Tagged lean-formalized
No description yet.
-
0 votes 0 replies 0 views
Paper Semialgebraic universal covers and bounded domains
Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products D\times\mathbb C^m\times F, with D bounded symmetric and F simply…
-
0 votes 0 replies 0 views
Paper Brennan's conjecture and the integral-means spectrum
Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, |\phi'|^s is area-integrable for 4/3\lt s\lt 4. The sharp universal integral-means identity is…
-
0 votes 0 replies 0 views
Paper The Euclidean Steinitz–Bergström bound
Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within C\sqrt d, independently of length. Consequently every zero-sum family can be reordered with the same…
-
0 votes 0 replies 0 views
Paper Riesz transforms and rectifiability in higher codimension
Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and 2\le n\le d-2, an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable whenever its n-dimensional…
-
0 votes 0 replies 0 views
Paper Sharp projection-body inequalities and a counterexample to simplex maximization
Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In…
-
0 votes 0 replies 0 views
Paper The Gaussian propeller conjecture in every dimension
Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most 9/(8\pi). In dimension at least two, three planar sectors of angle 2\pi/3, extended orthogonally, attain the…
-
0 votes 0 replies 0 views
Paper Hardness of coloring three-colorable graphs
It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed 0\lt \delta\lt 1/3, a deterministic polynomial-time reduction from 3SAT produces simple…
-
0 votes 0 replies 0 views
Paper Uniform sparsest cut: hardness and semidefinite gaps
Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality…
-
0 votes 0 replies 0 views
Paper Bin packing and unbounded configuration-LP gaps
Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant.…
-
0 votes 0 replies 0 views
Paper The Courtade–Kumar and Hellinger conjectures
Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized…
-
0 votes 0 replies 0 views
Paper A factor-two approximation for shortest common superstring
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…
-
0 votes 0 replies 0 views
Paper The second Kahn–Kalai conjecture with an edge-count bound
Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in G(n,p) is at most C p{\mathrm E}(n,H)(1+\log2 h), with universal C.…
-
0 votes 0 replies 0 views
Paper Hilbert's sixteenth problem: uniform bounds for limit cycles
Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree.…
-
0 votes 0 replies 0 views
Paper The entropy-rate dimension formula for self-similar measures
For every self-similar measure on the line generated by finitely many contracting similarities, proves \dim{\mathrm H}\mu=\min\{1,h{\mathrm{RW}}/\chi\}, where h{\mathrm{RW}} is the entropy rate of random…
-
0 votes 0 replies 0 views
Paper The Erdős–Gallai cycle-decomposition conjecture
Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most Cn simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai…
-
0 votes 0 replies 0 views
Paper Saxl’s conjecture and universal tensor squares
Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every Sn with n\notin\{2,4,9\}…
-
0 votes 0 replies 0 views
Paper Planar first-passage geometry and the absence of bigeodesics
Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape…
-
0 votes 0 replies 0 views
Paper Critical percolation on every quasi-transitive graph
Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with pc\lt 1: at the critical probability, there is almost surely no…
-
0 votes 0 replies 0 views
Paper Continuum phase transitions for radial pair potentials
Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout…
-
0 votes 0 replies 0 views
Paper The Partition Principle does not imply Choice
Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive…
-
0 votes 0 replies 0 views
Paper Shelah's eventual categoricity and the prescribed-threshold obstruction
Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply…
-
0 votes 0 replies 0 views
Paper Rigidity of the Turing degrees
Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability. () Manuscript September 24,…
-
0 votes 0 replies 0 views
Paper Single-fold Diophantine representations and undecidability under an at-most-one-solution promise
Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and…
-
0 votes 0 replies 0 views
Paper Separating choiceless counting from polynomial time and witnessed choice
Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽3 defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot…
-
0 votes 0 replies 0 views
Paper The ionization and generalized ionization conjectures
For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most Z+CM electrons. Neutral-atom…
-
0 votes 0 replies 0 views
Paper Classical capacity of generalized amplitude damping
Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by…
-
0 votes 0 replies 0 views
Paper The Grothendieck homotopy hypothesis
Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces. () Manuscript…
-
0 votes 0 replies 0 views
Paper Gigli’s characterization of Alexandrov curvature
Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support \mathop{\mathrm{RCD}}\nolimits ((n-1)\kappa,n) condition with reference measure…
-
0 votes 0 replies 0 views
Paper A three-manifold without conjugate points or nonpositive curvature
Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether…
-
0 votes 0 replies 0 views
Paper The Lane–Emden and Hénon–Lane–Emden conjectures
Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, p,q\gt 0 and real A, B, the system -\Delta u=|x|^A v^p, -\Delta v=|x|^B u^q has no positive entire solution when…
135 posts