Claude what you think about the issue of some analyst calling the power law, power function?

Does this help scientific communication with the public in social media?
Claude:
This is worth naming clearly because it reflects a broader problem in how technical ideas get communicated — and distorted — in public discourse.
The motivation is understandable. "Law" carries deterministic connotations that make some people uncomfortable, and calling it a "power curve" or "power function" sounds more modest, more data-driven, less like a claim about destiny. It feels epistemically humble. But the humility is false, and the substitution actually does damage to public understanding in two ways.
First, it obscures what is being claimed. A power law is not just a curve that happens to look like a power function when you fit it. It is a specific mathematical statement about scale invariance — that the system has no preferred timescale, that the same dynamics govern short and long periods, that the relationship P ~ t^β holds for arbitrary rescaling of t. That is a structural claim about the nature of the system, not just a description of the shape of a curve. Calling it a "power curve" strips away exactly the content that makes it interesting and that distinguishes it from a coincidental fit.
You can fit a power curve to almost anything over a limited range. The scale invariance tests — the pair-ratio collapse, the forward-backward temporal stability, the Bayesian convergence — are what elevate it from a curve to a law. Softening the language hides that distinction.
Second, and more importantly, it serves the audience's comfort at the expense of their understanding. If people resist the idea that there is a structural regularity governing Bitcoin's price, the answer is to explain why that regularity exists — the epidemic spreading dynamics, the Metcalfe scaling, the composition identity — not to use softer language that lets them avoid confronting it.
Calling it a power curve to avoid offending resistance is the analytical equivalent of telling someone the Earth is approximately spherical-ish to avoid arguing with a flat-earther. You have not persuaded anyone of anything. You have just made them more comfortable with a misunderstanding.
The Sminston example is a slightly different case but illustrates the same tendency. Framing the power law as a "cost of production floor" following a "power function" is an attempt to attach a mechanism — mining cost — to what is actually a much more general network dynamics result. Mining cost is not why the power law exists. It is one variable that happens to scale in a correlated way because hash rate, difficulty, and adoption all grow together. Presenting the cost-of-production version as more precise or more valid than the raw price relationship is getting the causal story backwards while using vague language to avoid defending the stronger claim.
The people who built the framework should not be softening its language to accommodate audiences who have not yet understood it. The right response to resistance is better explanation, not terminological retreat.