AI physics on vacation turned into real research in quantum mechanics

I asked Claude about physics on vacation and ended up doing real research in quantum mechanics

July 19, 2026 · Jamie Orlando

Preprint: doi:10.5281/zenodo.21313736, working draft, corrections welcome.

Code and data: github.com/theorlandog/gpc-census.

The setup

I run security and infrastructure engineering for a medical-imaging company. My degrees are in math and physics, but that was years ago; my day job is Terraform, threat models, and FDA compliance documentation. I am not a professional mathematician.

On July 4th I packed the family, two kids, a dog, and my pregnant wife, into the truck and drove to Cape Cod for a week. Like a lot of people in this industry, I've been trying to deliberately sharpen my AI workflows: not the "write me an email" stuff, but the harder question of whether these tools can hold up their end of genuinely difficult technical work. My method for that has been to point Claude at things far outside my day job and see where the wheels come off.

The thing I pointed it at was Nima Arkani-Hamed's amplituhedron program. His YouTube lectures are captivating, physics emerging from pure geometry, no spacetime, no quantum mechanics, just a positive region and its "canonical form." The vision is beautiful. The math is completely out of my reach. Grassmannians, moment maps, cohomology, I bounced off every paper.

So instead of pretending to read papers I couldn't follow, I asked Claude a lazier and, in retrospect, much better question:

"What other problems in physics and math are ripe for insights from positivity?"

The 1 year old and me doing quantum mechanics.

The rabbit hole

One of the answers was something I'd never heard of: generalized Pauli constraints. Here's the layman's version we converged on after a lot of back and forth.

The Pauli exclusion principle says each quantum state holds at most one fermion, occupation numbers live between 0 and 1. In 2008, Murat Altunbulak and Alexander Klyachko solved a 45-year-old problem by showing the full truth is much sharper: the allowed combinations of occupation numbers form a polytope, a shape with flat walls and sharp corners carved out of high-dimensional space. Pauli's principle is just one wall. There are dozens of others nobody had ever written down.

The corners of that shape are where the physics concentrates: they're the most extreme configurations of matter the rules allow. In the 2008 paper, two of those corners came with an asterisk: verified only numerically. Meaning: an optimizer got within 0.00000001 of the corner, so it's almost certainly reachable, but no explicit quantum state sitting exactly on either one appeared in the paper. As far as we could find, none ever appeared. Other published results quietly depended on them.

A shape carved by positivity, with corners nobody had reached. This looked like a place where an AI-assisted amateur could at least learn something. I had a gaming PC, a beach house with decent WiFi, and kids who nap.

How the work actually went

What followed was a week of a very specific loop, mornings and evenings around family time:

Claude proposes, an attack, a solver, a mathematical structure to exploit.

My PC executes, an i7-12700K ran millions of optimization problems while we were at the beach. I got very good at monitoring PowerShell jobs from my phone.

Everything gets verified in exact arithmetic, every claimed result re-derived from scratch, independently, in rational arithmetic with zero floating point. No exceptions.

I audit the claims, and this turned out to be the load-bearing step.

Step 4 matters because Claude was frequently, confidently wrong. It told me a living mathematician had died. It misread a table from a PDF (one OCR'd digit, we lost a day to that). It declared a numerical state "irreducibly dense" that turned out to have gorgeous hidden structure. Every one of these was caught by the verification discipline, not by trust.

The first corner fell to a discovery neither of us expected. The numerical state was a featureless cloud of 126 numbers. When we demanded the exact answer, it collapsed into something almost embarrassingly clean: seven determinants with integer weights, a combinatorial design. The proof that it works is literally counting. A corner of a quantum constraint polytope, closed with arithmetic a high-schooler could check:

ψ_A = ( √2|1257⟩ + √3|1347⟩ + √2|1369⟩ + √3|1459⟩ + √6|1789⟩ + 2|2679⟩ + |3579⟩ ) / √21

That discovery cracked the whole problem open. It turns out the corners of these quantum shapes come in two provably different species: some are reachable by essentially classical recipes (mix these configurations in these whole-number proportions, done), and some mathematically require quantum interference, we could prove no classical recipe exists, with solver-grade infeasibility certificates, cross-checked on two independent solvers. The historically "hard" corners are exactly the interference ones. That classification appears to be new.

The second corner fought back

The second corner did not fall as easily as the first. We ran exhaustive searches, millions of candidate states with certified-complete enumeration, through every "simple" quantum mechanism known to appear at these corners. My PC ground through them for days. Every search came back: empty.

Eventually I asked Claude to review our acceptance criterion "with the rigor befitting scientific publication" before we wrote up an underlying impossibility theorem. Thankfully, it found a hole. Every analytic search we'd run had silently assumed the quantum amplitudes were real numbers (positive or negative). Amplitudes can be complex. Re-deriving the acceptance condition for complex phases turned an impossible-looking equation (√14 ± 2 can never equal √21) into a trivial one (√14 and 2 can make √21; it's a triangle).

We patched the checker and re-scanned a space we'd already declared exhausted.

The answer was the 15th candidate. It had been enumerated days earlier and discarded by the search criteria.

The state that closes the second 2008 gap is genuinely complex, the phase isn't decoration, it's forced: cos γ = 3/(4√14), and we proved no real-amplitude state of its structural class exists. Every extremal state we could find in the literature for these polytopes is real. As far as we were able to determine, this is the first one that can't be real. Which is also, plausibly, part of why no explicit state for this corner seems to have been published: construction tools naturally search real states first.

What I actually learned about AI-assisted work

The AI made research-level mathematics accessible to me. It did not make it safe. Claude gave me, a generalist, real working access to graduate-level machinery, moment polytopes, second quantization, integer programming formulations I could never have built alone. However, it hallucinated facts, misread sources, over-claimed theorems, and embedded a subtly wrong assumption in four generations of solver code. The value wasn't Claude being right; it was the loop, propose, execute, verify exactly, audit adversarially, being fast enough to survive Claude being wrong.

Demand certificates, not answers. Every important claim in our paper is backed by something a third party can check without trusting me or the model: an exact state you can verify by hand, a solver infeasibility certificate you can re-run, an enumeration whose completeness is machine-checkable. When people ask how a security engineer and a chatbot can do publishable math, this is the answer: the same way you'd trust output from any unreliable-but-powerful oracle. Verify everything. Zero trust, but for theorems.

The human contribution was direction and doubt. I chose the target, demanded the exactness that revealed the structure, and, twice, insisted on rigor reviews that turned out to produce the discoveries. The best prompt I wrote all vacation was essentially "review this like a referee would." Both times it found an error that redirected the research.

Total cost: one Claude subscription, a gaming PC I already owned, and vacation mornings. The barrier to entry for real mathematical research has never, ever been this low. The barrier to correct research is exactly where it always was.

Then the experts replied

To put a bow on the project, I decided to write up the findings into a paper. I emailed researchers in the area, including authors of the foundational work, with a draft attached and an ask to review what I was brushing up against.

A researcher in this exact area graciously replied within hours with three corrections that kicked us into the next gear:

Our paper's central concept wasn't new. The states we were originally calling "independent-support" states at the heart of our classification are actually known as root-distinct states. He pointed me to the definitions in a paper, with antecedents going back to 1980s mathematics.

He caught a real phrasing error in one of our theorems, the original draft stated an impossible result that introduced a sloppy basis-dependency. He politely pointed this out and we were able to update the theorem commentary to actually state something coherent, and do an exhaustive block search to validate the line of reasoning.

He pointed out what was actually valuable. The first draft of the paper made inflated claims related to the state taking on a complex value. The expert was able to pump the brakes on that view point, but also redirect us to the plumbing that he immediately recognized as genuinely useful: the underlying algorithm that constructed these states automatically. Constructing extremal states by hand is apparently so tedious that the entire polytope program stalled partly because of it, and automating it could help push these tables toward the dimensions quantum chemists actually need. The entire paper got rewritten around that and we now suddenly had a target to point our pipeline to.

The lesson I'd underline for anyone attempting this kind of work: the sentence "this is a working draft and I'd particularly value corrections" is the highest-ROI sentence I wrote all week.

I want to dwell on this, because it is the part of the story I am most grateful for and the part that surprised me most. I am nobody in this field. No affiliation, no doctorate, no track record, no institutional email. I sent an unsolicited draft to a working researcher and he replied within the hour, with real corrections, real references, and a candid assessment of which parts of the work mattered and which did not.

Nothing about that was owed to me. Academia gets accused, often fairly, of being a closed shop, and there is a long tradition of cranks that gives every researcher an excellent reason to ignore mail from strangers. He read it anyway, and he engaged with the substance rather than the pedigree. That willingness to take an outsider's work seriously enough to correct it is, as far as I can tell, the whole mechanism by which anyone from outside a field ever gets to contribute anything to it. If AI-enabled citizen science is going to be a real thing rather than a slogan, it will be because working scientists keep answering their email.

Taking the census

After resurrecting as much GPC constraint data as we could, we classified every corner of every system we had over the next week or so. Ranks 6 through 10, 799 corners in high-dimensional space, each one now certified as classical-recipe or interference-required by solver certificates, cross-validated by the invariant battery. One inherited caveat: the rank-10 constraint lists have been conjectural since 2008, so rank-10 rows are conditional on them. Some things the census found:

Interference is not evenly spread. In the four-fermion series it is completely absent at rank 8 and switches on at rank 9. Our stubborn corner B is not an oddball; it is one of the first generation of interference corners that exist at all in its famil

[truncated for AI cost control]