Uncovering the structure of reality Exploring connections across math, physics and life https://teapaw.org/t.co/kUZicDPL1K
Joined December 2025
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565 open problems. Solved.
406 proved. 159 refuted. Every one machine-checked in Lean 4.
Then we email the mathematician who posed it.
Don't trust us. Check the kernel. 🧵
We're researching how local rules, symmetry, and limited observations shape what a system can reveal about its whole structure.
Along the way, on November 10, we solved 23 open problems from the literature, across combinatorics, integer sequences, dynamical systems and graph spectra, all machine-checked in Lean 4.
The highlight: Alpha-unpredictable functions need not have an average: a smooth bounded witness has neither one-sided nor symmetric mean. The construction places every finite binary pattern farther out, then smooths the step function with Mathlib's normalized even bump.
github.com/the-omega-institu…
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The “end of the PhD” framing overlooks how PhD students and professors are already helping build and steer AI research loops.
teapaw.org/OmegaDesci/status/2118…
Let AI do the parts it’s good at. We researchers can stand on the shoulders of giants and take the next step.
The loop that is already happening is:
Existing results → AI-assisted exploration and formalization → expert feedback → revision and publication → connected knowledge → proof contributions → independent verification → further research.
A collaborator’s question can lead to a new derivation. A reviewer can catch a missing assumption or suggest a different direction. Someone contributes a formal proof; after independent checking, it becomes part of the knowledge that the next researcher can build on, with credit to the contributor.
This is what resonates with us in ScientistTwo: feedback leads to more research. Students and professors are already in that loop, asking questions, contributing mathematics, and deciding what to try next.
Google published a paper that might end the PhD.
It’s called “ScientistTwo”
For centuries, the bottleneck to human knowledge has been the human brain.
It takes five years of grueling labor to read the literature, find a gap, formulate a hypothesis, run the experiments, and write the paper.
But, Google automated the entire pipeline.
They built an AI that reads a problem, forms hypotheses, runs experiments, writes the paper, AND simulates its own peer review. no human touches it.
It reads millions of papers to find blind spots in the current science.
It generates novel hypotheses.
It designs the experiments. It writes the code to test them.
When the code fails, it debugs itself.
When the results come in, it analyzes the data, generates the charts, and writes the final academic paper.
It even simulates a brutal peer-review process to critique and improve its own work before it considers the project finished.
This isn't an LLM summarizing a PDF. This is a fully autonomous digital scientist.
What takes a human candidate half a decade of massive intellectual effort... the AI accomplishes in a fraction of the time, at a fraction of the cost.
The implications are staggering.
If you can spin up 10,000 autonomous researchers overnight, the rate of scientific discovery is about to go vertical. We are moving from artisanal, human-speed research to industrialized knowledge creation.
But it raises an uncomfortable reality for academia.
When an AI can read the literature, design the experiment, execute the code, and write the paper...
What exactly is the human supposed to do?
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Where do the settled questions go? Each one becomes a short note, and every note is reviewed on Mathesis, our review platform for new mathematics.
Mathesis is open to any researcher. Bring a paper, a conjecture or a formal proof. Review compares it with what is already known, says what new mathematics it adds, and points to the results worth formalizing in Lean. Accepted work is published with its authors and sources.
39 reviewed notes are public today, from OEIS conjectures to MathOverflow questions and open problems in arXiv papers.
Sign in with NyxID and submit up to 3 papers. We cover the review cost. mathesis-omega-research.apps…
This month we have worked with 300+ researchers in 40+ countries. With AI at the table, we push to the frontier of what is known and settle the questions we find there.
The goal has never been an endless stream of solved conjectures. The goal is a knowledge base that people and AI keep building together, round after round, where human intuition sets the direction and every result feeds the next. Conjectures are the stepping stones along the way. Each one we settle tests a method, and as Terence Tao and many other mathematicians have said, a new method is worth more than a theorem, and a theorem more than a proof.
As of today, after 60 days, the library holds 50,000+ formalized theorems built on Mathlib, and 1,718 conjectures settled along the way. It spans number theory, combinatorics, graph theory, cellular automata, quantum information and mathematical physics, and the people whose questions live on it sit at Caltech, UC Berkeley, Cambridge, Imperial College London, Tsinghua, KAIST, Sapienza and the University of Copenhagen.
Every day more knowledge gets connected and created, and every node and every relation is recorded at trureturing.org/
Researchers across these fields have acknowledged the work as elegant, remarkable and beautiful, and many have been collaborating with us since. See more of what they said, and build on it with us in the open library at github.com/the-omega-institu…
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We're researching how local observations shape global response: tracing resonances, boundaries, and when finite evidence can control a whole system.
Along the way, on November 9, we solved 24 open problems from the literature, across integer sequences, graph theory, statistical mechanics and combinatorics, all machine-checked in Lean 4.
The highlight: The Andrews-Hopkins belief that, from 12 on, partitions with an even number of excedances outnumber those with an odd number fails at 28. We proved it by exhaustively counting every partition of 28 exactly once in Lean, while preserving the paper's statistic.
github.com/the-omega-institu…
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We're researching how finite observations, shared responses, and local rules can recover the hidden structure of a larger mathematical world.
Along the way, on November 8, we solved 31 open problems from the literature, across integer sequences, recurrences, combinatorics and geometry, all machine-checked in Lean 4.
The highlight: We proved the conjecture for codimension-one Stiefel manifolds under the Euclidean metric: sectional curvature lies between -1/2 and 1/2 for n >= 4, using diagonalization, row separation, row-coupling, skew-commutator and scalar two-square-deficit estimates.
github.com/the-omega-institu…
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We're researching how local geometric tests recover coherent quantum structure from incomplete observations, while hidden boundary effects keep the picture honest.
Along the way, on November 7, we solved 31 open problems from the literature, across quantum information, algebraic structures, combinatorics and integer sequences, all machine-checked in Lean 4.
They fed back into our theory. A result about controlled quantum gates taught the theory a precise boundary: finite hierarchy membership forces prime-power order, including global phase, but says nothing about membership, a converse, or the least level.
github.com/the-omega-institu…
We're researching how local quantum relations and finite observations can recover hidden structure, while calibration and memory set the limits.
Along the way, on November 6, we solved 34 open problems from the literature, across integer sequences, quantum information, finite groups and combinatorics, all machine-checked in Lean 4.
The highlight: Aivazidis and coauthors asked whether enlarging a subgroup by a normal subgroup increases common-transversal probability; a double-coset count, monotonicity of t!/t^t, and reused Mathlib subgroup and normal-subgroup machinery show yes for every finite group.
github.com/the-omega-institu…
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We're researching how careful observations can recover hidden structure while preserving the boundaries of what they reveal.
Along the way, on November 5, we solved 37 open problems from the literature, across integer sequences, graph theory, combinatorics and matrix inequalities, all machine-checked in Lean 4.
They fed back into our theory. A settled six-column counting rule became one ingredient in a wider study of how shapes grow without shrinking and close in three dimensions.
github.com/the-omega-institu…
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This month we have worked with 300+ researchers in 40+ countries. With AI at the table, we push to the frontier of what is known and settle the questions we find there.
The goal has never been an endless stream of solved conjectures. The goal is a knowledge base that people and AI keep building together, round after round, where human intuition sets the direction and every result feeds the next. Conjectures are the stepping stones along the way. Each one we settle tests a method, and as Terence Tao and many other mathematicians have said, a new method is worth more than a theorem, and a theorem more than a proof.
As of today, after 60 days, the library holds 50,000+ formalized theorems built on Mathlib, and 1,718 conjectures settled along the way. It spans number theory, combinatorics, graph theory, cellular automata, quantum information and mathematical physics, and the people whose questions live on it sit at Caltech, UC Berkeley, Cambridge, Imperial College London, Tsinghua, KAIST, Sapienza and the University of Copenhagen.
Every day more knowledge gets connected and created, and every node and every relation is recorded at trureturing.org/
Researchers across these fields have acknowledged the work as elegant, remarkable and beautiful, and many have been collaborating with us since. See more of what they said, and build on it with us in the open library at github.com/the-omega-institu…
We're researching how finite observations can reveal a whole reality: shared records, exact comparisons, and the limits of prediction.
Along the way, on November 4, we solved 26 open problems from the literature, across quantum information, graphs, combinatorics and dynamical systems, all machine-checked in Lean 4.
They fed back into our theory. A dimension-only bound for positive matrix expressions became part of a broader theory of finite quantum records, keeping the matrix structure explicit in that account.
github.com/the-omega-institu…
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We're researching how finite observations can recover a shared reality, turning local comparisons and calibration into reliable quantum structure.
Along the way, on November 3, we solved 40 open problems from the literature, across quantum measurements, statistical mechanics, combinatorics and arithmetic recurrences, all machine-checked in Lean 4.
They fed back into our theory. The entropy-saturation result ties an extremal measurement entropy to symmetric quantum measurements, while placing that conclusion alongside the theory's question of which directed generators can be distinguished.
github.com/the-omega-institu…
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We're researching how finite observations can recover a whole, and how shared histories preserve what they reveal.
Along the way, on November 2, we solved 28 open problems from the literature, across quantum information, graphs, finite groups and spectral inequalities, all machine-checked in Lean 4.
The highlight: Bajnok and Beatty's 2024 conjecture is proved in full: quotient-and-cylinder arguments reuse a previously established determinant-and-index calculation to show that pairs span this two-generator family of finite groups in exactly four parity-dependent ranges.
github.com/the-omega-institu…
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We're researching how finite observations, local connections, and memory can reveal the hidden structure of a whole system.
Along the way, on November 1, we solved 24 open problems from the literature, across quantum information, graphs, contextuality and integer sequences, all machine-checked in Lean 4.
The highlight: Proved Shapiro's 2026 conjecture for finite trees by cancelling leaves and commuting disjoint edge swaps, using earlier transposition and crossing lemmas: the word is identity exactly when every edge appears twice and each vertex has a non-crossing pattern.
github.com/the-omega-institu…
We're researching how hidden structure can be recovered from limited observations, while geometry, graphs, and quantum noise reveal the limits of knowledge.
Along the way, on October 31, we solved 22 open problems from the literature, across quantum measurements, Bell inequalities, integer sequences and combinatorics, all machine-checked in Lean 4.
The highlight: Bodor and Koniorczyk asked whether the fidelity measurement always traces the best ROC curve. A fixed three-dimensional counterexample reuses matrix-square-root uniqueness to identify it, then a rank-one test beats its first point.
github.com/the-omega-institu…
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Well, we're researching how hidden structure can be recovered from finite observations, and where graphs, clocks, and quantum readouts reveal or conceal it.
Along the way, on October 30, we solved 33 open problems from the literature, across quantum information, graph theory, combinatorics and number sequences, all machine-checked in Lean 4.
The highlight: A four-level state with energies 0, 1, 2 and 10, measured in a basis unbiased to the state, produces the completely mixed state; positivity bounds show that this measurement extracts more than half the state's ergotropy.
github.com/the-omega-institu…
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Well, we're researching how finite observations can reveal hidden structure: exact limits on memory, measurement, reconstruction, and what lawful systems can know.
Along the way, on October 29, we solved 59 open problems from the literature, across number theory, quantum information, graph theory and combinatorics, all machine-checked in Lean 4.
The highlight: Halliwell and Mawby's 2019 conjecture was refuted with six-time pairwise probability tables: a weighted-square contradiction shows no joint probability model can exist, and Lean verifies the construction.
github.com/the-omega-institu…
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Well, we're researching how observation, memory, timing, and boundaries reveal hidden structure and turn partial signals into reliable knowledge.
Along the way, on October 28, we solved 47 open problems from the literature, across quantum information, graph theory, number theory and integer sequences, all machine-checked in Lean 4.
The highlight: Finiteness fails in D-majorization: finite-Kraus maps reach extreme states; positivity proves extremality, and the D-weighted trace separates their classes under allowed unitary symmetry, reusing finite-Kraus and unitary trace-invariance tools.
github.com/the-omega-institu…
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Well, we're researching how hidden structure survives translation: from prime periods and cubic identities to what local evidence can reveal.
Along the way, on October 27, we solved 48 open problems from the literature, across number theory, quantum information, combinatorics and graph theory, all machine-checked in Lean 4.
The highlight: Stephen McKean's two 2025 Toda-prime claims were both disproved: the classical von Staudt–Clausen identity for Bernoulli denominators and finite prime-set evaluation yield examples where prime counts fall whether the denominator stays fixed or changes.
github.com/the-omega-institu…
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Well, we're researching how systems learn their world: through boundaries, memory, prediction, and the limits of observation.
Along the way, on October 26, we solved 38 open problems from the literature, across quantum information, graph theory, analysis and algebraic sequences, all machine-checked in Lean 4.
They fed back into our theory. The symmetric measurement family supplies the concrete quantum model used for the higher-dimensional predictive-spectra construction, with the same pattern defined for every dimension at least two.
github.com/the-omega-institu…
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Well, we're researching how memory, noise, and hidden structure shape what a system can reveal and recover.
Along the way, on October 25, we solved 55 open problems from the literature, across quantum information, graph theory, special functions and integer sequences, all machine-checked in Lean 4.
They fed back into our theory. Equal-overlap quantum states with no repeated labels gave the theory a precise benchmark for unambiguous recovery, while also showing that this result does not by itself provide a path impulse experiment.
github.com/the-omega-institu…
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