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An unconditional record, not a resolution: the Riemann hypothesis is untouched, and Anthropic states it does not expect these techniques to lead to a proof of it. The paper is explicit that these are lower bounds only - the remaining third of the zeros are not shown to be off the line, merely not reached by the certificate.
What it does settle is a question that was posed. Goldston and Suriajaya had reduced Montgom…
Assuming three explicit input statements, the project proves DHL[40,2]: every admissible 40-tuple contains at least two primes infinitely often after translation. An explicit admissible 40-tuple of diameter 186 then gives
n→∞liminf(pn+1−pn)≤186.
The Lean proof of the implication from the three inputs to the final theorem is kernel-checked. Two inputs are Kloosterman-type estimat…
H1≤212, improving Stadlmann's 240 of three days earlier and the 246 of Polymath8b that had stood since 2014. The twin prime conjecture, H1=2, is untouched. Held the record for hours at most: OpenAI's paper claiming 186 is dated 30 August, four days before this one, though its Lean development appeared on 2 September.
Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a C∞ object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.
Astra constructs a strictly stationary real process
Xt=ξt+K(ξt−1,ξt−2,…),
where the innovations ξt are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and K is bounded, continuous, and odd. The process is φ-mixing, centered, square-integrable, and satisfies
Var(Sn)→∞.
Nevertheless there are times nj→∞ such that…
The problem is open, and the repository says so: what is proved are criteria for it and two instances, not the general case.
Every compared statement that concludes Problem 10.61 does so for a quadratic setup - a real root α>1 of X2−aX−b whose conjugate has modulus below one - and both instances are quadratic. The arbitrary-degree material is conditional ingredients: for the family…
Answers Problem 3 and generalizes it: the classification m∈S⟺m=2 covers every square root, and a further theorem replaces parity by divisibility by any p≥2. Note the scope of the machine-checking, which is narrower than the paper: the author states that the case m=3 is what is verified in Lean, and the repository flags the thickness computation of section 4.1 and all of sectio…
No such pair exists: for every positive integer m and every n, (Dn,2m) is rate-monotonic (Theorem A), and more generally so is every inversion extension of a finite abelian group by an involution, a family containing the generalized dihedral, dicyclic and generalized quaternion groups (Theorem B). The picture is completed in the other direction: for every real p≥1 that is not an even integer there is an…
Let H be the Berlekamp–van Lint–Seidel graph on 243 vertices, the Cayley graph of Z35 with strongly regular parameters
(243,22,1,2),
and let G=H.
The proof establishes
γ(G)=3
because every pair has a common neighbor in H, while an H-triangle gives a dominating triple in G.
It then proves
γ∞(G)=3
by showing that the family of all dominating triples…
For pairwise distinct real numbers x1,…,xN and an integer D≥1, Ma proves the complete classification
det[(xj−xi)D]i,j=1N=0⟺D≥N−1and(N is even or D is even).
This fully resolves Colombo’s original conjecture for even N. The new part is the even-size, odd-exponent branch, strengthened to the strict Pfaffian…
New theorems, not a formalisation of known results. In base ten the bare existence of a square with any admissible digit sum was recorded in the recreational literature (Murthy–Ashbacher 2005); the theorem here is the quantitative all-base version — roots coprime to b (excluding their trailing-zero trick), exp(c_b√q) representations of controlled size — and that statement is what is machine-checked, with effective b…
New theorems, not a formalisation of previously known results. For every base b≥2 the Erdős–Kac law is established for the λ-digit base-b palindromes and for the base-b reversals of the λ-digit primes, for ω and Ω and for ωS,ΩS with any regular set S of primes; with normal order loglogn on both families, and, for ω, all moments of order up to…
For every integer-valued strongly b-additive g with gcd(g(1),…,g(b−1))=1 and digit mean μg≥0: g(p) is prime for infinitely many primes p. For μg>0, ∑1/p over p<X with g(p) prime is (dg/φ(dg))log3X+Cg,1+O(1/loglogX), likewise for the first j iterates. Also #{p≤x:g(p) prime}≪π(x)/loglogx, of that exact order on a large set of x, and…
VibeMathed recorded: Composites Among [ξ7n] and Right-Truncatable Primes in Base 7. Consult the linked registry entry for the original report and scope.
Answers the obstruction rather than the whole question: it says exactly when the Matrix-Tree Theorem can be applied and classifies the chambers, and gives a compact five-graviton formula outside the decay region. Simplifying the general solution, which is what the source paper left to future work, remains open.
VibeMathed recorded: The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture. Consult the linked registry entry for the original report and scope.
Finite-jet theorems about compiled truncated EMD equation certificates: the exact shear-orbit fiber classification of the complete first seed channels, an explicit collision family with one metric three-jet realized by an actual cubic metric germ (genuine Frechet Ricci value and first derivative), the compiled impossibility theorem, and the fourth-order recovery with equality fiber a=±b. Not settled here: promo…