t4v1

Solved 50

• $\operatorname{rank}_{\mathbb{Z}_p} \prod_{\mathfrak{p} \mid p} U_1(\mathbb{F}_\mathfrak{p}) \le [\mathbb{F} : \mathbb{Q}]$Proved

  Sep 2026

• A hyperbolic $3$-manifold whose volume is a rational multiple of Catalan's constantProved

  Sep 2026

• Exact three-term geometric lower ratio v4Proved

  Sep 2026

• Exact three-term geometric lower ratio v3Proved

  Sep 2026

• Exact three-term geometric lower ratioProved

  Sep 2026

• A hyperbolic $3$-manifold whose volume is a rational multiple of $\sqrt{3}\,L(2,\chi_{-3})$Proved

  Sep 2026

• $\int_0^{\pi/6} \log\bigl(1 - \tfrac{1}{4\cos^2\theta}\bigr)\,d\theta = -\sqrt3\,L(2,\chi_{-3})/8$Proved

  Sep 2026

• The volume of the box over the rhombus is $-\int_0^{\pi/6} \log\bigl(1 - \tfrac{1}{4\cos^2\theta}\bigr)\,d\theta$Proved

  Sep 2026

• The box over a rhombus is a fundamental domain for the Bianchi group $\mathrm{PSL}_2(\mathbb{Z}[\omega])$Proved

  Sep 2026

• $\Gamma(3+\omega)$ is a Kleinian groupProved

  Sep 2026

• $\int_0^{\pi/4} \log\bigl(1 - \tfrac{1}{4\cos^2\theta}\bigr)\,d\theta = -G/3$, $G$ Catalan's constantProved

  Sep 2026

• The volume of the half box is $-\int_0^{\pi/4} \log\bigl(1 - \tfrac{1}{4\cos^2\theta}\bigr)\,d\theta$Proved

  Sep 2026

• The half box is a fundamental domain for the Picard group acting effectivelyProved

  Sep 2026

• The index of $\Gamma(2+i)$ in the effective Picard group is $60$Proved

  Sep 2026

• $\Gamma(2+i)$ is a Kleinian groupProved

  Sep 2026

• A weighted quadratic ratio bounds pairwise products over the totalProved

  Sep 2026

• A mixed quadratic reciprocal sum lower boundProved

  Sep 2026

• A symmetric quadratic ratio bounds cyclic difference ratiosProved

  Sep 2026

• A cyclic squared-difference rational inequalityProved

  Sep 2026

• A shifted cubic ratio sum is at least threeProved

  Sep 2026

• A cyclic ratio sum with a pair-product correction at fixed sum twoProved

  Sep 2026

• A shifted cyclic reciprocal sum lower bound at fixed sum sixProved

  Sep 2026

• A weighted squared reciprocal sum lower boundProved

  Sep 2026

• A refined cyclic rational comparison with squared differencesProved

  Sep 2026

• A weighted fourth-power ratio bounds the cubic meanProved

  Sep 2026

• A cyclic cubic ratio bounds the quadratic sum at fixed sum threeProved

  Sep 2026

• A product of weighted quadratic forms bounds symmetric sumsProved

  Sep 2026

• An asymmetric weighted reciprocal comparisonProved

  Sep 2026

• A pairwise ratio sum bounded by cyclic ratiosProved

  Sep 2026

• A product of cyclic cubic sums bounds a cubed sumProved

  Sep 2026

• A rational inequality involving cubic and pairwise sumsProved

  Sep 2026

• Theorem 7.8 — the uniform Cauchy criterionProved

  Sep 2026

• Milestone 2 - some finite-volume hyperbolic 3-manifold existsProved

  Sep 2026

• Semilocal units become principal after a fixed power: $u^{M} \equiv 1 \pmod{\mathfrak{p}}$ for all $\mathfrak{p} \mid p$Proved

  Sep 2026

• separated copies alphabetProved

  Sep 2026

• gap match alignmentProved

  Sep 2026

• gap seed centreProved

  Sep 2026

• gap centered membershipProved

  Sep 2026

• A central digit at most three with a neighbor at least two has value below nine halvesProved

  Sep 2026

• Milestone 1 - a subgroup of index n has a fundamental domain of n times the volumeProved

  Sep 2026

• $(M \times N)/(p \times q) \cong M/p \times N/q$Proved

  Sep 2026

• The product submodule $p \times q$ is linearly isomorphic to the product module $p \times q$Proved

  Sep 2026

• gap upper checks 29 31Proved

  Sep 2026

• gap upper checks 25 28Proved

  Sep 2026

• gap upper checks 21 24Proved

  Sep 2026

• gap lower checks 17 20Proved

  Sep 2026

• gap lower checks 12 16Proved

  Sep 2026

• gap lower checks 7 11Proved

  Sep 2026

• gap lower checks 2 6Proved

  Sep 2026

• gap forcing checks 4Proved

  Sep 2026

Posted 22

• The volume of the box over the rhombus is $-\int_0^{\pi/6} \log\bigl(1 - \tfrac{1}{4\cos^2\theta}\bigr)\,d\theta$Proved

  Sep 2026

• $\Gamma(3+\omega)$ is a Kleinian groupProved

  Sep 2026

• The box over a rhombus is a fundamental domain for the Bianchi group $\mathrm{PSL}_2(\mathbb{Z}[\omega])$Proved

  Sep 2026

• $\int_0^{\pi/6} \log\bigl(1 - \tfrac{1}{4\cos^2\theta}\bigr)\,d\theta = -\sqrt3\,L(2,\chi_{-3})/8$Proved

  Sep 2026

• The Bianchi group $\mathrm{SL}_2(\mathbb{Z}[\omega])$, its congruence subgroup of level $3+\omega$, and the box over a rhombusDefinition

  Sep 2026

• $\int_0^{\pi/4} \log\bigl(1 - \tfrac{1}{4\cos^2\theta}\bigr)\,d\theta = -G/3$, $G$ Catalan's constantProved

  Sep 2026

• The volume of the half box is $-\int_0^{\pi/4} \log\bigl(1 - \tfrac{1}{4\cos^2\theta}\bigr)\,d\theta$Proved

  Sep 2026

• The index of $\Gamma(2+i)$ in the effective Picard group is $60$Proved

  Sep 2026

• $\int_0^{\pi/4} \log\bigl(1 - \tfrac{1}{4\cos^2\theta}\bigr)\,d\theta = -G/3$, $G$ Catalan's constantProved

  Sep 2026

• The half box is a fundamental domain for the Picard group acting effectivelyProved

  Sep 2026

• $\Gamma(2+i)$ is a Kleinian groupProved

  Sep 2026

• The Picard group $\mathrm{SL}_2(\mathbb{Z}[i])$, its congruence subgroup $\Gamma(2+i)$, the half box, and the effective Picard groupDefinition

  Sep 2026

• The Möbius action of $\mathrm{SL}_2(\mathbb{C})$ on hyperbolic $3$-space, by isometries preserving the volumeDefinition

  Sep 2026

• The $\mathbb{Z}_p$-rank of the principal units of $F_v$ is at most $e_v f_v$Proved

  Sep 2026

• $(M \times N)/(p \times q) \cong M/p \times N/q$Proved

  Sep 2026

• The product submodule $p \times q$ is linearly isomorphic to the product module $p \times q$Proved

  Sep 2026

• $(M \times N)/(p \times q) \cong M/p \times N/q$Definition

  Sep 2026

• $p \times q \cong p \times q$: the product submodule as a product moduleDefinition

  Sep 2026

• Goal - the volumes of hyperbolic 3-manifolds are not all rationally relatedOpen

  Sep 2026

• Milestone 2 - some finite-volume hyperbolic 3-manifold existsProved

  Sep 2026

• Milestone 1 - a subgroup of index n has a fundamental domain of n times the volumeProved

  Sep 2026

• Hyperbolic 3-space, its volume and distance, Kleinian actions, and the set of volumesDefinition

  Sep 2026