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003

Number theory 3 manuscripts, 257 pages Lean

Proves that every Dirichlet L-function, including $\zeta(s)$ , is zero-free in $\Re s\gt 7/8$ , resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over $\mathbb Q(\sqrt{-3})$ . A companion gives a different proof of the zero-free half-plane $\Re s\gt 11/12$ .

Closest manuscript: The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$

017

The irrationality exponent of π is 2

Number theory 1 manuscript, 23 pages Lean Reasoning summary

Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational $p/q$ satisfies $|\pi-p/q|\ge q^{-2-\varepsilon}$ . This also proves convergence of the Flint–Hills series $\sum_{n\ge1}1/(n^3\sin^2 n)$ , with angles in radians.

262

Sharp finite-matrix Lieb–Thirring inequalities and all equality cases

Mathematical physics 3 manuscripts, 73 pages Lean

Proves the sharp one-dimensional Lieb–Thirring inequality for $1/2\lt \gamma\lt 3/2$ and arbitrary finite-matrix potentials W ≥ 0 with $\int\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty$ : the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar sech² solitons with independent scales and centers, and zero channels.

Closest manuscript: Sharp one-dimensional Lieb–Thirring constants

007

Ordinary two-point correlations and the corrected Elliott conjecture

Number theory 1 manuscript, 87 pages Lean Reasoning summary

Proves the ordinary two-point Chowla conjecture, with a bound $O(X/(\log X)^c)$ for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times $n^{it}$ for $|t|\le X$ .

011

Prime-factor statistics of

p-1

Number theory 3 manuscripts, 187 pages

Proves that the normalized ordered logarithms of the prime factors of $p-1$ , counted with multiplicity, converge jointly to the Poisson–Dirichlet law $\mathrm{PD}(1)$ as p ranges uniformly over primes up to x and $x\to\infty$ . This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than $n^{1-\varepsilon}$ totient preimages, for every ε > 0.

Closest manuscript: Weighted dilation graphs, smooth shifted primes and totient fibers

022

The weak inhomogeneous Duffin–Schaeffer conjecture

Number theory 1 manuscript, 87 pages

Proves that for every real shift γ and finite-valued $\psi:\mathbb N\to[0,\infty)$ , divergence of $\sum_q\phi(q)\psi(q)/q$ implies $\|qx-\gamma\|\lt \psi(q)$ for infinitely many q, for almost every x. Here ϕ is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on γ is needed.

078

The three-dimensional Bochner–Riesz conjecture

Real and complex analysis 1 manuscript, 114 pages

Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on $L^p(\mathbb R^3)$ for every $1\le p\le\infty$ whenever $\delta\gt \max\{3|1/p-1/2|-1/2,0\}$ .

029

Primitive roots for every admissible integer base

Number theory 2 manuscripts, 113 pages

Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least $c_a x/(\log x)^2$ primes in every sufficiently large interval $(x,2x)$ have primitive root a, with $c_a\gt 0$ .

Closest manuscript: Primitive roots for every admissible integer base

075

The

L\log L

Fourier-convergence conjecture

Real and complex analysis 1 manuscript, 76 pages

Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in $L\log L(\mathbb T)$ converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the $L\log L$ scale.

023

Patterson's first moment for cubic Gauss sums

Number theory 1 manuscript, 52 pages Lean

Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order $X^{5/6}/\log X$ . Every fixed nonzero prime-angle Fourier mode has smaller order.

232

Gaussian fields and interfaces for triangular-lattice Lipschitz heights

Probability and statistical mechanics 3 manuscripts, 220 pages

Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments $0,\pm2$ and two-arc boundary values $\pm1$ , and zero-boundary integer Lipschitz heights weighted by fixed $x\in[1/\sqrt2,1]$ . Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal SLE₄, establishing Schramm’s real-field/interface predictions.

Closest manuscript: The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data

005

Irrationality of Catalan’s constant

Number theory 1 manuscript, 44 pages Lean

Proves that Catalan's constant $G=\sum_{j\ge0}(-1)^j/(2j+1)^2$ is irrational.

082

Annular variation and dyadic absolute bounds for the triangular Hilbert transform

Real and complex analysis 3 manuscripts, 75 pages Lean

Proves maximal and annular r-variation bounds, for every r > 2, from complex $L^3(\mathbb R^2)\times L^3(\mathbb R^2)$ to $L^{3/2}(\mathbb R^2)$ . The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $L^3\times L^3\times L^3$ point.

Closest manuscript: Annular variation of the triangular Hilbert transform at the symmetric point

153

Arithmetic classification and non-Pisot singularity for Bernoulli convolutions

Dynamical systems and ergodic theory 1 manuscript, 36 pages

Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every $\lambda\in(0,1)$ by an infinite, one-sided approximation condition using explicit finite sets of algebraic units. It also proves singularity at reciprocals of every quartic Salem number in $(1,2)$ , giving examples beyond reciprocal Pisot parameters.

072

Brennan's conjecture and the integral-means spectrum

Real and complex analysis 2 manuscripts, 46 pages Lean

Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, $|\phi'|^s$ is area-integrable for $4/3\lt s\lt 4$ . The sharp universal integral-means identity is $B_{\mathcal S}(t)=|t|-1$ for t ≤ −2. A strict bound $B_b(-1)\lt 1/4$ for bounded univalent functions disproves Kraetzer's prediction at that parameter.

217

The low-temperature Sherrington–Kirkpatrick fluctuation law

Probability and statistical mechanics 2 manuscripts, 206 pages

For every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to $c_\beta n^{1/3}$ , with $c_\beta\gt 0$ , confirming the predicted n^1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.

Closest manuscript: The low-temperature Sherrington–Kirkpatrick fluctuation scale

024

An asymptotic formula for the number of totients

Number theory 1 manuscript, 39 pages Lean

Gives an asymptotic equivalent for the number $V(x)$ of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, $V(cx)/V(x)\to c$ for every fixed c > 0, answering Erdős and Hall’s scaling question.

370

The Lane–Emden and Hénon–Lane–Emden conjectures

Partial differential equations 1 manuscript, 20 pages Lean

Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, $p,q\gt 0$ and real A, B, the system $-\Delta u=|x|^A v^p$ , $-\Delta v=|x|^B u^q$ has no positive entire solution when $(n+A)/(p+1)+(n+B)/(q+1)\gt n-2$ , with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for n ≥ 3 and $A,B\gt -2$ .

034

Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity

Algebraic and complex geometry 14 manuscripts, 781 pages

Using logarithmic Iitaka subadditivity, proves log abundance in every dimension for normal compact Kähler log canonical pairs with effective rational boundary: an analytically nef ℚ-Cartier adjoint is semiample. It also proves projective log abundance over every algebraically closed field of characteristic zero and the effective Iitaka fibration conjecture for smooth projective varieties of nonnegative Kodaira dimension in that setting. A further result resolves the finite-rational-coefficient index conjecture for connected projective semi-log-canonical log Calabi–Yau pairs over such fields in each fixed dimension and for each fixed finite set of rational boundary coefficients, with a uniform index independent of the number of components.

Closest manuscript: Log abundance in characteristic zero

027

Potential integral density on curve character varieties

Number theory 1 manuscript, 27 pages

Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SL_r character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.

076

Real ultraflat Littlewood polynomials and unbounded binary merit factors

Real and complex analysis 3 manuscripts, 70 pages Lean

Constructs polynomials with N consecutive coefficients in $\{-1,1\}$ whose modulus is $(1+o(1))\sqrt N$ uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.

Closest manuscript: Nearly minimal maxima and positive minima of Littlewood polynomials

006

Goldfeld’s conjecture: densities and mean analytic rank

Number theory 2 manuscripts, 171 pages

Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.

Closest manuscript: The mean analytic rank of quadratic twists of elliptic curves

216

Critical and near-critical XY scaling and BKT universality

Probability and statistical mechanics 6 manuscripts, 461 pages

For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations $C_{\beta_c}(r)\sim Ar^{-1/4}(\log r)^{1/8}$ and the Berezinskii–Kosterlitz–Thouless essential singularity $\sqrt{\beta_c-\beta}\log\xi(\beta)\to B$ , with $A,B\gt 0$ after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes. Critical center-magnetization and spin-field conclusions retain their stated height, renormalization, and field-input assumptions.

Closest manuscript: The Critical Spin Field of the Planar XY Model

012

Independent largest prime factors of consecutive integers

Number theory 1 manuscript, 84 pages Lean

Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and $n+1$ are asymptotically independent in ordinary natural density. In particular, the integers satisfying $P^+(n)\lt P^+(n+1)$ have density 1/2.

080

The exact Sobolev endpoint for Schrödinger convergence

Real and complex analysis 2 manuscripts, 179 pages

Proves almost-everywhere convergence $e^{it\Delta}f\to f$ as $t\downarrow0$ for every $f\in H^{n/(2(n+1))}(\mathbb R^n)$ and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H^1/3.

154

Pointwise multiple ergodic averages for mixing transformations

Dynamical systems and ergodic theory 4 manuscripts, 165 pages

Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.

Closest manuscript: Triple ergodic averages with distinct integer slopes

090

Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy

Convex and metric geometry 4 manuscripts, 199 pages Lean

Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for $0\lt s\lt 2$ and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.

Closest manuscript: An atomic certificate for triangular-lattice universal optimality

002

The full BSD formula from low Selmer corank

Number theory 3 manuscripts, 340 pages

Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.

Closest manuscript: The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one

170

Sharp logarithmic exponents for off-diagonal Ramsey numbers

Combinatorics 2 manuscripts, 94 pages Lean

For every fixed integer s ≥ 5, proves $r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)}$ as $t\to\infty$ , determining the logarithmic exponent and matching the classical upper bound at that scale. Here $r(s,t)$ is the least number of vertices forcing an s-clique or a t-vertex independent set.

Closest manuscript: Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers

369

The hot spots conjecture for simply connected planar domains

Partial differential equations 1 manuscript, 21 pages Lean

Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.