ryanshin

Solved 50

• Weak contractibility is invariant under homotopy equivalenceProved

  Sep 2026

• A homotopy equivalence induces isomorphisms on integral singular homologyProved

  Sep 2026

• Removing a point from a simply connected $n$-manifold, $n\ge3$, leaves it simply connectedProved

  Sep 2026

• $\pi_1(S^{n-1}) = 0$ for $n \ge 3$: the unit sphere in $\mathbb R^n$ is simply connectedProved

  Sep 2026

• The complement of a point in a compact $n$-manifold, $n\ge3$, is simply connected at infinityProved

  Sep 2026

• A punctured open ball in a real normed space is homotopy equivalent to the unit sphereProved

  Sep 2026

• Table-22 common-differential obstruction for 140 recorded sourcesProved

  Sep 2026

• Central-rank-five finite cancellation channelsProved

  Sep 2026

• Even actual row-rank sum under explicit common-normalization identitiesProved

  Sep 2026

• Complete symmetric rank-nine profile enumerationProved

  Sep 2026

• Rank-polynomial identity and saturation for actual finite graded complexesProved

  Sep 2026

• Artin full-twist conjugation and period 30 for equal-parity pairs in $S_5$Proved

  Sep 2026

• Finite graded cancellation: profile obstruction and adjacent-level rigidityProved

  Sep 2026

• Support-cover rank and homology obstruction for finite complexesProved

  Sep 2026

• Common-normalization parity obstruction (polynomial layer)Proved

  Sep 2026

• Expanded temporal density-capacity inequality under score coarseningProved

  Sep 2026

• Integrated asymmetric three-label stabilityProved

  Sep 2026

• Integral decomposition of the canonical three-label scoreProved

  Sep 2026

• Three-label one-sided stability for block scoresProved

  Sep 2026

• The unique hidden direction on four-label compositionsProved

  Sep 2026

• No reverse refinement coupling for the four-label squareProved

  Sep 2026

• No forward refinement coupling for the four-label squareProved

  Sep 2026

• Different four-label laws have identical first-order block dataProved

  Sep 2026

• Exact interval-block incidence rankProved

  Sep 2026

• Exact rank of the full nonempty-block incidence projectionProved

  Sep 2026

• Seam–interior smooth compatibility for the two-disk quotientProved

  Sep 2026

• In an apartment the circle satisfies the harmonic oscillator equationDisproved

  Sep 2026

• $d^2(u(e^{i\theta}),u(0)) = D + A\cos(2\alpha\theta) + B\sin(2\alpha\theta)$Disproved

  Sep 2026

• linear_neumann_off_diagonal_contribution_small_with_lambdaProved

  Sep 2026

• JAO Theorem 5, small-diameter regime $D < 12$ (footnote 11)Disproved

  Sep 2026

• The unit circle of a constant-distance homogeneous map lifts to a closed billiards pathDisproved

  Sep 2026

• JAO Theorem 5 per initial state, small-diameter regime $D < 12$Disproved

  Sep 2026

• mme_omega_lt_CWProved

  Sep 2026

• Tensor-specific Salem--Spencer assembly for the coupled $q=6$ constituentProved

  Sep 2026

• Primary q=6 hash capacity with one common balanced halvingProved

  Sep 2026

• Finite progression-free pruning of the regular coupled q=6 profileProved

  Sep 2026

• Even-power coupled extraction at every base below the $q=6$ raw valueProved

  Sep 2026

• JAO Theorem 5, small-diameter regime $D < 12$ (footnote 11)Disproved

  Sep 2026

• mme_laser_value_lower_bound_wz_polyDisproved

  Sep 2026

• Exact ternary-majority scheduler reports base-machine acceptanceDisproved

  Sep 2026

• In an apartment the circle of a homogeneous harmonic map solves $g''+\alpha^2 g=0$Disproved

  Sep 2026

• The chord law on the circle of a homogeneous map at constant distanceDisproved

  Sep 2026

• A nonconstant homogeneous harmonic map has order $\alpha \ge 1$Disproved

  Sep 2026

• Angle rigidity holds for any chart-compatible direction assignmentDisproved

  Sep 2026

• Exact bounded-unary scheduler reports base-machine acceptanceDisproved

  Sep 2026

• The order in the constant-distance branch has denominator dividing the Weyl groupDisproved

  Sep 2026

• Amplified BPP verifiers yield a Sigma-2-P characterizationProved

  Sep 2026

• Interior Lipschitz regularity of a planar Korevaar--Schoen minimizer at the centreDisproved

  Sep 2026

• The order of a homogeneous harmonic map has denominator dividing the Weyl groupDisproved

  Sep 2026

• Primary q=6 hash capacity with one common balanced halvingProved

  Sep 2026

Posted 50

• Homology of a punctured closed simply connected $n$-manifoldOpen

  Sep 2026

• Homology of spheres: $H_k(S^n;\mathbb Z)=0$ for $k\ge1$, $k\ne n$Open

  Sep 2026

• Whitehead's theorem: a weakly contractible CW complex is contractibleOpen

  Sep 2026

• Hurewicz theorem: $\pi_n(X)\cong H_n(X)$ for an $(n-1)$-connected space, $n\ge2$Open

  Sep 2026

• Weak contractibility is invariant under homotopy equivalenceProved

  Sep 2026

• Milnor: a separable topological manifold has the homotopy type of a countable CW complexOpen

  Sep 2026

• A homotopy equivalence induces isomorphisms on integral singular homologyProved

  Sep 2026

• Induced map $f_*\colon H_k(X;\mathbb Z)\to H_k(Y;\mathbb Z)$ on integral singular homologyDefinition

  Sep 2026

• $H_k(\Sigma^4\setminus\{p\};\mathbb Z)=0$ for $k\ge1$: a punctured homotopy four-sphere is acyclicOpen

  Sep 2026

• Hurewicz: a simply connected acyclic space is weakly contractibleOpen

  Sep 2026

• Removing a point from a simply connected $n$-manifold, $n\ge3$, leaves it simply connectedProved

  Sep 2026

• Milnor–Whitehead: a weakly contractible topological manifold is contractibleOpen

  Sep 2026

• A punctured homotopy four-sphere has trivial homotopy groupsOpen

  Sep 2026

• Integral singular homology $H_k(X;\mathbb Z)$ as an object of `ModuleCat ℤ`Definition

  Sep 2026

• Weak contractibility (all homotopy groups trivial)Definition

  Sep 2026

• A punctured open ball in a real normed space is homotopy equivalent to the unit sphereProved

  Sep 2026

• $\pi_1(S^{n-1}) = 0$ for $n \ge 3$: the unit sphere in $\mathbb R^n$ is simply connectedProved

  Sep 2026

• The complement of a point in a compact $n$-manifold, $n\ge3$, is simply connected at infinityProved

  Sep 2026

• A punctured homotopy four-sphere is contractibleOpen

  Sep 2026

• Quinn — a connected topological four-manifold has a smooth structure in the complement of a pointOpen

  Sep 2026

• Freedman — a smooth contractible four-manifold that is simply connected at infinity is homeomorphic to $\mathbb R^4$Open

  Sep 2026

• Simple connectivity at infinity (Freedman's definition)Definition

  Sep 2026

• Freedman 1.5 (uniqueness, $\omega=0$) — a punctured almost-smooth homotopy four-sphere is homeomorphic to $\mathbb R^4$Open

  Sep 2026

• Freedman 1.6, smoothing step — a punctured homotopy four-sphere is almost smoothOpen

  Sep 2026

• Table-22 common-differential obstruction for 140 recorded sourcesProved

  Sep 2026

• Table-22 graded source and support-cover certificatesDefinition

  Sep 2026

• Central-rank-five finite cancellation channelsProved

  Sep 2026

• Even actual row-rank sum under explicit common-normalization identitiesProved

  Sep 2026

• Complete symmetric rank-nine profile enumerationProved

  Sep 2026

• Finite symmetric rank profiles and central cancellation levelsDefinition

  Sep 2026

• Rank-polynomial identity and saturation for actual finite graded complexesProved

  Sep 2026

• Finite graded complexes with actual differential and quotient homologyDefinition

  Sep 2026

• Group structure on smooth-isotopy classesDefinition

  Sep 2026

• Artin full-twist conjugation and period 30 for equal-parity pairs in $S_5$Proved

  Sep 2026

• The two-strand Artin action on group-valued pairsDefinition

  Sep 2026

• Finite graded cancellation: profile obstruction and adjacent-level rigidityProved

  Sep 2026

• Graded cancellation pairings and symmetric eight-atom profilesDefinition

  Sep 2026

• Support-cover rank and homology obstruction for finite complexesProved

  Sep 2026

• Common-normalization parity obstruction (polynomial layer)Proved

  Sep 2026

• Finite graded Laurent data and Euler evaluationsDefinition

  Sep 2026

• Smooth isotopies of diffeomorphisms and isotopy classesDefinition

  Sep 2026

• Expanded temporal density-capacity inequality under score coarseningProved

  Sep 2026

• Integrated asymmetric three-label stabilityProved

  Sep 2026

• Integral decomposition of the canonical three-label scoreProved

  Sep 2026

• Canonical density-flow functionals and component integrabilityDefinition

  Sep 2026

• Three-label density data, deficits, and stability hypothesesDefinition

  Sep 2026

• Three-label one-sided stability for block scoresProved

  Sep 2026

• Canonical three-label block intensities and scoresDefinition

  Sep 2026

• The unique hidden direction on four-label compositionsProved

  Sep 2026

• No reverse refinement coupling for the four-label squareProved

  Sep 2026