 100% allocation constraint throughout the historical sample. Together these results indicate that Bitcoin has historically supported an unusually large growth-optimal allocation compared with conventional financial assets and that the finer log-time candles approach a stable continuous-time limit.

Log-log-log Candles

I introduced triple log candle charts for Bitcoin in a Substack article five months ago. Such a presentation is especially appropriate for long-term power law behavior. The two axes of price and Bitcoin age are logarithmic, but the duration of each candle is also logarithmic in time with the later candles much longer in duration than the earliest ones.

Each candle in the graphs below aggregates all daily prices that fall within a fixed interval of log10(age). As Bitcoin ages, each candle spans a longer period of calendar time, but the same fractional increase in age; this is a deliberate choice given the power law nature of Bitcoin. The plots are log10 in price on the y-axis and log10 in age on the x-axis; the current age of 17.5 years corresponds to a 1.24 log10 value.

The advantage of this triple log candle method:

* Calendar candles compress early history into only a handful of observations.

* Log-time candles distribute observations much more uniformly.

* Bubble and bear markets from different eras exhibit striking geometric similarity.

* Market structure becomes independent of Bitcoin’s age.

This chart style asks “What happened during one multiplicative stage of Bitcoin’s life?”

Below I present charts with increasing resolution starting from 0.1 in log10 of age, then 0.05, then 0.025, and finally 0.0125 in log10 terms. The final candle in these four cases is around 4 years long for the first chart, then 2 years, 1 year or 1/2 year in length respectively for the subsequent charts.

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For the first three charts I also display previously determined OLS and QR regressions of log price vs. log time appearing as straight lines on these log-log plots, with similar slopes of 5.67 and 5.86 respectively corresponding to their respective power law exponents. One notices that the ratio of green to red candles is highest for larger values of the candle duration and bubbles become more apparent with shorter duration candles.

For this next highest resolution shown fourth chart, instead I have plotted two candle-based regressions: (1) using the geometric mean value (log midpoint) for each candle, and (2) determining a best fit through the candles using sum of least squares weighting for candles outside the best regression line. These produce very similar results, on top of each other in the graph with slopes of 5.84 and 5.88 respectively.

Indeed, all four regression methods (OLS, quantile regression, and these two candle regression methods) recover essentially the same long-run power law, indicating that the observed scaling is not an artifact of the regression technique.

Kelly Criteria (three methods) vs. Log Time Window

Log-time candle width naturally defines an investor’s time horizon. As the candles become progressively finer, the corresponding investor archetype transitions from the Stoic investor, who views Bitcoin over multi-year adoption cycles, to the Speculator, who experiences predominantly short-term volatility.

I employ three Kelly methods, each has a different philosophy:

• Classical Kelly is deliberately conservative

• Empirical Kelly simply asks: “What allocation would have maximized realized geometric growth?”

• Continuous Kelly incorporates the first two moments of returns

Despite their differences, all indicate historically high optimal allocations relative to traditional asset classes.

The three approaches answer slightly different questions. Classical Kelly depends only on win probability and payoff ratio. Empirical Kelly directly maximizes realized geometric growth over the historical return sequence. Continuous Kelly approximates optimal allocation using only the first two moments of returns.

The results displayed are the full Kelly allocations and investors often look at fractional Kelly, e.g. half-Kelly would be determined by multiplying the entries in the table below by 0.5.

Comparing the classical and continuous results we see 0.8 to 0.9 (80% to 90%) full Kelly allocations for the longest duration candles falling to 0.36 on average for the shortest candle study. One should realize that approaching the full Kelly is a little like approaching the summit of a hill, that while optimal, can see returns that fall sharply as one exceeds those full Kelly allocation values. It is interesting however that even the shorter one year durations at present suggest allocating up to half of one’s capital to Bitcoin relative to cash or T-bills.

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All of the Kelly methods if put into practice assume regular reallocation to the new f* , which is to be updated as each candle closes. Here I have included partial (unclosed) candle results using the final candles of each series.

Table of Kelly results

Conclusions

The principal contribution of this work is the introduction of scale-invariant candlesticks. Once market history is represented in equal intervals of logarithmic age, Bitcoin’s geometric self-similarity becomes immediately apparent. The Kelly analysis follows naturally from this representation by treating each log-time candle as a standardized investment horizon.

The overarching contribution is not simply a different chart type. Triple log candlesticks provide a natural multiscale framework for analyzing Bitcoin. They preserve the network’s intrinsic scale invariance, produce robust estimates of the long-term power-law trend, and define investment horizons that map directly to growth-optimal portfolio allocation.

As candle widths become finer, the continuous Kelly estimates converge while the empirical optimization consistently favors full investment over the historical sample. Classical Kelly estimates are higher for investors with longer term outlooks. Taken together, these results suggest that Bitcoin’s long-run dynamics are unusually well suited to scale-invariant analysis and support substantially higher growth-optimal allocations than are typically observed for conventional financial assets.

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This article presents historical statistical analysis rather than investment advice. Stephen Perrenod is engaged in econophysics research as a founder and Associate Director at the Scientific Bitcoin Institute.

Bitcoin’s power-law growth has been extensively studied, but less attention has been paid to the structure of the residuals around that trend. Previous work identified a log-periodic component in Bitcoin price, suggesting the presence of discrete scale invariance (DSI) with a preferred scaling ratio near a log periodic spacing parameter λ ≈ 2.

And in this article last week

I identified clear power law behavior for both Realized Cap and Realized Cap per Bitcoin, with the latter having a power law exponent of 5.81 and R² = 0.965.

The new result presented here is that the same log-periodic structure seen in Bitcoin price residuals, after removing the power law, appears in Realized Capitalization per Bitcoin (RC/BTC), a fundamentally different measure representing accumulated capital commitment rather than market valuation.

Using daily Coin Metrics Community Data (Creative Commons Attribution License) data through 2026, RC/BTC residuals exhibit a dominant log-periodic frequency near ω ≈ 8.8, corresponding to λ ≈ 2.04, virtually identical to values previously measured in Bitcoin price. A simple log-periodic model explains roughly 44% of the residual variance around the RC/BTC power law.

Wavelet analysis further reveals that the dominant frequency evolves smoothly through time rather than remaining perfectly constant. Most importantly, cross-wavelet analysis shows that price residuals and RC/BTC residuals share the same DSI structure and remain strongly phase-locked. Short-term movements are led by price, while longer-horizon causality runs in both directions, consistent with a delayed feedback system linking market valuation and trust-capital accumulation.

The implication is that Bitcoin’s famous boom-bust cycles may not originate solely in price. They appear to be embedded within the capital formation process itself.

Introduction

Previous work established that Bitcoin price follows a power law and exhibits residual log-periodic structure consistent with discrete scale invariance (DSI). More recently, Realized Capitalization and Realized Capitalization per Bitcoin were shown to follow their own power laws, suggesting that Bitcoin’s accumulated trust capital grows according to a similar scaling process. The question addressed here is whether the oscillatory component is unique to market valuation or whether it is also embedded in the underlying capital formation process itself.

Table 1. Log-Periodic Parameters

Series ω λ R²

Price 8.80 2.04 0.44

RC Total 8.79 2.04 0.431

RC/BTC 8.80 2.04 0.439

Table 1. Compares the Lomb-Scargle ω, λ values for Bitcoin price, total realized cap and realized cap per Bitcoin. In each case the suggest the fundamental model explains nearly half of the residuals.

Key Findings

1. The Same DSI Frequency Appears in RC/BTC

Lomb-Scargle and nonlinear log-periodic fits both converge near:

ω ≈ 8.8

corresponding to:

λ ≈ 2.04

This is essentially identical to the Lomb-Scargle frequency previously observed in Bitcoin price. The cycle geometry appears to be shared across price and realized cap per coin.

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2. Nearly Half of RC/BTC Residual Variance Is Log-Periodic

For RC/BTC, the log periodic fundamental mode R² ≈ 0.44. A single oscillatory component explains roughly 44% of departures from the long-run trend. That is unusually large for a financial time series.

3. The Signal Appears In Capital Formation, Not Just Price

If log-periodicity existed only in price, it might be dismissed as a speculative market artifact.

However it also appears in:

• Total Realized Capitalization

• Realized Capitalization per Bitcoin

Both are measures of accumulated capital commitment. The oscillation therefore appears to be embedded within Bitcoin’s economic structure.

4. Wavelet Analysis Reveals Slowly Evolving DSI

The dominant frequency is not perfectly stationary. The wavelet ridge implies a slowly evolving local frequency. Interpreted as an instantaneous DSI scale ratio, λ drifts from approximately 2.0–2.1 toward values near 2.2 in later years. The global Lomb-Scargle estimate nevertheless remains remarkably stable near λ ≈ 2.04.

The ridge:

• begins near ω ≈ 8.0

• rises toward ω ≈ 8.8

• gradually declines back toward ω ≈ 8.0

The corresponding scale ratio broadens from roughly:

λ ≈ 2.04 to λ ≈ 2.23 as Bitcoin ages.

Bitcoin’s cycle spacing appears to lengthen slowly through time during the past decade.

5. Price And RC/BTC Share A Common Oscillatory Structure

Cross-wavelet analysis reveals strong coherence throughout the DSI band:

8 < ω < 9.5

Wavelet phase-locking yields a mean phase ≈ 26° (0.45 radians). The PLV ≈ 0.997 indicating extremely stable synchronization with a modest price lead.

The dominant structures observed in price and realized capital are statistically distinguishable from noise yet remain strongly synchronized. This suggests that both variables are responding to a common underlying scaling process rather than exhibiting independent oscillations that happen to share a similar frequency.

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6. The Coherence Survives Surrogate Testing

Phase-randomized surrogate tests preserve spectral structure while destroying timing relationships. Observed DSI-band coherence exceeds surrogate results of 100 randomized histories with:

p ≈ 0.01

for mean coherence. The synchronization therefore appears unlikely to arise from chance alignment of similar spectra.

7. Price Leads Short-Term, RC/BTC Matters Long-Term

Cross-correlation finds maximum alignment when price leads RC/BTC by a few days to a few weeks.

However, RC/BTC also exhibits statistically significant predictive content for future price residuals over horizons extending from weeks to many months.

8. The Evidence Supports A Feedback Loop

The results are inconsistent with a simple one-way causal story. Instead they support:

Price Shock

→ Realized Cap Adjustment

→ Trust Capital Accumulation

→ Future Price

Price appears to dominate short-term discovery, while realized capital accumulates more slowly and retains predictive information over longer horizons. The interaction is therefore consistent with a delayed positive feedback process rather than a purely unidirectional causal mechanism.

9. Implications For The Trust Capital Framework

In the Trust Capital framework, realized capitalization represents the cumulative economic commitment that network participants have made to Bitcoin through time. The discovery that RC/BTC exhibits the same λ≈2 log-periodic structure as price suggests that discrete scale invariance is not merely a property of speculative valuation. Instead it appears to be embedded within the process by which trust capital itself accumulates.

Under this interpretation, Bitcoin’s familiar boom-bust cycles may reflect episodic phases of trust formation and consolidation occurring at preferred scaling intervals. Market price responds immediately, while realized capitalization adjusts more slowly as coins change hands and capital is permanently committed to the network. The observed phase-locking between the two series is therefore consistent with trust capital acting as a slow state variable that both responds to and influences market valuation.

Summary

That Realized Cap per BTC follows a power law of index 5.8 was already established.

The discovery of the same λ≈2 log-periodic structure in Bitcoin Price, and now Total Realized Capitalization, and Realized Capitalization per Bitcoin, substantially strengthens the evidence for discrete scale invariance in Bitcoin. The phenomenon is no longer confined to market valuation alone. It appears within the accumulated capital base of the network itself.

If this interpretation is correct, Bitcoin’s recurring cycles are not simply speculative episodes superimposed upon a long-term trend. They are manifestations of a deeper scaling process governing the accumulation of trust capital throughout the Bitcoin economy. Price and realized capital then become two observable expressions of the same underlying dynamical system, linked through a delayed feedback loop that operates across timescales ranging from days to years.

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Stephen Perrenod is engaged in econophysics research of the Bitcoin network as Founder and Associate Director of the Scientific Bitcoin Institute.

Santostasi and Perrenod have described Bitcoin’s long-term price growth as a power law of age, arising from the combination of power-law adoption growth and generalized Metcalfe scaling.

The underlying mechanism remains elusive but scale invariance is expected for maturing communications networks, particularly one as autopoietic as Bitcoin, due to the strength of its protocol, its strong feedback loops and the development of fat nodes with preferential attachment. A new analysis suggests that several independent measures of adoption and capital commitment may all be manifestations of the same scaling process.

Using daily observations from Coin Metrics, Realized Capitalization per Bitcoin scales as Age^5.81 with R² = 0.965, while total Realized Capitalization scales as Age^6.40 with R² = 0.967. The exponent for Realized Capitalization per Bitcoin is strikingly close to the Bitcoin power law of price vs. age exponent of 5.67.

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Bitcoin’s returns have been declining for years. That should not be surprising. If Bitcoin follows a long-term power law, then as the network ages its growth rate naturally slows.

But what about risk? Is Bitcoin becoming more risky to invest in and save in? Or is it becoming less risky?

Many investors focus on returns and conclude that Bitcoin’s best days are behind it. Yet returns are only half of the equation. Volatility matters as well. If both expected return and volatility are changing with time, then a more important question emerges:

What is happening to Bitcoin’s risk-adjusted return?

In this article I estimate Bitcoin’s expected annual return using a power-law model and combine it with an age-dependent volatility model derived from the historical contraction of Bitcoin’s deviations around trend.

The results have direct implications for two widely used measures of investment performance:

* The Sharpe ratio, which measures expected return per unit of risk.

* The Kelly fraction, which estimates the theoretically optimal portfolio allocation for maximizing long-run capital growth.

The conventional view is that Bitcoin’s risk-adjusted return must decline as the network matures. The analysis below suggests a more nuanced conclusion. Under a power-law framework, the Sharpe ratio appears to stabilize at a respectable level while the Continuous Kelly fraction actually rises.

Measured volatility decline and fit

In https://stephenperrenod.substack.com/p/bitcoins-two-leashes I noted that the quite apparent trend of a fall in volatility over time can be fit with a declining envelope for the absolute value of the residuals of log price decreasing approximately as, using annual averages:

E(|r|) = 4.0 / (Age + 8.0) and with R² = 0.51, a surprisingly strong fit for a quantity as noisy as Bitcoin volatility.

That article went on to look at two-sided volatility, here I use an overall one-sided measure which is a conservative choice since the overall distribution of residuals skews positive. This form is a reciprocal of age with an offset and can be converted to a volatility under Gaussian assumptions.

The residual distribution is not perfectly Gaussian. In fact, a two-component mixture model provides a substantially better fit. Nevertheless, the Gaussian approximation used here is conservative because the observed tails are somewhat narrower than Gaussian expectations.

I also normalize to the current observed one-year historical volatility of σ = 42% per annum to build the graphs and table presented here of Sharpe ratio and the Continuous Kelly.

Measured return decline and fit

The expected decline of the return is also measured, and it comes straight from the fit to the power law for the price history, my latest regression yields a power law exponent β = 5.67. That yields a current mean return on trend of μ = 37%. At our present age of 17.4, μ and σ, the return and volatility, are currently of similar magnitude.

That wasn’t true historically, in its earlier history both were larger but μ even more so. Under a power law β the mean expected price ratio a year later goes as [(t+1)/t]^β and thus the expected return as:

μ = (1 + 1/t)^β - 1

Figure 1 contains the central observation of this article and the reason the subsequent Sharpe and Kelly results occur. Expected return and volatility decline together. Most investors think about these quantities separately.

Yet when plotted on the same chart, both appear to follow remarkably similar age-dependent scaling laws. Due to the large range covered, I have plotted both structural expected return and age-scaled volatility as the log percentage for each variable vs. Bitcoin age in years, projected out to age 50.

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Structural Sharpe ratio

For our Sharpe ratio calculations a risk-free rate is omitted because the objective is to compare the evolution of Bitcoin’s return and volatility over time rather than to estimate excess return relative to a particular Treasury yield environment. You could easily put in your own expected rate and see how the table below changes.

Sharpe = μ(t)/σ(t)

is the projected structural Sharpe ratio considering the power law measurements. There is no assumption here for log periodicity (discrete scale invariance).

While the Sharpe ratio declines drastically in the early years, it has flattened out into a slower decline, and in fact under this model is projected to asymptote toward a very respectable 0.53 in the long run. This occurs because both expected return and volatility decline approximately as reciprocals of age and in the limit their ratio becomes a constant. The current value is 0.89 = 37.3% / 42% given the values of μ, σ at present.

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Structural continuous Kelly criterion

Kelly and Sharpe are actually complementary, Sharpe tells you how attractive an asset is relative to other assets, and Kelly tells you how much to own.

The continuous Kelly criterion for portfolio allocation fraction is more aggressive than Gambler’s Kelly. In the latter we take a series of returns, on a pre-determined window, such as annually and for Bitcoin one finds that with annual updates a full Kelly fraction of a two-asset portfolio vs. T-bills is currently in the 60% to 70% range.

Continuous Kelly simply compares the return to the variance, which is the square of the volatility. And thus is calculated as μ/σ².

Why choose continuous Kelly rather than gambler’s Kelly? Gambler’s Kelly assumes a series of discrete bets with known probabilities and payoffs. Continuous Kelly, like the original gambler’s formula, is designed to maximize growth of log wealth also, assuming Brownian motion with a given mean return and volatility (which can be updated). With the gambler’s version you could lose everything if you went all in, with continuous maximum drawdown is constrained by some multiple of σ.

Bitcoin violates many of the assumptions underlying continuous Kelly, including Gaussian returns and stable market regimes. The metric is therefore used here as a structural indicator rather than a literal allocation recommendation. This is why half-Kelly and smaller fractions of Kelly are popular guidelines.

The continuous Kelly fraction is useful here not because investors should literally leverage Bitcoin 2-to-1, but because it reveals how the balance between expected return and variance evolves as the network matures. Gambler’s Kelly is more practical as a maximal allocation estimate.

The most surprising result is not that Kelly remains high. It is that the Kelly fraction appears to have already reached its minimum roughly five years ago and is now rising.

The Kelly criterion behaves very differently from the Sharpe ratio; volatility declines approximately as 1/t, implying that variance declines approximately as 1/t², while expected return declines approximately as 1/t. As a result, the Kelly optimal fraction f* grows with Bitcoin’s age. Under these assumptions, the full and fractional Kelly allocations bottom out at around age 12 and then begin to rise. The model implies a full Kelly fraction of approximately 2.1 today, increasing to roughly 3.0 by age 40. The implication is that Bitcoin’s risk-reward profile may continue to be more attractive for large pools of capital even as its growth rate slows.

Table of Sharpe, Kelly, drawdown projections

The table shows the values of return, volatility, Sharpe ratio and the Kelly fraction at selected ages, consistent with Figures 1, 2, and 3 and has two columns for one and two σ downside estimates over time. The downside magnitude at one (two) standard deviations drops from 34% (57%) currently to 20% (36%) by age 40 under this model.

Table 1 illustrates several aspects of Bitcoin’s maturation: expected return declines substantially with age, volatility declines as well, the structural Sharpe ratio remains surprisingly resilient and approaches a finite asymptotic value near 0.53, while the Kelly fraction gradually rises because volatility declines faster than expected return. At the same time, the magnitude of typical deviations below trend contracts markedly, with a two-sigma undervaluation falling from roughly 88% below trend at age two to approximately 36% below trend by age forty.

Summary

The steps required to build this thesis are:

1. Establishing the Bitcoin power law.

2. Showing residual stationarity for the power law (ADF, PP, KPSS, cointegration).

3. Developing the age-dependent volatility envelope.

4. Calibrating the envelope to actual volatility.

5. Realizing that both μ and σ scale approximately as reciprocals of age.

6. Recognizing the implication for Sharpe and Kelly.

The value is not in the algebra. The value is in realizing those two equations belong on the same chart. When expected return and volatility are plotted together, the projected behavior of Sharpe and Kelly become intuitive.

The Sharpe ratio asymptotes toward a respectable value above 0.5. The continuous Kelly fraction bottomed at around age 12 and is rising into the future.

Bitcoin’s expected return is substantially lower than it was a decade ago. That is the inevitable consequence of a maturing network. Yet the data suggest that volatility has been falling significantly as well. The remarkable consequence is that Bitcoin’s risk-adjusted characteristics may be changing far more slowly than its raw returns. Under this framework, Sharpe remains resilient, Kelly rises, and Bitcoin becomes increasingly compatible with the requirements of the world’s largest pools of capital.

This article treats the decline in volatility as an empirical observation. In a future article I will explore a deeper question: why should Bitcoin’s volatility decline at all, and what does that imply for future drawdowns, institutional adoption, and the maturation of a monetary network?

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This is not investment advice. The greatest uncertainty in these long-horizon projections is not the mathematics of the power-law model but the future monetary, macroeconomic and regulatory environment in which Bitcoin operates. The calculations should therefore be viewed as conditional forecasts assuming the continuation of the broad adoption and maturation process observed during Bitcoin’s first seventeen years.

Stephen Perrenod is engaged in econophysics research as Associate Director at the Scientific Bitcoin Institute.

Bitcoin trades today near $61,000, while the long-term power law implies $133,227. Many observers would conclude that Bitcoin is dramatically undervalued.

However, a 50% deviation from trend meant something very different for a two-year-old Bitcoin than for today’s seventeen-year-old Bitcoin.

To compare different eras fairly, we need a volatility model that evolves.

The Dog, the Owner, and the Leash

Imagine a dog walking beside its owner. The dog wanders left and right, sometimes far away, but remains attached by a leash.

Bitcoin’s price behaves similarly. The power law is the owner’s path. The residual is the distance between the dog and the owner.

One key question is: how long is the leash? A second one is: does the length vary with Bitcoin’s age? And a third question is: are there two leashes, one for uphill excursions and one for downhill, possibly of differing length?

Measuring Distance from the Power Law

Define:

r = log P - log P_trend

Where r is the log base 10 price residual, P is current price and P_trend is the power law trend price.

Then we have the residuals vs. calendar year shown in Figure 1. There are very strong bubbles that have declined in amplitude with time and clear bear markets or trough regions that also seem to be somewhat less deep with time. We also see persistence in the residuals on the time scale of weeks and months; it takes a while for bubbles and troughs to play out, of order a year is typical.

Bitcoin’s Two Leashes

The simplest approach is to fit the average residual magnitude. This reveals a declining envelope for the absolute value of the residuals approximately of the form, using annual averages:

E(|r|) = 4.0 / (Age + 8.0) and with R² = 0.51, a rather decent fit for a noisy series.

But the market is rather clearly not symmetric, with larger upside excursions than downside. The history is suggesting that we model using two separate leashes.

That can be an uphill leash with more slack, we don’t mind so much when the dog is ahead of us, and a downhill leash that is shorter. Currently the leash lengths are comparable, but earlier on, the uphill leash was about twice as long in log price terms.

I grouped the positive residuals and negative residuals separately and determined annual averages and then fit with

E( r⁺) = a⁺ / (Age + δ⁺)

and find a best fit a⁺ = 2.63, δ⁺ = 2.5, thus 0.58 at age 2 and +0.13 currently.

And,

E( r^-) = a^- / (Age + δ^-)

with a^- = -3.44, δ^- = 8.78, which yields -0.32 at age 2 and is -0.13 currently, the same amplitude as for the positive residuals, but with a negative sign. Both envelopes therefore contract quite substantially with age.

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For years, Giovanni Santostasi — who first realized the power law nature of Bitcoin — and I have argued that Bitcoin follows this remarkably simple long-term relationship:

P(t) ~ t^β

where P is price, t is Bitcoin’s age, and β is the power law exponent, which is measured to be close to 6. A link to our scientific paper “A Mechanistic Derivation of the Bitcoin Price Power Law”, demonstrating the relationship via multiple methods, and posted at the CERN preprint site, is here: https://scientificbitcoininstitute.org/research/publications

Critics often respond with a reasonable objection:

Of course Bitcoin price rises with age. Both variables increase over time. But is the apparent relationship merely a statistical artifact? As time moves along all prices increase in dollar terms. Is Bitcoin just leveraged exponential growth?

This criticism deserves a serious answer.

To investigate, I analyzed the complete daily Bitcoin price history from Coin Metrics, spanning nearly sixteen years, from July 2010 through May 2026, comprising 5,789 daily observations.

The results were surprisingly strong.

The Power Law Fit

Using the entire daily history, the best-fit relationship for the dollar price is:

log (P) = -1.917 + 5.674 log (Age)

which corresponds to:

P = 0.0121 * Age^{5.674}

The model explains over 96% of the variation in Bitcoin’s logarithmic price history (all logs in this article are log base 10).

Attempts at an exponential fit for Bitcoin’s price history fail spectacularly, see Table 3 from the recent Baquero and Menezes paper https://arxiv.org/pdf/2605.21316 . Their fit of the power law with two parameters strongly outperforms an exponential fit with two parameters by an identical R² = 0.96 vs. 0.87 for exponential, lower residuals of 0.30 in log price vs. 0.53, and most tellingly, by an enormous margin of 6480 points on the AIC (Akaike information criterion). Even a 100 point improvement is clear evidence of model superiority

Table 1. Power Law Fit (this article)

Quantity : Value

Observations : 5,789

Start Date : July 18, 2010

End Date : May 19, 2026

Power Law Exponent β : 5.674

R² : 0.962

An R² of 0.962 is impressive, but by itself it is not enough. Two variables can exhibit a high correlation while having no genuine long-term relationship. The key question is whether Bitcoin remains tied to the power law over time.

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Dog Walking analogy

Imagine a dog walking beside its owner. The dog wanders left and right, sometimes far away, but remains attached by a leash.

Bitcoin’s price appears to behave similarly. The power law is the owner’s path, while the distance between price and the power law is the leash. During speculative bubbles the dog may sprint far ahead. During severe bear markets it may lag well behind. The owner is focused on the path forward.

The critical question is whether the leash actually exists. Does Bitcoin eventually return toward its long-term trajectory, or are the apparent power-law relationships merely a statistical illusion created by two quantities that both happen to increase through time?

Modern econometrics has tools specifically designed to test this idea.

The Statistical Tests

I used separate approaches on price and residuals.

Augmented Dickey-Fuller (ADF)

A test that asks whether a series tends to wander indefinitely or eventually returns toward a stable range.

Phillips-Perron (PP)

A more robust version of the same idea that better handles correlated noise and changing volatility.

Engle-Granger Cointegration Test

A test that determines whether two variables share a common long-term equilibrium even though both wander over time.

Step One: Is Bitcoin Price Itself Stable?

The answer is no. Both the Augmented Dickey-Fuller (ADF) test and the Phillips-Perron test indicate that Bitcoin’s logarithmic price is non-stationary.

A stationary series fluctuates around a relatively stable range. While it may move up and down, its statistical character remains broadly unchanged over time. A non-stationary series behaves differently. It can drift, trend, or evolve over long periods, so that its average level today may be very different from its average level years earlier.

Bitcoin clearly belongs in the second category.

Table 2. Bitcoin Price Tests

Test: Result

ADF on Log Price: Non-stationary

Phillips-Perron on Log Price: Non-stationary

First Difference of Log Price: Stationary

These tests indicate that Bitcoin’s price history is not simply oscillating around a fixed equilibrium value. Instead, the price exhibits a persistent long-term tendency to evolve over time.

Importantly, these tests do not tell us what form that evolution takes. They do not distinguish between exponential growth, power-law growth, random-walk behavior, technological adoption, or other possibilities. They simply tell us that Bitcoin’s price contains a powerful long-term trend.

The critical question is therefore not whether Bitcoin trends, but whether the particular power-law relationship observed between Bitcoin price and Bitcoin age is statistically meaningful.

If Bitcoin and age are merely two variables that happen to increase through time, the apparent power law could be a statistical illusion. To investigate that possibility, we remove the power-law trend and examine what remains.

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Step Two: What Happens After Removing the Power Law?

After fitting the power-law model and subtracting it from the data, I examined the remaining residuals—the deviations above and below the trend.

If the power law were spurious, these residuals would continue to wander indefinitely. In that case, the apparent relationship between Bitcoin price and age would have little statistical meaning.

Instead, the residuals become stationary.

Table 3. Residual Tests

Test : p-value, Result

ADF Residuals: 0.006, Stationary

Phillips-Perron Residuals: 0.024, Stationary

Engle-Granger: 0.026, Cointegrated

All three tests point in the same direction, and all three produce p-values below the conventional 5% significance threshold.

If Bitcoin’s apparent power law were merely the result of two unrelated quantities increasing through time, we would expect the deviations from the fitted trend to continue drifting without bound. Instead, the residuals pass multiple stationarity tests, while the Engle-Granger test finds a statistically significant long-term relationship between Bitcoin price and Bitcoin age.

In practical terms, Bitcoin does not appear to wander arbitrarily far from its power-law trajectory. Bubbles and bear markets occur, sometimes dramatically, but the departures remain bounded rather than exhibiting an ever-growing random drift. The evidence therefore supports the existence of a persistent long-term relationship between Bitcoin price and age rather than a spurious correlation between two unrelated trending variables.

The power-law trend acts as an attractor around which bubbles and bear markets occur.

This is the central result of the analysis. Bitcoin itself is non-stationary, but the deviations from a power-law function of age appear stationary. Multiple independent statistical tests support the conclusion that the observed power-law relationship is unlikely to be a spurious correlation between two unrelated variables that merely happen to trend upward over time.

A Note on Autocorrelation

The residuals are highly autocorrelated. In simple terms, today’s deviation from the power law is strongly related to yesterday’s deviation.

This should not be surprising. Consider a rocket following a trajectory to the Moon. If it is slightly above its planned path today, it will probably still be above that path tomorrow. Its position evolves continuously rather than resetting randomly each day.

Bitcoin behaves similarly. Adoption, capital flows, investor sentiment, and macroeconomic conditions evolve over months and years rather than changing instantaneously. As a result, deviations from the long-term power-law path exhibit persistence and memory.

Importantly, persistence is not the same thing as drift. A pendulum may have a highly predictable position from one moment to the next, yet remain confined within a limited range. In statistical language, such a process can be stationary even while exhibiting strong autocorrelation.

That appears to be the case for Bitcoin. The residuals are highly autocorrelated, but the stationarity and cointegration tests indicate that they remain bounded over the long run. Bitcoin may wander above or below its power-law trajectory for extended periods, but the deviations do not grow without limit.

Why This Matters

This does not prove that Bitcoin’s future price is predetermined. Nor does it prove that the power law must continue indefinitely.

What it does show is that the relationship observed over the last sixteen years is not easily dismissed as a coincidence produced by two variables that both happen to rise over time.

The power-law model survives multiple standard econometric tests designed specifically to identify spurious regressions. In particular, the Engle-Granger test rejects the hypothesis that Bitcoin price and Bitcoin age are unrelated long-term processes.

That result is important because it addresses one of the most common criticisms of power-law modeling.

What the Tests Do Not Say

These tests do not predict next month’s price. They do not identify cycle peaks. They do not explain every bubble and crash.

They simply address a foundational question: Is there evidence for a stable long-term relationship between Bitcoin price and Bitcoin age?

Based on sixteen years of daily data, the answer appears to be yes.

⸻

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Stephen Perrenod researches the network economic and econophysics properties of Bitcoin, as Associate Director of the Scientific Bitcoin Institute.

It is self-evident with the bubbles and troughs around Bitcoin’s long term power law trajectory that there is additional structure to be explained. The question is: is it purely ad hoc and exogenous, entirely due to macro and market influences, or is there a strong endogenous component, as a consequence of the scale invariant nature of the network?

Recently Carlos Baquero and Raquel Menezes posted a preprint “Bitcoin’s Power Law: Weak Structure, Strong Forecasts” https://arxiv.org/pdf/2605.21316. They stated that they can explain Bitcoin’s price history more accurately with a set of three sigmoid functions corresponding to, as they infer, major adoption waves.

However, their model is entirely ad hoc, and requires 10 parameters. The Bitcoin power law requires only two. They also evaluated two other 10 parameter models: a polynomial fit and a cubic B-spline model, and make the same general claim, although their three sigmoid model is best of the group, with R² of 0.97 vs. 0.96 for the power law, a standard deviation of 0.24 (rather than 0.30) in log10 price terms and a superior AIC of -19 vs. the power law’s 2528, very much improved (since smaller is better).

Essentially they say that Bitcoin possesses structural features that the power law model is not capturing, hence the title terminology “weak structure”. And yet, when they tested the ability of their sigmoid model to forecast Bitcoin it clearly failed, the power law performed much better, as they conclude in their paper.

This article shows that including log-periodic behavior also flips their structural conclusion, and does it with fewer parameters than their 10 parameter sigmoid, polynomial, and B-spline models. A power law plus a single log-periodic mode is statistically preferred by the price history.

And I show that adding log-periodicity to the power law improves forecasting for walk-forward models with a horizon of up to three years.

The central result is that the structure identified by Baquero and Menezes is real, but it is not best described by a sequence of logistic adoption waves. A power law plus a single log-periodic mode achieves a lower AIC with half the parameter count of the three sigmoid model, while additional harmonics improve the fit further. More importantly, the sigmoid model performs poorly in walk-forward forecasting, whereas log-periodic models improve predictive performance relative to a pure power law over horizons from six months to roughly three years.

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Why Log-periodicity?

The log-periodicity addition to the Bitcoin power law is physically motivated. It is a natural consequence of continuous scale invariance (the power law) interacting with external influences on the network, and represents the endogenous discrete scale invariance (DSI) structure for Bitcoin.

While a sigmoid model assumes Bitcoin evolves through a finite sequence of independent primary adoption waves, a log-periodic model assumes Bitcoin is a scale-invariant network whose growth exhibits both continuous scale invariance (the power law) and discrete scale invariance (log-periodic oscillations in logarithmic time). The latter is not merely a statistical fit; it is a candidate dynamical description of the system.

I have presented evidence here: https://stephenperrenod.substack.com/p/bitcoin-vs-gold-and-the-geometry and here:

and here: https://stephenperrenod.substack.com/p/bitcoin-as-a-sandworm-ii-twist-dominance and in an additional 18 Substack articles dating from November 2025.

Bitcoin’s log-periodicity has been established by a number of techniques evaluating both Bitcoin vs. the dollar and Bitcoin vs. gold, and with methods including:

• FFT

• Lomb-Scargle periodogram

• Wavelet scalogram

• Timing of peaks

• Timing of troughs

• And regression fits to the full history of price residuals after removing the power law

The fundamental log-periodicity parameter has been repeatedly found to be in the range λ = 2.01 to 2.08 throughout these various tests.

My Fits of Sigmoids and Power Laws plus Log-periodicity

I fit 5 models, the power law, a three sigmoid model, and power law (PL) plus one or more log-periodic (LP) modes. In table 1, the LP1, LP2, LP3 indicate fundamental mode only, or one or two additional harmonics, respectively and K is the number of model parameters.

Figure 1 shows my best fit models to daily price history (Bitcoin vs. dollar) using power law (PL) only, three sigmoid functions, the power law with one fundamental log-periodicity mode, and fits adding one or two harmonics. Visually the red line of the PL plus log-periodic fundamental appears superior to the 3 sigmoids, especially early on, but we leave it to the detailed statistical evaluation for definitive determination. Adding harmonics improves the fits visually. For my triple sigmoid model the best fit inflection dates at ages 4.1, 8.29, and 12.0 years align with the build-up phases of three major bubbles.

Figure 2 shows the RMSE (root mean square error) values for each of the five models and we see that the three sigmoid model does indeed outperform the power law only model, with a lower average error. However, once we add the fundamental log-periodicity we outperform the sigmoids, and the fits continue to improve when adding one or two harmonics.

Figure 3 shows the significant superiority of the three sigmoids vs. power law only for the fit over price history, the differential is similar to that seen by Baquero and Menezes. However, once again, as we add log-periodic terms, the AIC of the power law plus log-periodic fits is superior to the three sigmoids, by as much as an overwhelming 1433 improvement in AIC (again, smaller is better).

Table 2 shows the numbers represented in the bar charts for RMSE and AIC. It also shows K, the number of parameters for each model. The model designation is PL plus LP3 or LP2 or LP1 corresponding to the number of log-periodic modes (LP1 has fundamental only) and in each case the number of parameters is smaller than that for the three sigmoid model.

Inclusion of log-periodicity is able to lower the RMSE of the unexplained residual behavior from 0.3 to 0.2 or less in log price terms.

Our general assertion for the structure is that it is better explained by Bitcoin possessing a power law continuous scale invariance and also a log-periodicity reflecting discrete scale invariance. In fact we assert the two are coupled in a manner consistent with renormalization group theory.

Bitcoin adoption is not best modeled as a finite sequence of saturating logistic curves.

Instead:

* the process is recursively scale-structured, with a power law core and log-periodic oscillations

* it is oscillatory in logarithmic time, and from a physics perspective

* closer to renormalization/cascade dynamics than standard diffusion-adoption models.

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Walk-Forward Forecasting

In walk-forward forecasting we consider only prior history at a given time, determine the best fit parameters for the model at that particular time in the past, and then compare its forward forecasting results (out-of-sample relative to the fitting procedure) with the actual observed Bitcoin price history.

Figure 4 above is the walk--forward forecast of Baquero and Menezes, showing their out of sample RMSE values with horizons up to 24 months in duration. They found that a Naive model (tomorrow’s price equals today’s price) outperforms on average up to 6 months, there is a crossover at ~7 months after which the power law model (red solid line) is the best performing, with the lowest RMSE values. It outperforms a broad range of models, including the triple sigmoid (orange dashed line). It also has a flat curve, maintaining reasonable error up until 24 months.

I have performed a similar test, using 12 independent walk-forward origins from January 3, 2014 stepping a year at a time through January 3, 2025, thus ages 5 through 16 for Bitcoin. At each checkpoint I determined the best fit power law exponent and log-periodic spacing parameters β, λ for the power law model as well as associated amplitudes and phases for the log-periodic modes, only using data up until the checkpoint date, for each year in question. Where a harmonic was included the algorithm chose the best fit from among 4/3, 5/3 and 2 times the angular frequency of the fundamental, ω = 2 π / ln(λ) corresponding to the best fit λ for that series. Amplitude decay scaling as a reciprocal of age with offset δ = 2 years was used in two of the models with log-periodicity.

Then I averaged the outcomes for the overall RMSE estimates as shown in Figure 5. I looked at forecast horizons of 3, 6, 12, 24, and 48 months. My results for the power law and three sigmoid models are broadly similar. The sigmoid error (purple) rises very quickly, reaching nearly a multiplicative factor of 2.5 at one year and exceeding a factor of 3 in actual/forecast ratio (0.5 in log10) at the two year horizon. The power law (blue) improves with age. It is also outperformed by a model with only a fundamental log-periodic mode or fundamental and one harmonic (red, green curves) up until a horizon of three years.

Log-periodicity with one or two modes improves forecasting out to a three year horizon relative to a pure power law model that in turn is useful and well behaved.

Conclusion

“All models are wrong, some models are useful”. The three sigmoids model is ad hoc, has no theoretical motivation, and turns out to not be useful for forecasting. The power law model is quite useful, especially at longer time frames, and is directly motivated by Metcalfe’s law and the observed power law seen for adoption. See the Santostasi and Perrenod paper “A Mechanistic Derivation of the Bitcoin Price Power Law” available at https://scientificbitcoininstitute.org/research/publications.

The log-periodic oscillations (DSI) around the power law are a natural adjunct and endogeneous consequence of the continuous scale invariance and response of the Bitcoin network to adoption waves and appear to be useful enhancements to forecasting in intermediate time frames from 1/2 year to 2.5 years.

Spurious oscillatory models can collapse when resampling, changing resolution, and changing spacing. In contrast, here and elsewhere I have demonstrated DSI with a broad range of methods, using both daily and weekly data for Bitcoin denominated both in the US dollar and in gold terms, and found in each case similar values for the log-periodic parameter (and as a result, the fundamental angular frequency). And I have shown here that inclusion of log-periodicity is statistically useful for forecasting over intermediate time frames.

These results materially strengthen the claim that:

* the log-periodic structure is endogenous,

* persistent,

* and scale-related rather than sampling-related.

Baquero and Menezes are correct that Bitcoin contains structure beyond a pure power law. However, the evidence presented here indicates that this structure is not best described by a finite sequence of sigmoid adoption waves. Log-periodic modes provide a superior explanation of the residual behavior, achieve better information criteria with fewer parameters, and improve walk-forward forecasting. The evidence therefore favors a Bitcoin model exhibiting both continuous and discrete scale invariance rather than one governed by a series of unrelated logistic adoption events.

The important point is not merely that sigmoid models fail at forecasting. The deeper result is that Bitcoin appears to possess a reproducible hierarchy of scales. The power law captures the continuous scale invariance of the network, while the log-periodic modes capture discrete scale invariance. Together they explain both the long-term growth and a substantial fraction of the oscillatory behavior around that growth. The same characteristic spacing parameter, λ ≈ 2, emerges repeatedly across Bitcoin denominated in dollars and gold, across daily and weekly data, and across multiple independent analytical methods.

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Stephen Perrenod engages in econophysics research as a founder and Associate Director of the Scientific Bitcoin Institute (.org).

Currently Bitcoin priced in gold ounces is far below the 50 ounces expected according to its long term behavior. However, it has recovered noticeably from a low around 13 ounces to a present value of 17 oz. Since the American-Iran war began, Bitcoin has reversed upward and risen against gold by over 30%. Some of this may reflect a broader repricing of geopolitical and monetary risk, particularly around reserve assets and the long-term stability of the petrodollar framework.

Relative to the power law alone (without superposed log periodic structure), the residual standard deviation is σ = 0.323 in log 10 terms, corresponding to a multiplicative factor of 2.1. The trough near 13 ounces represented a Z = -1.8 deviation, while the current level is Z = -1.45. While this fall vs. gold is not statistically errant, it is one of the most prominent downdrafts that has occurred over the long term history of Bitcoin vs. gold behaving as a T^5.2 power law of Bitcoin age T. (Equivalently, log Price = constant + 5.2 log T).

Best Models including Log Periodicity

In my prior article I updated the Bitcoin - gold power law analysis and examined the log periodicity in some detail. I also demonstrated that log periodicity provides a highly superior fit relative to the best linear cycle uncovered. The improvement over the best linear-cycle competitor is dramatic: the log-periodic models outperform the best fixed-period linear cycle by more than ΔAIC = 1400, decisively favoring discrete scale invariance.

The fundamental mode, with log periodic spacing λ = 2.08 corresponds to angular frequency ω = 8.61 as determined from the wavelet ridge analysis, was found to dominate, but harmonics contribute some significant signal as well. In particular some fractional harmonics in thirds have much better R² and AIC scores.

Interestingly, the dominant higher-order structure is not found at integer harmonics (2ω, 3ω), but instead at fractional harmonics near 4/3 ω and 5/3 ω, suggesting a more complex nonlinear mode coupling structure. The best three models are all based on a dominant fundamental mode (possessing R² = 0.41 alone) plus contributions from two fractional modes with 4/3 ω and 5/3 ω.

What we are finding is that BTC/gold residuals exhibit persistent discrete scale invariance with stable fractional harmonic structure.

For this article I ran a few additional models and chose the best three, allowing for phase drift and/or allowance amplitude decay. The amplitude decay was found to be quite slow, especially when phase drift was added, while phase drift is modest but statistically consequential.

For the three models with a fundamental and the two fractional modes and with amplitude decay only, or phase drift only, or both amplitude decay and phase drift, the respective R² values cluster at 0.57 to 0.58 and the AICs are in a range of -1042 to -1107 (more negative is better for AIC). Thus the models are explaining nearly 60% of the observed residuals after removing the power law trend.

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Seven years ago, Giovanni Santostasi noticed the possibility of log periodic behavior of Bitcoin superimposed on its overall power law of price vs. age. The latter is continuous scale invariance, the former discrete scale invariance. The idea had not been revisited in detail; last year I confirmed its viability first in Bitcoin vs. gold terms, and then also Bitcoin vs. the dollar.

Bitcoin’s long-term power law has by now been studied extensively in fiat terms. A persistent criticism, however, is that the apparent scaling structure might simply reflect, to some significant degree, dollar debasement, global liquidity expansion, or monetary inflation.

The power law nature vs. gold is established in our scientific paper “A Mechanistic Derivation of the Bitcoin Price Power Law: Network Adoption Dynamics and Generalised Metcalfe Scaling“ posted to zenodo.org that is available also at https://scientificbitcoininstitute.org/research/publications. Here I give a fuller treatment including the latest data that examines in depth the log-periodic (DSI) oscillations around the power law core of Bitcoin vs. gold.

If Bitcoin’s scale-invariant and log-periodic structure survives denomination into gold ounces, then the phenomenon is unlikely to be merely a fiat artifact. Instead it would suggest that the structure is endogenous to Bitcoin itself: its network growth, reflexive adoption, and recursive market dynamics.

In this article I analyze daily BTC/gold data from January 2011 through May 2026 using two methods for continuous scale invariance and four methods for discrete scale invariance.

The results are unexpectedly strong. Bitcoin vs. gold clearly follows a power law. And the dominant log-periodic mode survives almost unchanged in BTC/gold space, with remarkable agreement across independent methods:

* FFT: ω ≈ 8.73

* Lomb–Scargle: ω ≈ 8.28

* Wavelet ridge: ω ≈ 8.61

corresponding to a preferred scaling ratio:

λ ~ 2.05 – 2.14

Moreover, log-periodic fractional modes outperform ordinary linear cycles by enormous statistical margins.

Bitcoin’s discrete scale invariance appears intrinsic to the network dynamics themselves, not merely to fiat liquidity expansion.

Bitcoin Measured in Gold Still Follows a Power Law

The BTC/gold ratio remains extremely well described by a power law in Bitcoin age.

The OLS fit gives:

with:

* R² = 0.939

* residual standard deviation in log10 price = 0.323

* p-value effectively zero.

The median quantile regression gives almost the identical scaling exponent:

This is important because quantile regression is much less sensitive to blowoff tops and extreme bubbles. The persistence of the scaling exponent under QR detrending suggests the core BTC/gold scaling structure is not an artifact of a few mania episodes.

The root-transform test also supports continuous scale invariance (CSI):

with:

R² = 0.912, although with substantial uncertainty in β.

The age-doubling tests are similarly remarkable. Over the 2→4 year and 8→16 year Bitcoin age octaves, the actual BTC/gold multiplier remained extremely close to the scale-invariant prediction. Only the 2013–2017 octave substantially overshot trend.

This is exactly what one expects from a scale-invariant system periodically amplified by log-periodic bubbles.

The Current BTC/Gold Regime is Historically Depressed

One surprising result is that Bitcoin measured in gold currently remains significantly below its long-term OLS power-law trajectory.

The latest anti-bubble regime began:

* 2025-08-25 → 2026-05-05

* duration: 175 days

* energy: 45.8

making it one of the larger BTC/gold undervaluation episodes in the entire series.

Residual quartile analysis also confirms that the latest age regime remains persistently below the historical trend. The latest quartile has:

* mean residual: −0.191

* median residual: −0.165

with almost no positive excursions above trend.

This suggests Bitcoin’s recent relative underperformance against gold is not merely short-term noise but part of a coherent anti-bubble structure.

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The Fundamental Log-Periodic Mode Survives in BTC/Gold Space

After removing the power-law backbone, the residuals were analyzed in log-age coordinates.

The FFT spectrum is extraordinarily clean. The dominant frequency is:

ω = 8.73

and therefore:

The remarkable result is that this is almost identical to what has previously been found in BTC/USD studies.

That matters greatly.

If the log-periodic structure were mainly driven by fiat liquidity or dollar debasement, one would expect it to weaken substantially when Bitcoin is denominated in gold. Instead the dominant mode remains almost unchanged.

Independent methods confirm the same structure:

Method ω λ

FFT 8.73 2.05

Lomb–Scargle 8.28 2.14

CWT ridge 8.61 2.08

The agreement across methods is unusually strong.

Harmonics Exist — But Ordinary Harmonics are Weak

The FFT reveals a genuine harmonic ladder, however, ordinary harmonics turn out not to be the dominant higher-order structure. The second harmonic improves the fit only marginally:

with only a small AIC improvement.

Instead, the dominant higher-order structure comes from fractional modes:

These dramatically outperform ordinary harmonics.

The best compact model is:

with:

R² = 0.574

and an enormous improvement over the single-mode model:

This is not a subtle effect.

The fractional modes are not small corrections. Their amplitudes are large percentages of the fundamental mode itself.

This suggests Bitcoin’s log-periodic structure behaves less like a simple Fourier oscillator and more like:

* nonlinear mode coupling,

* weak turbulence,

* resonant splitting,

* or interacting scale hierarchies.

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Why the Strong Fractional Modes Matter

Ordinary Fourier systems tend to favor integer harmonics:

2ω, 3ω, 4ω

But Bitcoin/gold strongly prefers:

This is physically suggestive.

The spectrum resembles:

* coupled nonlinear oscillators,

* sideband generation,

* beat structures,

* or resonant mode splitting.

The continuous wavelet transform makes this especially clear.

The fundamental mode persists across almost the entire Bitcoin history as a coherent ridge near:

while the higher modes appear intermittently, strengthening during certain bubble epochs and fading during others.

This is much more characteristic of a relatively new and evolving nonlinear dynamical system than of ordinary market cycles.

The system appears to possess:

* a dominant coherent DSI backbone,

* evolving phase structure,

* intermittent harmonic enrichment,

* and nonlinear waveform sharpening during bubbles.

Modes are Multiplicative, Fractional

A key conceptual point is that the log-periodic structure should not be treated as a small additive perturbation.

The correct form is multiplicative in price space:

not

The additive form is only a first-order approximation valid for small A. Bitcoin’s deviations are often factors of several times above or below the power-law core, so the multiplicative formulation is the physically and mathematically coherent description.

A standard motivation for the exponential form is that nonlinear multiplicative modulation naturally generates spectral structure through a Bessel expansion, which produces an integer harmonic ladder ω , 2ω , 3 ω, …

However, the empirical Bitcoin spectrum is more subtle. The strongest higher-order modes are not primarily integer harmonics, but instead appear near fractional frequencies such as

This suggests that the observed structure is not simply a textbook harmonic expansion of a single oscillator. Instead, the log-periodic carrier itself appears to undergo modulation, splitting, or phase structuring in log-time.

In this picture, the observed fractional modes are not external artifacts and not ad hoc synthetic additions. They are likely the signature of nonlinear multiplicative uplift and suppression around the scale-invariant backbone, together with slow endogenous modulation of the log-periodic phase itself.

Rather than a single perfectly stationary discrete oscillator, Bitcoin appears to exhibit a structured hierarchy of interacting log-periodic modes embedded within the broader continuous scale-invariant power-law evolution.

Linear Cycles Perform Poorly

The best linear cycle fit occurs near:

* 3.5 years

with:

R² = 0.371

But the best synthetic log-periodic model achieves:

R² = 0.574

with:

relative to the best linear cycle.

That statistical gap is enormous. The data strongly prefer evolving discrete-scale structure over ordinary calendar-time cyclicality.

Phase Drift Matters More Than Amplitude Decay

Initially the residual amplitudes appeared to decay approximately as:

However, once the fractional modes are included, the need for strong reciprocal damping largely disappears, with delta large compared to Bitcoin’s current age.

Instead, phase drift becomes the dominant improvement.

Allowing the fractional modes to evolve slowly in phase produces further gains:

R² = 0.582

for the best model, with another large AIC improvement.

This suggests Bitcoin is not governed by static log periodic oscillations, but by evolving coherent modal structures in log time.

The system behaves more like:

* a nonlinear wave medium,

* a coupled resonant network,

* or a weakly turbulent scale-coupled system,

than a stationary financial oscillator.

Residual Structure After Removing the Three-Mode System

After removing:

the residual standard deviation falls substantially:

The remaining residuals are close to Gaussian but still retain moderate heavy tails.

A Student-t distribution outperforms a Gaussian with ν = 17.7, suggesting some remaining exogenous structure and volatility clustering.

GARCH testing shows near-IGARCH persistence:

indicating long-memory volatility remains even after removing the dominant DSI structure.

This is important conceptually: the log-periodic system removes a very large fraction of the coherent structure, but Bitcoin/gold still retains residual stochastic dynamics.

Conclusion

Like Bitcoin measured in dollar terms, Bitcoin measured in gold also exhibits:

* strong continuous scale invariance,

* robust log-periodic structure,

* coherent wavelet ridges,

* synthetic modal hierarchies,

* and statistically decisive superiority over linear cycles.

The dominant mode:

survives:

* OLS detrending,

* quantile regression detrending,

* FFT analysis,

* Lomb–Scargle analysis,

* and continuous wavelet analysis.

That robustness is difficult to dismiss as accidental.

The most important implication is probably this:

The key empirical result is that the same preferred scaling ratio,

λ ~ 2.1,

persists across BTC/USD and BTC/gold, across FFT, Lomb–Scargle, wavelets, OLS detrending, and quantile detrending. That degree of structural persistence is difficult to explain as a fiat-denominated artifact. Bitcoin’s discrete scale invariance appears endogenous to the Bitcoin network itself.

The fractional modes are intriguing. Bitcoin/gold does not behave like a simple harmonic oscillator. It behaves more like a nonlinear evolving resonant system with interacting scale structures.

Bitcoin may be better understood not as a conventional financial asset exhibiting cycles, but as a scale-coupled nonlinear network undergoing recursive phase organization across logarithmic time.

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Stephen Perrenod engages in econophysics research as the Associate Director of the Scientific Bitcoin Institute.