Light that mixes gently, proved at last?
An AI preprint claims the entropy photon-number inequality
Covers result 273 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
3:28. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Add two independent random noises, and Shannon’s entropy power inequality says the sum is at least as disordered as if both had been Gaussian with the same entropies. Light poses a quantum version of the question. Mix two independent beams on a beam splitter that keeps a fraction η of one and the rest of the other: how disordered must the output be?
In 2007, Guha, Erkmen and Shapiro proposed an answer. Measure each beam’s entropy in photons: its entropy photon number is the mean photon number of thermal light with the same entropy. They conjectured that the output’s entropy photon number is at least the weighted average of the inputs’; thermal beams meet it exactly. A proof would settle the capacities of several optical channels. Some were later settled by other routes, but the broadcast channel, one sender and two receivers over a lossy fibre, still waited. A survey in 2018 still called the inequality an open conjecture, and a paper in late September 2026 still did.
Result 273 is one OpenAI preprint, 32 pages, dated September 24, 2026. It claims the inequality for two independent inputs with finite average photon number, in any finite number of modes, with any entanglement inside each input. As corollaries it gives the least output entropy of identical noisy attenuators at a given input entropy, and the exact capacity region of the degraded lossy broadcast channel.
What it doesn’t claim
Inputs with infinite energy, amplifiers or other noisy channels, more than two inputs, or the lossy wiretap and quantum capacities, which had already been settled without it. Independence between the two inputs matters: two correlated beams, each looking like half a photon of thermal light, can recombine into darkness.
Where it stands
- Posted as a preprint
- OpenAI, 32 pages, dated September 24, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- The main theorem, not the corollaries (docs)
Last checked October 19, 2026. The preprint has not been revised. The main theorem is formalized in Lean on the full infinite-dimensional space, about 28,000 lines with no unproved steps, which we did not build. The two corollaries are not formalized. OSVerify has not reviewed it, and we found no comment from specialists; two curated lists still mark the problem open, one because inputs of infinite energy are left out.
How the proof goes
- The best counterexample is thermal. The proof argues by contradiction. Suppose some pair of inputs beats thermal light. The paper adds four regularizing terms, takes a best possible counterexample, and uses its equilibrium equations. A new interpolation inequality, built from the function x/sinh x, then forces that best counterexample to turn thermal in the limit, and thermal light cannot beat itself.
- Checking the formal statement. The beam splitter written out photon by photon in Lean matches the textbook optics formula to 4 parts in 1015 (our check).
The numbers in the video
- Ours: thermal beams of 3 and 0.2 photons meet the inequality exactly; the exponential and linear quantum entropy power inequalities fall short by up to 0.02 and 0.37 nats.
- Ours: about 1,200 input pairs (single photons, cat states, random mixtures) with no violation.
- Ours: a correlated pair with half a photon each recombines to an output of about a millionth of a photon.
Read the source
- The Entropy Photon-Number Inequality (PDF), 32 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 273.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Add two independent random noises, and Shannon’s entropy power inequality says the sum is at least as disordered as if both had been Gaussian with the same entropies. Light poses a quantum version of the question. Mix two independent beams on a beam splitter that keeps a fraction η of one and the rest of the other: how disordered must the output be? In 2007, Guha, Erkmen and Shapiro proposed an answer. Measure each beam’s entropy in photons: its entropy photon number is the mean photon number of thermal light with the same entropy. They conjectured that the output’s entropy photon number is at least the weighted average of the inputs’; thermal beams meet it exactly. Our computer checks equality for thermal beams of 3 and 0.2 photons, while two older quantum versions fall short. A proof would settle the capacities of several optical channels. Some were later settled by other routes, but the broadcast channel, one sender and two receivers over a lossy fibre, still waited. A survey in 2018 still called the inequality an open conjecture, and a paper in late September 2026 still did.
1:11The claims
Result 273 is one OpenAI preprint, 32 pages, dated September 24, 2026. It claims the inequality for two independent inputs with finite average photon number, in any finite number of modes, with any entanglement inside each input. As corollaries it gives the least output entropy of identical noisy attenuators at a given input entropy, and the exact capacity region of the degraded lossy broadcast channel. Our computer tries it on single photons, cat states and random mixtures, about 1,200 pairs, and finds no violation. Independence matters: two beams correlated by running a beam splitter backwards, each looking like half a photon of thermal light, recombine into darkness. Not claimed: inputs with infinite energy, amplifiers or other noisy channels, more than two inputs, or the lossy wiretap and quantum capacities, which had already been settled without it.
2:14How the proof goes
The proof argues by contradiction. Suppose some pair of inputs beats thermal light. The paper adds four regularizing terms, takes a best possible counterexample, and uses its equilibrium equations. A new interpolation inequality, built from the function x/sinh x, then forces that best counterexample to turn thermal in the limit, and thermal light cannot beat itself. Our check of the formal statement: the beam splitter written out photon by photon in Lean matches the textbook optics formula to 4 parts in 10¹⁵.
2:50Where it stands
The preprint is unrefereed and unchanged since release. The main theorem is formalized in Lean on the full infinite-dimensional space, about 28,000 lines with no unproved steps, which we did not build. The two corollaries are not formalized. OSVerify has not reviewed it, and we found no comment from specialists; two curated lists still mark the problem open, one because inputs of infinite energy are left out. If it holds, no two independent beams of finite energy mix more gently than thermal light with the same entropies.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.