When the Same Pattern Shows Up in Unconnected Places, Is It a Law or a Coincidence?

• A regularity across enough unconnected fields is presumed to point past any single one of them, into its shared parameter.
• The recurring number buys a shared constraint, not a shared mechanism.
• The mechanism has to be demonstrated separately, in each field.

Drop a stone, and its pull weakens with the square of the distance. Move a charge, and its field weakens with the square of the distance. Step back from a lamp, and its brightness falls with the square of the distance. The same number, an exponent of two, keeps turning up in places that have nothing to do with each other. Gravity does not know about electric charge. A lamp does not know about either. So why do they agree?

This is a question about method, and it comes up the moment you notice any regularity in more than one place. A ratio that recurs in finance and in biology. A curve that fits both traffic and neurons. A threshold that shows up in three experiments that were never meant to be compared. Each time, you face the same fork, and how you walk it decides whether you are doing science or telling yourself a story.

Law or coincidence: how the two readings differ in cost

When one regularity appears in several domains that share no causal channel, there are two ways to read it. The first: each domain happens to show the pattern for its own unrelated local reasons, and the agreement is a pile of coincidences. The second: the pattern does not belong to any one domain, and each place is just one instance of it.

Here is the part worth being precise about. Suppose each domain has some modest chance of showing the pattern by accident, and the domains really are independent. Then the chance that all of them show it by accident is that small chance multiplied by itself once per domain. Each extra domain multiplies the cost again, so if that per-domain chance is modest, the coincidence reading gets steadily more expensive the more unconnected places agree, while the other reading pays no such penalty.

So the sensible default flips as unconnected domains pile up. Past a few of them, the reasonable starting assumption is that the regularity is shared, and that the local details are the thing you still have to explain. This does not prove any particular law. It just moves the burden onto whoever wants to call the agreement a coincidence, and that person now owes you one independent accident per domain.

Call this the presumption of nonlocality: a pattern across enough unconnected places should be presumed to point past any one of them.

Shared constraint vs shared mechanism: the safeguard that keeps the rule honest

The presumption is easy to overspend. Someone who has earned it for a recurring number quietly spends it on a recurring story about why, and those are two different purchases.

Keep them apart:

• The parameter is the thing that recurs you can actually point at: a number, an exponent, a ratio, a shape of curve. Its recurrence is a fact, and the argument above says to presume that fact points to something shared.
• The mechanism is the process you claim produces it. That is a much bigger claim, and noticing the recurring number does not hand it to you.

The recurring number buys you a presumption of a shared constraint: maybe several domains are all subject to the same geometry, the same conservation law, the same bit of arithmetic. It does not buy you a shared process. The process has to be earned separately in each domain, because the same number can be reached by many different roads.

Watch the two cases side by side.

The inverse-square law pays off. Gravity, electric fields, light, and sound all thin out as the inverse square of distance, and those domains do not talk to each other. The presumption says: treat the exponent of two as shared. And here you can collect on it. The shared thing is geometry. Anything that spreads out from a source has to cover the surface of a sphere, and a sphere's area grows as the square of its radius, so whatever you are measuring gets diluted as the inverse square. The number recurs because a constraint recurs, and you can show the constraint. The presumption turns into a proof.

Power laws are the cautionary case. The same shape of curve, a power law, has been reported for the sizes of cities, the magnitudes of earthquakes, the frequencies of words, the energies of solar flares, and the sizes of fortunes (Newman, 2005). The shape recurs across places that share nothing causally, so the shape itself is a real fact that wants explaining. The tempting move is to announce one law of nature with one mechanism behind all of it.

That move is the error. Power laws of the same kind are produced by several unrelated processes: rich-get-richer dynamics, systems that tune themselves to a critical point, optimization under a budget, random multiplicative growth, and more. Newman's review walks through the main families. Stumpf and Porter (2012) put the knife in twice. First, a lot of the reported power laws are not even solid power laws: over the limited range measured, a log-normal or a stretched curve fits about as well, and most claims never ran the statistical test that would settle which shape it really is. Second, even when the shape is real, finding it is rarely the scientific win, because the shape on its own tells you very little about what produced it. Grant a real power law and the recurring exponent is still only a shared number, not a shared mechanism. Reading one process off the shared curve is exactly the overspend the safeguard blocks.

The rule in one breath

A regularity shared across enough unconnected domains should be presumed nonlocal in its parameter: the regularity points past any single domain, and the local realizations are what you explain. But do not promote a shared parameter to a shared mechanism without showing the mechanism, separately, in each domain. The number buys a constraint. The process is a different bill.

A bet you can settle

First, one thing the bet is not about. Whether the parameter-versus-mechanism split is valid is already settled, and not by any survey: the power-law case alone shows a recurring number that several unrelated processes can produce, so a recurring number can never, by itself, hand you the mechanism. No tally of other cases can undo that. Even if shared mechanisms turned out to be common, you would still need the separate per-domain check to spot the exceptions.

What a survey can settle is how often the safeguard actually bites. So here is the prediction that makes this more than advice. Fix the reference class to the documented scaling-exponent recurrences in Newman's catalogue, and for each pair of domains, judged blind, count it as sharing a constraint when the same dimensionality, conservation law, or normalization provably forces the number, and as sharing a process when the best-supported models are the same one. I expect constraint-only matches to come out above 70 percent. If they instead fall below half, the safeguard rarely bites in practice and you can mostly skip it, though it stays valid where it does apply. Either way the how-often claim is checkable by the pattern across many cases, which is the only honest place to settle a question about method.