Let $C_n$ be a cyclic group of order $n$. We prove that if $(n,6)=1$, then every minimal zero-sum sequence of length four over $C_n$ has index one, thereby resolving the length-four index conjecture. After the gcd reduction, the nonunit case follows from the theorem of Shen-Xia-Li, and the remaining unit case is solved by a new multiplicative Fourier argument. The index-two residue identity yields a character-moment relation, and the odd characters with vanishing first moment form an exceptional spectrum of size at most $157\varphi(n)/1440<\varphi(n)/9$. A finite-group uncertainty principle then forces the four-term multiset to be invariant under negation, contradicting minimality. Apart from standard facts about primitive Dirichlet $L$-functions, the remaining argument is finite and requires neither asymptotic estimates nor computational verification.
We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every dimension $n\geq3$. We also give a six-term cubic example in four variables, which was found first and already proves failure for every $n\geq4$. Both examples follow from the same coefficient identity. The search was prompted by Levent Alp\"oge's public announcement of an explicit three-dimensional counterexample to the Jacobian Conjecture. Although the main theorem of Derksen, van den Essen, and Zhao is stated globally in dimension, its proof has fixed-dimensional content: a noninvertible cubic-homogeneous Keller map in $r$ variables forces the failure of ${\mathrm GMC}(2r)$. Tracking a standard Bass--Connell--Wright reduction of the announced map gives a conservative cubic-homogeneous counterexample in $79$ variables, and hence a route-based failure of ${\mathrm GMC}(158)$. That route is nonconstructive at the final Gaussian step and does not furnish explicit polynomials $P,Q$. The much smaller explicit failures in dimensions $4$ and $3$ below were not derived from the announced Jacobian map.
For positive integers $n$ and $M$, Sundaram defined $ F_{n,M}=\sum_{\substack{d\mid n\\d\le M}} Lie_{n/d}[p_d] $ and conjectured that every coefficient in its Schur expansion is a nonnegative integer. We prove the conjecture. The power-sum expansion isolates the contribution of the identity class, and the proof divides Young diagrams at the intrinsic majority threshold. If the first row or first column contains more than half of the boxes, the Bernstein creation formula separates each rectangular-cycle character value into a hook constant and a remainder supported only on short cycles. The restrictions of the trivial and sign characters to the cyclic subgroup evaluate the constants exactly, whereas a uniform binomial contraction controls every distance from the boundary at once. If neither a row nor a column has a majority, Swanson's opposite-hook estimate gives a Catalan-scale lower bound for the dimension. The Fomin--Lulov rectangular-character estimate, together with uniform bounds for the truncated M\"obius weights, then shows that the identity class dominates. Sundaram's plethystic identity consequently implies that $\prod_{r=1}^M(1-p_r)^{-1}$ is Schur-positive for every fixed $M$. We also record a degree-$12$ counterexample to a separate congruence-class product conjecture attributed to Richard Stanley.
Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities $1,2,4,\ldots,2^{m-1}$, normalized by $2^m-1$. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density at least 1/3. After reducing the partition to primitive common-period data, Fourier cancellation produces a finite inverse-sine system. We prove that no such system can exist when every density is below 1/3. The proof combines a divisor-concentration identity with uniform analytic estimates and three exact finite verifications, all carried out with integer or rational arithmetic. The component supplied by the density bound has mean spacing at most three. Deleting it preserves balance, and every surviving periodic balanced set is again a rational Beatty set. Induction determines the surviving binary scales, while a two-sequence disjointness criterion forces the deleted density to be the next binary scale. This yields the asserted density pattern.
Let $X^4_m=\{x_0^m+\dots+x_5^m=0\}\subset\mathbb{P}^5$ be the Fermat fourfold of degree $m$. We give a computer-assisted proof of the Hodge conjecture for $X^4_m$ for every odd $m\le 199$: three geometric closure criteria combined with an exhaustive machine census of the Hodge $(2,2)$-orbits, with a completeness proof and re-verifiable certificates. Criteria: (1) if the character multiset splits into two zero-sum triples, the rational Hodge block is transported from a $(1,1)$-substructure of a product of Fermat curves, hence algebraic; (2) algebraicity follows for characters that, after adjoining two vanishing pairs, decompose into an Aoki standard sextuple and a grade-$2$ Hodge quadruple; (3) the exceptional class at $m=33$ has a quasi-decomposable lift to level $66$, whose algebraicity descends along $X^4_{66}\to X^4_{33}$. The census covers the $89$ levels $21\le m\le 199$, $m\ne 23$, classifies all $78{,}299$ Galois-orbit representatives, and isolates thirteen orbits beyond decomposability, quasi-decomposability and Aoki's standard cycles: six close by the $*$-split criterion, seven by the two-pair and level-lifted closures, leaving none. Every orbit carries a machine-checked witness (negative screenings for the terminal ones), and the census is reproduced by an algorithmically independent implementation and brute force through $m=143$. Seven of the thirteen are gap classes outside Aoki's lattice calculus, new to the author's knowledge; for the other six, algebraicity is also derivable from that calculus, the explicit presentations being the new content. An exact Jacobi-sum computation at $p=67$ shows no cycle defined over $\mathbb{Q}(\zeta_{33})$ projects nontrivially onto the exceptional $m=33$ block; more generally, over any finite extension of $\mathbb{Q}(\zeta_{33})$ carrying a certifying cycle, every residue degree above $67$ is divisible by $6$.
Let $\mathrm{IMS}_n(m)$ count the $n\times n$ nonnegative integer matrices whose row sums, column sums, main-diagonal sum, and antidiagonal sum are all $m$. We determine the Ehrhart series $F_n(q)=\sum_{m\geq0}\mathrm{IMS}_n(m)q^m$ as reduced rational functions for $n=7$ and $n=8$. Their numerator--denominator degrees are respectively $(366,373)$ and $(540,548)$. Both numerators have positive integer coefficients, are palindromic and strictly unimodal. The proofs share one finite architecture: a signed SimpCone decomposition is evaluated by quotient characters over finite fields, a certified common denominator and Ehrhart reciprocity reduce the rational identity to finitely many coefficients, an explicit counting bound lifts modular congruences to integer equalities, and exact gcd computations prove reducedness. For order eight, a face-index pole certificate gives a degree-$598$ common denominator without enumerating the full face lattice, leaving $296$ independent coefficients in degrees $0$ through $295$. Once the candidate rational function is known, the first six production primes certify this finite prefix by the same bounded-coefficient argument used for order seven.
We prove a nine-distance theorem for Kronecker sequences on flat three-tori. That is, we show that among the first $N$ orbit points, at most nine distinct positive nearest-neighbour distances occur. This proves the conjecture of Haynes and Marklof. An example of Dettmann shows that nine is optimal. More generally, we prove that on a flat $d$-dimensional torus the number of such distances is at most $2^d+1$. The main tool is a new growth theorem for the denominators $q_1<q_2<\cdots$ of best simultaneous approximations in a $d$-dimensional inner-product space, which is of independent interest. We prove that, whenever $q_{n+2^d}$ is defined, either $q_{n+2^d}\ge2q_{n+1}$, or the indices $1,\ldots,2^d$ can be partitioned into disjoint pairs $\{j,k\}$, $j<k$, such that $q_{n+k}=q_n+q_{n+j}$. In particular, $$ q_{n+2^d}\ge \min\{2q_{n+1},q_n+q_{n+2^{d-1}}\}\ge q_n+q_{n+1}. $$
N. Shulga· 3 citations· ⚡2
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