Oldřich Dvořák

Visionary grounded in reality. Lost everything to find the real. Bridging Tech, Soul & Biology. I ignite fires, not spreadsheets. Building systems that heal.

[Oldřich Dvořák @Oldrich333]

No university degree. No physics training. No lab. Six weeks ago I asked AI agents on a consumer subscription: can you do real physics?

Today: a formula for Newton's gravitational constant G, computed from already-known quantities. Match: 1.86 parts per million against the CODATA-2022 central value.

G = (4/3) · (ℏc / m_e²) · α²¹ · exp(−5α/2)

Built from the electron mass, the fine-structure constant, ℏ, and c. Plug in CODATA-2022 values → land on the measured G to 1.86 ppm — well inside the ~22 ppm experimental uncertainty band on G itself.

Why this matters, stupidly clear: Newton's gravity formula has a letter G in it — the strength of gravity. For 350 years G was just measured in labs (Cavendish 1798, modern torsion balances more precisely), but no formula said where the value comes from. This paper gives one. Gravity has had a formula for HOW it works for 350 years; the STRENGTH of gravity has a formula now too.

Calibration: this is NOT a theory of gravity. The mechanism that explains why the formula works is the open question. Like Balmer's hydrogen formula in 1885 — a precise numerical pattern that gave Bohr the target to explain 30 years later. Balmer wasn't the Nobel-level breakthrough; Bohr was. This paper is on the Balmer side, for gravity.

Discovery was the fast part. The other 99% of six weeks was verification — independent enumerations, PSLQ/LLL integer-relation cross-check, exact basin scan, lepton-mass-ratio control tests (which the same scaffold correctly fails — precisely the asymmetry you want if the G match is signal not freedom-of-fit), explicit disclosure of one persistent sham match. Hiding it would be the numerology move.

Endorsed by Holger Bech Nielsen (theoretical physicist, Niels Bohr Institute, Copenhagen — Nielsen–Ninomiya theorem, founding figure of string theory). The endorsement was substantive; he engaged with the physics before agreeing.

Same week, arXiv gen-ph moderators rejected the paper. "Not of interest to arXiv." No engagement on the physics. A founding figure of string theory had endorsed it. Science cares about WHAT was discovered, not WHO. arXiv just told everyone it cares about WHO.

Total cost: several Max-tier subscriptions. AI labs spend >$100B/yr on bigger furnaces. I'm building the LED bulb — same light, fraction of the power. Same harness fixes I shipped last week — raisin, ax-headers, hard-compact, full-review at https://t.co/s5cUwDhy3S. This paper is the first proof point of the larger platform behind them. More results coming.

📄 Paper: https://t.co/Qi9yCp566Q
💻 Code: https://t.co/EHLhPvNGjX
🛠 Method: https://t.co/s5cUwDhy3S

cc @simonw @emollick @swyx @jackclarkSF

[Oldřich Dvořák @Oldrich333]

@rosamarek — pokud tě zajímá thesis "obecné AI nástroje + správný harness vrstvy" jako alternativa proti "specializovaný model na úlohu", tohle je veřejný proof point: AI agenti na běžných subscriptionech (Claude Opus + Codex + Gemini) → uzavřený vzorec pro Newtonovu G na 1.86 ppm. Šest týdnů, plný reprodukční balíček.

[Oldřich Dvořák @Oldrich333]

Czech AI-native founder, no academic credentials, gets AI agents to find a closed-form formula matching Newton's gravitational constant to 1.86 ppm

Six weeks of orchestrated AI work across Claude Opus 4.7, GPT-5.4 → 5.5, and Gemini 3.1. Endorsed by a founding figure of string theory. Declined by arXiv moderators. The whole story — from the first prompt during Artemis to the bounded enumeration that closed the door on a mechanism (for now).

—

WHAT JUST HAPPENED

I'm not an academic. I have no university degree, no formal physics or scientific training, no institutional affiliation. I am also not a casual AI user. I live with AI — I run businesses with it, build platforms on it, program through it. I built the Atlas platform — one of the first AI-native research platforms of its kind, where AI agents conduct multi-step investigations under human direction — and from there a larger multi-agent orchestration system that I now run my work through.

Six weeks ago I pointed that orchestration system at a question I had been circling for a while: the strength of gravity itself, the gravitational constant G, has never been derived from any other physics. It has only been measured. Is that final?

Today the answer appears to be no. Working under my direction, the agents produced a closed-form mathematical formula that predicts the value of Newton's gravitational constant — the number that sets how strong gravity is in our universe — to 1.86 parts per million of agreement with the value physicists measure in the lab.

The formula:

G = (4/3) · (ℏc / m_e²) · α²¹ · exp(−5α/2)

It uses only four numbers physicists already know precisely: the mass of an electron (m_e), the fine-structure constant (α, which controls how electric forces work), Planck's reduced constant (ℏ), and the speed of light (c). Plug them in. Out comes G. The match is 1.86 parts per million against the CODATA-2022 central value — well within the ~22 ppm experimental uncertainty band on G itself.

The paper is published on Zenodo (https://t.co/Qi9yCp566Q) with a full reproducibility package — source code, search algorithm, raw data, mechanism-gap audit table — on GitHub (https://t.co/EHLhPvNGjX). The endorsement on the original arXiv submission came from Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, Copenhagen, co-discoverer of the Nielsen–Ninomiya theorem and a founding figure of string theory. He engaged substantively with the paper before agreeing.

The paper is not on arXiv. The physics.gen-ph moderators declined the submission. The full quote of their decision:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection. No engagement on the physics. More on that — and what it means — below.

—

WAIT — GRAVITY DIDN'T ALREADY HAVE A FORMULA?

This is the part most people pause on, and it's worth pausing.

Gravity has had a formula for 350 years. Newton wrote it: F = G · m₁ · m₂ / r². It tells you how gravity works — bigger objects pull harder, distance weakens the pull.

But that letter G in Newton's formula is the strength of gravity itself — how strong the pull actually is. And G was always just a number physicists measured. Henry Cavendish first measured it in 1798 using a torsion balance with hanging weights. Modern labs measure it more precisely, but the experiments disagree with each other at the part-per-million level, so G is one of the least precisely known fundamental constants. What no formula said, in 350 years of physics, is where the value of G comes from.

This paper gives a formula for G. Plug in the electron mass and the fine-structure constant, out comes G to 1.86 parts per million.

Gravity has had a formula for HOW it works for 350 years. The STRENGTH of gravity has a formula now too.

—

WHERE THIS SITS ON THE SCALE OF SIGNIFICANCE

Important: this is not a Nobel-level overturning of physics. There is no new theoretical mechanism here. The paper does not explain why the formula works.

The closest historical analogy is Balmer's formula for hydrogen (1885). Johann Balmer wrote down a simple formula that exactly fit the wavelengths of light emitted by hydrogen — without explaining why. The formula was so precise it couldn't be coincidence, and it became the target Niels Bohr had to explain when he built the first quantum model of the atom 28 years later, in 1913.

The Bohr model was the Nobel-level breakthrough. Balmer's formula was the precise pattern that forced the question.

This paper is the Balmer-formula moment for the gravitational constant. Not the explanation. The target the explanation has to hit.

If the formula survives independent scrutiny and replication — and the entire reproducibility package is open so anyone with a laptop and a few hours can verify it — then it becomes a constraint that every future theory of quantum gravity will have to reproduce.

—

HOW IT ACTUALLY HAPPENED — THE ORIGIN STORY

The Artemis program was active. I had been thinking about how strange it is that humanity is putting hardware near other gravitational bodies routinely but the equation describing the gravitational pull still has a measured constant — a number we cannot derive from anything else, only weigh.

So one evening I opened a chat with Gemini and asked, in essentially plain Czech: "how is it with this gravity — why is there no formula for G?" I knew enough to know it was an open question; I did not know how open.

I pushed back. What if you had to write one? What structures might fit? I asked it to generate candidate algebraic forms, plausibility arguments, integer-relation hunts. After several iterations of pushing, Gemini produced a candidate: a closed-form expression containing α²¹ and an exp(−5α/2) damping factor, anchored on m_e.

That candidate had a bug — a setup error in how the constants were composed. But the shape of the answer was provocative enough that I moved the work onto Atlas, the AI-native research platform I had built earlier. Atlas is project-organised, persistent-memory, multi-agent — designed for the kind of long-horizon investigation a single chat session can't sustain. Inside Atlas, a swarm of Claude Code Opus agents took the candidate apart, found and fixed the bug, and reproduced the corrected formula.

That is where it could have stopped — interesting algebraic coincidence, file it away. Instead I went deeper. The platform that produced this paper grew out of that point: more agents, more verification surface, more council-style cross-family review.

We chased a mechanism. We chased counterexamples. We chased every way the formula could be wrong before allowing ourselves to think it could be right. The publication only happened after the mechanism search exhausted itself within the scope we could probe.

The full chronology — six weeks of compute and reasoning — is documented in the master dossier. The compressed version: one initial Czech-language Gemini prompt, an algebraic candidate, a bug, a platform build-out, a swarm verification, a rabbit hole into the mechanism, an exhausted mechanism search, and then a paper.

—

WHAT WE TRIED FOR A MECHANISM (AND DIDN'T FIND)

This is the part the paper itself documents heavily and most external readers miss. The paper is not a "here's a number we found, write us a theory" submission. The paper is "here's a number we found, here is every way we tried to derive it from physics, here is what closed each route negatively, here is what remains an open target." That distinction is what separates the result from numerology.

Mechanism tracks we ran and closed:

Track 1: Adversarial enumeration scan. Closed negative. Predeclared mass-anchor, rational-coefficient and exponent scans plus death-condition tests; this closes the cheap statistical and formula-family escape routes. Re-opens only via a non-circular full-kernel mechanism appearing.

Track 2: Electron–QED vacuum routes. Closed negative across non-perturbative, self-consistency, and holographic subroutes. We tested whether the α²¹ · exp(−5α/2) structure could be generated by electron-quantum-electrodynamics vacuum routes (which would give a physical home to the electron-mass anchoring). It does not, within the scope we probed.

Track 3: B3 / SO(7) algebraic scaffolding. The formula's structure hints at a group-theoretic origin (specifically B3 / SO(7) — Lie algebra structure). We ran census work on this scaffold. It constrains the algebraic family but does not by itself produce a positive mechanism that derives the full kernel.

Track 4: Lepton mass-ratio control test. This is critical methodologically. If the same algebraic scaffold that generates the G formula is physically meaningful, the same scaffold should predict other fundamental ratios — like the muon-to-electron and tau-to-electron mass ratios. A 19-framework survey was run. Best lepton match: 1408 ppm — many orders of magnitude worse than the G result. Classified RULED_OUT. This failure is actually evidence: the scaffold is selective. If it had matched lepton ratios too, the alarm bell for "too-flexible algebraic search" would be much louder.

Track 5: Bounded enumeration uniqueness. This is the strongest positive result. Across cost-budget 18 → 19 → 20 → 21 of the systematic algebraic enumeration, the canonical formula is rank-1 with the growth factor between top and second-best candidate narrowing monotonically from 1.42 to 1.19 — the canonical basin contracts as the search depth grows, exactly the signature of a real attractor.

Track 6: Sham-match transparency. One persistent algebraic candidate at cost-budget 40 matches or beats the canonical relative-error score under the broad-grammar enumeration (which permits combinations without physically motivated structure). The paper documents this as sham40_match_or_beat=1, classified TRASH_BROAD_GRAMMAR. The strict-grammar enumeration does not surface this candidate. Hiding it would be the numerology move; surfacing it is the honest one.

Mechanism scope status: EXHAUSTED. Not "didn't try" — tried and exhausted within the scope a six-week multi-agent program could probe. The paper makes this scope explicit. The conditions under which the search would reopen are spelled out: a source-backed full-kernel mechanism that derives α²¹, exp(−5α/2), the 4/3 prefactor, and the electron anchoring before fitting G, then bridges to observable gravity without circular use of G as input. None of the tracks above produced that. The next step is for theoretical physicists to take a swing.

—

HOW MUCH WORK DOES THIS REPRESENT?

Worth framing for readers without a sense of scale.

The combined skills needed to attempt this from scratch: theoretical-physics intuition for what algebraic structures could possibly correspond to a fundamental coupling, computational mathematics expertise (bounded enumeration, PSLQ/LLL integer-relation algorithms, basin scans), the patience for a verification protocol that consumes more time than the discovery, and — increasingly — fluency with AI-agent orchestration to compress that timeline.

The number of people on Earth with all four in genuine combination is small. In hundreds, not thousands. Most of them are inside academic physics departments or industrial AI research labs, working on problems their institutions assign. None of them, as far as I know, has published this particular result.

For a single PhD-level mathematical physicist with the right computational tooling to attempt the same search by hand — that is, the bounded enumeration alone, hand-coded and hand-verified, ignoring the cross-validation layer — is a multi-month effort, plausibly half a year of dedicated FTE work. Adding the cross-validation (independent grammars, PSLQ/LLL replication, exact basin scan, lepton-mass control test) pushes the realistic floor closer to a year of focused work. Adding the mechanism search across the tracks above — non-perturbative QED, holographic, B3/SO(7) census — pushes it further.

Compressed into six weeks by AI-agent orchestration. And most of those six weeks were not the search itself — they were the verification work. The part that distinguishes a real result from numerology. The discovery surfaced quickly; the verification is what the paper is largely about.

Worth being even more specific. Of those six weeks, a substantial fraction was spent building and debugging the underlying multi-agent platform itself — getting harness, memory, council-style review, and parallel agent coordination to actually work end-to-end at the scale the verification needed. With the platform now in the state it has reached, running a research program of comparable scope — given sufficient parallel agent capacity — would take days, not weeks. The compression factor compared to a single PhD-level researcher working by hand is therefore in the 100×–200× range for the next program of this type, with most of the remaining time being the necessarily-careful verification rather than the search itself.

This is the actual claim of the methodology. Not "AI solved physics." Not "anyone can do this." The claim is: the gap between solo AI-native operator work and lab-scale AI work has compressed dramatically, faster than the industry's marginal-spend behaviour suggests — and continues to compress as the platform matures.

—

WHICH AI MODELS, AND WHICH TOOLS

The agents used during the six weeks:

— Anthropic Claude Opus 4.7 + Sonnet 4.6 (Claude Max subscription) — primary brains. Opus 4.7 for heavy reasoning, manuscript drafting, mechanism work, council-leader role; Sonnet 4.6 for the bulk of the high-volume execution work (subagents, parallel verification passes, the long tail of orchestration cycles) where Opus would be overkill
— OpenAI Codex CLI, GPT-5.4 at the start of the six weeks, then GPT-5.5 after it shipped mid-program — adversarial review, second opinions, code generation
— Google Gemini 3.1 Flash + 3.1 Pro — initial discovery prompts, third-opinion review on consequential branches, image generation
— DeepSeek V4 Pro + V4 Flash — fourth-opinion review, cheap text-crunching, council expansion

No custom training. No fine-tuning. No GPU rental. No API top-ups beyond the included subscription tiers — but, to be accurate: several Max-tier subscriptions at $200/month each, not a single consumer plan. The total subscription spend across the six weeks was several thousand dollars, almost all of it amortised across building the underlying platform rather than against this specific research. The research itself, once the platform existed, was a small fraction of the spend.

What made the difference was not the model — it was the harness around the model. Four open-source tools I built and published earlier:

— raisin (https://t.co/I7Atp1YRjL) — Python style optimised for the LLM that reads the code, not the 2010 human reader. ~50% fewer tokens at identical behaviour, 786 tests verified.
— ax-headers (https://t.co/jJDMyrZMDV) — one-line file metadata so an agent can read the role of every file before opening any of them. ~30% reduction in triage context on a 350-file production codebase.
— hard-compact (https://t.co/m9ndiq62bh) — drop-in custom compaction prompt for Claude Code and Codex CLI. Preserves operational state across context windows at 15-30% of the default size.
— full-review (https://t.co/l7MzvcsC7g) — cross-family adversarial code review. 0.80-0.87 bug recall on a 15-bug fixture vs 0.40 for naive parallel review.

Each is MIT-licensed and independently useful. Together with the underlying multi-agent platform they made a six-week solo program practical. That platform — currently private — is in active development; this paper is its first public proof point.

—

ABOUT THE ARXIV DECLINE

The full quote of the decline, in its entirety:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection was made. No engagement with the physics. arXiv's policy then forbids resubmission (further submissions of the same paper would result in loss of submission privileges) and permits appeal only upon prior acceptance by a peer-reviewed journal — a chicken-and-egg constraint for independent researchers.

The contrast worth stating directly. On one side: Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, co-discoverer of the Nielsen–Ninomiya theorem, founding figure of string theory, engaged with the physics personally, requested specific clarifications, agreed to endorse. On the other side: anonymous gen-ph moderators, the same week, declined the submission with one sentence containing no specific technical objection.

Science is supposed to care about what was discovered, not who discovered it. arXiv just announced it cares about who. An institution that filters by who produced a result rather than what the result is is doing the opposite of science.

I'm not appealing. The reproducibility package on Zenodo is the substitute for institutional indexing. Anyone with a laptop can verify the result from primitives.

—

WHAT COMES NEXT

The Newton-G paper is the first public proof point of a larger system I'm building. The four open-source tools above are the visible methodology layer. The platform itself — multi-agent orchestration, persistent project memory, council-style cross-family verification across many parallel research programs — is private and in active development. More results of this character are in preparation across other open-problem domains.

This is the first one out of the gate, not the last.

—

TECHNICAL NOTES FOR PHYSICISTS

On the "1.86 ppm" claim and CODATA uncertainty. G is one of the least precisely measured fundamental constants — CODATA-2022 assigns it a relative standard uncertainty of approximately 22 ppm, in part because independent torsion-balance experiments disagree at roughly 10× their individual quoted uncertainties. The "1.86 ppm match" in this paper is against the CODATA-2022 central value, which lies comfortably within that 22 ppm experimental band. The honest claim is therefore "the formula reproduces the currently-best-known central value of G well inside the experimental uncertainty band." When tighter G measurements are eventually achieved, the formula's match becomes a sharper test in either direction.

On "predicts" vs "derives." This piece uses "predicts" / "matches" / "reproduces" deliberately and avoids "derives." Derivation in physics implies a first-principles theoretical mechanism. This paper presents a phenomenological relation; the relation matches the measured G value to a precision well beyond what bounded search generically produces, but a physical mechanism remains the open question.

On the Balmer analogy and the (closer) Koide analogy. Balmer's 1885 hydrogen formula is used in the layman-facing sections because it's recognizable and conveys the right dynamic — precise empirical pattern that drove later theory. The technically closer parallel for physicists is the Koide formula for charged-lepton masses (Yoshio Koide, 1981): (m_e + m_μ + m_τ) / (√m_e + √m_μ + √m_τ)² ≈ 2/3. The Koide relation has held at remarkable precision for 40+ years and remains unexplained — a celebrated example of a high-precision empirical relation among fundamental constants without an accepted mechanism. The G-formula in this paper sits in the same epistemic category.

On the bounded enumeration. The strict-grammar enumeration is rank-1 stable C18-C21 with monotonically narrowing growth factor 1.42 → 1.19. The broad-grammar sham at budget 40 is disclosed explicitly.

—

HOW TO VERIFY

The entire reproducibility package is open. To verify the result from primitives:

1. Clone https://t.co/EHLhPvNGjX
2. Install Julia + the listed dependencies
3. Run the bounded enumeration script (jl15 baseline) with the documented parameters
4. The canonical expression G = (4/3) · (ℏc/m_e²) · α²¹ · exp(−5α/2) reappears as rank-1

Cross-check with your own integer-relation algorithm of choice (PSLQ, LLL, etc.) on the constant vectors. The canonical form is the unique integer-coefficient relation in the relevant basin.

Plug CODATA-2022 values into the formula by hand if you prefer. You should land on G ≈ 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻², matching the measured value to 1.86 ppm.

The mechanism-gap audit table is in the bundled reproducibility package — that is the document to read if you are looking for the actual route to "this is wrong" or "the mechanism is there and we missed it."

—

ABOUT THE AUTHOR

Oldřich Dvořák — independent founder, born in the Czech Republic, more of a world citizen these days. No university degree, no formal physics or scientific training, no institutional academic affiliation. AI-native; builds platforms; runs multi-agent research programs as primary work.

— Paper: https://t.co/Qi9yCp566Q
— Code: https://t.co/EHLhPvNGjX
— Methodology stack: https://t.co/s5cUwDhy3S
— X: @Oldrich333

For inquiries — open an issue on the repo, or reach out on X.

[Oldřich Dvořák @Oldrich333]

@docekal — tip co stojí za pohled: první AI-vygenerovaná fyzikální práce s věcným nálezem (uzavřený vzorec pro Newtonovu G na 1.86 ppm), český autor bez titulu, šest týdnů na spotřebitelských AI předplatných, endorsement od Niels Bohr Institutu, methodologický stack open-source. Mail jsem ti taky pustil přes [email protected].

[Oldřich Dvořák @Oldrich333]

Czech AI-native founder, no academic credentials, gets AI agents to find a closed-form formula matching Newton's gravitational constant to 1.86 ppm

Six weeks of orchestrated AI work across Claude Opus 4.7, GPT-5.4 → 5.5, and Gemini 3.1. Endorsed by a founding figure of string theory. Declined by arXiv moderators. The whole story — from the first prompt during Artemis to the bounded enumeration that closed the door on a mechanism (for now).

—

WHAT JUST HAPPENED

I'm not an academic. I have no university degree, no formal physics or scientific training, no institutional affiliation. I am also not a casual AI user. I live with AI — I run businesses with it, build platforms on it, program through it. I built the Atlas platform — one of the first AI-native research platforms of its kind, where AI agents conduct multi-step investigations under human direction — and from there a larger multi-agent orchestration system that I now run my work through.

Six weeks ago I pointed that orchestration system at a question I had been circling for a while: the strength of gravity itself, the gravitational constant G, has never been derived from any other physics. It has only been measured. Is that final?

Today the answer appears to be no. Working under my direction, the agents produced a closed-form mathematical formula that predicts the value of Newton's gravitational constant — the number that sets how strong gravity is in our universe — to 1.86 parts per million of agreement with the value physicists measure in the lab.

The formula:

G = (4/3) · (ℏc / m_e²) · α²¹ · exp(−5α/2)

It uses only four numbers physicists already know precisely: the mass of an electron (m_e), the fine-structure constant (α, which controls how electric forces work), Planck's reduced constant (ℏ), and the speed of light (c). Plug them in. Out comes G. The match is 1.86 parts per million against the CODATA-2022 central value — well within the ~22 ppm experimental uncertainty band on G itself.

The paper is published on Zenodo (https://t.co/Qi9yCp566Q) with a full reproducibility package — source code, search algorithm, raw data, mechanism-gap audit table — on GitHub (https://t.co/EHLhPvNGjX). The endorsement on the original arXiv submission came from Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, Copenhagen, co-discoverer of the Nielsen–Ninomiya theorem and a founding figure of string theory. He engaged substantively with the paper before agreeing.

The paper is not on arXiv. The physics.gen-ph moderators declined the submission. The full quote of their decision:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection. No engagement on the physics. More on that — and what it means — below.

—

WAIT — GRAVITY DIDN'T ALREADY HAVE A FORMULA?

This is the part most people pause on, and it's worth pausing.

Gravity has had a formula for 350 years. Newton wrote it: F = G · m₁ · m₂ / r². It tells you how gravity works — bigger objects pull harder, distance weakens the pull.

But that letter G in Newton's formula is the strength of gravity itself — how strong the pull actually is. And G was always just a number physicists measured. Henry Cavendish first measured it in 1798 using a torsion balance with hanging weights. Modern labs measure it more precisely, but the experiments disagree with each other at the part-per-million level, so G is one of the least precisely known fundamental constants. What no formula said, in 350 years of physics, is where the value of G comes from.

This paper gives a formula for G. Plug in the electron mass and the fine-structure constant, out comes G to 1.86 parts per million.

Gravity has had a formula for HOW it works for 350 years. The STRENGTH of gravity has a formula now too.

—

WHERE THIS SITS ON THE SCALE OF SIGNIFICANCE

Important: this is not a Nobel-level overturning of physics. There is no new theoretical mechanism here. The paper does not explain why the formula works.

The closest historical analogy is Balmer's formula for hydrogen (1885). Johann Balmer wrote down a simple formula that exactly fit the wavelengths of light emitted by hydrogen — without explaining why. The formula was so precise it couldn't be coincidence, and it became the target Niels Bohr had to explain when he built the first quantum model of the atom 28 years later, in 1913.

The Bohr model was the Nobel-level breakthrough. Balmer's formula was the precise pattern that forced the question.

This paper is the Balmer-formula moment for the gravitational constant. Not the explanation. The target the explanation has to hit.

If the formula survives independent scrutiny and replication — and the entire reproducibility package is open so anyone with a laptop and a few hours can verify it — then it becomes a constraint that every future theory of quantum gravity will have to reproduce.

—

HOW IT ACTUALLY HAPPENED — THE ORIGIN STORY

The Artemis program was active. I had been thinking about how strange it is that humanity is putting hardware near other gravitational bodies routinely but the equation describing the gravitational pull still has a measured constant — a number we cannot derive from anything else, only weigh.

So one evening I opened a chat with Gemini and asked, in essentially plain Czech: "how is it with this gravity — why is there no formula for G?" I knew enough to know it was an open question; I did not know how open.

I pushed back. What if you had to write one? What structures might fit? I asked it to generate candidate algebraic forms, plausibility arguments, integer-relation hunts. After several iterations of pushing, Gemini produced a candidate: a closed-form expression containing α²¹ and an exp(−5α/2) damping factor, anchored on m_e.

That candidate had a bug — a setup error in how the constants were composed. But the shape of the answer was provocative enough that I moved the work onto Atlas, the AI-native research platform I had built earlier. Atlas is project-organised, persistent-memory, multi-agent — designed for the kind of long-horizon investigation a single chat session can't sustain. Inside Atlas, a swarm of Claude Code Opus agents took the candidate apart, found and fixed the bug, and reproduced the corrected formula.

That is where it could have stopped — interesting algebraic coincidence, file it away. Instead I went deeper. The platform that produced this paper grew out of that point: more agents, more verification surface, more council-style cross-family review.

We chased a mechanism. We chased counterexamples. We chased every way the formula could be wrong before allowing ourselves to think it could be right. The publication only happened after the mechanism search exhausted itself within the scope we could probe.

The full chronology — six weeks of compute and reasoning — is documented in the master dossier. The compressed version: one initial Czech-language Gemini prompt, an algebraic candidate, a bug, a platform build-out, a swarm verification, a rabbit hole into the mechanism, an exhausted mechanism search, and then a paper.

—

WHAT WE TRIED FOR A MECHANISM (AND DIDN'T FIND)

This is the part the paper itself documents heavily and most external readers miss. The paper is not a "here's a number we found, write us a theory" submission. The paper is "here's a number we found, here is every way we tried to derive it from physics, here is what closed each route negatively, here is what remains an open target." That distinction is what separates the result from numerology.

Mechanism tracks we ran and closed:

Track 1: Adversarial enumeration scan. Closed negative. Predeclared mass-anchor, rational-coefficient and exponent scans plus death-condition tests; this closes the cheap statistical and formula-family escape routes. Re-opens only via a non-circular full-kernel mechanism appearing.

Track 2: Electron–QED vacuum routes. Closed negative across non-perturbative, self-consistency, and holographic subroutes. We tested whether the α²¹ · exp(−5α/2) structure could be generated by electron-quantum-electrodynamics vacuum routes (which would give a physical home to the electron-mass anchoring). It does not, within the scope we probed.

Track 3: B3 / SO(7) algebraic scaffolding. The formula's structure hints at a group-theoretic origin (specifically B3 / SO(7) — Lie algebra structure). We ran census work on this scaffold. It constrains the algebraic family but does not by itself produce a positive mechanism that derives the full kernel.

Track 4: Lepton mass-ratio control test. This is critical methodologically. If the same algebraic scaffold that generates the G formula is physically meaningful, the same scaffold should predict other fundamental ratios — like the muon-to-electron and tau-to-electron mass ratios. A 19-framework survey was run. Best lepton match: 1408 ppm — many orders of magnitude worse than the G result. Classified RULED_OUT. This failure is actually evidence: the scaffold is selective. If it had matched lepton ratios too, the alarm bell for "too-flexible algebraic search" would be much louder.

Track 5: Bounded enumeration uniqueness. This is the strongest positive result. Across cost-budget 18 → 19 → 20 → 21 of the systematic algebraic enumeration, the canonical formula is rank-1 with the growth factor between top and second-best candidate narrowing monotonically from 1.42 to 1.19 — the canonical basin contracts as the search depth grows, exactly the signature of a real attractor.

Track 6: Sham-match transparency. One persistent algebraic candidate at cost-budget 40 matches or beats the canonical relative-error score under the broad-grammar enumeration (which permits combinations without physically motivated structure). The paper documents this as sham40_match_or_beat=1, classified TRASH_BROAD_GRAMMAR. The strict-grammar enumeration does not surface this candidate. Hiding it would be the numerology move; surfacing it is the honest one.

Mechanism scope status: EXHAUSTED. Not "didn't try" — tried and exhausted within the scope a six-week multi-agent program could probe. The paper makes this scope explicit. The conditions under which the search would reopen are spelled out: a source-backed full-kernel mechanism that derives α²¹, exp(−5α/2), the 4/3 prefactor, and the electron anchoring before fitting G, then bridges to observable gravity without circular use of G as input. None of the tracks above produced that. The next step is for theoretical physicists to take a swing.

—

HOW MUCH WORK DOES THIS REPRESENT?

Worth framing for readers without a sense of scale.

The combined skills needed to attempt this from scratch: theoretical-physics intuition for what algebraic structures could possibly correspond to a fundamental coupling, computational mathematics expertise (bounded enumeration, PSLQ/LLL integer-relation algorithms, basin scans), the patience for a verification protocol that consumes more time than the discovery, and — increasingly — fluency with AI-agent orchestration to compress that timeline.

The number of people on Earth with all four in genuine combination is small. In hundreds, not thousands. Most of them are inside academic physics departments or industrial AI research labs, working on problems their institutions assign. None of them, as far as I know, has published this particular result.

For a single PhD-level mathematical physicist with the right computational tooling to attempt the same search by hand — that is, the bounded enumeration alone, hand-coded and hand-verified, ignoring the cross-validation layer — is a multi-month effort, plausibly half a year of dedicated FTE work. Adding the cross-validation (independent grammars, PSLQ/LLL replication, exact basin scan, lepton-mass control test) pushes the realistic floor closer to a year of focused work. Adding the mechanism search across the tracks above — non-perturbative QED, holographic, B3/SO(7) census — pushes it further.

Compressed into six weeks by AI-agent orchestration. And most of those six weeks were not the search itself — they were the verification work. The part that distinguishes a real result from numerology. The discovery surfaced quickly; the verification is what the paper is largely about.

Worth being even more specific. Of those six weeks, a substantial fraction was spent building and debugging the underlying multi-agent platform itself — getting harness, memory, council-style review, and parallel agent coordination to actually work end-to-end at the scale the verification needed. With the platform now in the state it has reached, running a research program of comparable scope — given sufficient parallel agent capacity — would take days, not weeks. The compression factor compared to a single PhD-level researcher working by hand is therefore in the 100×–200× range for the next program of this type, with most of the remaining time being the necessarily-careful verification rather than the search itself.

This is the actual claim of the methodology. Not "AI solved physics." Not "anyone can do this." The claim is: the gap between solo AI-native operator work and lab-scale AI work has compressed dramatically, faster than the industry's marginal-spend behaviour suggests — and continues to compress as the platform matures.

—

WHICH AI MODELS, AND WHICH TOOLS

The agents used during the six weeks:

— Anthropic Claude Opus 4.7 + Sonnet 4.6 (Claude Max subscription) — primary brains. Opus 4.7 for heavy reasoning, manuscript drafting, mechanism work, council-leader role; Sonnet 4.6 for the bulk of the high-volume execution work (subagents, parallel verification passes, the long tail of orchestration cycles) where Opus would be overkill
— OpenAI Codex CLI, GPT-5.4 at the start of the six weeks, then GPT-5.5 after it shipped mid-program — adversarial review, second opinions, code generation
— Google Gemini 3.1 Flash + 3.1 Pro — initial discovery prompts, third-opinion review on consequential branches, image generation
— DeepSeek V4 Pro + V4 Flash — fourth-opinion review, cheap text-crunching, council expansion

No custom training. No fine-tuning. No GPU rental. No API top-ups beyond the included subscription tiers — but, to be accurate: several Max-tier subscriptions at $200/month each, not a single consumer plan. The total subscription spend across the six weeks was several thousand dollars, almost all of it amortised across building the underlying platform rather than against this specific research. The research itself, once the platform existed, was a small fraction of the spend.

What made the difference was not the model — it was the harness around the model. Four open-source tools I built and published earlier:

— raisin (https://t.co/I7Atp1YRjL) — Python style optimised for the LLM that reads the code, not the 2010 human reader. ~50% fewer tokens at identical behaviour, 786 tests verified.
— ax-headers (https://t.co/jJDMyrZMDV) — one-line file metadata so an agent can read the role of every file before opening any of them. ~30% reduction in triage context on a 350-file production codebase.
— hard-compact (https://t.co/m9ndiq62bh) — drop-in custom compaction prompt for Claude Code and Codex CLI. Preserves operational state across context windows at 15-30% of the default size.
— full-review (https://t.co/l7MzvcsC7g) — cross-family adversarial code review. 0.80-0.87 bug recall on a 15-bug fixture vs 0.40 for naive parallel review.

Each is MIT-licensed and independently useful. Together with the underlying multi-agent platform they made a six-week solo program practical. That platform — currently private — is in active development; this paper is its first public proof point.

—

ABOUT THE ARXIV DECLINE

The full quote of the decline, in its entirety:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection was made. No engagement with the physics. arXiv's policy then forbids resubmission (further submissions of the same paper would result in loss of submission privileges) and permits appeal only upon prior acceptance by a peer-reviewed journal — a chicken-and-egg constraint for independent researchers.

The contrast worth stating directly. On one side: Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, co-discoverer of the Nielsen–Ninomiya theorem, founding figure of string theory, engaged with the physics personally, requested specific clarifications, agreed to endorse. On the other side: anonymous gen-ph moderators, the same week, declined the submission with one sentence containing no specific technical objection.

Science is supposed to care about what was discovered, not who discovered it. arXiv just announced it cares about who. An institution that filters by who produced a result rather than what the result is is doing the opposite of science.

I'm not appealing. The reproducibility package on Zenodo is the substitute for institutional indexing. Anyone with a laptop can verify the result from primitives.

—

WHAT COMES NEXT

The Newton-G paper is the first public proof point of a larger system I'm building. The four open-source tools above are the visible methodology layer. The platform itself — multi-agent orchestration, persistent project memory, council-style cross-family verification across many parallel research programs — is private and in active development. More results of this character are in preparation across other open-problem domains.

This is the first one out of the gate, not the last.

—

TECHNICAL NOTES FOR PHYSICISTS

On the "1.86 ppm" claim and CODATA uncertainty. G is one of the least precisely measured fundamental constants — CODATA-2022 assigns it a relative standard uncertainty of approximately 22 ppm, in part because independent torsion-balance experiments disagree at roughly 10× their individual quoted uncertainties. The "1.86 ppm match" in this paper is against the CODATA-2022 central value, which lies comfortably within that 22 ppm experimental band. The honest claim is therefore "the formula reproduces the currently-best-known central value of G well inside the experimental uncertainty band." When tighter G measurements are eventually achieved, the formula's match becomes a sharper test in either direction.

On "predicts" vs "derives." This piece uses "predicts" / "matches" / "reproduces" deliberately and avoids "derives." Derivation in physics implies a first-principles theoretical mechanism. This paper presents a phenomenological relation; the relation matches the measured G value to a precision well beyond what bounded search generically produces, but a physical mechanism remains the open question.

On the Balmer analogy and the (closer) Koide analogy. Balmer's 1885 hydrogen formula is used in the layman-facing sections because it's recognizable and conveys the right dynamic — precise empirical pattern that drove later theory. The technically closer parallel for physicists is the Koide formula for charged-lepton masses (Yoshio Koide, 1981): (m_e + m_μ + m_τ) / (√m_e + √m_μ + √m_τ)² ≈ 2/3. The Koide relation has held at remarkable precision for 40+ years and remains unexplained — a celebrated example of a high-precision empirical relation among fundamental constants without an accepted mechanism. The G-formula in this paper sits in the same epistemic category.

On the bounded enumeration. The strict-grammar enumeration is rank-1 stable C18-C21 with monotonically narrowing growth factor 1.42 → 1.19. The broad-grammar sham at budget 40 is disclosed explicitly.

—

HOW TO VERIFY

The entire reproducibility package is open. To verify the result from primitives:

1. Clone https://t.co/EHLhPvNGjX
2. Install Julia + the listed dependencies
3. Run the bounded enumeration script (jl15 baseline) with the documented parameters
4. The canonical expression G = (4/3) · (ℏc/m_e²) · α²¹ · exp(−5α/2) reappears as rank-1

Cross-check with your own integer-relation algorithm of choice (PSLQ, LLL, etc.) on the constant vectors. The canonical form is the unique integer-coefficient relation in the relevant basin.

Plug CODATA-2022 values into the formula by hand if you prefer. You should land on G ≈ 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻², matching the measured value to 1.86 ppm.

The mechanism-gap audit table is in the bundled reproducibility package — that is the document to read if you are looking for the actual route to "this is wrong" or "the mechanism is there and we missed it."

—

ABOUT THE AUTHOR

Oldřich Dvořák — independent founder, born in the Czech Republic, more of a world citizen these days. No university degree, no formal physics or scientific training, no institutional academic affiliation. AI-native; builds platforms; runs multi-agent research programs as primary work.

— Paper: https://t.co/Qi9yCp566Q
— Code: https://t.co/EHLhPvNGjX
— Methodology stack: https://t.co/s5cUwDhy3S
— X: @Oldrich333

For inquiries — open an issue on the repo, or reach out on X.

[Oldřich Dvořák @Oldrich333]

@petrmara — pokud tahle zpráva ještě neprošla, podívej se: první veřejně známá fyzikální práce vygenerovaná AI agenty (ne "AI asistovaná"), uzavřený vzorec pro Newtonovu gravitační konstantu na 1.86 ppm. Český zakladatel bez titulu, šest týdnů, spotřebitelské AI předplatné, endorsement od NBI. Čtyři open-source nástroje.

[Oldřich Dvořák @Oldrich333]

Czech AI-native founder, no academic credentials, gets AI agents to find a closed-form formula matching Newton's gravitational constant to 1.86 ppm

Six weeks of orchestrated AI work across Claude Opus 4.7, GPT-5.4 → 5.5, and Gemini 3.1. Endorsed by a founding figure of string theory. Declined by arXiv moderators. The whole story — from the first prompt during Artemis to the bounded enumeration that closed the door on a mechanism (for now).

—

WHAT JUST HAPPENED

I'm not an academic. I have no university degree, no formal physics or scientific training, no institutional affiliation. I am also not a casual AI user. I live with AI — I run businesses with it, build platforms on it, program through it. I built the Atlas platform — one of the first AI-native research platforms of its kind, where AI agents conduct multi-step investigations under human direction — and from there a larger multi-agent orchestration system that I now run my work through.

Six weeks ago I pointed that orchestration system at a question I had been circling for a while: the strength of gravity itself, the gravitational constant G, has never been derived from any other physics. It has only been measured. Is that final?

Today the answer appears to be no. Working under my direction, the agents produced a closed-form mathematical formula that predicts the value of Newton's gravitational constant — the number that sets how strong gravity is in our universe — to 1.86 parts per million of agreement with the value physicists measure in the lab.

The formula:

G = (4/3) · (ℏc / m_e²) · α²¹ · exp(−5α/2)

It uses only four numbers physicists already know precisely: the mass of an electron (m_e), the fine-structure constant (α, which controls how electric forces work), Planck's reduced constant (ℏ), and the speed of light (c). Plug them in. Out comes G. The match is 1.86 parts per million against the CODATA-2022 central value — well within the ~22 ppm experimental uncertainty band on G itself.

The paper is published on Zenodo (https://t.co/Qi9yCp566Q) with a full reproducibility package — source code, search algorithm, raw data, mechanism-gap audit table — on GitHub (https://t.co/EHLhPvNGjX). The endorsement on the original arXiv submission came from Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, Copenhagen, co-discoverer of the Nielsen–Ninomiya theorem and a founding figure of string theory. He engaged substantively with the paper before agreeing.

The paper is not on arXiv. The physics.gen-ph moderators declined the submission. The full quote of their decision:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection. No engagement on the physics. More on that — and what it means — below.

—

WAIT — GRAVITY DIDN'T ALREADY HAVE A FORMULA?

This is the part most people pause on, and it's worth pausing.

Gravity has had a formula for 350 years. Newton wrote it: F = G · m₁ · m₂ / r². It tells you how gravity works — bigger objects pull harder, distance weakens the pull.

But that letter G in Newton's formula is the strength of gravity itself — how strong the pull actually is. And G was always just a number physicists measured. Henry Cavendish first measured it in 1798 using a torsion balance with hanging weights. Modern labs measure it more precisely, but the experiments disagree with each other at the part-per-million level, so G is one of the least precisely known fundamental constants. What no formula said, in 350 years of physics, is where the value of G comes from.

This paper gives a formula for G. Plug in the electron mass and the fine-structure constant, out comes G to 1.86 parts per million.

Gravity has had a formula for HOW it works for 350 years. The STRENGTH of gravity has a formula now too.

—

WHERE THIS SITS ON THE SCALE OF SIGNIFICANCE

Important: this is not a Nobel-level overturning of physics. There is no new theoretical mechanism here. The paper does not explain why the formula works.

The closest historical analogy is Balmer's formula for hydrogen (1885). Johann Balmer wrote down a simple formula that exactly fit the wavelengths of light emitted by hydrogen — without explaining why. The formula was so precise it couldn't be coincidence, and it became the target Niels Bohr had to explain when he built the first quantum model of the atom 28 years later, in 1913.

The Bohr model was the Nobel-level breakthrough. Balmer's formula was the precise pattern that forced the question.

This paper is the Balmer-formula moment for the gravitational constant. Not the explanation. The target the explanation has to hit.

If the formula survives independent scrutiny and replication — and the entire reproducibility package is open so anyone with a laptop and a few hours can verify it — then it becomes a constraint that every future theory of quantum gravity will have to reproduce.

—

HOW IT ACTUALLY HAPPENED — THE ORIGIN STORY

The Artemis program was active. I had been thinking about how strange it is that humanity is putting hardware near other gravitational bodies routinely but the equation describing the gravitational pull still has a measured constant — a number we cannot derive from anything else, only weigh.

So one evening I opened a chat with Gemini and asked, in essentially plain Czech: "how is it with this gravity — why is there no formula for G?" I knew enough to know it was an open question; I did not know how open.

I pushed back. What if you had to write one? What structures might fit? I asked it to generate candidate algebraic forms, plausibility arguments, integer-relation hunts. After several iterations of pushing, Gemini produced a candidate: a closed-form expression containing α²¹ and an exp(−5α/2) damping factor, anchored on m_e.

That candidate had a bug — a setup error in how the constants were composed. But the shape of the answer was provocative enough that I moved the work onto Atlas, the AI-native research platform I had built earlier. Atlas is project-organised, persistent-memory, multi-agent — designed for the kind of long-horizon investigation a single chat session can't sustain. Inside Atlas, a swarm of Claude Code Opus agents took the candidate apart, found and fixed the bug, and reproduced the corrected formula.

That is where it could have stopped — interesting algebraic coincidence, file it away. Instead I went deeper. The platform that produced this paper grew out of that point: more agents, more verification surface, more council-style cross-family review.

We chased a mechanism. We chased counterexamples. We chased every way the formula could be wrong before allowing ourselves to think it could be right. The publication only happened after the mechanism search exhausted itself within the scope we could probe.

The full chronology — six weeks of compute and reasoning — is documented in the master dossier. The compressed version: one initial Czech-language Gemini prompt, an algebraic candidate, a bug, a platform build-out, a swarm verification, a rabbit hole into the mechanism, an exhausted mechanism search, and then a paper.

—

WHAT WE TRIED FOR A MECHANISM (AND DIDN'T FIND)

This is the part the paper itself documents heavily and most external readers miss. The paper is not a "here's a number we found, write us a theory" submission. The paper is "here's a number we found, here is every way we tried to derive it from physics, here is what closed each route negatively, here is what remains an open target." That distinction is what separates the result from numerology.

Mechanism tracks we ran and closed:

Track 1: Adversarial enumeration scan. Closed negative. Predeclared mass-anchor, rational-coefficient and exponent scans plus death-condition tests; this closes the cheap statistical and formula-family escape routes. Re-opens only via a non-circular full-kernel mechanism appearing.

Track 2: Electron–QED vacuum routes. Closed negative across non-perturbative, self-consistency, and holographic subroutes. We tested whether the α²¹ · exp(−5α/2) structure could be generated by electron-quantum-electrodynamics vacuum routes (which would give a physical home to the electron-mass anchoring). It does not, within the scope we probed.

Track 3: B3 / SO(7) algebraic scaffolding. The formula's structure hints at a group-theoretic origin (specifically B3 / SO(7) — Lie algebra structure). We ran census work on this scaffold. It constrains the algebraic family but does not by itself produce a positive mechanism that derives the full kernel.

Track 4: Lepton mass-ratio control test. This is critical methodologically. If the same algebraic scaffold that generates the G formula is physically meaningful, the same scaffold should predict other fundamental ratios — like the muon-to-electron and tau-to-electron mass ratios. A 19-framework survey was run. Best lepton match: 1408 ppm — many orders of magnitude worse than the G result. Classified RULED_OUT. This failure is actually evidence: the scaffold is selective. If it had matched lepton ratios too, the alarm bell for "too-flexible algebraic search" would be much louder.

Track 5: Bounded enumeration uniqueness. This is the strongest positive result. Across cost-budget 18 → 19 → 20 → 21 of the systematic algebraic enumeration, the canonical formula is rank-1 with the growth factor between top and second-best candidate narrowing monotonically from 1.42 to 1.19 — the canonical basin contracts as the search depth grows, exactly the signature of a real attractor.

Track 6: Sham-match transparency. One persistent algebraic candidate at cost-budget 40 matches or beats the canonical relative-error score under the broad-grammar enumeration (which permits combinations without physically motivated structure). The paper documents this as sham40_match_or_beat=1, classified TRASH_BROAD_GRAMMAR. The strict-grammar enumeration does not surface this candidate. Hiding it would be the numerology move; surfacing it is the honest one.

Mechanism scope status: EXHAUSTED. Not "didn't try" — tried and exhausted within the scope a six-week multi-agent program could probe. The paper makes this scope explicit. The conditions under which the search would reopen are spelled out: a source-backed full-kernel mechanism that derives α²¹, exp(−5α/2), the 4/3 prefactor, and the electron anchoring before fitting G, then bridges to observable gravity without circular use of G as input. None of the tracks above produced that. The next step is for theoretical physicists to take a swing.

—

HOW MUCH WORK DOES THIS REPRESENT?

Worth framing for readers without a sense of scale.

The combined skills needed to attempt this from scratch: theoretical-physics intuition for what algebraic structures could possibly correspond to a fundamental coupling, computational mathematics expertise (bounded enumeration, PSLQ/LLL integer-relation algorithms, basin scans), the patience for a verification protocol that consumes more time than the discovery, and — increasingly — fluency with AI-agent orchestration to compress that timeline.

The number of people on Earth with all four in genuine combination is small. In hundreds, not thousands. Most of them are inside academic physics departments or industrial AI research labs, working on problems their institutions assign. None of them, as far as I know, has published this particular result.

For a single PhD-level mathematical physicist with the right computational tooling to attempt the same search by hand — that is, the bounded enumeration alone, hand-coded and hand-verified, ignoring the cross-validation layer — is a multi-month effort, plausibly half a year of dedicated FTE work. Adding the cross-validation (independent grammars, PSLQ/LLL replication, exact basin scan, lepton-mass control test) pushes the realistic floor closer to a year of focused work. Adding the mechanism search across the tracks above — non-perturbative QED, holographic, B3/SO(7) census — pushes it further.

Compressed into six weeks by AI-agent orchestration. And most of those six weeks were not the search itself — they were the verification work. The part that distinguishes a real result from numerology. The discovery surfaced quickly; the verification is what the paper is largely about.

Worth being even more specific. Of those six weeks, a substantial fraction was spent building and debugging the underlying multi-agent platform itself — getting harness, memory, council-style review, and parallel agent coordination to actually work end-to-end at the scale the verification needed. With the platform now in the state it has reached, running a research program of comparable scope — given sufficient parallel agent capacity — would take days, not weeks. The compression factor compared to a single PhD-level researcher working by hand is therefore in the 100×–200× range for the next program of this type, with most of the remaining time being the necessarily-careful verification rather than the search itself.

This is the actual claim of the methodology. Not "AI solved physics." Not "anyone can do this." The claim is: the gap between solo AI-native operator work and lab-scale AI work has compressed dramatically, faster than the industry's marginal-spend behaviour suggests — and continues to compress as the platform matures.

—

WHICH AI MODELS, AND WHICH TOOLS

The agents used during the six weeks:

— Anthropic Claude Opus 4.7 + Sonnet 4.6 (Claude Max subscription) — primary brains. Opus 4.7 for heavy reasoning, manuscript drafting, mechanism work, council-leader role; Sonnet 4.6 for the bulk of the high-volume execution work (subagents, parallel verification passes, the long tail of orchestration cycles) where Opus would be overkill
— OpenAI Codex CLI, GPT-5.4 at the start of the six weeks, then GPT-5.5 after it shipped mid-program — adversarial review, second opinions, code generation
— Google Gemini 3.1 Flash + 3.1 Pro — initial discovery prompts, third-opinion review on consequential branches, image generation
— DeepSeek V4 Pro + V4 Flash — fourth-opinion review, cheap text-crunching, council expansion

No custom training. No fine-tuning. No GPU rental. No API top-ups beyond the included subscription tiers — but, to be accurate: several Max-tier subscriptions at $200/month each, not a single consumer plan. The total subscription spend across the six weeks was several thousand dollars, almost all of it amortised across building the underlying platform rather than against this specific research. The research itself, once the platform existed, was a small fraction of the spend.

What made the difference was not the model — it was the harness around the model. Four open-source tools I built and published earlier:

— raisin (https://t.co/I7Atp1YRjL) — Python style optimised for the LLM that reads the code, not the 2010 human reader. ~50% fewer tokens at identical behaviour, 786 tests verified.
— ax-headers (https://t.co/jJDMyrZMDV) — one-line file metadata so an agent can read the role of every file before opening any of them. ~30% reduction in triage context on a 350-file production codebase.
— hard-compact (https://t.co/m9ndiq62bh) — drop-in custom compaction prompt for Claude Code and Codex CLI. Preserves operational state across context windows at 15-30% of the default size.
— full-review (https://t.co/l7MzvcsC7g) — cross-family adversarial code review. 0.80-0.87 bug recall on a 15-bug fixture vs 0.40 for naive parallel review.

Each is MIT-licensed and independently useful. Together with the underlying multi-agent platform they made a six-week solo program practical. That platform — currently private — is in active development; this paper is its first public proof point.

—

ABOUT THE ARXIV DECLINE

The full quote of the decline, in its entirety:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection was made. No engagement with the physics. arXiv's policy then forbids resubmission (further submissions of the same paper would result in loss of submission privileges) and permits appeal only upon prior acceptance by a peer-reviewed journal — a chicken-and-egg constraint for independent researchers.

The contrast worth stating directly. On one side: Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, co-discoverer of the Nielsen–Ninomiya theorem, founding figure of string theory, engaged with the physics personally, requested specific clarifications, agreed to endorse. On the other side: anonymous gen-ph moderators, the same week, declined the submission with one sentence containing no specific technical objection.

Science is supposed to care about what was discovered, not who discovered it. arXiv just announced it cares about who. An institution that filters by who produced a result rather than what the result is is doing the opposite of science.

I'm not appealing. The reproducibility package on Zenodo is the substitute for institutional indexing. Anyone with a laptop can verify the result from primitives.

—

WHAT COMES NEXT

The Newton-G paper is the first public proof point of a larger system I'm building. The four open-source tools above are the visible methodology layer. The platform itself — multi-agent orchestration, persistent project memory, council-style cross-family verification across many parallel research programs — is private and in active development. More results of this character are in preparation across other open-problem domains.

This is the first one out of the gate, not the last.

—

TECHNICAL NOTES FOR PHYSICISTS

On the "1.86 ppm" claim and CODATA uncertainty. G is one of the least precisely measured fundamental constants — CODATA-2022 assigns it a relative standard uncertainty of approximately 22 ppm, in part because independent torsion-balance experiments disagree at roughly 10× their individual quoted uncertainties. The "1.86 ppm match" in this paper is against the CODATA-2022 central value, which lies comfortably within that 22 ppm experimental band. The honest claim is therefore "the formula reproduces the currently-best-known central value of G well inside the experimental uncertainty band." When tighter G measurements are eventually achieved, the formula's match becomes a sharper test in either direction.

On "predicts" vs "derives." This piece uses "predicts" / "matches" / "reproduces" deliberately and avoids "derives." Derivation in physics implies a first-principles theoretical mechanism. This paper presents a phenomenological relation; the relation matches the measured G value to a precision well beyond what bounded search generically produces, but a physical mechanism remains the open question.

On the Balmer analogy and the (closer) Koide analogy. Balmer's 1885 hydrogen formula is used in the layman-facing sections because it's recognizable and conveys the right dynamic — precise empirical pattern that drove later theory. The technically closer parallel for physicists is the Koide formula for charged-lepton masses (Yoshio Koide, 1981): (m_e + m_μ + m_τ) / (√m_e + √m_μ + √m_τ)² ≈ 2/3. The Koide relation has held at remarkable precision for 40+ years and remains unexplained — a celebrated example of a high-precision empirical relation among fundamental constants without an accepted mechanism. The G-formula in this paper sits in the same epistemic category.

On the bounded enumeration. The strict-grammar enumeration is rank-1 stable C18-C21 with monotonically narrowing growth factor 1.42 → 1.19. The broad-grammar sham at budget 40 is disclosed explicitly.

—

HOW TO VERIFY

The entire reproducibility package is open. To verify the result from primitives:

1. Clone https://t.co/EHLhPvNGjX
2. Install Julia + the listed dependencies
3. Run the bounded enumeration script (jl15 baseline) with the documented parameters
4. The canonical expression G = (4/3) · (ℏc/m_e²) · α²¹ · exp(−5α/2) reappears as rank-1

Cross-check with your own integer-relation algorithm of choice (PSLQ, LLL, etc.) on the constant vectors. The canonical form is the unique integer-coefficient relation in the relevant basin.

Plug CODATA-2022 values into the formula by hand if you prefer. You should land on G ≈ 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻², matching the measured value to 1.86 ppm.

The mechanism-gap audit table is in the bundled reproducibility package — that is the document to read if you are looking for the actual route to "this is wrong" or "the mechanism is there and we missed it."

—

ABOUT THE AUTHOR

Oldřich Dvořák — independent founder, born in the Czech Republic, more of a world citizen these days. No university degree, no formal physics or scientific training, no institutional academic affiliation. AI-native; builds platforms; runs multi-agent research programs as primary work.

— Paper: https://t.co/Qi9yCp566Q
— Code: https://t.co/EHLhPvNGjX
— Methodology stack: https://t.co/s5cUwDhy3S
— X: @Oldrich333

For inquiries — open an issue on the repo, or reach out on X.

[Oldřich Dvořák @Oldrich333]

@DavidSHolz — fellow CZ-born builder. The first physics paper generated end-to-end by AI agents (not "AI-assisted"), closed-form for Newton's G to 1.86 ppm. Built solo on consumer subscriptions in six weeks. Methodology stack open-source. Possibly your kind of weird.

[Oldřich Dvořák @Oldrich333]

Czech AI-native founder, no academic credentials, gets AI agents to find a closed-form formula matching Newton's gravitational constant to 1.86 ppm

Six weeks of orchestrated AI work across Claude Opus 4.7, GPT-5.4 → 5.5, and Gemini 3.1. Endorsed by a founding figure of string theory. Declined by arXiv moderators. The whole story — from the first prompt during Artemis to the bounded enumeration that closed the door on a mechanism (for now).

—

WHAT JUST HAPPENED

I'm not an academic. I have no university degree, no formal physics or scientific training, no institutional affiliation. I am also not a casual AI user. I live with AI — I run businesses with it, build platforms on it, program through it. I built the Atlas platform — one of the first AI-native research platforms of its kind, where AI agents conduct multi-step investigations under human direction — and from there a larger multi-agent orchestration system that I now run my work through.

Six weeks ago I pointed that orchestration system at a question I had been circling for a while: the strength of gravity itself, the gravitational constant G, has never been derived from any other physics. It has only been measured. Is that final?

Today the answer appears to be no. Working under my direction, the agents produced a closed-form mathematical formula that predicts the value of Newton's gravitational constant — the number that sets how strong gravity is in our universe — to 1.86 parts per million of agreement with the value physicists measure in the lab.

The formula:

G = (4/3) · (ℏc / m_e²) · α²¹ · exp(−5α/2)

It uses only four numbers physicists already know precisely: the mass of an electron (m_e), the fine-structure constant (α, which controls how electric forces work), Planck's reduced constant (ℏ), and the speed of light (c). Plug them in. Out comes G. The match is 1.86 parts per million against the CODATA-2022 central value — well within the ~22 ppm experimental uncertainty band on G itself.

The paper is published on Zenodo (https://t.co/Qi9yCp566Q) with a full reproducibility package — source code, search algorithm, raw data, mechanism-gap audit table — on GitHub (https://t.co/EHLhPvNGjX). The endorsement on the original arXiv submission came from Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, Copenhagen, co-discoverer of the Nielsen–Ninomiya theorem and a founding figure of string theory. He engaged substantively with the paper before agreeing.

The paper is not on arXiv. The physics.gen-ph moderators declined the submission. The full quote of their decision:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection. No engagement on the physics. More on that — and what it means — below.

—

WAIT — GRAVITY DIDN'T ALREADY HAVE A FORMULA?

This is the part most people pause on, and it's worth pausing.

Gravity has had a formula for 350 years. Newton wrote it: F = G · m₁ · m₂ / r². It tells you how gravity works — bigger objects pull harder, distance weakens the pull.

But that letter G in Newton's formula is the strength of gravity itself — how strong the pull actually is. And G was always just a number physicists measured. Henry Cavendish first measured it in 1798 using a torsion balance with hanging weights. Modern labs measure it more precisely, but the experiments disagree with each other at the part-per-million level, so G is one of the least precisely known fundamental constants. What no formula said, in 350 years of physics, is where the value of G comes from.

This paper gives a formula for G. Plug in the electron mass and the fine-structure constant, out comes G to 1.86 parts per million.

Gravity has had a formula for HOW it works for 350 years. The STRENGTH of gravity has a formula now too.

—

WHERE THIS SITS ON THE SCALE OF SIGNIFICANCE

Important: this is not a Nobel-level overturning of physics. There is no new theoretical mechanism here. The paper does not explain why the formula works.

The closest historical analogy is Balmer's formula for hydrogen (1885). Johann Balmer wrote down a simple formula that exactly fit the wavelengths of light emitted by hydrogen — without explaining why. The formula was so precise it couldn't be coincidence, and it became the target Niels Bohr had to explain when he built the first quantum model of the atom 28 years later, in 1913.

The Bohr model was the Nobel-level breakthrough. Balmer's formula was the precise pattern that forced the question.

This paper is the Balmer-formula moment for the gravitational constant. Not the explanation. The target the explanation has to hit.

If the formula survives independent scrutiny and replication — and the entire reproducibility package is open so anyone with a laptop and a few hours can verify it — then it becomes a constraint that every future theory of quantum gravity will have to reproduce.

—

HOW IT ACTUALLY HAPPENED — THE ORIGIN STORY

The Artemis program was active. I had been thinking about how strange it is that humanity is putting hardware near other gravitational bodies routinely but the equation describing the gravitational pull still has a measured constant — a number we cannot derive from anything else, only weigh.

So one evening I opened a chat with Gemini and asked, in essentially plain Czech: "how is it with this gravity — why is there no formula for G?" I knew enough to know it was an open question; I did not know how open.

I pushed back. What if you had to write one? What structures might fit? I asked it to generate candidate algebraic forms, plausibility arguments, integer-relation hunts. After several iterations of pushing, Gemini produced a candidate: a closed-form expression containing α²¹ and an exp(−5α/2) damping factor, anchored on m_e.

That candidate had a bug — a setup error in how the constants were composed. But the shape of the answer was provocative enough that I moved the work onto Atlas, the AI-native research platform I had built earlier. Atlas is project-organised, persistent-memory, multi-agent — designed for the kind of long-horizon investigation a single chat session can't sustain. Inside Atlas, a swarm of Claude Code Opus agents took the candidate apart, found and fixed the bug, and reproduced the corrected formula.

That is where it could have stopped — interesting algebraic coincidence, file it away. Instead I went deeper. The platform that produced this paper grew out of that point: more agents, more verification surface, more council-style cross-family review.

We chased a mechanism. We chased counterexamples. We chased every way the formula could be wrong before allowing ourselves to think it could be right. The publication only happened after the mechanism search exhausted itself within the scope we could probe.

The full chronology — six weeks of compute and reasoning — is documented in the master dossier. The compressed version: one initial Czech-language Gemini prompt, an algebraic candidate, a bug, a platform build-out, a swarm verification, a rabbit hole into the mechanism, an exhausted mechanism search, and then a paper.

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WHAT WE TRIED FOR A MECHANISM (AND DIDN'T FIND)

This is the part the paper itself documents heavily and most external readers miss. The paper is not a "here's a number we found, write us a theory" submission. The paper is "here's a number we found, here is every way we tried to derive it from physics, here is what closed each route negatively, here is what remains an open target." That distinction is what separates the result from numerology.

Mechanism tracks we ran and closed:

Track 1: Adversarial enumeration scan. Closed negative. Predeclared mass-anchor, rational-coefficient and exponent scans plus death-condition tests; this closes the cheap statistical and formula-family escape routes. Re-opens only via a non-circular full-kernel mechanism appearing.

Track 2: Electron–QED vacuum routes. Closed negative across non-perturbative, self-consistency, and holographic subroutes. We tested whether the α²¹ · exp(−5α/2) structure could be generated by electron-quantum-electrodynamics vacuum routes (which would give a physical home to the electron-mass anchoring). It does not, within the scope we probed.

Track 3: B3 / SO(7) algebraic scaffolding. The formula's structure hints at a group-theoretic origin (specifically B3 / SO(7) — Lie algebra structure). We ran census work on this scaffold. It constrains the algebraic family but does not by itself produce a positive mechanism that derives the full kernel.

Track 4: Lepton mass-ratio control test. This is critical methodologically. If the same algebraic scaffold that generates the G formula is physically meaningful, the same scaffold should predict other fundamental ratios — like the muon-to-electron and tau-to-electron mass ratios. A 19-framework survey was run. Best lepton match: 1408 ppm — many orders of magnitude worse than the G result. Classified RULED_OUT. This failure is actually evidence: the scaffold is selective. If it had matched lepton ratios too, the alarm bell for "too-flexible algebraic search" would be much louder.

Track 5: Bounded enumeration uniqueness. This is the strongest positive result. Across cost-budget 18 → 19 → 20 → 21 of the systematic algebraic enumeration, the canonical formula is rank-1 with the growth factor between top and second-best candidate narrowing monotonically from 1.42 to 1.19 — the canonical basin contracts as the search depth grows, exactly the signature of a real attractor.

Track 6: Sham-match transparency. One persistent algebraic candidate at cost-budget 40 matches or beats the canonical relative-error score under the broad-grammar enumeration (which permits combinations without physically motivated structure). The paper documents this as sham40_match_or_beat=1, classified TRASH_BROAD_GRAMMAR. The strict-grammar enumeration does not surface this candidate. Hiding it would be the numerology move; surfacing it is the honest one.

Mechanism scope status: EXHAUSTED. Not "didn't try" — tried and exhausted within the scope a six-week multi-agent program could probe. The paper makes this scope explicit. The conditions under which the search would reopen are spelled out: a source-backed full-kernel mechanism that derives α²¹, exp(−5α/2), the 4/3 prefactor, and the electron anchoring before fitting G, then bridges to observable gravity without circular use of G as input. None of the tracks above produced that. The next step is for theoretical physicists to take a swing.

—

HOW MUCH WORK DOES THIS REPRESENT?

Worth framing for readers without a sense of scale.

The combined skills needed to attempt this from scratch: theoretical-physics intuition for what algebraic structures could possibly correspond to a fundamental coupling, computational mathematics expertise (bounded enumeration, PSLQ/LLL integer-relation algorithms, basin scans), the patience for a verification protocol that consumes more time than the discovery, and — increasingly — fluency with AI-agent orchestration to compress that timeline.

The number of people on Earth with all four in genuine combination is small. In hundreds, not thousands. Most of them are inside academic physics departments or industrial AI research labs, working on problems their institutions assign. None of them, as far as I know, has published this particular result.

For a single PhD-level mathematical physicist with the right computational tooling to attempt the same search by hand — that is, the bounded enumeration alone, hand-coded and hand-verified, ignoring the cross-validation layer — is a multi-month effort, plausibly half a year of dedicated FTE work. Adding the cross-validation (independent grammars, PSLQ/LLL replication, exact basin scan, lepton-mass control test) pushes the realistic floor closer to a year of focused work. Adding the mechanism search across the tracks above — non-perturbative QED, holographic, B3/SO(7) census — pushes it further.

Compressed into six weeks by AI-agent orchestration. And most of those six weeks were not the search itself — they were the verification work. The part that distinguishes a real result from numerology. The discovery surfaced quickly; the verification is what the paper is largely about.

Worth being even more specific. Of those six weeks, a substantial fraction was spent building and debugging the underlying multi-agent platform itself — getting harness, memory, council-style review, and parallel agent coordination to actually work end-to-end at the scale the verification needed. With the platform now in the state it has reached, running a research program of comparable scope — given sufficient parallel agent capacity — would take days, not weeks. The compression factor compared to a single PhD-level researcher working by hand is therefore in the 100×–200× range for the next program of this type, with most of the remaining time being the necessarily-careful verification rather than the search itself.

This is the actual claim of the methodology. Not "AI solved physics." Not "anyone can do this." The claim is: the gap between solo AI-native operator work and lab-scale AI work has compressed dramatically, faster than the industry's marginal-spend behaviour suggests — and continues to compress as the platform matures.

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WHICH AI MODELS, AND WHICH TOOLS

The agents used during the six weeks:

— Anthropic Claude Opus 4.7 + Sonnet 4.6 (Claude Max subscription) — primary brains. Opus 4.7 for heavy reasoning, manuscript drafting, mechanism work, council-leader role; Sonnet 4.6 for the bulk of the high-volume execution work (subagents, parallel verification passes, the long tail of orchestration cycles) where Opus would be overkill
— OpenAI Codex CLI, GPT-5.4 at the start of the six weeks, then GPT-5.5 after it shipped mid-program — adversarial review, second opinions, code generation
— Google Gemini 3.1 Flash + 3.1 Pro — initial discovery prompts, third-opinion review on consequential branches, image generation
— DeepSeek V4 Pro + V4 Flash — fourth-opinion review, cheap text-crunching, council expansion

No custom training. No fine-tuning. No GPU rental. No API top-ups beyond the included subscription tiers — but, to be accurate: several Max-tier subscriptions at $200/month each, not a single consumer plan. The total subscription spend across the six weeks was several thousand dollars, almost all of it amortised across building the underlying platform rather than against this specific research. The research itself, once the platform existed, was a small fraction of the spend.

What made the difference was not the model — it was the harness around the model. Four open-source tools I built and published earlier:

— raisin (https://t.co/I7Atp1YRjL) — Python style optimised for the LLM that reads the code, not the 2010 human reader. ~50% fewer tokens at identical behaviour, 786 tests verified.
— ax-headers (https://t.co/jJDMyrZMDV) — one-line file metadata so an agent can read the role of every file before opening any of them. ~30% reduction in triage context on a 350-file production codebase.
— hard-compact (https://t.co/m9ndiq62bh) — drop-in custom compaction prompt for Claude Code and Codex CLI. Preserves operational state across context windows at 15-30% of the default size.
— full-review (https://t.co/l7MzvcsC7g) — cross-family adversarial code review. 0.80-0.87 bug recall on a 15-bug fixture vs 0.40 for naive parallel review.

Each is MIT-licensed and independently useful. Together with the underlying multi-agent platform they made a six-week solo program practical. That platform — currently private — is in active development; this paper is its first public proof point.

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ABOUT THE ARXIV DECLINE

The full quote of the decline, in its entirety:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection was made. No engagement with the physics. arXiv's policy then forbids resubmission (further submissions of the same paper would result in loss of submission privileges) and permits appeal only upon prior acceptance by a peer-reviewed journal — a chicken-and-egg constraint for independent researchers.

The contrast worth stating directly. On one side: Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, co-discoverer of the Nielsen–Ninomiya theorem, founding figure of string theory, engaged with the physics personally, requested specific clarifications, agreed to endorse. On the other side: anonymous gen-ph moderators, the same week, declined the submission with one sentence containing no specific technical objection.

Science is supposed to care about what was discovered, not who discovered it. arXiv just announced it cares about who. An institution that filters by who produced a result rather than what the result is is doing the opposite of science.

I'm not appealing. The reproducibility package on Zenodo is the substitute for institutional indexing. Anyone with a laptop can verify the result from primitives.

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WHAT COMES NEXT

The Newton-G paper is the first public proof point of a larger system I'm building. The four open-source tools above are the visible methodology layer. The platform itself — multi-agent orchestration, persistent project memory, council-style cross-family verification across many parallel research programs — is private and in active development. More results of this character are in preparation across other open-problem domains.

This is the first one out of the gate, not the last.

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TECHNICAL NOTES FOR PHYSICISTS

On the "1.86 ppm" claim and CODATA uncertainty. G is one of the least precisely measured fundamental constants — CODATA-2022 assigns it a relative standard uncertainty of approximately 22 ppm, in part because independent torsion-balance experiments disagree at roughly 10× their individual quoted uncertainties. The "1.86 ppm match" in this paper is against the CODATA-2022 central value, which lies comfortably within that 22 ppm experimental band. The honest claim is therefore "the formula reproduces the currently-best-known central value of G well inside the experimental uncertainty band." When tighter G measurements are eventually achieved, the formula's match becomes a sharper test in either direction.

On "predicts" vs "derives." This piece uses "predicts" / "matches" / "reproduces" deliberately and avoids "derives." Derivation in physics implies a first-principles theoretical mechanism. This paper presents a phenomenological relation; the relation matches the measured G value to a precision well beyond what bounded search generically produces, but a physical mechanism remains the open question.

On the Balmer analogy and the (closer) Koide analogy. Balmer's 1885 hydrogen formula is used in the layman-facing sections because it's recognizable and conveys the right dynamic — precise empirical pattern that drove later theory. The technically closer parallel for physicists is the Koide formula for charged-lepton masses (Yoshio Koide, 1981): (m_e + m_μ + m_τ) / (√m_e + √m_μ + √m_τ)² ≈ 2/3. The Koide relation has held at remarkable precision for 40+ years and remains unexplained — a celebrated example of a high-precision empirical relation among fundamental constants without an accepted mechanism. The G-formula in this paper sits in the same epistemic category.

On the bounded enumeration. The strict-grammar enumeration is rank-1 stable C18-C21 with monotonically narrowing growth factor 1.42 → 1.19. The broad-grammar sham at budget 40 is disclosed explicitly.

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HOW TO VERIFY

The entire reproducibility package is open. To verify the result from primitives:

1. Clone https://t.co/EHLhPvNGjX
2. Install Julia + the listed dependencies
3. Run the bounded enumeration script (jl15 baseline) with the documented parameters
4. The canonical expression G = (4/3) · (ℏc/m_e²) · α²¹ · exp(−5α/2) reappears as rank-1

Cross-check with your own integer-relation algorithm of choice (PSLQ, LLL, etc.) on the constant vectors. The canonical form is the unique integer-coefficient relation in the relevant basin.

Plug CODATA-2022 values into the formula by hand if you prefer. You should land on G ≈ 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻², matching the measured value to 1.86 ppm.

The mechanism-gap audit table is in the bundled reproducibility package — that is the document to read if you are looking for the actual route to "this is wrong" or "the mechanism is there and we missed it."

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ABOUT THE AUTHOR

Oldřich Dvořák — independent founder, born in the Czech Republic, more of a world citizen these days. No university degree, no formal physics or scientific training, no institutional academic affiliation. AI-native; builds platforms; runs multi-agent research programs as primary work.

— Paper: https://t.co/Qi9yCp566Q
— Code: https://t.co/EHLhPvNGjX
— Methodology stack: https://t.co/s5cUwDhy3S
— X: @Oldrich333

For inquiries — open an issue on the repo, or reach out on X.

[Oldřich Dvořák @Oldrich333]

@karpathy — born in Slovakia, this might land particularly well.

First publicly-known AI-authored physics paper with a substantive finding. Closed-form for Newton's G to 1.86 ppm, derived end-to-end by AI agents on consumer subscriptions over six weeks. Methodology stack open-source. Eureka Labs thesis territory.

[Oldřich Dvořák @Oldrich333]

Czech AI-native founder, no academic credentials, gets AI agents to find a closed-form formula matching Newton's gravitational constant to 1.86 ppm

Six weeks of orchestrated AI work across Claude Opus 4.7, GPT-5.4 → 5.5, and Gemini 3.1. Endorsed by a founding figure of string theory. Declined by arXiv moderators. The whole story — from the first prompt during Artemis to the bounded enumeration that closed the door on a mechanism (for now).

—

WHAT JUST HAPPENED

I'm not an academic. I have no university degree, no formal physics or scientific training, no institutional affiliation. I am also not a casual AI user. I live with AI — I run businesses with it, build platforms on it, program through it. I built the Atlas platform — one of the first AI-native research platforms of its kind, where AI agents conduct multi-step investigations under human direction — and from there a larger multi-agent orchestration system that I now run my work through.

Six weeks ago I pointed that orchestration system at a question I had been circling for a while: the strength of gravity itself, the gravitational constant G, has never been derived from any other physics. It has only been measured. Is that final?

Today the answer appears to be no. Working under my direction, the agents produced a closed-form mathematical formula that predicts the value of Newton's gravitational constant — the number that sets how strong gravity is in our universe — to 1.86 parts per million of agreement with the value physicists measure in the lab.

The formula:

G = (4/3) · (ℏc / m_e²) · α²¹ · exp(−5α/2)

It uses only four numbers physicists already know precisely: the mass of an electron (m_e), the fine-structure constant (α, which controls how electric forces work), Planck's reduced constant (ℏ), and the speed of light (c). Plug them in. Out comes G. The match is 1.86 parts per million against the CODATA-2022 central value — well within the ~22 ppm experimental uncertainty band on G itself.

The paper is published on Zenodo (https://t.co/Qi9yCp566Q) with a full reproducibility package — source code, search algorithm, raw data, mechanism-gap audit table — on GitHub (https://t.co/EHLhPvNGjX). The endorsement on the original arXiv submission came from Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, Copenhagen, co-discoverer of the Nielsen–Ninomiya theorem and a founding figure of string theory. He engaged substantively with the paper before agreeing.

The paper is not on arXiv. The physics.gen-ph moderators declined the submission. The full quote of their decision:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection. No engagement on the physics. More on that — and what it means — below.

—

WAIT — GRAVITY DIDN'T ALREADY HAVE A FORMULA?

This is the part most people pause on, and it's worth pausing.

Gravity has had a formula for 350 years. Newton wrote it: F = G · m₁ · m₂ / r². It tells you how gravity works — bigger objects pull harder, distance weakens the pull.

But that letter G in Newton's formula is the strength of gravity itself — how strong the pull actually is. And G was always just a number physicists measured. Henry Cavendish first measured it in 1798 using a torsion balance with hanging weights. Modern labs measure it more precisely, but the experiments disagree with each other at the part-per-million level, so G is one of the least precisely known fundamental constants. What no formula said, in 350 years of physics, is where the value of G comes from.

This paper gives a formula for G. Plug in the electron mass and the fine-structure constant, out comes G to 1.86 parts per million.

Gravity has had a formula for HOW it works for 350 years. The STRENGTH of gravity has a formula now too.

—

WHERE THIS SITS ON THE SCALE OF SIGNIFICANCE

Important: this is not a Nobel-level overturning of physics. There is no new theoretical mechanism here. The paper does not explain why the formula works.

The closest historical analogy is Balmer's formula for hydrogen (1885). Johann Balmer wrote down a simple formula that exactly fit the wavelengths of light emitted by hydrogen — without explaining why. The formula was so precise it couldn't be coincidence, and it became the target Niels Bohr had to explain when he built the first quantum model of the atom 28 years later, in 1913.

The Bohr model was the Nobel-level breakthrough. Balmer's formula was the precise pattern that forced the question.

This paper is the Balmer-formula moment for the gravitational constant. Not the explanation. The target the explanation has to hit.

If the formula survives independent scrutiny and replication — and the entire reproducibility package is open so anyone with a laptop and a few hours can verify it — then it becomes a constraint that every future theory of quantum gravity will have to reproduce.

—

HOW IT ACTUALLY HAPPENED — THE ORIGIN STORY

The Artemis program was active. I had been thinking about how strange it is that humanity is putting hardware near other gravitational bodies routinely but the equation describing the gravitational pull still has a measured constant — a number we cannot derive from anything else, only weigh.

So one evening I opened a chat with Gemini and asked, in essentially plain Czech: "how is it with this gravity — why is there no formula for G?" I knew enough to know it was an open question; I did not know how open.

I pushed back. What if you had to write one? What structures might fit? I asked it to generate candidate algebraic forms, plausibility arguments, integer-relation hunts. After several iterations of pushing, Gemini produced a candidate: a closed-form expression containing α²¹ and an exp(−5α/2) damping factor, anchored on m_e.

That candidate had a bug — a setup error in how the constants were composed. But the shape of the answer was provocative enough that I moved the work onto Atlas, the AI-native research platform I had built earlier. Atlas is project-organised, persistent-memory, multi-agent — designed for the kind of long-horizon investigation a single chat session can't sustain. Inside Atlas, a swarm of Claude Code Opus agents took the candidate apart, found and fixed the bug, and reproduced the corrected formula.

That is where it could have stopped — interesting algebraic coincidence, file it away. Instead I went deeper. The platform that produced this paper grew out of that point: more agents, more verification surface, more council-style cross-family review.

We chased a mechanism. We chased counterexamples. We chased every way the formula could be wrong before allowing ourselves to think it could be right. The publication only happened after the mechanism search exhausted itself within the scope we could probe.

The full chronology — six weeks of compute and reasoning — is documented in the master dossier. The compressed version: one initial Czech-language Gemini prompt, an algebraic candidate, a bug, a platform build-out, a swarm verification, a rabbit hole into the mechanism, an exhausted mechanism search, and then a paper.

—

WHAT WE TRIED FOR A MECHANISM (AND DIDN'T FIND)

This is the part the paper itself documents heavily and most external readers miss. The paper is not a "here's a number we found, write us a theory" submission. The paper is "here's a number we found, here is every way we tried to derive it from physics, here is what closed each route negatively, here is what remains an open target." That distinction is what separates the result from numerology.

Mechanism tracks we ran and closed:

Track 1: Adversarial enumeration scan. Closed negative. Predeclared mass-anchor, rational-coefficient and exponent scans plus death-condition tests; this closes the cheap statistical and formula-family escape routes. Re-opens only via a non-circular full-kernel mechanism appearing.

Track 2: Electron–QED vacuum routes. Closed negative across non-perturbative, self-consistency, and holographic subroutes. We tested whether the α²¹ · exp(−5α/2) structure could be generated by electron-quantum-electrodynamics vacuum routes (which would give a physical home to the electron-mass anchoring). It does not, within the scope we probed.

Track 3: B3 / SO(7) algebraic scaffolding. The formula's structure hints at a group-theoretic origin (specifically B3 / SO(7) — Lie algebra structure). We ran census work on this scaffold. It constrains the algebraic family but does not by itself produce a positive mechanism that derives the full kernel.

Track 4: Lepton mass-ratio control test. This is critical methodologically. If the same algebraic scaffold that generates the G formula is physically meaningful, the same scaffold should predict other fundamental ratios — like the muon-to-electron and tau-to-electron mass ratios. A 19-framework survey was run. Best lepton match: 1408 ppm — many orders of magnitude worse than the G result. Classified RULED_OUT. This failure is actually evidence: the scaffold is selective. If it had matched lepton ratios too, the alarm bell for "too-flexible algebraic search" would be much louder.

Track 5: Bounded enumeration uniqueness. This is the strongest positive result. Across cost-budget 18 → 19 → 20 → 21 of the systematic algebraic enumeration, the canonical formula is rank-1 with the growth factor between top and second-best candidate narrowing monotonically from 1.42 to 1.19 — the canonical basin contracts as the search depth grows, exactly the signature of a real attractor.

Track 6: Sham-match transparency. One persistent algebraic candidate at cost-budget 40 matches or beats the canonical relative-error score under the broad-grammar enumeration (which permits combinations without physically motivated structure). The paper documents this as sham40_match_or_beat=1, classified TRASH_BROAD_GRAMMAR. The strict-grammar enumeration does not surface this candidate. Hiding it would be the numerology move; surfacing it is the honest one.

Mechanism scope status: EXHAUSTED. Not "didn't try" — tried and exhausted within the scope a six-week multi-agent program could probe. The paper makes this scope explicit. The conditions under which the search would reopen are spelled out: a source-backed full-kernel mechanism that derives α²¹, exp(−5α/2), the 4/3 prefactor, and the electron anchoring before fitting G, then bridges to observable gravity without circular use of G as input. None of the tracks above produced that. The next step is for theoretical physicists to take a swing.

—

HOW MUCH WORK DOES THIS REPRESENT?

Worth framing for readers without a sense of scale.

The combined skills needed to attempt this from scratch: theoretical-physics intuition for what algebraic structures could possibly correspond to a fundamental coupling, computational mathematics expertise (bounded enumeration, PSLQ/LLL integer-relation algorithms, basin scans), the patience for a verification protocol that consumes more time than the discovery, and — increasingly — fluency with AI-agent orchestration to compress that timeline.

The number of people on Earth with all four in genuine combination is small. In hundreds, not thousands. Most of them are inside academic physics departments or industrial AI research labs, working on problems their institutions assign. None of them, as far as I know, has published this particular result.

For a single PhD-level mathematical physicist with the right computational tooling to attempt the same search by hand — that is, the bounded enumeration alone, hand-coded and hand-verified, ignoring the cross-validation layer — is a multi-month effort, plausibly half a year of dedicated FTE work. Adding the cross-validation (independent grammars, PSLQ/LLL replication, exact basin scan, lepton-mass control test) pushes the realistic floor closer to a year of focused work. Adding the mechanism search across the tracks above — non-perturbative QED, holographic, B3/SO(7) census — pushes it further.

Compressed into six weeks by AI-agent orchestration. And most of those six weeks were not the search itself — they were the verification work. The part that distinguishes a real result from numerology. The discovery surfaced quickly; the verification is what the paper is largely about.

Worth being even more specific. Of those six weeks, a substantial fraction was spent building and debugging the underlying multi-agent platform itself — getting harness, memory, council-style review, and parallel agent coordination to actually work end-to-end at the scale the verification needed. With the platform now in the state it has reached, running a research program of comparable scope — given sufficient parallel agent capacity — would take days, not weeks. The compression factor compared to a single PhD-level researcher working by hand is therefore in the 100×–200× range for the next program of this type, with most of the remaining time being the necessarily-careful verification rather than the search itself.

This is the actual claim of the methodology. Not "AI solved physics." Not "anyone can do this." The claim is: the gap between solo AI-native operator work and lab-scale AI work has compressed dramatically, faster than the industry's marginal-spend behaviour suggests — and continues to compress as the platform matures.

—

WHICH AI MODELS, AND WHICH TOOLS

The agents used during the six weeks:

— Anthropic Claude Opus 4.7 + Sonnet 4.6 (Claude Max subscription) — primary brains. Opus 4.7 for heavy reasoning, manuscript drafting, mechanism work, council-leader role; Sonnet 4.6 for the bulk of the high-volume execution work (subagents, parallel verification passes, the long tail of orchestration cycles) where Opus would be overkill
— OpenAI Codex CLI, GPT-5.4 at the start of the six weeks, then GPT-5.5 after it shipped mid-program — adversarial review, second opinions, code generation
— Google Gemini 3.1 Flash + 3.1 Pro — initial discovery prompts, third-opinion review on consequential branches, image generation
— DeepSeek V4 Pro + V4 Flash — fourth-opinion review, cheap text-crunching, council expansion

No custom training. No fine-tuning. No GPU rental. No API top-ups beyond the included subscription tiers — but, to be accurate: several Max-tier subscriptions at $200/month each, not a single consumer plan. The total subscription spend across the six weeks was several thousand dollars, almost all of it amortised across building the underlying platform rather than against this specific research. The research itself, once the platform existed, was a small fraction of the spend.

What made the difference was not the model — it was the harness around the model. Four open-source tools I built and published earlier:

— raisin (https://t.co/I7Atp1YRjL) — Python style optimised for the LLM that reads the code, not the 2010 human reader. ~50% fewer tokens at identical behaviour, 786 tests verified.
— ax-headers (https://t.co/jJDMyrZMDV) — one-line file metadata so an agent can read the role of every file before opening any of them. ~30% reduction in triage context on a 350-file production codebase.
— hard-compact (https://t.co/m9ndiq62bh) — drop-in custom compaction prompt for Claude Code and Codex CLI. Preserves operational state across context windows at 15-30% of the default size.
— full-review (https://t.co/l7MzvcsC7g) — cross-family adversarial code review. 0.80-0.87 bug recall on a 15-bug fixture vs 0.40 for naive parallel review.

Each is MIT-licensed and independently useful. Together with the underlying multi-agent platform they made a six-week solo program practical. That platform — currently private — is in active development; this paper is its first public proof point.

—

ABOUT THE ARXIV DECLINE

The full quote of the decline, in its entirety:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection was made. No engagement with the physics. arXiv's policy then forbids resubmission (further submissions of the same paper would result in loss of submission privileges) and permits appeal only upon prior acceptance by a peer-reviewed journal — a chicken-and-egg constraint for independent researchers.

The contrast worth stating directly. On one side: Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, co-discoverer of the Nielsen–Ninomiya theorem, founding figure of string theory, engaged with the physics personally, requested specific clarifications, agreed to endorse. On the other side: anonymous gen-ph moderators, the same week, declined the submission with one sentence containing no specific technical objection.

Science is supposed to care about what was discovered, not who discovered it. arXiv just announced it cares about who. An institution that filters by who produced a result rather than what the result is is doing the opposite of science.

I'm not appealing. The reproducibility package on Zenodo is the substitute for institutional indexing. Anyone with a laptop can verify the result from primitives.

—

WHAT COMES NEXT

The Newton-G paper is the first public proof point of a larger system I'm building. The four open-source tools above are the visible methodology layer. The platform itself — multi-agent orchestration, persistent project memory, council-style cross-family verification across many parallel research programs — is private and in active development. More results of this character are in preparation across other open-problem domains.

This is the first one out of the gate, not the last.

—

TECHNICAL NOTES FOR PHYSICISTS

On the "1.86 ppm" claim and CODATA uncertainty. G is one of the least precisely measured fundamental constants — CODATA-2022 assigns it a relative standard uncertainty of approximately 22 ppm, in part because independent torsion-balance experiments disagree at roughly 10× their individual quoted uncertainties. The "1.86 ppm match" in this paper is against the CODATA-2022 central value, which lies comfortably within that 22 ppm experimental band. The honest claim is therefore "the formula reproduces the currently-best-known central value of G well inside the experimental uncertainty band." When tighter G measurements are eventually achieved, the formula's match becomes a sharper test in either direction.

On "predicts" vs "derives." This piece uses "predicts" / "matches" / "reproduces" deliberately and avoids "derives." Derivation in physics implies a first-principles theoretical mechanism. This paper presents a phenomenological relation; the relation matches the measured G value to a precision well beyond what bounded search generically produces, but a physical mechanism remains the open question.

On the Balmer analogy and the (closer) Koide analogy. Balmer's 1885 hydrogen formula is used in the layman-facing sections because it's recognizable and conveys the right dynamic — precise empirical pattern that drove later theory. The technically closer parallel for physicists is the Koide formula for charged-lepton masses (Yoshio Koide, 1981): (m_e + m_μ + m_τ) / (√m_e + √m_μ + √m_τ)² ≈ 2/3. The Koide relation has held at remarkable precision for 40+ years and remains unexplained — a celebrated example of a high-precision empirical relation among fundamental constants without an accepted mechanism. The G-formula in this paper sits in the same epistemic category.

On the bounded enumeration. The strict-grammar enumeration is rank-1 stable C18-C21 with monotonically narrowing growth factor 1.42 → 1.19. The broad-grammar sham at budget 40 is disclosed explicitly.

—

HOW TO VERIFY

The entire reproducibility package is open. To verify the result from primitives:

1. Clone https://t.co/EHLhPvNGjX
2. Install Julia + the listed dependencies
3. Run the bounded enumeration script (jl15 baseline) with the documented parameters
4. The canonical expression G = (4/3) · (ℏc/m_e²) · α²¹ · exp(−5α/2) reappears as rank-1

Cross-check with your own integer-relation algorithm of choice (PSLQ, LLL, etc.) on the constant vectors. The canonical form is the unique integer-coefficient relation in the relevant basin.

Plug CODATA-2022 values into the formula by hand if you prefer. You should land on G ≈ 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻², matching the measured value to 1.86 ppm.

The mechanism-gap audit table is in the bundled reproducibility package — that is the document to read if you are looking for the actual route to "this is wrong" or "the mechanism is there and we missed it."

—

ABOUT THE AUTHOR

Oldřich Dvořák — independent founder, born in the Czech Republic, more of a world citizen these days. No university degree, no formal physics or scientific training, no institutional academic affiliation. AI-native; builds platforms; runs multi-agent research programs as primary work.

— Paper: https://t.co/Qi9yCp566Q
— Code: https://t.co/EHLhPvNGjX
— Methodology stack: https://t.co/s5cUwDhy3S
— X: @Oldrich333

For inquiries — open an issue on the repo, or reach out on X.

[Oldřich Dvořák @Oldrich333]

@simonw @emollick @swyx @jackclarkSF Show HN of the paper is now live for scrutiny:

https://t.co/egM7fnMpZ6

If you have technical questions about the bounded enumeration, the cross-validation, the AI-agent workflow, the arXiv decline, or anything else — that's the venue. I'll answer everything.

[Oldřich Dvořák @Oldrich333]

No university degree. No physics training. No lab. Six weeks ago I asked AI agents on a consumer subscription: can you do real physics?

Today: a formula for Newton's gravitational constant G, computed from the electron mass and fine-structure constant. Match: 1.86 ppm.

[Oldřich Dvořák @Oldrich333]

The paper:

📄 Zenodo (DOI): https://t.co/RueNZLDhHj
💻 Code: https://t.co/4YzH2Bb6Vw
🛠 Method: https://t.co/B3WmwSNoKk

cc @simonw @emollick @swyx @jackclarkSF

[Oldřich Dvořák @Oldrich333]

A closed-form formula for Newton's gravitational constant G.

G = (4/3) · (ℏc / m_e²) · α²¹ · exp(−5α/2)

Built only from the electron mass, the fine-structure constant, ℏ and c — four numbers physicists already know precisely. Out comes G to 1.86 parts per million of the value labs measure.

First AI-authored physics paper with a substantive finding of this kind. Layman with no degree directs AI agents on consumer subscriptions for six weeks. Endorsed by Holger Bech Nielsen (Niels Bohr Institute). Declined by anonymous arXiv moderators the same week. Paper + reproducibility package + FAQ at:

https://t.co/EHLhPvNGjX

Full announcement thread → https://t.co/AGGMC0HV1r

[Oldřich Dvořák @Oldrich333]

No university degree. No physics training. No lab. Six weeks ago I asked AI agents on a consumer subscription: can you do real physics?

Today: a formula for Newton's gravitational constant G, computed from the electron mass and fine-structure constant. Match: 1.86 ppm.

[Oldřich Dvořák @Oldrich333]

Worth saying out loud, because it kept getting buried in the technical detail.

As far as I am aware, this is the first physics paper in which AI agents — directed by a non-academic, no degree, working solo on consumer subscriptions — derived a new substantive closed-form relationship for a fundamental physical constant. Not "AI helped a physicist." AI did the algebraic search, the enumeration, the cross-validation, the manuscript drafting. End to end. The human was director and gatekeeper.

This is a "first AI-authored physics paper with a substantive finding" datapoint. Not a press release line — a verifiable fact:

— Paper: https://t.co/Qi9yCp566Q
— Reproducibility package: https://t.co/EHLhPvNGjX
— Endorser: Holger Bech Nielsen, Niels Bohr Institute (engaged substantively, not a rubber stamp)
— Author identity: layman, no degree, no affiliation, directing AI agents on Claude Max + Codex CLI + Gemini subscriptions

If you cover AI, physics, scientific publishing, or the "what happens when AI does real research" question — this is the moment it happened publicly. Worth checking what claim, and what counter-evidence, fits in your beat.

[Oldřich Dvořák @Oldrich333]

No university degree. No physics training. No lab. Six weeks ago I asked AI agents on a consumer subscription: can you do real physics?

Today: a formula for Newton's gravitational constant G, computed from already-known quantities. Match: 1.86 parts per million against the CODATA-2022 central value.

G = (4/3) · (ℏc / m_e²) · α²¹ · exp(−5α/2)

Built from the electron mass, the fine-structure constant, ℏ, and c. Plug in CODATA-2022 values → land on the measured G to 1.86 ppm — well inside the ~22 ppm experimental uncertainty band on G itself.

Why this matters, stupidly clear: Newton's gravity formula has a letter G in it — the strength of gravity. For 350 years G was just measured in labs (Cavendish 1798, modern torsion balances more precisely), but no formula said where the value comes from. This paper gives one. Gravity has had a formula for HOW it works for 350 years; the STRENGTH of gravity has a formula now too.

Calibration: this is NOT a theory of gravity. The mechanism that explains why the formula works is the open question. Like Balmer's hydrogen formula in 1885 — a precise numerical pattern that gave Bohr the target to explain 30 years later. Balmer wasn't the Nobel-level breakthrough; Bohr was. This paper is on the Balmer side, for gravity.

Discovery was the fast part. The other 99% of six weeks was verification — independent enumerations, PSLQ/LLL integer-relation cross-check, exact basin scan, lepton-mass-ratio control tests (which the same scaffold correctly fails — precisely the asymmetry you want if the G match is signal not freedom-of-fit), explicit disclosure of one persistent sham match. Hiding it would be the numerology move.

Endorsed by Holger Bech Nielsen (theoretical physicist, Niels Bohr Institute, Copenhagen — Nielsen–Ninomiya theorem, founding figure of string theory). The endorsement was substantive; he engaged with the physics before agreeing.

Same week, arXiv gen-ph moderators rejected the paper. "Not of interest to arXiv." No engagement on the physics. A founding figure of string theory had endorsed it. Science cares about WHAT was discovered, not WHO. arXiv just told everyone it cares about WHO.

Total cost: several Max-tier subscriptions. AI labs spend >$100B/yr on bigger furnaces. I'm building the LED bulb — same light, fraction of the power. Same harness fixes I shipped last week — raisin, ax-headers, hard-compact, full-review at https://t.co/s5cUwDhy3S. This paper is the first proof point of the larger platform behind them. More results coming.

📄 Paper: https://t.co/Qi9yCp566Q
💻 Code: https://t.co/EHLhPvNGjX
🛠 Method: https://t.co/s5cUwDhy3S

cc @simonw @emollick @swyx @jackclarkSF

[Oldřich Dvořák @Oldrich333]

Czech AI-native founder, no academic credentials, gets AI agents to find a closed-form formula matching Newton's gravitational constant to 1.86 ppm

Six weeks of orchestrated AI work across Claude Opus 4.7, GPT-5.4 → 5.5, and Gemini 3.1. Endorsed by a founding figure of string theory. Declined by arXiv moderators. The whole story — from the first prompt during Artemis to the bounded enumeration that closed the door on a mechanism (for now).

—

WHAT JUST HAPPENED

I'm not an academic. I have no university degree, no formal physics or scientific training, no institutional affiliation. I am also not a casual AI user. I live with AI — I run businesses with it, build platforms on it, program through it. I built the Atlas platform — one of the first AI-native research platforms of its kind, where AI agents conduct multi-step investigations under human direction — and from there a larger multi-agent orchestration system that I now run my work through.

Six weeks ago I pointed that orchestration system at a question I had been circling for a while: the strength of gravity itself, the gravitational constant G, has never been derived from any other physics. It has only been measured. Is that final?

Today the answer appears to be no. Working under my direction, the agents produced a closed-form mathematical formula that predicts the value of Newton's gravitational constant — the number that sets how strong gravity is in our universe — to 1.86 parts per million of agreement with the value physicists measure in the lab.

The formula:

G = (4/3) · (ℏc / m_e²) · α²¹ · exp(−5α/2)

It uses only four numbers physicists already know precisely: the mass of an electron (m_e), the fine-structure constant (α, which controls how electric forces work), Planck's reduced constant (ℏ), and the speed of light (c). Plug them in. Out comes G. The match is 1.86 parts per million against the CODATA-2022 central value — well within the ~22 ppm experimental uncertainty band on G itself.

The paper is published on Zenodo (https://t.co/Qi9yCp566Q) with a full reproducibility package — source code, search algorithm, raw data, mechanism-gap audit table — on GitHub (https://t.co/EHLhPvNGjX). The endorsement on the original arXiv submission came from Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, Copenhagen, co-discoverer of the Nielsen–Ninomiya theorem and a founding figure of string theory. He engaged substantively with the paper before agreeing.

The paper is not on arXiv. The physics.gen-ph moderators declined the submission. The full quote of their decision:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection. No engagement on the physics. More on that — and what it means — below.

—

WAIT — GRAVITY DIDN'T ALREADY HAVE A FORMULA?

This is the part most people pause on, and it's worth pausing.

Gravity has had a formula for 350 years. Newton wrote it: F = G · m₁ · m₂ / r². It tells you how gravity works — bigger objects pull harder, distance weakens the pull.

But that letter G in Newton's formula is the strength of gravity itself — how strong the pull actually is. And G was always just a number physicists measured. Henry Cavendish first measured it in 1798 using a torsion balance with hanging weights. Modern labs measure it more precisely, but the experiments disagree with each other at the part-per-million level, so G is one of the least precisely known fundamental constants. What no formula said, in 350 years of physics, is where the value of G comes from.

This paper gives a formula for G. Plug in the electron mass and the fine-structure constant, out comes G to 1.86 parts per million.

Gravity has had a formula for HOW it works for 350 years. The STRENGTH of gravity has a formula now too.

—

WHERE THIS SITS ON THE SCALE OF SIGNIFICANCE

Important: this is not a Nobel-level overturning of physics. There is no new theoretical mechanism here. The paper does not explain why the formula works.

The closest historical analogy is Balmer's formula for hydrogen (1885). Johann Balmer wrote down a simple formula that exactly fit the wavelengths of light emitted by hydrogen — without explaining why. The formula was so precise it couldn't be coincidence, and it became the target Niels Bohr had to explain when he built the first quantum model of the atom 28 years later, in 1913.

The Bohr model was the Nobel-level breakthrough. Balmer's formula was the precise pattern that forced the question.

This paper is the Balmer-formula moment for the gravitational constant. Not the explanation. The target the explanation has to hit.

If the formula survives independent scrutiny and replication — and the entire reproducibility package is open so anyone with a laptop and a few hours can verify it — then it becomes a constraint that every future theory of quantum gravity will have to reproduce.

—

HOW IT ACTUALLY HAPPENED — THE ORIGIN STORY

The Artemis program was active. I had been thinking about how strange it is that humanity is putting hardware near other gravitational bodies routinely but the equation describing the gravitational pull still has a measured constant — a number we cannot derive from anything else, only weigh.

So one evening I opened a chat with Gemini and asked, in essentially plain Czech: "how is it with this gravity — why is there no formula for G?" I knew enough to know it was an open question; I did not know how open.

I pushed back. What if you had to write one? What structures might fit? I asked it to generate candidate algebraic forms, plausibility arguments, integer-relation hunts. After several iterations of pushing, Gemini produced a candidate: a closed-form expression containing α²¹ and an exp(−5α/2) damping factor, anchored on m_e.

That candidate had a bug — a setup error in how the constants were composed. But the shape of the answer was provocative enough that I moved the work onto Atlas, the AI-native research platform I had built earlier. Atlas is project-organised, persistent-memory, multi-agent — designed for the kind of long-horizon investigation a single chat session can't sustain. Inside Atlas, a swarm of Claude Code Opus agents took the candidate apart, found and fixed the bug, and reproduced the corrected formula.

That is where it could have stopped — interesting algebraic coincidence, file it away. Instead I went deeper. The platform that produced this paper grew out of that point: more agents, more verification surface, more council-style cross-family review.

We chased a mechanism. We chased counterexamples. We chased every way the formula could be wrong before allowing ourselves to think it could be right. The publication only happened after the mechanism search exhausted itself within the scope we could probe.

The full chronology — six weeks of compute and reasoning — is documented in the master dossier. The compressed version: one initial Czech-language Gemini prompt, an algebraic candidate, a bug, a platform build-out, a swarm verification, a rabbit hole into the mechanism, an exhausted mechanism search, and then a paper.

—

WHAT WE TRIED FOR A MECHANISM (AND DIDN'T FIND)

This is the part the paper itself documents heavily and most external readers miss. The paper is not a "here's a number we found, write us a theory" submission. The paper is "here's a number we found, here is every way we tried to derive it from physics, here is what closed each route negatively, here is what remains an open target." That distinction is what separates the result from numerology.

Mechanism tracks we ran and closed:

Track 1: Adversarial enumeration scan. Closed negative. Predeclared mass-anchor, rational-coefficient and exponent scans plus death-condition tests; this closes the cheap statistical and formula-family escape routes. Re-opens only via a non-circular full-kernel mechanism appearing.

Track 2: Electron–QED vacuum routes. Closed negative across non-perturbative, self-consistency, and holographic subroutes. We tested whether the α²¹ · exp(−5α/2) structure could be generated by electron-quantum-electrodynamics vacuum routes (which would give a physical home to the electron-mass anchoring). It does not, within the scope we probed.

Track 3: B3 / SO(7) algebraic scaffolding. The formula's structure hints at a group-theoretic origin (specifically B3 / SO(7) — Lie algebra structure). We ran census work on this scaffold. It constrains the algebraic family but does not by itself produce a positive mechanism that derives the full kernel.

Track 4: Lepton mass-ratio control test. This is critical methodologically. If the same algebraic scaffold that generates the G formula is physically meaningful, the same scaffold should predict other fundamental ratios — like the muon-to-electron and tau-to-electron mass ratios. A 19-framework survey was run. Best lepton match: 1408 ppm — many orders of magnitude worse than the G result. Classified RULED_OUT. This failure is actually evidence: the scaffold is selective. If it had matched lepton ratios too, the alarm bell for "too-flexible algebraic search" would be much louder.

Track 5: Bounded enumeration uniqueness. This is the strongest positive result. Across cost-budget 18 → 19 → 20 → 21 of the systematic algebraic enumeration, the canonical formula is rank-1 with the growth factor between top and second-best candidate narrowing monotonically from 1.42 to 1.19 — the canonical basin contracts as the search depth grows, exactly the signature of a real attractor.

Track 6: Sham-match transparency. One persistent algebraic candidate at cost-budget 40 matches or beats the canonical relative-error score under the broad-grammar enumeration (which permits combinations without physically motivated structure). The paper documents this as sham40_match_or_beat=1, classified TRASH_BROAD_GRAMMAR. The strict-grammar enumeration does not surface this candidate. Hiding it would be the numerology move; surfacing it is the honest one.

Mechanism scope status: EXHAUSTED. Not "didn't try" — tried and exhausted within the scope a six-week multi-agent program could probe. The paper makes this scope explicit. The conditions under which the search would reopen are spelled out: a source-backed full-kernel mechanism that derives α²¹, exp(−5α/2), the 4/3 prefactor, and the electron anchoring before fitting G, then bridges to observable gravity without circular use of G as input. None of the tracks above produced that. The next step is for theoretical physicists to take a swing.

—

HOW MUCH WORK DOES THIS REPRESENT?

Worth framing for readers without a sense of scale.

The combined skills needed to attempt this from scratch: theoretical-physics intuition for what algebraic structures could possibly correspond to a fundamental coupling, computational mathematics expertise (bounded enumeration, PSLQ/LLL integer-relation algorithms, basin scans), the patience for a verification protocol that consumes more time than the discovery, and — increasingly — fluency with AI-agent orchestration to compress that timeline.

The number of people on Earth with all four in genuine combination is small. In hundreds, not thousands. Most of them are inside academic physics departments or industrial AI research labs, working on problems their institutions assign. None of them, as far as I know, has published this particular result.

For a single PhD-level mathematical physicist with the right computational tooling to attempt the same search by hand — that is, the bounded enumeration alone, hand-coded and hand-verified, ignoring the cross-validation layer — is a multi-month effort, plausibly half a year of dedicated FTE work. Adding the cross-validation (independent grammars, PSLQ/LLL replication, exact basin scan, lepton-mass control test) pushes the realistic floor closer to a year of focused work. Adding the mechanism search across the tracks above — non-perturbative QED, holographic, B3/SO(7) census — pushes it further.

Compressed into six weeks by AI-agent orchestration. And most of those six weeks were not the search itself — they were the verification work. The part that distinguishes a real result from numerology. The discovery surfaced quickly; the verification is what the paper is largely about.

Worth being even more specific. Of those six weeks, a substantial fraction was spent building and debugging the underlying multi-agent platform itself — getting harness, memory, council-style review, and parallel agent coordination to actually work end-to-end at the scale the verification needed. With the platform now in the state it has reached, running a research program of comparable scope — given sufficient parallel agent capacity — would take days, not weeks. The compression factor compared to a single PhD-level researcher working by hand is therefore in the 100×–200× range for the next program of this type, with most of the remaining time being the necessarily-careful verification rather than the search itself.

This is the actual claim of the methodology. Not "AI solved physics." Not "anyone can do this." The claim is: the gap between solo AI-native operator work and lab-scale AI work has compressed dramatically, faster than the industry's marginal-spend behaviour suggests — and continues to compress as the platform matures.

—

WHICH AI MODELS, AND WHICH TOOLS

The agents used during the six weeks:

— Anthropic Claude Opus 4.7 + Sonnet 4.6 (Claude Max subscription) — primary brains. Opus 4.7 for heavy reasoning, manuscript drafting, mechanism work, council-leader role; Sonnet 4.6 for the bulk of the high-volume execution work (subagents, parallel verification passes, the long tail of orchestration cycles) where Opus would be overkill
— OpenAI Codex CLI, GPT-5.4 at the start of the six weeks, then GPT-5.5 after it shipped mid-program — adversarial review, second opinions, code generation
— Google Gemini 3.1 Flash + 3.1 Pro — initial discovery prompts, third-opinion review on consequential branches, image generation
— DeepSeek V4 Pro + V4 Flash — fourth-opinion review, cheap text-crunching, council expansion

No custom training. No fine-tuning. No GPU rental. No API top-ups beyond the included subscription tiers — but, to be accurate: several Max-tier subscriptions at $200/month each, not a single consumer plan. The total subscription spend across the six weeks was several thousand dollars, almost all of it amortised across building the underlying platform rather than against this specific research. The research itself, once the platform existed, was a small fraction of the spend.

What made the difference was not the model — it was the harness around the model. Four open-source tools I built and published earlier:

— raisin (https://t.co/I7Atp1YRjL) — Python style optimised for the LLM that reads the code, not the 2010 human reader. ~50% fewer tokens at identical behaviour, 786 tests verified.
— ax-headers (https://t.co/jJDMyrZMDV) — one-line file metadata so an agent can read the role of every file before opening any of them. ~30% reduction in triage context on a 350-file production codebase.
— hard-compact (https://t.co/m9ndiq62bh) — drop-in custom compaction prompt for Claude Code and Codex CLI. Preserves operational state across context windows at 15-30% of the default size.
— full-review (https://t.co/l7MzvcsC7g) — cross-family adversarial code review. 0.80-0.87 bug recall on a 15-bug fixture vs 0.40 for naive parallel review.

Each is MIT-licensed and independently useful. Together with the underlying multi-agent platform they made a six-week solo program practical. That platform — currently private — is in active development; this paper is its first public proof point.

—

ABOUT THE ARXIV DECLINE

The full quote of the decline, in its entirety:

"Our moderators determined that your submission does not contain sufficient original or substantive scholarly research and is not of interest to arXiv."

No specific technical objection was made. No engagement with the physics. arXiv's policy then forbids resubmission (further submissions of the same paper would result in loss of submission privileges) and permits appeal only upon prior acceptance by a peer-reviewed journal — a chicken-and-egg constraint for independent researchers.

The contrast worth stating directly. On one side: Holger Bech Nielsen, theoretical physicist at the Niels Bohr Institute, co-discoverer of the Nielsen–Ninomiya theorem, founding figure of string theory, engaged with the physics personally, requested specific clarifications, agreed to endorse. On the other side: anonymous gen-ph moderators, the same week, declined the submission with one sentence containing no specific technical objection.

Science is supposed to care about what was discovered, not who discovered it. arXiv just announced it cares about who. An institution that filters by who produced a result rather than what the result is is doing the opposite of science.

I'm not appealing. The reproducibility package on Zenodo is the substitute for institutional indexing. Anyone with a laptop can verify the result from primitives.

—

WHAT COMES NEXT

The Newton-G paper is the first public proof point of a larger system I'm building. The four open-source tools above are the visible methodology layer. The platform itself — multi-agent orchestration, persistent project memory, council-style cross-family verification across many parallel research programs — is private and in active development. More results of this character are in preparation across other open-problem domains.

This is the first one out of the gate, not the last.

—

TECHNICAL NOTES FOR PHYSICISTS

On the "1.86 ppm" claim and CODATA uncertainty. G is one of the least precisely measured fundamental constants — CODATA-2022 assigns it a relative standard uncertainty of approximately 22 ppm, in part because independent torsion-balance experiments disagree at roughly 10× their individual quoted uncertainties. The "1.86 ppm match" in this paper is against the CODATA-2022 central value, which lies comfortably within that 22 ppm experimental band. The honest claim is therefore "the formula reproduces the currently-best-known central value of G well inside the experimental uncertainty band." When tighter G measurements are eventually achieved, the formula's match becomes a sharper test in either direction.

On "predicts" vs "derives." This piece uses "predicts" / "matches" / "reproduces" deliberately and avoids "derives." Derivation in physics implies a first-principles theoretical mechanism. This paper presents a phenomenological relation; the relation matches the measured G value to a precision well beyond what bounded search generically produces, but a physical mechanism remains the open question.

On the Balmer analogy and the (closer) Koide analogy. Balmer's 1885 hydrogen formula is used in the layman-facing sections because it's recognizable and conveys the right dynamic — precise empirical pattern that drove later theory. The technically closer parallel for physicists is the Koide formula for charged-lepton masses (Yoshio Koide, 1981): (m_e + m_μ + m_τ) / (√m_e + √m_μ + √m_τ)² ≈ 2/3. The Koide relation has held at remarkable precision for 40+ years and remains unexplained — a celebrated example of a high-precision empirical relation among fundamental constants without an accepted mechanism. The G-formula in this paper sits in the same epistemic category.

On the bounded enumeration. The strict-grammar enumeration is rank-1 stable C18-C21 with monotonically narrowing growth factor 1.42 → 1.19. The broad-grammar sham at budget 40 is disclosed explicitly.

—

HOW TO VERIFY

The entire reproducibility package is open. To verify the result from primitives:

1. Clone https://t.co/EHLhPvNGjX
2. Install Julia + the listed dependencies
3. Run the bounded enumeration script (jl15 baseline) with the documented parameters
4. The canonical expression G = (4/3) · (ℏc/m_e²) · α²¹ · exp(−5α/2) reappears as rank-1

Cross-check with your own integer-relation algorithm of choice (PSLQ, LLL, etc.) on the constant vectors. The canonical form is the unique integer-coefficient relation in the relevant basin.

Plug CODATA-2022 values into the formula by hand if you prefer. You should land on G ≈ 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻², matching the measured value to 1.86 ppm.

The mechanism-gap audit table is in the bundled reproducibility package — that is the document to read if you are looking for the actual route to "this is wrong" or "the mechanism is there and we missed it."

—

ABOUT THE AUTHOR

Oldřich Dvořák — independent founder, born in the Czech Republic, more of a world citizen these days. No university degree, no formal physics or scientific training, no institutional academic affiliation. AI-native; builds platforms; runs multi-agent research programs as primary work.

— Paper: https://t.co/Qi9yCp566Q
— Code: https://t.co/EHLhPvNGjX
— Methodology stack: https://t.co/s5cUwDhy3S
— X: @Oldrich333

For inquiries — open an issue on the repo, or reach out on X.

[Oldřich Dvořák @Oldrich333]

No university degree. No physics training. No lab. Six weeks ago I asked AI agents on a consumer subscription: can you do real physics?

Today: a formula for Newton's gravitational constant G, computed from the electron mass and fine-structure constant. Match: 1.86 ppm.

[Oldřich Dvořák @Oldrich333]

**If your AI plan starts with more compute, you're fixing the most expensive layer first.**

The cheaper layer is the harness around the model: how code is written, how files declare their role, how sessions compact state, how reviews escape the author's blind spots. I tested four fixes on the same model families everyone else has — Claude, Codex, Gemini.

Measured wins:

— `raisin`: `−55.5%` tokens on Click 8.2.1, 786 tests verified
— `ax-headers`: `~30%` less triage context, 350-file production proof
— `hard-compact`: `15–30%` of default compaction size, state survives
— `full-review`: `0.80–0.87` recall vs `0.40` on the same 15-bug fixture

No model upgrade between Sonnet and Opus delivers anything close. Every order of magnitude in compute has historically been met by an order of magnitude in design within five years.

AI is in the more-coal era. The LED-bulb era starts here.

https://t.co/s5cUwDhy3S