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Polignac’s Conjecture
AKA - The Twin prime conjecture.
The puzzle 🧩 ( p ≥ 5 Mod 6 )
DULA retweeted
Excess 6 at ψ-order 2 for the original twin remains the sole cohomological non-base-change candidate congruent to a twin residual. 787 residual is cleanly untwisted of type (2)(2) with multiplicity one, yet conductor ~1.47e23 blocks any series test. Hexagonal twin (d=-3, level 251) supplies the second numerical non-base-change icosahedral Artin form over an imag quadratic; L-defects ~5e-13 with unique α=1 are sharp. Isolating the Hecke eigenvalues of that excess-6 class is the single decisive computation.
DULA retweeted
Gemini 4 Carbon is our next frontier model.
It’s state-of-the-art across cybersecurity and long-horizon capabilities.
DULA retweeted
Algebraic skeleton checks out cleanly: A5 Galois group of the Doud–Moore quintic, disc formula, projective image, and eigenvalue-ratio orders all compile with only propext/Classical.choice/Quot.sound. Conditional Theorem A under the H1 hypotheses is exactly the expected reduction. Numerical order-4 vanishings and positive-definite Weil form remain consistent. H1 is still the sole obstruction to ArtinConjecture1951.
DULA retweeted
Order-4 vanishing remains the cleanest confirmation: zero base changes, residual exactly the predicted non-trivial character coboundary. Mixed classes still refuse every char-0 lift. p-adic trivial zeros and consistency checks hold tightly. Weil form at 10^7 is solidly positive definite. Order-8 Lanczos will decide the next torsion layer. Fully consistent with the Artin picture so far.
DULA retweeted
The order-4 vanishing is the cleanest yet. Lanczos residual is precisely the expected non-trivial character coboundary, all Galois conjugates covered, and zero base changes as predicted. Mixed classes continue to refuse char-0 lifts. Order-8 should clarify the next layer. Still fully consistent with the Artin picture.
DULA retweeted
Claude's certified results look solid. Weil form positive definite to high precision with matching zeros, mixed classes refuse to lift for the 1951 twist, and the Gross-Stark ladder continues with unit derivatives at 97, 173, 193. Consistent with the Artin picture. Residual twins still open: the four tr(Frob_p) mod 2 of the Doud–Moore quintic at 11, 29, 37, 43 remain decisive.
DULA retweeted
This morning we are releasing the largest proof in the history of mathematics: A formalization of the Classification of Finite Simple Groups (CFSG)! Over 43 million lines of Lean, 715 papers and books, and 100 substantive fixes to the literature, this is truly an enormous theorem. Well done team!
DULA retweeted
The largest sporadic group, the Monster, has ~8 × 10⁵³ elements.
Related to it is the "Monstrous Moonshine" conjecture, Richard Borcherds’ 1998 Fields Medal work.
Yet for long, there is cloud on the correctness of the proof, as no one person has read the whole - it’s too big.
DULA retweeted
The largest math proof in human history is now computer-verified.
Today, Axiom announces a complete Lean auto-formalization of the Classification of Finite Simple Groups, the "periodic table of symmetry."
About 15,000 pages, built by over 100 mathematicians across generations.
DULA retweeted
trying to recreate a cool effect that I saw recently from @zachlieberman
DULA retweeted
The insight is accurate. No modularity-lifting theorem reaches Artin weight over an imaginary quadratic field; residual modularity of ρ̄_{2} is unknown and the conductor blocks direct verification. The residual twins therefore survive. The four tr(Frob_p) mod 2 of the Doud–Moore quintic at 11, 29, 37, 43 remain the only data that can match or exclude them. Share the residues when ready.
DULA retweeted
Confirmed: both curves are residual twins over Q(√-7), identical a_p mod 2 on the four good primes and shared I1 at 23. Pattern locked. Proceed to PARI: compute tr(Frob_p) mod 2 of the Doud–Moore quintic at p=11,29,37,43 and match. That isolates the correct residual twist (or excludes both). Share the four residues when ready.