STOP fumbling conjectures

a framework for generating the questions worth asking

In 1900, David Hilbert listed 23 unsolved problems in front of the International Congress of Mathematicians. He didn’t solve anything that day. He pointed. That act of pointing shaped the next century of mathematics.

The problems were conjectures. The solutions came decades later. The value was in the asking.

A conjecture is a question about an unknown. A tangent in the idea space that attempts to expand it. A quest for information. An attempt your mind makes to fill the vacuum created by curiosity.

Why build a science of conjectures?

We’ve always had conjectures. They come to us in the shower, on walks, while reading papers at 2am. Why formalize any of this?

Because the bottleneck is shifting.

For most of history, the hard part was solving. You had a question and the constraint was the years of grinding labor it took to find the answer. Intelligence in abundance changes the math. When you can throw compute at any problem and expect a reasonable answer, solving gets cheaper by the month. The scarce resource becomes knowing what to solve.

The person who can generate the right conjecture, the one that cracks open a field, becomes more valuable than the person who can grind through the proof. We don’t have a playbook for this.

Solving is about to get very easy. Asking is about to become the whole game.

The conjecture engine

I’m sitting with pen and paper, astrophysics on my mind. But the only questions landing are the biggest, baddest ones. Why isn’t quantum gravity solved? Is there a theory of everything? Is there relativity beyond special relativity?

These aren’t conjectures. They’re vibes. This is what fumbling conjectures looks like.

Most random, unorganized conjectures aren’t productive. There’s value in reading widely and letting conjectures come to you. But a disciplined method takes you farther. Categories and axes aren’t just a way to label conjectures after you’ve had them. They’re a generator. You use them to force a subject to reveal the conjectures you’re missing.

Map the frame first

Before generating anything, write down four things about your subject. The objects in the domain. The observables, what can actually be measured. The current framework, the dominant model. And the silent assumptions, what everyone takes for granted without questioning.

That last one matters most. The silent assumptions are where the best conjectures hide. Take each one and flip it. Not the debated assumptions. The invisible ones. Ask: what if this is false? What new observable would matter? What old observable would become misleading?

This is probably the single best generative device for conjectures.

Run a category sweep

Each category is a prompt, not a label. Each one forces a different kind of question out of the subject.

Existence. What might exist here that the framework treats as negligible?

Structural. Could two descriptions here be the same structure in different languages?

Degeneracy. Could two very different mechanisms produce the same observable?

Negation. What might be impossible here, even in principle?

Measurement-theoretic. What cannot be inferred from current measurements, no matter how good the data get?

Robustness. What would remain true if the model details changed?

Reframing. What if the current observable or problem statement is the wrong one entirely?

Sharpen with axes

Raw conjectures are vague. Three axes make them real.

Resolution mode. How would this be settled? Analytic proof, simulation, observation, impossibility argument? Some conjectures don’t appear until you ask what would count as settling them.

Framework distance. Near-field (within the current paradigm), bridge (connecting two frameworks), or far-field (requiring new language entirely). This prevents you from generating only conservative conjectures.

Epistemic function. What job would this conjecture do if true? Unify, discriminate, bound, prune, reframe, or open a new measurement channel?

Photon rings: a worked example

I’ve been researching photon rings around black holes. Here’s what the engine produces.

Frame. Photon rings are nested sub-rings of light around a black hole’s shadow, created by photons orbiting one, two, three or more times before escaping. The EHT captured M87’s shadow, but the finer sub-ring structure remains unresolved. Dominant framework: Kerr metric. Key observables: ring diameter, brightness profile, asymmetry.

Silent assumptions. The spacetime is stationary. Environmental factors are negligible. Classical observables are sufficient. The photon ring is an imaging target.

Category sweep.

Existence. Do horizon-scale quantum corrections leave any imprint in the ring?

Degeneracy. Can a Kerr black hole and a horizonless compact object produce indistinguishable ring observables? If yes, the entire program of using rings to probe horizons has a ceiling nobody’s talking about.

Negation. Is some class of near-horizon physics causally inaccessible to ring measurements? Not with better telescopes. Ever.

Reframing. What if the photon ring is the wrong observable? What if the real information is in polarization statistics or time variability?

Sharpen one. Take the degeneracy conjecture and run the axes. Resolution mode: impossibility proof, not observation. Framework distance: bridge, connecting GR black hole physics with exotic compact object literature. Epistemic function: it establishes a bound on what photon ring science can deliver.

Result: “No classical photon ring measurement can distinguish Kerr from a certain class of horizonless objects over a nontrivial parameter region.”

One sweep. From “astrophysics is cool” to a conjecture with a resolution path.

The matrix is a provocation tool, not a vending machine. Don’t fill every cell mechanically. But do run the sweep. The categories you’d skip are exactly where the non-obvious questions live.

(It’s tempting to hand this framework to Claude and let it generate conjectures. Don’t. Not yet. Your brain makes connections an LLM won’t, because you have taste, intuition about what feels off, the itch of “something isn’t right here” that comes from doing the reading yourself. Use AI to pressure-test. The conjecture itself should be yours.)

Popper and Deutsch

If you’ve paid any attention to conjectures before, you’ve heard these names. They’re load-bearing pieces of the framework. Let’s stop fumbling them.

Popper

Karl Popper spent a lot of time thinking about the difference between Einstein’s physics and Freud’s psychology.

Einstein’s general relativity made a specific, risky prediction: light from distant stars would bend by a precise amount near the sun. In 1919, Eddington measured it during a solar eclipse. The theory could have died in a single afternoon. That’s what made it powerful.

Freud and Adler could explain everything. A man drowns a child, Freud explains it through repression, Adler through inferiority complexes. A man saves a child, same thing, different narrative. No matter what happened, the theory had an answer.

Popper realized this wasn’t a strength. It was the signature of a theory that says nothing.

The filters for our toolkit:

Science advances by disconfirmation, not confirmation. The moment you only look for supporting evidence, you’ve left science and entered storytelling.

Confirmation is trivially easy to find, and irrefutable theories aren’t valid ones. Horoscopes have mountains of confirming evidence. So do conspiracy theories. If your conjecture can explain any possible outcome, it explains nothing. Science disconfirms. Pseudoscience confirms.

A good conjecture must be testable and falsifiable. What experiment would kill it? If you can’t answer that, you don’t have a conjecture. You have a belief.

Conjectures Popper would toss:

“Consciousness is a quantum phenomenon that collapses into awareness through a mechanism we cannot yet measure.” The “cannot yet measure” clause is doing all the work. Built-in escape hatch. No experiment touches it.

“My trading strategy works, but only in market conditions I can identify in hindsight.” If you can only identify the conditions after the fact, you have a story, not a model.

The test is always the same. Does the conjecture stick its neck out? If not, move on.

Deutsch

Popper gave us the filter: falsifiability. Deutsch asks a harder question. What makes one falsifiable conjecture better than another?

You never have just one conjecture. You have dozens. They’re all technically testable. You can’t run all the experiments. You need a way to rank them.

Deutsch’s argument, in The Beginning of Infinity: what separates a good conjecture from a mediocre one is explanatory reach. A good conjecture doesn’t just predict what will happen. It explains why. And good explanations are hard to vary.

It’s not enough that your conjecture is falsifiable. You shouldn’t be able to tweak its details without destroying the explanation. If you can swap out components freely and it still “works,” those components aren’t doing real explanatory work. They’re decorative.

Newton’s gravity: change “inverse square” to “inverse cube” and orbits fail, the solar system destabilizes. Every piece is load-bearing. That’s a hard-to-vary explanation. Compare: “the universe wants things to be in harmony.” Swap “harmony” for “balance” or “equilibrium.” Nothing changes. Infinitely malleable. Not an explanation.

Where Popper gives a binary, Deutsch gives a spectrum. When you’re sitting with five photon ring conjectures deciding which to pursue, pick the one that would explain the most and could be varied the least.

Between Popper and Deutsch, two filters. Popper: can this be wrong? Deutsch: if it’s right, does it actually explain anything?

A conjecture that passes both is worth your time.

Most people fumble conjectures because they never learn to generate them on purpose. They wait for inspiration, chase vibes, or skip the filters entirely. Now you have the engine and the filters. Stop fumbling.