Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb{F}_p)$ be the Heisenberg group of order $p^3$ and exponent $p$. We prove $\mathsf{d}(H_{p^3})=3p-3$. The main ingredient of the proof is an order-value growth theorem. If $B$ is a noncollinear zero-sum sequence of $n$ nonzero vectors in $\mathbb{F}_p^2$, then the alternating areas obtained by ordering $B$ assume at least $\min(p,n-1)$ distinct values. Its proof is a short contraction induction: contract a suitable independent pair, replace the contracted vector in both orders, and apply Cauchy--Davenport. A polynomial relative-subsum theorem and a sharp representation-rigidity lemma then turn this local growth into a uniform spread bound. Combined with the standard product-one criterion for $H_{p^3}$, the spread bound yields the upper bound; the usual sequence $x^{p-1}y^{p-1}v^{p-1}$ gives the lower bound.
The small Davenport constant $\mathsf{d}(G)$ of a finite group $G$ is the maximal length of a product-one-free sequence over $G$. For the exponent-$p$ Heisenberg group $H_{p^3}$ of order $p^3$, Godara and Sarkar proved $\mathsf{d}(H_{27})=6$ and posed $\mathsf{d}(H_{p^3})=3p-3$ for every odd prime $p$, leaving $p\ge5$ open. We settle the first open case: $\mathsf{d}(H_{125})=12$. The lower bound is the explicit product-one-free sequence $x^4y^4v^4$. For the upper bound we record a product-one criterion that reduces the non-commutative problem to additive combinatorics over $\mathbb{F}_5^2$, and then reduce"every length-13 sequence has a product-one subsequence"to a single finite statement -- a spread bound on quotient multisets -- which we verify by an exhaustive, memory-flat search in C, its verdict independently reproduced by a second search with a different pruning strategy. Every auxiliary lemma is machine-checked. The argument is genuinely $p$-specific: we identify the exact step that fails for $p\ge7$ (a Chevalley-Warning shortcut whose forced block need not be wide), exhibit the obstructing multiset for $p=7$, and leave only $18\le\mathsf{d}(H_{343})\le24$. The techniques -- the Cauchy-Davenport theorem, Chevalley-Warning, and Olson's value of the Davenport constant of $C_p^2$ -- are standard; the contribution is their assembly against a new non-abelian target and the finite verification that closes it.
For a finite group $G$, let $\mathsf{d}(G)$ denote the maximum length of a sequence having no nonempty subsequence whose terms can be ordered to have product one. For an odd prime $p$, let $H_{p^3}=\operatorname{UT}*3(\mathbb{F}*p)$. Godara and Sarkar proved $\mathsf{d}(H*{27})=6$ and conjectured $\mathsf{d}(H*{p^3})=3p-3$; in a recent preprint, White proved the next case $\mathsf{d}(H_{125})=12$ and left $18\leq\mathsf{d}(H_{343})\leq24$. We prove $\mathsf{d}(H_{343})=18$. We adopt White's product-one criterion and spread framework and develop a $p=7$-specific direction stratification. An explicit product-one-free sequence gives the lower bound. For the upper bound, we stratify a hypothetical product-one-free sequence of length $19$ by the number of central terms and by the occupied projective directions of its quotient multiset. Supports on at most two directions are excluded by a theoretical argument whose finite auxiliary statements are exhaustively checked; the three-direction case and the case of five central terms are settled by exact finite computations. The remaining thirty strata are encoded by a counterexample-guided SAT procedure. A separately implemented checker verifies all $9{,}920{,}815$ seed cuts and all $27{,}207$ learned cuts, and each final unsatisfiable instance is accompanied by a checked LRAT certificate. A separate implementation-level audit verifies the master encoding, the proof archives, and the lower-bound witness.
For coprime $1<a<b$, let $M_n^{a,b}(\mathbb{F}_q)$ be the set of commuting pairs of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ with $X^a=Y^b$. Huang, Jiang, and Oblomkov assembled their orders as an Eulerian $q$-series $Z_{a,b}(q)$. They conjectured that it is an explicit product $P_{a,b}(q)$ involving Jacobi's theta function and Dedekind's eta-function, implying the threefold equality $$\underbrace{\prod_{n\ge1}(1-q^n)\cdot\Biggl(\sum_{n=0}^{\infty}\frac{|M_n^{a,b}(\mathbb{F}_q)|}{|\mathrm{GL}_n(\mathbb{F}_q)|}\Biggr)\Biggr|_{q\mapsto q^{-1}}}_{\text{point count}}\;=\;\underbrace{Z_{a,b}(q)}_{q\text{-series}}\;=\;\underbrace{P_{a,b}(q)}_{\text{theta quotient}}$$ If true, the point count on $X^a=Y^b$ is essentially a modular function on $\Gamma(a+b)$. The conjecture is layered in $a$, with an identity for each $b$. The $a=2$ layer is classical, including identities of Rogers--Ramanujan and Andrews--Gordon. For $a\geq3,$ nothing was known. We prove the $a=3$ layer in full: a new infinite family of Rogers--Ramanujan identities, and a geometric origin for Warnaar's products. AxiomProver verified these new identities in Lean assuming existing literature.
Let $\Gamma$ be an $n\times m$ matrix with independent standard Gaussian entries and let $G_m = \Gamma(B_1^m)$ be the associated Gaussian Gluskin polytope. In the regime $m = n^3$ we prove that, with probability at least $1-C/n$, $$ d_{\mathrm{BM}}(G_m,B_1^n) \ge c n^{5/8}(\log n)^{-1/4}. $$ This improves the polynomial exponent $4/7$ obtained in the author's preceding work and gives an explicit logarithmic factor. The proof retains the discretization and conditioning/powering framework, but replaces the earlier split into two coefficient regimes by two uniform quotient events. One controls successive directions of the big-coordinate parts; the other compresses the entire small-coordinate cloud near a low-dimensional subspace after every admissible quotient. Suppression, a local Maurey argument, and Gram-Schmidt volume estimates then combine these two forms of control.
Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity, and put $P_2$={$m$ $\ge$ 1:$\Omega(m)$ $\le$ 2}. We prove that the sumset $P_2$+{$a^a$: a$\ge$ 1} has positive lower density. The proof uses the Romanoff second moment method, in the spirit of Li and Pan's theorem on $P_2$+$2^{\mathcal P}$. The main new ingredient is the following average estimate for the singular factor \[ \frac{1}{K(K-1)} \sum_{\substack{1\le a,b\le K\\a\ne b}} \prod_{p\mid a^a-b^b}\left(1+\frac{\kappa}{p}\right) \le C_\kappa \] for some constant $C_\kappa>0$, which is valid for all $K \ge 2$ and any fixed $\kappa>0$. This estimate controls the average arithmetic correlation among the shifts $a^a$ and allows the Romanoff argument to be carried out.
Let $K$ be an imaginary quadratic number field, let $p$ be an odd prime ($K\neq\mathbb Q(\sqrt{-3})$ if $p=3$), and let $G$ be the Galois group of the maximal everywhere unramified pro-$p$ extension of $K$. To each mod-$p$ character $x$ of $G$ we associate a linear map $D_x$ from $\mathrm{Cl}(K)[p]$ to $\mathrm{Cl}(K)/p$; a formula of Ahlqvist and Carlson expresses it through the class of a norm ideal in the unramified cyclic degree-$p$ extension attached to $x$. These maps determine all triple Massey products on $H^1(G,\mathbb F_p)$, and with them the cubic initial relations of $G$. Suppose the $p$-class rank $d$ of $K$ is at least three. The $(d-1)\times(d-1)$ minors of the family $x\mapsto D_x$ define a subscheme $\Sigma_D$ of $\mathbb P^{d-1}_{\mathbb F_p}$, the norm-degeneracy scheme of $K$. We prove: if the rank condition $\mathrm{rk}\,D_x=d-2$ holds transversally at a point of $\Sigma_D$, over some finite extension of $\mathbb F_p$, then $G$ is mild, and hence of cohomological dimension 2. For $p>3$ transversality means that $\Sigma_D$ is smooth of dimension $d-3$ at the point; at $p=3$ the kernel of the Bockstein map enters as an additional constraint. We treat every imaginary quadratic field of $p$-class rank at least three with $|D_K|<2^{30}$, for every odd prime $p$; only $p=3,5,7$ occur. We prove, unconditionally, that the $p$-class tower group is mild for 864 of the 12749 rank-three fields at $p=3$, for 203 of the 204 fields at $p=5$, and for all three fields at $p=7$; the single rank-four field remains undecided. To the best of our knowledge, these are the first number fields for which the full maximal everywhere unramified pro-$p$ Galois group is proved to be mild; in particular they provide explicit infinite $p$-class towers whose Galois groups have cohomological dimension 2.
