If we have so much noise in the system how do we know that Bitcoin follows a power law and it is scale invariant?

The answer is that noise and signal operate at different timescales, and the tests that establish the power law are specifically designed to separate them.
The residual noise of ±0.30 dex and the cycle-to-cycle variation in β of ±0.57 are both real. But they are oscillations around a stable attractor, not evidence that the attractor does not exist.
Think of it this way: a pendulum has a well-defined equilibrium position even though it is never at rest at that position. The amplitude of the swing does not tell you the equilibrium is uncertain — it tells you the system has energy. Bitcoin's halving cycles are the energy. The power law is the equilibrium.
The more precise answer has four parts.
First, R² = 0.961 across 5,696 observations spanning six orders of magnitude in price. Noise that is truly random averages out over large samples. If the residuals were not oscillating around a fixed line — if the underlying relationship were not stable — the cumulative R² would not grow monotonically toward 0.96 as we add more data. It does. That monotonic growth is direct evidence that there is a signal beneath the noise.
Second, the noise itself has structure that confirms the power law. If β were genuinely unstable — if the power law were breaking down — the residuals would show a secular trend: systematically drifting upward or downward over time. They do not. The residuals are stationary. They oscillate with the four-year halving cycle and return to zero. Structured, mean-reverting noise around a stable line is not evidence against the line. It is evidence for it.
Third, the scale invariance tests bypass the noise entirely. The pair-ratio test does not ask "does a regression fit well?" It asks a model-independent question: does P(λt)/P(t) = λ^β hold for arbitrary λ? We tested this with 5,298 directly measured price ratios across 300 anchor times and 25 multipliers. The answer is yes, to within 2% across three independent estimators. This test is immune to the distributional assumptions that OLS critics invoke — it requires no normality, no homoscedasticity, no independence of errors. It just asks whether the functional identity holds in the data. It does.
Fourth, the Bayesian sequential analysis shows the noise is bounded and the signal is stable. After 1,899 local β estimates the posterior is β = 5.729 ± 0.013 and the uncertainty shrinks precisely as σ/√n with no structural breaks at any halving event. If the power law were not real — if it were an artifact of OLS assumptions — the posterior would not converge. It would plateau or reverse as contradictory data accumulated. It does neither.
So the answer to the skeptic is this: the large noise tells you Bitcoin is volatile.
The stable attractor beneath the noise tells you that volatility is oscillation, not drift. These are not contradictory statements.
They are the two defining properties of a dynamical system with a power law attractor — and both are confirmed independently by the data.