The power law emerges even when we use addresses with different wallet balances. This is another signature of scale invariance. Three address tiers were constructed: •Shrimps = total non-zero balance addresses (the full dataset) •Crabs = addresses holding ≥1 BTC = (1–10 BTC) + (10–100 BTC) •Dolphins = addresses holding ≥10 BTC = (10–100 BTC) only ________________________________________ Panel 1 — N(t) vs time, log-log Each tier is plotted as log₁₀(addresses) vs log₁₀(t_days). An OLS linear regression on these log-transformed values gives the power law exponent n for each tier — the slope of the best-fit line. The dashed lines are those fits. The x-axis ticks are converted back to calendar years for readability. Panel 2 — Generalized Metcalfe, log-log Price vs addresses for each tier, both log-transformed. OLS regression gives the Metcalfe exponent α — how steeply price scales with the number of addresses in that tier. Since larger holders are rarer and harder to add, their α is steeper. Panel 3 — Combined price model, log-log The key result. Because P ∝ N^α and N ∝ t^n, substituting gives P ∝ t^(n·α). So each tier produces an independent price-vs-time prediction using only its own address data — no direct price fitting. The intercept is ic_combined = ic_Metcalfe + α × ic_time. All three lines are plotted against the actual price (white line) on log-log axes. Tiern (time)α (Metcalfe)n × α Shrimps3.060 1.831 5.604 Crabs (≥1 BTC) 1.383 4.021 5.564 Dolphins (≥10 BTC) 0.462 11.080 5.116 The convergence emerges because n and α trade off against each other across tiers. When you use a harder-to-reach tier (larger holders), n drops (those addresses grow more slowly) but α rises (price is more sensitive to each additional whale). The product n·α stays approximately constant at ~5.5–5.6 across all three tiers — which is also the global Bitcoin power law exponent from the direct price fit. This is the generalized Metcalfe theorem: the price exponent is invariant to which address tier you use as the adoption proxy.