The position level changes sharply when the rolling window changes. Using the same price history and estimator, alpha/sigma_0^2 is 0.7441 on the four-year window and 1.3701 on the one-year window, a factor of about 1.84. These are reference levels at system age one year. The model's current prescription is different. At age 16, our own arithmetic using the same two calibrations gives roughly 0.85 and roughly 1.08, bringing the spread nearer 1.3x. The level depends on calibration either way. The exponent likely to attract the attention explains little of that gap.
A disclosure belongs next to any figures of our own. Our crypto price coverage starts around 2018, near system age 9, leaving roughly three quarters of the log-age range in the paper's price regression outside our data. We also lack the on-chain fee and network-participation series. We therefore could not replicate the variance-modifier scenario, the extension where fee dispersion multiplies variance. The unconstrained continuous-time fraction also cannot be traded as written. An investable version would require caps, discrete rebalancing and costs, making it a different object from the paper's idealised allocation.
Vera-Marun states the setup in two lines of algebra. The risky asset follows dS/S = mu dt + sigma dW. A deterministic long-run price trend supplies the drift: P(t) = At^alpha, where t is measured in years from the 3 January 2009 genesis block. Proportional growth is therefore mu(t) = alpha/t, while instantaneous variance decays as sigma_0^2 t^-2gamma. With cash set to zero, the unconstrained log-optimal fraction becomes K*(t) = (alpha/sigma_0^2) t^(2gamma-1). System age leaves that fraction constant if and only if the volatility-decay exponent equals one half. The entire age-invariance result depends on this equality.
The paper motivates the equality through a chain of scaling assumptions, explicitly presenting them as assumptions rather than a derivation. Cumulative wallets grow as t^eta, with the benchmark eta = 3 drawn from a cited adoption study. Active participation, taken as the time derivative, then scales as t^2. Liquidity depth rises with active participation to the power delta. Setting delta at one half makes depth proportional to t, and volatility falls with the inverse square root of depth. This produces gamma = delta(eta-1)/2 = 1/2. No network-activity or liquidity data appear in the paper. The author directly calls for testing these relationships with activity and microstructure data, then repeats that request in the closing section.
The empirical work uses daily Bitcoin closes from CoinGecko, accessed September 2026. Age begins at genesis, and the first post-genesis year is removed from the log-log regressions. Missing calendar dates are forward-filled with the last close before daily log returns are calculated. Across the full sample, the price fit estimates alpha = 5.185 plus or minus 0.023 and an R2 of 0.9126. The paper next calculates realised annualised variance on overlapping rolling windows of one to twelve years, then regresses it on log age.
What survives the window choice?
The volatility exponent is fairly stable across the selected windows. It is 0.4576 for a one-year window, 0.5240 at four years, 0.5577 at eight and 0.4786 at nine. Across the four-to-nine band, the arithmetic mean is 0.53 and the cross-window standard deviation is near 0.03.
After nine years, the estimate breaks away: 0.6188, 0.9151 and 1.4037, based on remaining regression spans of 3.36, 2.36 and 1.36 years respectively. The author compares this behaviour with a finite-size effect. Table I reports T=10, 11 and 12 while omitting those windows from the summary range. Nor does the paper treat the rise in R2 from 0.3518 to 0.9377 as evidence of greater precision. The author attributes part of that increase to mechanical smoothing from overlapping windows. That restraint is warranted.
Window choice has little effect on the invariance claim. Between two ages, the weight changes by (t2/t1)^(2gamma-1). At 0.4786, the low end of the band, doubling system age reduces the position by about 3%. At the high end of 0.5577, it raises the position by about 8%. The two adjacent windows reverse the direction while keeping the size of the change small.
The position size is the fragile part.
At unit age, K* equals alpha/sigma_0^2. The fitted log variance intercept is 1.9414 with the four-year window and 1.3310 with the one-year window. That intercept explains the difference between the paper's reference calibrations of 0.7441 and 1.3701. The paper says it "should be interpreted as a calibration-dependent parameter rather than a unique structural quantity". It also describes the four-to-nine window criterion as diagnostic rather than optimal. Both qualifications fit the evidence.
The author then disclaims the numerical level directly: "The principal result of the analysis concerns the temporal scaling of K*(t), rather than the particular numerical level obtained from one calibration." The abstract makes the corresponding point about window choice, describing the 0.03 spread as model sensitivity rather than a formal confidence interval. That treatment is fair on the paper's terms. Yet a position is sized by the level, which is fixed by the intercept and alpha = 5.185 plus or minus 0.023. The paper still presents 0.74 as a benchmark and plots it as a reference line in Figure 3.
Four judgment calls remain
- The drift. The estimate alpha = 5.185 comes from the full sample. As the author writes, the exercise is "an in-sample calibration and consistency check, rather than an out-of-sample test". The paper claims no backtest. Since K* varies linearly with alpha, implementation requires an expanding-window estimator that the paper does not specify. We have previously written about growth-optimal allocation designed to avoid drift forecasts altogether.
- The zero rate. The paper declares r(t) = 0 to isolate age dependence. A constant positive rate makes excess return alpha/t minus r, which loses the power-law form. Under the paper's own proposition, the matching-exponent condition then cannot hold. At age 16, model drift is 5.185/16, about 32% a year. Using the four-year parameters and adding a 5% cash rate, our assumption because the paper sets r(t) = 0, our arithmetic lowers the weight from roughly 0.85 to roughly 0.72.
- The forward fill. Forward-filled dates create zero returns, lowering realised variance and raising K*. The paper says the number of affected observations and the sensitivity to removing them should be reported as a check. We did not find those figures in the paper.
- The annualisation convention. The author lists it among the choices affecting the 0.74 estimate, together with data treatment and the span of price history.
Fee dispersion and the broken clock
The final section contains no fee data, and its response parameter remains unestimated. Variance is written as sigma_0^2 t^-1 g(t), combining an age-decay term with a modifier. The resulting Kelly fraction is (alpha/sigma_0^2) g(t)^-1. As the paper puts it, "Exact temporal invariance is recovered only when g(t) is constant."
For Bitcoin, relative transaction-fee dispersion supplies the illustration, although the response parameter is never fitted. The broader statement carries more weight than this example. Any risk driver operating on a clock other than t^-2gamma breaks the invariance. The author's own list also excludes leverage and position constraints, transaction costs, parameter uncertainty, jumps, drawdown limits and estimation risk from the derivation.
A volatility exponent near one half, estimated from non-overlapping variance observations, would move me. Evidence from a second asset with its own age clock would as well, provided the window were fixed before seeing the estimates. Until then, the paper offers clean algebra attached to a calibrated constant.