Bitguns⚡️

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What Happens When a Power Law Learns to Flow? For fifteen years, one number has quietly governed Bitcoin. Plot price against time since the genesis block on log-log axes and the whole history collapses onto a straight line: price ∝ time^5.65. We usually draw that line and stop there. But a line is a frozen thing, and Bitcoin is not frozen. So let's do what Kolmogorov did to Markov in 1931 — stop asking where the line sits, and watch the entire distribution move. Change coordinates. Let x = log price and let the clock be τ = log t. In this frame the power law is no longer a curve; it is a constant-velocity drift. The cloud of every possible price — the probability law p — keeps a fixed width (the corridor) while its center glides upward at a constant speed. And that speed is the exponent itself. So 5.65 is not a slope. It is a velocity — the rate at which probability streams up the corridor. Written as motion: ∂p/∂τ = −b·∂ₓp + D·∂²ₓp, with net current J = b·p. The bright layer is that current. The fog is the evolving law. The particles are sample paths of the very same law. And the gold line is the real Bitcoin — one realized path, one draw from the fog, spiking and crashing but never leaving the channel the current carries it through. How was the flow built? Not invented — measured. From real daily prices, 2010 to 2026. The fit gives b ≈ 5.65. The jitter around the trend has heavy, Student-t tails (that's where the violent moves live), and its volatility decays like ~1/time — which is exactly what cancels the growth and holds the corridor's width constant in log-time. The animated paths are simulated from that fitted generator; the fog is its law; the arrows are J = b·p. Here is why it matters for the theory. Bitcoin never relaxes to a fixed "fair price." The fog holds its shape while flowing, forever. Physicists have a name for stable-and-moving-at-once: a nonequilibrium steady state. Detailed balance is broken; a permanent probability current flows. In that light the power law is not a lucky curve-fit — it is the signature of a steady current. And the quantity I call the local slope, n = log(P₂/P₁)/log(t₂/t₁), is simply the pathwise velocity of that current: n = b + noise. Its distribution has stayed stationary for years. That stationarity is the fingerprint of a constant flux — the macroscopic law and the microscopic wiggle turning out to be the same object. One law. Two lenses. Microscopic paths and macroscopic flow — the Power Law is probability itself, moving.