https://evidence-press.pages.dev/ Press releases for new research, published with the evidence attached en-gb Thu, 30 Jul 2026 12:00:00 GMT https://evidence-press.pages.dev/releases/exact-low-length-recht-re-inequalities/ https://evidence-press.pages.dev/releases/exact-low-length-recht-re-inequalities/ Thu, 30 Jul 2026 12:00:00 GMT This paper asks a basic question behind many machine-learning algorithms: is it better to shuffle a dataset and use each example once, or keep sampling examples at random? It gives the first exact account of where a leading mathematical justification for reshuffling works, where it fails, and where it recovers. It also finds something more surprising: the usual theoretical measure can say reshuffling is worse even when it actually reduces error. The proof is computer-assisted, exact, and fully replayable. https://evidence-press.pages.dev/releases/z20-equals-6/ https://evidence-press.pages.dev/releases/z20-equals-6/ Tue, 28 Jul 2026 12:00:00 GMT How few colours do you need to partition any 20-vertex graph so that every colour class is either a clique (all pairs connected) or an independent set (no pairs connected)? The answer for up to 19 vertices was known; at 20 it was either 6 or 7. This computer-assisted proof establishes that the answer is 6. The lower bound is a short, hand-checkable argument about the Paley graph on 17 vertices; the upper bound reduces all possible counterexamples to two SAT problems whose impossibility is certified with proof objects checked by four independent checkers, including a formally verified one. https://evidence-press.pages.dev/releases/vr2-k4-equals-20/ https://evidence-press.pages.dev/releases/vr2-k4-equals-20/ Tue, 28 Jul 2026 12:00:00 GMT The classical Ramsey number R(4,4) = 18 says that among 18 points with red/blue connections there is always a single-colour foursome. This release asks for more: how many points guarantee two such foursomes sharing no points at all? The claimed answer is exactly 20. A hand-checkable 19-vertex colouring (a 'Paley twin') has no two disjoint monochromatic K4s, while the impossibility of avoiding them at 20 vertices reuses the SAT certificates from the companion z(20) = 6 release. https://evidence-press.pages.dev/releases/reducible-incidence-divisors/ https://evidence-press.pages.dev/releases/reducible-incidence-divisors/ Tue, 28 Jul 2026 12:00:00 GMT This paper classifies the reducible members of a marked-common-root incidence linear system for binary forms: for adjacent degrees, a divisor becomes reducible precisely when its defining functional lies on the tangent developable of the rational normal curve of evaluation functionals. It proves that the corresponding affine slices are never isomorphic to affine space, computes their precise motivic and Hodge-theoretic defects, obstructs non-adjacent-degree slices via finite cyclic actions, and — conditionally on an upstream classification — isolates a unique slice that could in principle carry a nonzero-constant-Jacobian polynomial map. https://evidence-press.pages.dev/releases/exotic-affine-three-spheres/ https://evidence-press.pages.dev/releases/exotic-affine-three-spheres/ Tue, 28 Jul 2026 12:00:00 GMT Some algebraic shapes impersonate familiar ones: they are indistinguishable by the tools of smooth topology yet are provably different as algebraic varieties. This paper argues that a natural slice of the space of polynomial factorisations is exactly such an impostor — a known exotic affine three-sphere (the Dubouloz–Finston torsor) rather than the standard SL2 quadric it resembles. It further claims that every normalised quadratic–cubic slice fails to be affine five-space, by an exact Grothendieck-class formula and a rank-by-rank Hodge–Deligne obstruction, and that in characteristic 3 the tangent slice remains affine three-space with an Artin–Schreier collision in the induced Keller map. https://evidence-press.pages.dev/releases/erdos-848-all-n/ https://evidence-press.pages.dev/releases/erdos-848-all-n/ Tue, 28 Jul 2026 12:00:00 GMT Erdős and Sárközy asked: how large can a set A of integers from 1 to N be if the product of any two members (including a member with itself), plus one, is never squarefree? Recent work resolved the question for all sufficiently large N. This release establishes the exact answer f(N) = ⌊(N+18)/25⌋ for every positive integer N, stitching together exact colouring certificates for small N, structural decompositions and exact-rational envelope arguments for intermediate ranges, and a pinned explicit-threshold analytic theorem for N beyond 2.64 × 10^17. https://evidence-press.pages.dev/releases/degree-difference-affine-slices/ https://evidence-press.pages.dev/releases/degree-difference-affine-slices/ Tue, 28 Jul 2026 12:00:00 GMT Multiply two polynomials together and, alongside the product, record their resultant — a single number measuring whether they share a root. This paper proves an exact formula for the Jacobian determinant of that combined operation: up to sign, it is the degree difference of the two polynomials times the square of their resultant. From this 'degree-difference principle' the paper develops the geometry of natural affine slices of binary-form factorisation spaces — the setting in which the project's related Jacobian-conjecture investigations take place — including classifications of low-degree slices, explicit coordinates for Keller maps, and Euler-characteristic obstructions in higher degrees.