What we found when we asked a physical constant, instead of what it is, who is looking at it
There’s a number that shows up in almost every physics book wearing an air of mystery: 137. It’s approximately the inverse of what’s called the fine-structure constant — the number that measures how strong the electromagnetic force is, the one that holds atoms together and makes light behave the way it does. Physicists of an entire era, Richard Feynman among them, pointed to it as one of nature’s great unexplained numbers: there’s no theory that says why it has to be 137 and not 136, or 140.
For the last fifteen years I’ve been working on a theoretical framework of my own, ArXe, that starts from a specific bet: that these numbers aren’t arbitrary, but instead register an underlying logical structure — layers of increasing complexity, each with its own associated number. The work consists of factoring physical constants into those numbers and seeing whether the factorization makes sense, not just whether it “adds up.” Only in the last two years, with help from artificial intelligence, was I able to move with the speed and verification rigor this kind of work needs.
This week, while reviewing that work with Claude (Anthropic’s assistant), we found something that changes how I understand the whole project. It wasn’t a new formula. It was a different question.
Before going further: what a number is, here
It’s worth pausing here, because otherwise this can sound like numerology — the habit of finding meaning in the fact that some date adds up to some number. It isn’t that, and the difference matters.
In ArXe, a number isn’t a quantity that shows up by chance. It’s an arity: the minimum number of mutually distinct states needed for a certain kind of distinction to be possible. It’s not a label we stick on the phenomenon afterward — it’s a condition of possibility, something that has to be available before the phenomenon can occur.
A few examples, from simplest to most complex:
Binary logic (2). A switch: on or off. It’s the simplest distinction that exists — there’s nothing smaller than “this, or that, but not both at once.” Any system that needs to distinguish something from its absence needs, at minimum, this structure.
Ternary logic (3). Rock, paper, scissors: three states where each one beats one and loses to another, closing a cycle without any of them being “the strongest” outright. Two states aren’t enough to express that kind of circular, non-hierarchical relationship — a third is needed.
Quaternary logic (4). The four suits in a deck of cards, or the four cardinal directions: it isn’t simply “double the binary” — it’s a distinction that crosses two independent axes at once (for example, north/south combined with east/west). It needs four slots so that no case is left without a place.
And so on, n-ary in general. A die needs six faces because there are six equally possible outcomes to distinguish; it can’t work with five. Each arity is the minimum size of the “alphabet of states” that a particular kind of distinction requires in order to function without ambiguity.
When a formula for a physical constant uses the number 13, the claim isn’t “nature has a fixation on 13.” It’s a structural claim: this phenomenon, to be what it is, needs at minimum a logic capable of distinguishing 13 simultaneous states. It’s a hypothesis about the minimum logical complexity behind the phenomenon — testable, arguable, maybe wrong — but not a numerical coincidence. With that clarified, it’s worth returning to 137.
The question we’d been asking wrong
For a long time, the implicit question was: what’s the correct formula for this constant? We searched, for 137, for the combination of small numbers that explained it best — and when we found several that worked equally well, we treated that as a problem to solve, not as information.
This week we flipped the question. Instead of looking for the formula, we asked: what if every formula that works is, in some sense, true — but not of the same phenomenon?
The analogy I use to think about it: we know waves exist. Wave-like behavior is a real, common pattern. But a sound wave propagates through air, and an electromagnetic wave propagates through vacuum — they share structure, not substance. Maybe the different formulas for 137 aren’t candidates competing to be the one true one. Maybe they describe distinct, related aspects of the same behavior — visible from different vantage points.
To make this concrete: using only the numbers 2, 3, 5, and 7 (with their powers), we found eight distinct combinations that give exactly 137:
2⁷ + 3² = 128 + 9 = 137
2² + 2³ + 5³ = 4 + 8 + 125 = 137
3¹ + 3² + 5³ = 3 + 9 + 125 = 137
5¹ + 5³ + 7¹ = 5 + 125 + 7 = 137
3⁴ + 7¹ + 7² = 81 + 7 + 49 = 137(plus three more). None is more “correct” than the others in the arithmetic sense — all eight give exactly the same number. The question that mattered stopped being which one to choose, and became what happens to each one when we ask it for more precision.
What “position” means, here
That word — position — ended up being the key to the whole session. When you factor a constant and find a combination of small numbers that reproduces it, you’re not necessarily describing what the physical phenomenon is. You might instead be describing from where you’re looking at it: what instrument you chose, what measurement convention you accepted, what question you asked first.
Put differently: the formula isn’t a portrait of the phenomenon. It’s more like a return address — it tells you from what place on the map the observation was made, not what’s in the landscape being observed.
This isn’t cheap relativism (“anything goes, there’s no truth”). It’s more precise than that: some positions are more consistent than others, some formulas hold up better under scrutiny than others, and some are simply wrong (while reviewing earlier work, we found a formula for 137 that was off by a factor of almost nine — nobody had checked it numerically before accepting it). But among the ones that do work, picking one as “the” true one is a choice, not a discovery.
The detail that made it testable, not just philosophical
This is where the session stopped being a nice reflection and started having teeth.
If a formula describes an observer’s position, then looking for more precision — adding one more decimal to the number — is also a choice made from that same position. And that choice has a rule, I suspected, almost before I could prove it: the correction that adds precision can’t contradict the position you already adopted. It can add to it, extend it — but it can’t ask the same piece to mean two different things at once.
We tested it against real numbers. The real, measured value of the constant is 137.035999206. Starting from one of the baseline readings —
11² − 7² + 5×13 = 137— we looked for the small fraction, built from the same allowed numbers, that would need to be added to get closer to the real value. The best one we found was extraordinary:
137 + 9/250 = 137.036000
error: 0.0000006 %But looked at closely, that correction uses the number 5 cubed (5³, inside 250 = 2×5³) — and the 5 in the original formula appeared unraised, to no power (5×13). It’s the same number playing two different roles in the same formula: technically perfect, but contradicting its own starting point.
So we looked for a second correction, requiring that any reused number keep the same role it already had:
137 + 13/361 = 137.036011
error: 0.0000086 %Fourteen times less precise than the first — but still extraordinary, and this time it didn’t contradict anything.
(One quick clarification: the fact that the more precise one turns out to be the contradictory one isn’t some quirk of the universe — requiring coherence shrinks the pool of candidates being searched, and searching a smaller pool can never beat searching the full one. What is real information is how much is lost by requiring that coherence: here, quite a bit; in other constants, almost nothing.)
We repeated the exercise with three other constants: the universe’s dark energy density, the electroweak mixing angle, and the amplitude of matter fluctuations in the cosmos. The same pattern showed up every time: a more precise but contradictory correction, and a compatible correction that lost little.
And then, something that can actually be wrong
There’s a classic trap in any search for patterns in numbers: you can always tweak something after knowing the data and make it look like a discovery. That doesn’t prove anything.
What makes this last part interesting is that three of the four constants we worked with — dark energy, the electroweak angle, the fluctuation amplitude — aren’t yet measured as precisely as what our formulas propose. Today’s experimental data has a wider margin of error than the prediction. That means we’re not explaining an already-known number — we’re anticipating figures that the experiment can’t yet confirm or rule out.
If future measurements — there are observatories working on this right now — converge toward those values, that would be real evidence in favor of there being something genuine here. If they converge elsewhere, that would be real evidence against it. Either way, we won’t be able to adjust the theory after seeing the result. That’s what separates a speculative idea from one that’s willing to risk something.
The three predictions, with their number and their implicit expiration date:
Matter fluctuation amplitude (σ₈)
prediction: 0.8111080
current data: 0.8111 ± 0.0060 (confirms only 2 digits)
Dark energy density (Ω_Λ)
prediction: 0.6847096
current data: 0.6847 ± 0.0073 (confirms only 2 digits)
Electroweak mixing angle (sin²θ_W)
prediction: 0.2312518
current data: 0.23122 ± 0.00004 (confirms 4 digits)In all three cases, the number we’re proposing is compatible with what’s measured today — it falls within the margin of error — but current data isn’t enough to confirm or rule it out. It needs more experimental precision than currently exists. When it arrives, the comparison will be direct: either the number that came out of this reasoning resembles what the next survey measures, or it doesn’t.
Where this leaves things
I don’t have a closed theory. I have a hunch that’s been tested a good deal better than it was a week ago, three concrete predictions with an expiration date, and a question that’s still open: if physical constants aren’t just facts about the universe but also traces of whoever measured them, how much of what we call “the laws of physics” is the landscape — and how much is, really, the map of how we chose to look at it?
I don’t know yet. But for the first time in this project, it’s a question a future data point can answer for me.
This article summarizes an exploratory work session within the ArXe project. The formulas, numbers, and three predictions mentioned are documented in more technical detail in the full research record.