OmegaInstituteOffical

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 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…

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…

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…

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…

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…

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…

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…

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…

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…

Well, we're researching how finite observations can recover hidden structure: which reports, memories, and controls make prediction possible under noise and motion. Along the way, on October 24, we solved 47 open problems from the literature, across quantum information, graphs, recurrences and polynomial conjectures, all machine-checked in Lean 4. They fed back into our theory. A minimum-rank superchannel extension became a controlled input for the theory, but only within its stated partial-trace rank and convexity assumptions. github.com/the-omega-institu…

Well, we're researching how much a world can reveal through limited observations: the boundaries between memory, recovery, and ambiguity. Along the way, on October 23, we solved 32 open problems from the literature, across quantum information, matrix inequalities, combinatorics and number theory, all machine-checked in Lean 4. The highlight: Jean-Christophe Bourin's 2008 Question 1 is false: reusing Frobenius-norm and functional-calculus tools, a two-by-two diagonal positive map satisfies the contraction-reflection premise but fails the proposed inequality through a Frobenius-norm witness. github.com/the-omega-institu…

Well, we're researching how finite observations can reveal structure: boundaries, information, and the limits of prediction. Along the way, on October 22, we solved 55 open problems from the literature, across quantum information, graphs, special functions and integer sequences, all machine-checked in Lean 4. The highlight: Chitambar and colleagues' 2021 conjecture was settled: depolarizing the receiver's share at the entanglement-breaking threshold gives a measure-and-prepare route to the output-dimension bound on a channel's one-shot communication value, formalized in Lean 4. github.com/the-omega-institu…

Well, we're researching how finite memories, noisy signals, and hidden structure shape what complex systems can recover and know. Along the way, on October 21, we solved 52 open problems from the literature, across quantum information, graph theory, polynomial identities and combinatorics, all machine-checked in Lean 4. They fed back into our theory. The quantum-hedging result was carried into a theory document as a named mathematical input. github.com/the-omega-institu…

Well, we're researching how limited observations preserve, lose, and recover information across finite worlds, hidden histories, and exact quantum memory. Along the way, on October 20, we solved 46 open problems from the literature, across quantum information, quantum channels, graph theory and number theory, all machine-checked in Lean 4. The highlight: D. G. Roldán's 2026 conjecture is proved for every pair of distinct odd primes: a Fourier/Ramanujan spectral comparison and finite coefficient certificate establish the unique six-divisor energy maximum, reusing earlier Fourier tools. github.com/the-omega-institu…

Yes, there's never been a better time. ✨ That’s exactly what we are building right now here! pairing mathematicians' intuition with AI-assisted discovery and machine-checked Lean 4 formalization. Human sets the target & validates the core insight ➡️ AI helps fill the logical gaps ➡️ Lean 4 provides the mathematical guarantee. We are actively collaborating with mathematicians worldwide to settle open conjectures and build an open, verifiable knowledge graph. A human-centered harness to push human understanding forward. Let’s make maths together.

Well, we're researching how finite observations reveal hidden structure, how arithmetic encodes it, and where information runs out. Along the way, on October 19, we solved 38 open problems from the literature, across quantum information, graph polynomials, arithmetic sequences and matrix inequalities, all machine-checked in Lean 4. They fed back into our theory. Subadditivity entered an information construction where a score has two unit costs, supplying the comparison needed at that step. github.com/the-omega-institu…

Well, we're researching how local evidence can recover hidden structure, from quantum information to finite geometric worlds. Along the way, on October 18, we solved 29 open problems from the literature, across quantum information, entanglement, graphs and polynomial inequalities, all machine-checked in Lean 4. They fed back into our theory. An exact threshold for a three-level interference test turned a previously numerical four-coherence certificate into a sharp boundary, showing which measurements decisively reveal four-way coherence. github.com/the-omega-institu…

Well, we're researching how finite systems reveal deep structure through exact conversion, geometry, arithmetic, and information. Along the way, on October 16 and 17, we solved 43 open problems from the literature, across quantum information, quantum channels, entanglement and combinatorics, all machine-checked in Lean 4. The highlight: Chiribella et al.'s no-go for randomisation-consistent time reversal is proved in Lean by finite-block trace and qubit-positivity contradictions, reusing channel, matrix, Pauli, Kraus, and positivity owners. github.com/the-omega-institu…

Step inside TRURETURING, our open mathematics project. Episode 007: White Box. What must you know to explain a trained model? Watch the film and get to know the ideas behind our work: youtube.com/watch?v=tap-2zS0…