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1. 109
   Integer multiplication below $n\log n$

   Theoretical computer science 1 manuscript, 73 pages

   Multiplies two n-bit integers exactly at every input length in deterministic worst-case time $O(n(\log n)^{1-\kappa})$, with $\kappa=2^{-182}$, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen $n\log n$ optimality conjecture in the ordinary multitape bit model.

2. 107
   Matrix multiplication with exponent at most 9/4

   Theoretical computer science 3 manuscripts, 131 pages Lean

   Proves $\omega\le9/4$ over ℂ, giving $O_\varepsilon(n^{9/4+\varepsilon})$ arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension n^a with a > 0.465 permits $n^{2+o(1)}$ rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.

   Closest manuscript: Complex Matrix Multiplication Below 2.258 and Rectangular Bounds

3. 142
   Deterministic polynomial factorization over prime fields

   Theoretical computer science 1 manuscript, 48 pages

   Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field 𝔽_p, including multiplicities, in bit complexity polynomial in $(n+1)\log p$. The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.

4. 130
   Exact Fourier transforms below $n\log n$

   Theoretical computer science 2 manuscripts, 80 pages Lean

   Gives a deterministic length-n discrete Fourier transform algorithm using $O(n(\log n)^{1-\delta})$ operations for every n, with explicit $\delta=10^{-13}$. The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing.

   Closest manuscript: An explicit power saving for the exact discrete Fourier transform

5. 135
   Homogeneous depth-five lower bounds for iterated matrix multiplication

   Theoretical computer science 1 manuscript, 22 pages Lean

   Over every characteristic-zero field, the $(1,1)$ entry of a product of n independent $n\times n$ variable matrices requires $n^{\Theta(\sqrt n)}$ gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.

6. 116
   Uniform black-box noncommutative identity testing across characteristics

   Theoretical computer science 3 manuscripts, 52 pages Lean

   For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation.

   Closest manuscript: Uniform Matrix Hitting Points in Every Positive Characteristic

7. 124
   Polynomial-time scheduling on three identical machines

   Theoretical computer science 1 manuscript, 22 pages Lean

   Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule.

8. 279
   Exact quantum factoring over a fixed finite gate set

   Mathematical physics 1 manuscript, 24 pages Lean

   Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices.

9. 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 $|t|\le X$.

10. 238
    Optimal logarithmic mixing of the Thorp shuffle

    Probability and statistical mechanics 12 manuscripts, 683 pages Lean

    Proves that the Thorp shuffle randomizes $N=2^d$ labeled cards in $\Theta(\log N)$ physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions.

11. 138
    Subset Sum in $O(2^{0.49n})$ time

    Theoretical computer science 2 manuscripts, 69 pages

    Gives a uniform randomized classical algorithm for worst-case Subset Sum in ordinary $O(2^{0.49n})$ word-RAM time on polynomial-bit inputs, where n counts the integers. The time bound holds on every execution and success probability is at least 2/3 on every input. Inputs may repeat positive integers; words have $O(n+b)$ bits for maximum input bit length b.

    Closest manuscript: Subset Sum in Time $O(2^{0.49n})$

12. 104
    Quasipolynomial algorithms for mean-payoff, stochastic and parity games

    Theoretical computer science 4 manuscripts, 102 pages Lean

    Gives deterministic algorithms using $2^{O((\log(L+2))^2)}$ bit operations, for complete binary input length L, for ordinary mean-payoff games and two separate extensions. They compute exact values and optimal positional strategies in ordinary games, the nonnegative expectation-of-liminf value set in turn-based stochastic games, and the winning set for nonnegative liminf mean payoff conjoined with parity. Signed rewards, rational chance probabilities, and parity priorities are unrestricted and binary-encoded.

13. 021
    A quadratic bound for Jacobsthal’s function

    Number theory 1 manuscript, 83 pages Lean

    Answers Jacobsthal's quadratic-bound question: every interval of $Ck^2$ consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.

14. 112
    Beyond the square-root exponent for depth-three circuits

    Theoretical computer science 1 manuscript, 20 pages Lean

    Constructs a single language in deterministic polynomial time whose n-bit membership function requires $2^{\omega(\sqrt n)}$ total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length. This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds.

15. 120
    Almost-linear-time exact matching and prescribed-degree factors in general graphs

    Theoretical computer science 1 manuscript, 86 pages

    Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in $(n+m)^{1+o(1)}$ word time, with success probability at least 2/3. The time bound holds on every computation path. The same guarantees apply to finding a spanning subgraph with prescribed admissible vertex degrees, or deciding that none exists.

16. 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.

17. 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: The Poisson-Dirichlet law for prime predecessors

18. 182
    Power savings for intersective polynomial differences and prime arguments

    Combinatorics 3 manuscripts, 137 pages Lean

    For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of $\{1,\ldots,N\}$ avoiding nonzero values $h(1),h(2),\ldots$ as differences has size $O_h(N^{1-c_k})$, with $c_k\gt 0$ depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h.

19. 179
    The circulant Hadamard and Barker-sequence conjectures

    Combinatorics 1 manuscript, 15 pages Lean

    Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n > 1 exactly when $n\in\{2,3,4,5,7,11,13\}$.

20. 134
    Generalized star height at most three

    Theoretical computer science 3 manuscripts, 116 pages Lean

    Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem.

    Closest manuscript: Finite Monoid Computations and a Uniform Generalized Star-Height Bound

21. 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 $c_a\gt 0$.

22. 114
    Approximate counting of common integer polymatroid bases

    Theoretical computer science 2 manuscripts, 68 pages Lean

    Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution. For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ.

    Closest manuscript: Approximate counting of common bases of two matroids

23. 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.

24. 087
    The Mahler conjectures, functional inequalities and polar-product symplectic width

    Convex and metric geometry 3 manuscripts, 117 pages Lean Reasoning summary

    Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product $K\times K^\circ$ in dimension $2n$ has Gromov width 4.

    Closest manuscript: The symmetric Mahler conjecture and its equality cases

25. 025
    Short Egyptian fractions

    Number theory 1 manuscript, 33 pages Lean

    Every rational $a/b$ with $1\le a\lt b$ is a sum of $O(\log\log b)$ distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.

26. 115
    Sampling and counting contingency tables with arbitrary margins

    Theoretical computer science 2 manuscripts, 78 pages Lean

    For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution.

    Closest manuscript: An FPRAS for Cell-Bounded Contingency Tables

27. 110
    Optimal-order randomized k-server on arbitrary metrics

    Theoretical computer science 2 manuscripts, 87 pages Lean

    Establishes a randomized competitive ratio $O(\log^2(k+1))$ for k-server on every metric space, matching the worst-case lower-bound order. One policy serves every finite oblivious request sequence, including on infinite unbounded metrics. On finite rational metrics, a uniform implementation has polynomial preprocessing and per-request bit cost in the input length and $\log(t+1)$ at request t, with a finite instance-dependent additive movement constant.

    Closest manuscript: Uniform computation of the squared-logarithmic k-server bound

28. 194
    Lech’s multiplicity conjecture

    Algebra 1 manuscript, 41 pages Lean

    Proves $e(R)\le e(S)$ for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic.

29. 128
    A factor-two approximation for shortest common superstring

    Theoretical computer science 1 manuscript, 19 pages Lean

    Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full encoded input length, including symbol labels.

30. 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: Pointwise Multiple Ergodic Averages for Mixing Transformations