A Markoff number is the largest coordinate of a positive integer solution of
a2+b2+c2=3abc. Frobenius conjectured in 1913 that every Markoff number
is the largest coordinate of exactly one such solution with a≤b≤c.
We prove that the conjecture holds for almost all Markoff numbers: the
proportion of exceptions among the Markoff numbers up to X tends to 0
as X→∞.
Two such solutions with the same largest coordinate give simple closed
geodesics of equal length on the modular torus, that is, primitive homology
classes v,w with equal values N(v)=N(w) of the McShane--Rivin stable
norm N. Estimates for the boundary of the unit ball of N reduce the
problem to the equations N(Gv)=N(v) with G a rational linear map. For
each G that is not a symmetry of the torus, a zero-one law for the
3-adic analytic dynamics of the mapping class group on the Markoff
surface shows that the solutions v have density zero.
Primarily AI-generated textHuman understanding: some partsMarkoff triplesmapping class group dynamicspunctured torus
Let β(s)=∑m≥0(−1)m(2m+1)−s be Dirichlet's beta function. We prove that at least one of the seventeen numbers β(4),β(6),…,β(36) is irrational. All previous explicit results of this kind that we know of include β(2), Catalan's constant; our statement does not involve β(2) at all. The proof uses the linear forms Ln=∑t≥0(−1)tRn′′(t) attached to a very-well-poised rational function Rn with triple integer zeros and half-integer poles. The second derivative, a β-analogue of a device of Rivoal, removes β(2), and a symmetry removes the odd values. The zeros of Rn run through the region of its poles, which makes every prime p>n harmless. The common denominators are controlled by a p-adic analysis and an arithmetic integral that we bound with an exact rational certificate. The size of Ln is bounded by an elementary estimate on the real axis, and its nonvanishing for infinitely many n follows from a saddle-point analysis along a certified contour. The whole proof, including every numerical certificate, has been formalised in Lean 4 with Mathlib and checked by the Lean kernel, using only Lean's three standard axioms.
Primarily AI-generated textHuman understanding: no partsDirichlet beta functionLeanformal verificationirrationalitylinear forms in zeta-type valuessaddle-point method
We develop a finite-word obstruction framework for the accelerated Syracuse map on positive odd integers. A finite valuation word determines an exact iterate formula and a unique realization class modulo a power of two; compatible infinite prefixes define a natural two-adic inverse-limit object together with least positive representatives. The first part records the exact finite-word congruence machinery, finite-descent equivalence, terminal-surplus identity, terminal-ceiling break criterion, bad-prefix windows, activated density collapse, stabilization criterion, and strict-jump bookkeeping. The second part uses these tools to isolate terminal obstruction classes for a hypothetical counterexample: a bounded branch leading to a nontrivial accelerated cycle, and an unbounded eventual no-jump branch, successively reduced through a chain of geometric and combinatorial normal forms to a packet-surplus compensation target. The resulting reduction spine is lossless: Collatz follows from exclusion of nontrivial accelerated cycles together with a packet-surplus, or stronger record-ladder, compensation theorem for the unbounded branch. As a closed module, we include a closed class-exclusion theorem showing that no eventually periodic infinite low-surplus valuation chain is realized by a positive odd integer. On the bounded branch, we further record, as an explicit unproved roadmap step, that the terminal-ceiling strict-jump obstruction reduces to a single recurrent valuation-mismatch statement. This manuscript proves the architecture stated in its theorems, propositions, lemmas, and corollaries; it states the remaining terminal exclusions explicitly as problems, conjectures, or roadmap steps, and does not claim a completed proof of the full Collatz conjecture.
Primarily AI-generated textHuman understanding: no parts3x+1 problemCollatz conjectureaccelerated Syracuse maplow-surplus chainsterminal obstructionsvaluation words
We construct a fourteen-dimensional Lie bialgebra for which both the
underlying Lie algebra and its dual are perfect. The construction is
defined over Q and is self-dual over C. It gives an affirmative
answer to Question 4.1(3) of Etingof--Gelaki--Guralnick--Saxl, arXiv:9912068.
Its twenty-eight-dimensional Drinfeld double is quasitriangular with
injective cobracket, also answering Question 4.1(4).
Primarily AI-generated textHuman understanding: some partsLie bialgebrasPoisson-Lie groups
We construct a finitely presented group with 13 relators that contains an isomorphic copy of every finitely presented group. This improves Valiev's bound of 21 relators for universal groups. The proof combines Valiev's construction, subgroup calculations of Kegel--Li--Ren, and small-cancellation theory.
Primarily AI-generated textHuman understanding: some partsfinitely presented groupuniversal group
Contributed by Dylan T. Costa, Alvaro Lozano-Robledo
In this article, we find a quadratic twist over Q of rank ≥2 (and ≥3 assuming parity) of an elliptic curve E/Q of rank ≥32, recently found by Elkies, Klagsbrun, and Li. We use a family of quadratic twists of rank ≥2 over Q(t) constructed by Mestre to search for twists ED with root number −1, computing all root numbers without factoring the (very large) twisting parameter D. As a result, we exhibit an explicit integer D, with 411 digits (a square times an integer with 381 digits), such that ED has root number −1 and two independent rational points. In particular, the rank of E over Q(D) is at least 34, unconditionally, and at least 35 if we assume the Parity Conjecture.
A mix of human-written and AI-generated textHuman understanding: all partselliptic curveparity conjecturequadratic fieldrank
We prove that the only primitive positive integer solutions of x4+y4=82z4 are (1,3,1) and (3,1,1). An arithmetic reduction lifts every other putative solution to an elliptic curve over K=Q(41,2). An explicit 2-descent determines its rank as 3. A logarithmic functional then excludes all nonzero parameters in the relevant 2-adic disc, including both relative sign patterns at the two places above 2. The same argument determines every point of the auxiliary quartic whose parameter is rational: only the two points at infinity occur. Exact square identities also determine the quotient modulo 2, prove that the displayed subgroup has odd index, and give the vanishing of the 2-primary part of the Tate–Shafarevich group. Two reductions determine the torsion. The finite arithmetic certificates are supplied in two self-contained Python programs; no height cutoff or conjectural rank computation is used.
Contributed by Deep Bhattacharjee, Ushashi Bhattacharya
We prove unconditionally that every primitive Dirichlet character χ modulo q≥3 has
character sums of size at most Cqloglogq. Montgomery and Vaughan proved this bound under
the generalised Riemann hypothesis, and Paley showed that it is optimal. The proof uses the
zero-free half-plane ℜs>7/8, which also gives ∣L′/L(1,χ)∣≪loglogq and two-sided
bounds for the Euler--Kronecker constants of the cyclotomic fields Q(ζq).
A mix of human-written and AI-generated textHuman understanding: some partsEuler--Kronecker constantsP\'olya--Vinogradov inequalitySelberg sievecharacter sumszero-free regions
Let Xi(t) = xi(1/2 + it) be the Riemann Xi-function, let
Xi(t) = sum_{k>=0} (-1)^k c_k t^{2k} be its even Taylor
expansion at the origin, and let D(n,r) = det[c_{n+i-j}]
(i,j = 0..r-1) be the Toeplitz minors whose positivity for
all n >= 0, r >= 1 is equivalent to the Riemann Hypothesis
(RH). We prove a sharp and quantitative obstruction in the
resulting hierarchy of criteria.
Parametrising the zeros of G(z) = sum c_k z^k by
beta_j = -1/z_j, we establish by the dual Jacobi-Trudi
identity the exact identity D(n,r) = s_{(r^n)}(beta), the
Schur function of the rectangular partition (r^n) in the
variables beta. A zero of zeta off the critical line
contributes to beta a conjugate pair rho e^{+-i theta} whose
angle theta is exactly the vertical angle of the violating
pair of zeros, while zeros on the critical line contribute
positive real variables. For two variables we prove the
closed form
s_{(a,b)}(rho e^{i theta}, rho e^{-i theta})
= rho^{a+b} sin((a+1-b) theta) / sin theta
= rho^{a+b} U_{a-b}(cos theta),
U_k being the Chebyshev polynomial of the second kind, so
that the sign is governed by the classical criterion
U_k(cos theta) > 0 if and only if (k+1) theta < pi.
Since a Schur function in two variables can be expanded
through the coproduct s_lambda(X sqcup Y) = sum_mu s_mu(X)
s_{lambda/mu}(Y), and since s_mu(X) > 0 for positive real X,
positivity reduces to the two-variable factors, whose index
k = a - b is bounded by the number of columns r, not by the
shift n. This yields:
Blind Zone Theorem. If a violating pair subtends a vertical
angle theta < pi/(r+1), then every Toeplitz minor D(n,r')
with r' <= r is strictly positive, notwithstanding the
presence of non-real zeros.
The critical strip together with gamma_1 = 14.134725141...
forces theta <= theta_max = 2 arctan(1/(2 gamma_1)) =
0.07071826... rad, whence pi/theta_max = 44.424... and
therefore every order r <= 43 is completely
non-discriminating.
We emphasise that the exponential bound is on the order r
and not on the shift n; we exhibit the mechanism and show
the contrary n-based bound is false. We also record the
refutation of a "universal covering law" previously
conjectured by the author, and a floating-point artefact
that nearly produced a false refutation of the main theorem.
No claim of a proof of RH is made.
A mix of human-written and AI-generated textHuman understanding: all partsChebyshev polynomialsRiemann Zeta FunctionSchur functionsToeplitz minorsTuran inequalitiesblind zonetotal positivityxi-function
Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník
Every Hermitian positive semidefinite contraction A satisfies per(A)≤exp(−tr(I−A)/68). This proves the linear trace-deficit bound conjectured by Ebrahimnejad, Nagda, and Oveis Gharan, whose estimate had quadratic dependence on the normalized deficit. The proof combines a size-biased complex Gaussian representation, a block version of Hanson–Wright decoupling, and a rational Loewner majorant. The constant is uniform in the dimension and is not optimized.
Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník
For all finite simple graphs, the strong product and both ordered lexicographic products have the same Shannon capacity, answering a question of Sason. An explicit code conversion delays one component by a single block, with rate loss tending to zero. The equality extends to arbitrary mixtures of the two products and survives adjoining an additional strong-product factor.
Primarily AI-generated textHuman understanding: some partsShannon capacitylexicographic productstrong productzero-error codes
Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník
If an α fraction of the (d−k+1)-tuples of closed convex sets in Rd have an intersection containing a whole affine k-flat, then a fraction 1−(1−α)1/(d−k+1) of the sets share such a flat. This coefficient is best possible, resolving the sharp fractional De Santis question of Ludwigson and Soberón. The geometric result behind the bound is (d−k)-collapsibility of the common-flat complex, proved by coordinating convex sweeps in varying quotient spaces.
Primarily AI-generated textHuman understanding: some partsLeray complexesaffine flatsfractional Hellysharp bounds
Let K be a field of characteristic zero and let P,Q in A_1(K) satisfy [Q,P]=1. We prove that P,Q generate A_1(K) if either has mass at most six, meaning at most six nonzero homogeneous components for the grading deg X=1, deg Y=-1, with no degree bound on either element. This extends the four-component theorem of Guccione, Guccione and Valqui. The proof combines their Newton-polygon restrictions with a classification of polynomial pairs r,f satisfying an associated scalar differential equation under the condition that r^2 has at most six monomial terms. The scalar argument uses a two-dimensional Vandermonde kernel, Rolle's theorem, and a sign and parity obstruction. The scalar classification and the mass bound reduce the crossing cases to a binomial-power face. We exclude that face using at most one Newton cut, without a mass bound on the transformed pair. We also exclude a family of pure-power crossing faces with prime outer exponent, without a mass bound. A seven-term scalar solution shows that the scalar classification is sharp, although its associated Weyl face is excluded by the pure-power theorem. Theorem 1.1 is formalized in Lean 4 and registered as Palomar entry PALOMAR-2026-10-05-000006, version 2.
A mix of human-written and AI-generated textHuman understanding: all partsDixmier conjectureNewton polygonWeyl algebrahomogeneous componentssparse polynomials
Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník
Wagner asked whether Gaussian kernel density estimation in Rd admits query time O(d+ε−2) without dependence on the diameter of the data. We answer this affirmatively for each fixed query: a randomized data structure answers a query to additive error ε with probability 1−δ in time O((d+ε−2)log(d+1)log(2/δ)), independently of the number of data points and of their diameter. The key ingredient is a variance bound for a Fastfood-type structured Fourier block that holds uniformly in the norm of the displacement.
Primarily AI-generated textHuman understanding: some partsFastfoodGaussian kernel density estimationpointwise approximationrandom Fourier featuresstructured random features
Starting from the two trace functions p1 = Tr(A) and p2 = Tr(A^2) for semisimple A in GL2(C), we develop the associated projective source geometry. The coordinate y = t^2/d - 1 identifies the adjoint character, and normalized Haar measure on a specified closed subgroup H of SO(3) pushes forward to a probability measure nu_H. We describe this measure through harmonic multiplicities, moments, and its Stieltjes transform, and prove that it determines H up to conjugacy within the closed-subgroup class of SO(3).
A positive Abel kernel resums the projective character tower and gives a resolvent formula whose endpoint limit recovers the mass of the identity atom. We give explicit Abel transforms for the finite rotation groups and positive-dimensional benchmark sources, and derive a finite-image Frobenius-prime Abel limit from Chebotarev. For central-jet interpolation, we prove two exact degree laws: the cyclic-versus-torus minimum degree is max(n, M+1), while the normalized tetrahedral detector against S4 and A5 has minimum degree 3 for M = 1, 2 and M+2 for M >= 3. We also compute the integral image of the tetrahedral averaging functional, obtaining 2Z for every M >= 2.
All results are stated with their hypotheses. The paper does not claim a general infinite-spectrum Beyond Endoscopy trace formula or transfer of central-jet vanishing to orbital kernels.
Primarily AI-generated textHuman understanding: some partsAbel resolventsChebotarev density theoremGL2PGL2character measurescharacter theorycompact subgroups of SO(3)finite rotation groupspolynomial interpolationprojective characters
Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník
For a nonempty finite graph, let F be its fractional DAG treewidth, maximized over all acyclic orientations. Its DAG treewidth is at most 160F3ln(16F). This answers the boundedness question of Bressan, Lanzinger, and Roth. Reorienting the graph converts shattered source profiles into projective-plane obstructions, which bound their VC dimension by 5F2. Classical epsilon-net rounding then converts fractional bag covers to integral ones without changing the decomposition.
Let H(d,q)=Kqd be the Hamming graph and K(H(d,q)) its sandpile group (critical group, Jacobian). For primes p∤q its p-part was determined by Jacobson, Niedermaier and Reiner in 2003, who wrote that the other primes appear "to be difficult, due to bad behavior of its p-primary structure". We determine the Sylow p-subgroup of K(H(d,q)) for every prime p dividing q and every d, by an explicit rule. After the splitting of Jacobson, Niedermaier and Reiner, the group is a sum of cokernels of operators q(c+H)+U on the Boolean lattice (Theorem A). These split over the indecomposable tilting modules T(λ) of SL2 in characteristic p that occur in a tensor power of the natural module. The contribution of T(λ) consists of explicit small cyclic factors and of 2∣I∣ large ones, where I is the set of positions of the nonzero base-p digits of λ+1 below the leading one. The large ones are obtained by pooling explicit integers, built from binomial valuations and base-p carries, into a non-increasing sequence (Theorems I and J). The proof writes down explicit Zp-lattices for all tilting modules of SL2, one move per base-p digit; it reduces each contribution to the Smith form of an explicit 2∣I∣×2∣I∣ matrix, and computes that form as in our proof for the hypercube (The Dancing Sand Theorem), the case p=2. We also give closed formulas that need no pooling: for p>d with ballot numbers (B), for the exponent with every prime (C), and for the number of cyclic factors (D). This proves Conjecture I of version 3 of this paper. Every result of the paper, Theorems A–J, is checked in Lean 4 with Mathlib, for every prime, and the Lean project was rebuilt from its sources on GitHub's machines, on Mathlib's published build. This work has not been refereed by a human expert.
Primarily AI-generated textHuman understanding: some partsHamming graphLean 4Smith normal formcritical groupsandpile grouptilting modules
The notion of exceptional maps in arbitrary dimension was recently introduced by Ji, Song, and Xie. In this paper, we introduce exceptional extensions of polarized dynamical systems: a family of systems admitting a positive-dimensional exceptional factor whose generic fiber dynamics have simultaneous local product coordinates with an isotrivial system. We then prove that an algebraic family of polarized endomorphisms of normal projective varieties over a quasi-projective curve is J-stable on a nonempty Zariski open subset if and only if it is isotrivial or it is an exceptional extension after a finite base change. This extends McMullen's theorem for stable algebraic families of rational maps to higher dimension. One ingredient is inspired by Ji and Xie's proof of McMullen's theorem: the use of homoclinic orbits and the multipliers of adjoint periodic points to control local return maps. The new ingredients are a higher-dimensional algebraization method and an inductive rigidity argument for the fiber dynamics, which together yield the exceptional extension structure.
Primarily AI-generated textHuman understanding: some partsComplex Dynamics; Arithmetic Dynamics
A finite Euclidean point set is distance-separated if the lengths of any two distinct unordered edges differ by at least one. Counting gives a diameter lower bound of binom(n,2), whereas Erdős asked for a lower bound with leading term n^2. We give two constructions in fixed dimension. For every 0 < epsilon <= 1/2, put m0 = ceil(8192/epsilon). For every n >= 4m0 there is a distance-separated n-point set in dimension 10m0-1 with diameter < (1/2+epsilon)n^2. A six-cluster construction gives arbitrarily large configurations in R^29 with diameter < 0.998n^2, and an explicit prime-interval bound gives diameter < 0.999n^2 for every n >= 400000. The constructions use two orthogonal circle families to control distances within and between clusters separately. Their common geometric argument reduces to a uniform operator estimate for the first result and three rational positive-definiteness certificates for the second. Lean 4 formalizations accompany the tolerance family and the dimension-29 result for arbitrarily large sizes; the all-size corollary is a written consequence.
Primarily AI-generated textHuman understanding: no partsErdős Problem 670Lean 4fixed dimensionseparated Euclidean distances
We present a uniform randomized classical algorithm for Subset Sum with running time O(2^{0.4877n}) in the word-RAM model, for inputs whose integer bit lengths are polynomially bounded in the number n of input integers. The algorithm halts on every execution and answers correctly with probability at least 2/3.
The construction builds on OpenAI’s “Subset Sum in Time O(2^{0.49n})” (4 October 2026), retaining its framework of isolation, a capped preliminary search, a modular checksum, and alias subtraction. Two components are replaced: an alias procedure that computes phase sums using a smaller modulus, and a family of 43 checksum filters selected according to the input pattern.
The analysis combines an averaged small-layer lemma, grouping by reduced and unreduced subset sums, and rigorous numerical certificates. Verification scripts using Arb ball arithmetic check the alias constraints and 430 moment inequalities. The argument relies on results from the earlier paper, which are restated in an appendix; the numerical certificates verify specified inequalities rather than formally verifying the entire proof.
We formulate a time-series game in which each player chooses a hidden Markov model (HMM) as a mixed strategy. The HMM generates a finite, variable-length sequence of probability vectors, and the payoff is assigned to the completed game as the sum of its stage rewards. We prove a conditional Nash-equilibrium existence result for games with a common finite bound on output lengths. The proof applies a fixed-point argument to the best-response correspondence when each player's admissible HMM-density set is nonempty, convex, and compact in (L1), and the completed-game payoff is bounded and measurable. A common BV bound together with (L1)-closedness can ensure compactness; closure under suitable disjoint-state mixtures can ensure convexity. These properties do not follow from the HMM representation or BV regularity alone and must be verified for a specified model. Consistent estimation of HMM densities may support computation, but by itself does not establish equilibrium existence or replace the theorem's assumptions.
A mix of human-written and AI-generated textHuman understanding: some parts
For a nonempty permutation β of length m, let g2(β) denote the number of distinct permutations of length m+2 containing β. Ray and West showed that
$$
g_2(\beta)=\frac{m^4+2m^3+m^2+4m+4-2j(\beta)}{2},
\qquad 0\le j(\beta)\le m-1,
$$
but their correction term was not expressed as a statistic of β. This was posed explicitly as Problem 3.7 by Vatter. We give a polynomial-time formula defined entirely from β: j(β)=L(β)+R(β), where L counts licensed gaps and R counts anchor-free two-deletion classes containing a collision whose canonical alignment support is a permutation interval. We also derive the Ray–West bound intrinsically and characterize its upper-bound equality cases as precisely the layered or reverse-layered permutations. The proof is combinatorial. Exact computation is used only for independent corroboration; the accompanying proof-and-replay archive contains the full proof supplement and bounded verification. This preliminary technical report was first deposited on Zenodo on August 9, 2026: https://doi.org/10.5281/zenodo.21857882.
Let g be a finite-dimensional simple complex Lie algebra with root lattice Q and lacing number t, equipped with an invariant bilinear form normalized so that long roots have squared length two. We prove C2-cofiniteness of the
Feigin--Tipunin logarithmic W-algebra $\W_p(Q)$ for
p=tm≥2 and $m\geq h^\vee_{{}^L\g}-1$, where
$h^\vee_{{}^L\g}$ is the dual Coxeter number of the Langlands dual.
Drinfeld--Sokolov reduction of induced affine Weyl modules gives
graded generators for the multiplicity spaces of the conformal group
action. Their degree bounds show that the intrinsic C2-algebra
is a finite module over the polynomial C2-algebra of its principal
invariant subalgebra. Adjoint contractions produce a finite relation
ideal in this polynomial ring. Heisenberg normal ordering and a
constant-coefficient differential equation show that its common zero
is the origin. The argument gives explicit sufficient bounds and
strong fin
Primarily AI-generated textHuman understanding: no parts
We study adjoining a half-bandwidth signed cubic trace and a spectral count ladder to a bandwidth-one trace/Frobenius model motivated by zeta-zero certificates. In a finite circle model with mass N = 64, flat windows, and explicit budget centers, we prove exact absorption over the full atom-only class: the augmented and baseline optima agree for both the distinct-count and simple-fraction objectives. Three explicitly specified integer-marked grid columns attain the lower bounds. Their mixing weights are defined algebraically, and an Arb certificate proves all strict feasibility and inertia inequalities. The witness satisfies the ladder at every positive threshold, not just on a finite threshold grid. We also give the exact conjugate-pair spectrum, the isolated-block cubic identity, grid Parseval, and the sharp conditional spectral-capacity constant 2 sqrt(2). The result is a finite model theorem, not an unconditional statement about zeta zeros. Version 2 corrects an infeasible witness, a cubic-tolerance mismatch, and an incorrect rationality claim in version 1. Global absorption in the pair-containing or pinned classes remains unproved; equality on a searched column dictionary and heuristic pricing do not establish it.
Primarily AI-generated textHuman understanding: some parts
We prove two exact coherence results for marked handles in Walker's extended cobordism category. At fixed disjoint sites, a matching pair of forward and reverse phases contributes one local S1×S2 summand, whereas a mismatch contributes an identity cylinder. Thus the resulting double depends only on the total Walker weight and the number of phase matches, not on the orders of attachment. For positive handleslides supported in pairwise-disjoint protected sectors, we construct the complete marked Boolean cube and prove that every two-face commutes with weight zero; consequently, every finite route has the same composite and the endpoint identities telescope. An optional odd-prime SO(3)-TQFT corollary evaluates the fixed-site double as κpN(DpGM)b times the identity. Overlapping slides and their pentagon are treated by a separate companion theorem.
We prove that the kissing numbers in 19 and 21 dimensions are at least 12460 and 31560, improving the best published bounds 11948 (Ho) and 29768 (Cohn and Li). Both constructions start from the odd-sign configurations of Cohn and Li, which leave room around the sign vectors they add. We use this room to reshape the added vectors so that codewords closer than n/4 can be used together. In 21 dimensions each added vector has entries ±2 on one line of the projective plane of order 4 formed by the weight-5 words of the punctured Golay code, and ±3 elsewhere; a linear labelling of the codewords by lines gives 3840 new points, and the proof is by hand. In 19 dimensions we cut the 20-dimensional configuration by a hyperplane and project the sign vectors of the 20-dimensional code into it. The projections have a 16-coordinate head and a three-dimensional tail, the tails separate words whose heads are close, and one point for each of the 1792 heads with a nonzero tail can be added, where Ho adds 1280.
Primarily AI-generated textHuman understanding: all partsKissing Number
Lech Mazur proved that a positive proportion of integers n reach 1 under the Collatz map in at most 10.46logn steps. We give a quantitative refinement of his construction. For any X≥222287640, at least X/22287640 positive integers n<X reach 1 within $\floor{10.46\log n}$ ordinary steps, and $\floor{3.4881\log n}$ accelerated steps. We also obtain results for paths that include any prescribed integer t not divisible by 3. AxiomProver was used to obtain and formalize this result in Lean.
A mix of human-written and AI-generated textHuman understanding: some partsCollatz conjecturearithmetic dynamics
Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník
Sah, Sawhney, Stoner and Zhao conjectured that for every spin model whose interaction matrix has exactly one positive eigenvalue, the partition function of any graph is at most a product of biclique partition functions determined by its edge degrees. We disprove this conjecture with a strictly positive six-spin model on a triangle with three pendant edges; the violation persists on an open set of nonsingular models. The mechanism is a negative mixed covariance trace at vertices of degrees two, three and four: the triangle contributes a favourable cubic term, whereas all biclique corrections start at fourth order.
Primarily AI-generated textHuman understanding: some partsantiferromagnetic modelsgraph homomorphismspartition functions
Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník
Let M be a uniformly random matching of a finite graph, let D=ν−E∣M∣ be its expected deficiency from a maximum matching, and let V=Var(∣M∣). Kahn showed that D and V tend to infinity together, proved D=O(V4) when V is bounded below, and conjectured D=O(V3). We prove D≤41V+104V2+64V3 for every graph, confirming the conjecture, and show that the exponent is attained by connected bipartite graphs with perfect matchings. The proof combines alternating-path identities for monomer probabilities with the sharp estimate ∣V(G)−V(G−v)∣≤1/4.
Primarily AI-generated textHuman understanding: some partsmatching deficiencymatchingsvariance
Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník
A connectivity code on a graph H is a family of edge sets of H whose pairwise symmetric differences are connected spanning subgraphs. Alon asked whether a nonlinear connectivity code can ever be larger than every linear one on the same graph. We answer this affirmatively: we construct a 4-regular graph on fifteen vertices on which the largest connectivity code has nine members and the largest linear connectivity code has eight. The code is obtained by lifting a quaternary code of length six with pairwise agreement one through clique gadgets.
Primarily AI-generated textHuman understanding: some partsconnectivity codesgraph codesnonlinear codes
Bannai, Munemasa and Venkov, and later Nebe and Venkov, excluded tight spherical designs — among them strongly regular graphs of a special kind — by studying the lattice spanned by the design: they bound its discriminant group, pass to an even (over)lattice, and compare Gauss sums by the Milgram–Braun formula. We extend this method to arbitrary strongly regular graphs with integral eigenvalues. For an integral representation G=b(A-ρ I)+cJ that is positive semidefinite with even diagonal, an elementary identity for dual vectors ties every class of the discriminant form of a maximal even overlattice to the class of a rescaled sum of the vertex vectors. This gives a necessary condition on the parameters (v,k,λ,μ) alone: a finite quadratic module with a short list of properties must exist. Its simplest case applies the textbook congruence det≡(-1)^(n/2) (mod 4) for even lattices of odd determinant to the centred representation. A sweep of Brouwer's table (v≤1300) finds 70 complementary pairs of parameter sets listed as open that fail the condition, among them (96,35,10,14), the triangle-free set (176,25,0,4) and (169,42,5,12); the grade of evidence is given for each (hand proof, or exact computation by at least two independent programs). No existing parameter set of the table fails in any representation tested, real graphs satisfy every condition with their actual discriminant forms, and the method gives short independent second proofs of the known nonexistence of srg(75,32,10,16) and srg(289,54,1,12). The simplest case also yields an infinite family: if q is a power of a prime p≡1 (mod 4) and m is odd, there is no strongly regular graph with the parameters (q^2, m(q+1), -q+m^2+3m, m(m+1)) of the negative-Latin-square type NL_m(q). This family, the parity case and the nonexistence of srg(169,42,5,12) are formalized in Lean 4 with Mathlib. Also deposited on Zenodo with the full data archive (the citable record): https://doi.org/10.5281/zenodo.23281359
Primarily AI-generated textHuman understanding: some parts
This paper organizes finite tensor–quiver reconstruction for reductive source data. For a fixed reductive Zariski-closed subgroup H of a reductive complex group G and a faithful finite-dimensional representation, the standard mixed-tensor reconstruction theorem and Noetherianity imply that finitely many H-invariant tensors have common stabilizer H. Classical quiver invariant theory then implies that finitely many traces of oriented cycles separate closed orbits, including semisimple realizations.
For GL₂/ℚ, we give a restricted application of Cheng’s symmetric-square limiting distribution. After a finite-level cuspidal block has been specified and a target constituent selected by an interpolating Hecke test function built from that block’s Hecke characters, ratios of trace-formula distributions recover the first two Satake power sums at auxiliary unramified places, provided Cheng’s hypotheses hold and the full selected coefficient is nonzero. This is post-selection recovery, not a method for identifying the target constituent, and applies only to the nonzero-coefficient visible subset.
Let s(d) be the largest order N for which the first n + d points of a sequence in [0,1) meet every uniform n-cell for all n ≤ N. We prove s(d) = d/(1 − log 2) + O(√d), determining the limit asked for by Anholcer et al. and disproving their conjectured coefficient. For strong piercing, which requires meeting every interval of length 1/n, we obtain an exact finite harmonic capacity and rational constructions. A Dirichlet covering lemma and a packing estimate for disjoint gaps transfer the capacity bound to ordinary grids. We also solve the finite largest-gap minimax problem, establish rigidity for infinite heap refinements, and extend the asymptotic estimate to affine arrival budgets. Ordinary near-extremizers satisfy universal laws for root sizes and first-split times, with a corresponding entropy correction. An explicit finite obstruction shows why root constraints alone do not characterize ordinary piercing.
A mix of human-written and AI-generated textHuman understanding: some partsSteinhaus problementropyharmonic capacityinterval refinementirregularities of distributionpiercing sequences
Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník
Erdős and Straus asked whether, for each fixed n≥2, every sufficiently large integer m admits 1≤k≤m−n with (nn+k)∣(km+k). We prove that it does; previously this was known only for a set of m of density one. The witness can be taken with k=On,ε(mε) for every ε>0: a factorial identity reduces the problem to finding a small value of a fixed polynomial coprime to (nm), which an elementary sieve provides. Conversely, the least witness is at least logarithmic in m along an infinite sequence.
Primarily AI-generated textHuman understanding: some partsErdős–Strausbinomial divisibilitysmall witnesses
Haglund introduced the partial sums ΞN=∑n≤NΦn of Riemann's expansion of the Ξ-function in incomplete gamma functions and conjectured that for every N the zeros of ΞN in the closed first quadrant, listed by increasing real part, have nondecreasing imaginary parts. The conjecture implies the Riemann hypothesis; Haglund's computations supported a weak form of it for N≤10. We show that the conjecture is false for N=27: Ξ27 has a real zero in (3144.8946,3144.8947) and a non-real zero z0=3143.2206824215…+0.3152587993…i in the first quadrant, whose real part is smaller. The weak form fails as well. Both facts are certified twice, independently, in interval arithmetic (Arb ball arithmetic; outward-rounded intervals with proved truncation bounds), and z0 is enclosed in a square of half-width 10−48. The same violation occurs at N=24 for the companion approximants ξN=ΞN+cN. The failure has a mechanism: on the critical line ΞN<Ξ<ξN, so the real zeros of ΞN lie in positive lobes of Ξ that rise above a positive level, and a low lobe followed by a higher one near the height 4(N+1)2, where the approximant departs from Ξ, can leave a non-real zero below real zeros. The same structure shows that the number of positive real zeros of ΞN, counted with multiplicity, is odd, and that x2ΞN(x) tends to a negative limit on the real axis for every N.
Primarily AI-generated textHuman understanding: some parts
We prove that for every real ξ ̸= 0 the sequence of nearest integers [ξ(3/2)^n + 1/2], n ∈ N, contains infinitely many elements divisible by 2 or by 5. In particular, for ξ > 0 infinitely many of these nearest integers are composite. This is the nearest-integer counterpart of the theorem of Forman and Shapiro on [ξ(3/2)^n], for which Dubickas and Novikas give the unavoidable set {2, 5, 7, 11}; for nearest integers two primes suffice. The proof follows the method of Dubickas and Novikas: if 2 and 5 divide only finitely many terms, the sequence of operations is forced to become (A^2B^2)^∞, which is impossible by their Lemma 2. All results are machine-verified in Lean 4.
Primarily AI-generated textHuman understanding: some partsuniform distribution
Massera and Schaffer [\textit{Ann. Math. (2), 1958}] derived a breakthrough upper bound for the Clarkson angle between two nonzero vectors in a normed linear space, which was later improved by Maligranda [\textit{Am. Math. Mon., 2006}]. Pecaric and Rajic [\textit{Math. Inequal. Appl., 2007}] extended Maligranda's inequality to finitely many nonzero vectors. We derive a non-Archimedean version of Massera-Schaffer-Maligranda-Pecaric-Rajic inequality.
Primarily human-written textHuman understanding: all partsNon-Archimedean linear spaceNormed linear spaceTriangle inequalityUltra-norm.
Let f be a random completely multiplicative function with independent symmetric signs at the primes. We study rare constant runs and prescribed words at logarithmic lengths, despite long-range multiplicative identities. A uniform weighted two-window square-relation estimate, with critical bound Oε(N5/3+ε), separates exact rational relations from residual components.
At critical intensity, this estimate gives Poisson comparisons retaining positions, exact excess lengths and signs. For dictionaries, uniform bounds under size and overlap conditions are complemented by typical bounds obtained by averaging the distance after fixing the dictionary. In a quantitatively controlled regime of diverging intensity, an exact one-word coupling by largest-odd-prime pivots compares the full signed run field under prescribed small-prime conditioning. For fixed admissible window parameters and almost every fixed realization of f, run-start counts in prescribed near-macroscopic windows satisfy an empirical Poisson law along dyadic scales.
In the exceptionally rare regime, a relative marked comparison separates the boundary event from bulk occurrences. Conditional on existence, the first location, sign and overshoot have a two-source lattice law with scale-dependent weights. The two overshoots use different clocks: prime rank at the boundary and integer distance in the bulk.
Let K=Q(−3). We prove that the Weil classes of every polarized abelian sixfold of Weil type for K with Hermitian discriminant −2 are algebraic. This is a non-split discriminant, to which the theorems of Schoen and of Markman on split Weil classes do not apply. The proof uses Prym--Tyurin varieties of covers of P1 with monodromy the complex reflection group G32, and combines known mechanisms in a new configuration. First, on every such variety the Weil classes of the six-dimensional isotypic factor are algebraic. They are obtained, by a Lefschetz-class correspondence supplied by Milne's theorem, from Schoen's algebraic Weil classes on fifty-one cyclic intermediate quotients; the choice of quotients is governed by an integral identity among characters of a torus, verified by an exact computation. Second, the ten-dimensional family of these covers dominates the nine-dimensional moduli space: we exhibit an explicit cover at which the period map has rank nine, by a certified computation. Third, a Hilbert-scheme and Baire-category argument spreads algebraicity from an open set to the whole moduli space. The discriminant of the family is identified exactly. All computations are supplied as reproducible certificates. For the members whose special Mumford--Tate group is the full special unitary group, an invariant-theoretic lemma then gives the Hodge conjecture for every power. This version, with its computational certificates and review bundle, is archived on Zenodo: https://doi.org/10.5281/zenodo.23377274
Primarily AI-generated textHuman understanding: some partsHodge conjecturePrym-Tyurin varietiesWeil classesabelian varieties of Weil typecomplex reflection groupsperiod maps
Contributed by Mohammed Abouzaid, Nikhil Srivastava, Rachel Ward, Lauren Williams
To assess the ability of current AI systems to correctly solve research-level mathematics problems, we tested several AI systems on our ``third batch'' of ten problems in a broad range of mathematical fields; these problems arose naturally in the research process of the contributors. This document includes the problems, our methodology, and the results of our testing. We assessed the AI-generated solutions based on mathematical correctness, quality of exposition, and quality
of attribution and citations. We provide links to supplementary documents including the human solutions, the AI-generated solutions, and the referee reports and logs for the AI-generated solutions.
Primarily human-written textHuman understanding: all partsAI systemsbenchmarklarge language models
Let W:X→Ygeom be a completed oriented compression body with N≥3 ordered product 1-handles and a supplied protected three-handle sector. We realize one positive handleslide pentagon by five diffeomorphisms with compatible complete endpoint markings. An isotopy fixed on the entire protected boundary collar identifies the two-edge and three-edge route composites, so the comparison takes place at the level of marked maps rather than homology matrices.
The construction fixes the marked disk systems at all five vertices before choosing any edge map, so incident edges use the same endpoint data. The cut-ball maps intertwine all fifteen disk pairings on common full-normal intervals and therefore descend smoothly through the paired faces. Extending these quotient maps by the identity gives global representatives. The same maps satisfy the finite compatibility conditions, including at N=3, when the unused-handle domains are empty but remain part of the typed construction.
The resulting boundary maps preserve the embedded meridian and product-framed dual-longitude rows, so their homological actions have blocks Me and Me−T with no mixed shear. They also preserve the same rational disk-kernel Lagrangian. Walker's forward multiplication law lifts the relative isotopy to a literal weight-zero equality of mapping cylinders, including the seam and route weights.
A mix of human-written and AI-generated textHuman understanding: some partscompression bodymarked handlesliderelative isotopy
Dick, Goda and Suzuki asked for low-discrepancy digital nets that are quasi-uniform, with minimum distance δ and covering radius both of the optimal order N−1/s, and stated the problem as open in dimension s≥3. We construct such nets in every dimension. For every s≥3, every base q whose prime factors are at least s−1, and every k≥1, we give explicit digital (0,sk,s)-nets over Z/qZ with N=qsk points, covering radius at most sN−1/s, and δN1/s≥1−as∗/q once q≥5as∗+4. The constant as∗≤eslogs+O(s) is computed from s by a deterministic polynomial-time lattice algorithm. In dimension three every N=pm, m≥3, is covered for each prime p≥37, and, with computer assistance, every N=2m and N=3m. In prime bases these families are well separated in the sense of Dick, Goda and Suzuki, and they stay so under every digital shift. One mechanism underlies all constructions. For N=qsk it reads as follows. The first k base-q digits of a point's coordinates select a cube of side q−k, and a (0,sk,s)-net has exactly one point in each such cube. Digit k+1 of each coordinate is a linear function of the first k digits of all coordinates, because the net is digital. We choose this linear function so that, between the planes where this digit wraps around modulo q or where a carry occurs among the first k digits, the points lie on a translate of one lattice; for the main theorem this lattice is a slightly distorted copy of Zs. All later digits move a point by less than 1/q of the cube side. Part~II studies the constant K in δN1/s=1−K/q+O(q−2). We prove lower bounds on K when the linear maps have bounded coefficients. At N=q3 we find the optimal K among nets whose points lie on lines parallel to the main diagonal. With the cube-tiling theorems of Haj\'os and Keller we show that idealized lattices with one point per cube, whose points sit independently inside their cubes, cannot exceed δN1/s=1. When the positions inside the cubes are coupled, the value 1 can be exceeded: δN1/3→281/16 at N=p3, δN1/5>1 for (0,5,5)-nets from p=37 on, and δN1/s→min(s/2,2) at N=psk, which is the E8 value at s=8 (limits as p→∞). The results are formalized in Lean~4, apart from some finite computations, which are checked by scripts.
We classify one-dimensional families of endomorphisms on P2 that are algebraically stable, generalizing a theorem of McMullen. The non-isotrivial families are either homogeneous lifts of flexible Lattes maps on P1, or a family of Lattes maps arising from abelian surfaces.
Primarily AI-generated textHuman understanding: some partsComplex dynamicsalgebraic dynamics
A characteristic-zero counterexample to the Little Finitistic Dimension Conjecture is given, arguably providing a simpler variant of OpenAI's construction.
Primarily AI-generated textHuman understanding: some partsLittle Finitistic Dimension Conjecture
We construct a bijection between tensor equivalence classes of unipotent tensor categories over
$\C$ and continuous Lie quasibialgebras with linearly compact pronilpotent underlying Lie algebra,
modulo continuous isomorphisms and classical twists. The bijection is defined using a fixed
universal quantization functor and solves the problem raised in Remark~4.6 of arXiv:0708.1487 by Etingof and Gelaki. The proof combines reconstruction with filtered dequantization modulo twists and
convergence of universal formulas. A category admits a tensor fiber functor precisely when a
classical twist removes its curvature.
Primarily AI-generated textHuman understanding: some partsTensor categoriesquasi-Hopf algebras
Let Sm⊂P3 be the complex Fermat surface x0m+x1m+x2m+x3m=0 of prime degree m≥5. Its Néron–Severi lattice NS(Sm) is generated by the classes of the 3m2 lines on Sm (Schütt–Shioda–van Luijk, Degtyarev). We prove that ∣discNS(Sm)∣=m3(m−3)2 (the Watermark Theorem), which answers, for prime degree, the determinant question asked by Shioda in 1987, and we determine the discriminant group itself: NS(Sm)∗/NS(Sm)≅(Z/m)3m2−24m+59×(Z/m2)3m−16 for m≥7, and (Z/5)10×Z/25 for m=5 (the Double Ladder Theorem). The transcendental lattice of Sm has the same discriminant group. Both proofs work with the explicit integer intersection matrix G of the lines and use only elementary lattice arithmetic: the action of G on functions pulled back along the m+1 pencils of directions of (Z/m)2, an explicit sublattice of kerG of index m12, and two index computations for the order; an integer identity GYG=6m2G for the exponent, and a factorization of G modulo m through twelve one-variable "shadows" for the number of cyclic factors of order m2. The lattice statements of both theorems are checked in the proof assistant Lean 4 with the Mathlib library, with no axiom beyond the three standard ones; the identification of G with the intersection matrix of the lines is proved by hand and checked by computer. This work has not been refereed by a human expert.
Primarily AI-generated textHuman understanding: some partsFermat surfacesLean 4Néron–Severi latticediscriminant grouplatticeslines on surfaces
Contributed by Mark R. Diamond, Angela O'Brien-Malone
A continuous quantity, observable only after nearest-lattice rounding, is
estimated by the mean of n rounded readings of an imperfect instrument.
Averaged over the position of the true value within a rounding cell, the
mean-squared error of this estimator is a function of one dimensionless
parameter: the pre-rounding standard deviation s, measured in cell widths.
A perfectly repeatable instrument (s=0) returns the same rounded value on
every reading, so averaging provides no additional information; its
cell-averaged error is the rounding floor 1/12. We show that this floor is
never optimal once more than one reading is taken. For \emph{any} centred
pre-rounding law with finite, non-zero variance and any sample size n>1,
some strictly positive dispersion yields a smaller cell-averaged error than
zero, so the global minimizer lies away from the no-scatter endpoint; and each
of the three hypotheses---n>1, finite variance, non-degeneracy---is
individually necessary, so existence is \emph{exactly} characterized. The
mechanism is elementary: the autocorrelation of the rounding error has a
\emph{corner} at zero lag, so a little scatter relieves the rounding bias at
first order in s while its variance contribution grows only at second order.
We give the short corner-based proof and then derive the spectral form of the
criterion, using it to compute the optimum for four laws. The Gaussian
optimizer is unique and solves a Jacobi-theta equation. For the Laplace law
the bias term has a hyperbolic closed form, and the optimizer is unique and
characterized by the corresponding derivative equation. For the uniform and
two-point laws the optimizer is explicit. The uniform law cancels every bias
harmonic exactly at its first positive cancellation scale, which saturates
the natural localization bound when n=2; and the two-point law, whose
criterion has several local wells once n≥4, shows that finite variance
ensures existence but not a single-well objective.
A mix of human-written and AI-generated textHuman understanding: all partsFourier seriescharacteristic functionsmean-squared errorquantization errorrounded sample meanrounding error
Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník
We show that for every fixed prime power q and 0<p<1, a uniformly random Fq-linear rank-metric code in Fqn×n of rate (1−p)2−ε is, with high probability, average-radius list-decodable at radius pn with list size Op,q(1/ε). This answers a question of Guruswami, Li and Singhal; the case q=2 was known. The failure probability is explicit and remains small when ε depends on n with εn2→∞. The proof combines a weighted affine extraction, span estimates for matrices from nearby rank shells, and Gaussian summation over rank profiles.
Primarily AI-generated textHuman understanding: some partsaverage-radius list decodingcapacityrandom linear codesrank metric
We construct a singleton-labeled arbor on 243 vertices whose Ehrhart polynomial has a nonreal root. This disproves the conjecture that the Ehrhart roots of every arbor polytope are real and lie in [−1,0]. The polynomial factors as (m+1)230(7m+2)8f(m)/6144, where f is an explicit quintic. We derive this identity by counting lattice points for every nonnegative integer dilation, using three moments of subtree deficits. A negative value of f′2−ff′′ at −3/10 gives an exact nonreal-root certificate. The lattice-count identity and the resulting counterexample have been formalized and checked in Lean. The source release and reproduction evidence are archived at https://doi.org/10.5281/zenodo.23435787.
Primarily AI-generated textHuman understanding: no partsEhrhart polynomialsLean formalizationarbor polytopescounterexamplereal-rootedness