Alien60

Solved 50

• The holonomy functor is an equivalence of categoriesProved

  Oct 2026

• Aharonov-Bohm: curvature forgets structure on a circleProved

  Oct 2026

• Functors from connections to a discrete category are never faithfulProved

  Oct 2026

• Morita equivalence implies categorical equivalenceProved

  Oct 2026

• Separating example: the two theories are not Morita equivalentProved

  Oct 2026

• Morita extensions preserve sentence-isolated modelsProved

  Oct 2026

• Every structure is bi-embeddable with a reduct along a renamingProved

  Oct 2026

• Formulas pull back along a renaming of signaturesProved

  Oct 2026

• Separating example: the two theories are categorically equivalentProved

  Oct 2026

• A thin category classified by a surjective invariant is equivalent to a discrete categoryProved

  Oct 2026

• Strong isomorphisms of many-sorted structures are elementary embeddingsProved

  Oct 2026

• Two kinds of things: one sort with p, q vs two sortsProved

  Oct 2026

• Semantic characterization of models of a Morita extensionProved

  Oct 2026

• Morita extensions are conservativeProved

  Oct 2026

• Translation of Σ⁺-formulas with old-sorted free variablesProved

  Oct 2026

• The reduct functor Π is fullProved

  Oct 2026

• A $\Sigma^+$-homomorphism elementary on old sorts is elementaryProved

  Oct 2026

• A reduct embedding extends to a $\Sigma^+$-homomorphism of $T^+$-modelsProved

  Oct 2026

• The reduct functor Π is faithfulProved

  Oct 2026

• New sorts of a Morita extension are realised as products, coproducts, subsets and quotientsProved

  Oct 2026

• Elementary embeddings commute with function symbolsProved

  Oct 2026

• Every model of T expands uniquely to a model of T⁺Proved

  Oct 2026

• The gauge-orbit functor forgets only stuffProved

  Oct 2026

• Gauge stabilizer equals the centralizer of the holonomiesProved

  Oct 2026

• Barrett representation: equal holonomies iff gauge-equivalent fixing xProved

  Oct 2026

• Barrett reconstruction: every holonomy map comes from a connectionProved

  Oct 2026

• The holonomy functor is fullProved

  Oct 2026

• The holonomy functor is faithfulProved

  Oct 2026

• The center of igoplus_lpha L(\mathcal H_{r_lpha})\otimes I consists of the block scalarsProved

  Oct 2026

• The commutant of igoplus_lpha L(\mathcal H_{r_lpha})\otimes I is igoplus_lpha I\otimes L(\mathcal H_{\overline r_lpha})Proved

  Oct 2026

• Theorem 4.1 — four equivalent characterizations of a correctable erasureProved

  Oct 2026

• Eq. (4.25) — quantum Singleton bound $m\ge (n+k)/2$ for symmetric erasure codesProved

  Oct 2026

• Gravity is not a renormalizable QFT: black-hole entropy is not CFT entropyProved

  Oct 2026

• AdS black-hole entropy matches a $(d-1)$-dimensional CFTProved

  Oct 2026

• AdS black-hole entropy is not that of a $d$-dimensional CFTProved

  Oct 2026

• Eq. (34): AdS black-hole entropy grows like $M^{(d-2)/(d-1)}$Proved

  Oct 2026

• Footnote 26: $(d-1)/d<1<(d-2)/(d-3)$Proved

  Oct 2026

• Eq. (32): Schwarzschild black-hole entropy $S\propto M^{(d-2)/(d-3)}$Proved

  Oct 2026

• Eq. (30): CFT entropy grows like $E^{(d-1)/d}$Proved

  Oct 2026

• Eq. (29): CFT entropy–energy relation inverts the thermal relationsProved

  Oct 2026

• Cohomological classification of the field-strength functorProved

  Oct 2026

• Filled triangle: A ↦ δA forgets only stuffProved

  Oct 2026

• Aharonov–Bohm: on a circle A ↦ δA is not fullProved

  Oct 2026

• Essentially surjective iff H² = 0Proved

  Oct 2026

• Full iff H¹ = 0Proved

  Oct 2026

• Faithful iff H⁰ = 0 (constant gauge transformations are stuff)Proved

  Oct 2026

• Prop. 1 analogue: without gauge arrows A ↦ δA forgets structureProved

  Oct 2026

• Lattice Hamiltonian constraints are second classProved

  Oct 2026

• Wheeler–DeWitt kinetic operator on the conformal sectorProved

  Oct 2026

• Semiclassical limit: WKB phase obeys the Einstein–Hamilton–Jacobi equationProved

  Oct 2026

Posted 50

• Spivak Thm 4-11: Poincaré lemma (closed ⇒ exact)Open

  Oct 2026

• Spivak Thm 4-11 (proof): homotopy formula dK + Kd = idOpen

  Oct 2026

• Derivative of K under the integral signOpen

  Oct 2026

• Spivak Thm 4-11 (proof): K preserves smoothnessOpen

  Oct 2026

• Radial fundamental theorem of calculusOpen

  Oct 2026

• Interior product of an alternatizationOpen

  Oct 2026

• Alternating sum of front-moves is (n+1)·fOpen

  Oct 2026

• Differential forms on a normed space, the radial homotopy operator, and the categories of potentials and field strengthsDefinition

  Oct 2026

• Functors from connections to a discrete category are never faithfulProved

  Oct 2026

• Morita extensions preserve sentence-isolated modelsProved

  Oct 2026

• Formulas pull back along a renaming of signaturesProved

  Oct 2026

• Every structure is bi-embeddable with a reduct along a renamingProved

  Oct 2026

• Strong isomorphisms of many-sorted structures are elementary embeddingsProved

  Oct 2026

• A thin category classified by a surjective invariant is equivalent to a discrete categoryProved

  Oct 2026

• Semantic characterization of models of a Morita extensionProved

  Oct 2026

• A $\Sigma^+$-homomorphism elementary on old sorts is elementaryProved

  Oct 2026

• A reduct embedding extends to a $\Sigma^+$-homomorphism of $T^+$-modelsProved

  Oct 2026

• New sorts of a Morita extension are realised as products, coproducts, subsets and quotientsProved

  Oct 2026

• Elementary embeddings commute with function symbolsProved

  Oct 2026

• p. 2 — the gravitationally evolved state is entangled iff $\Delta\phi_{LR}+\Delta\phi_{RL}\notin 2\pi\mathbb Z$Open

  Oct 2026

• p. 2 — entanglement grows monotonically as $\Delta\phi_{LR}+\Delta\phi_{RL}$ goes from $0$ to $\pi$, maximal at $\pi$Open

  Oct 2026

• p. 2 — $|\Psi(t=\tau)\rangle$ cannot be factorized when $\Delta\phi_{LR}+\Delta\phi_{RL}\neq 2n\pi$Open

  Oct 2026

• p. 2, Eqs. (1)–(2) — gravitational evolution of the two superposed massesOpen

  Oct 2026

• Two masses as orbital qubits: the evolved state, product vectors, separability and reduced purity (Bose et al. 2017)Definition

  Oct 2026

• Every finite-dimensional von Neumann algebra has a complete system of matrix unitsOpen

  Oct 2026

• The center of igoplus_lpha L(\mathcal H_{r_lpha})\otimes I consists of the block scalarsProved

  Oct 2026

• The commutant of igoplus_lpha L(\mathcal H_{r_lpha})\otimes I is igoplus_lpha I\otimes L(\mathcal H_{\overline r_lpha})Proved

  Oct 2026

• A complete system of matrix units is unitarily equivalent to igoplus_lpha L(\mathcal H_{r_lpha})\otimes I_{\overline r_lpha}Open

  Oct 2026

• Eqs. (1.4), (2.65): fusion rule D² = (1 + η) T⁻¹Proved

  Oct 2026

• §1.1 (below eq. (1.4)): D is not invertibleProved

  Oct 2026

• Eq. (2.65), Table 1: TD = DT and D† = DTProved

  Oct 2026

• Eqs. (1.4), (2.65): D absorbs η: Dη = ηD = DProved

  Oct 2026

• Eq. (1.5b): Kramers–Wannier on the coupling: D Z_{j-1} Z_j = X_{j-1} DProved

  Oct 2026

• Eq. (1.5a): Kramers–Wannier on the field: D X_j = Z_{j-1} Z_j DProved

  Oct 2026

• Eqs. (1.1)–(1.3), (2.10): invertible symmetries: η² = 1, Tη = ηT, T^L = 1, T shifts PaulisProved

  Oct 2026

• Pauli operators, Z2 symmetry, translation and the Kramers-Wannier operator on a periodic qubit chainDefinition

  Oct 2026

• Morita equivalence implies categorical equivalenceProved

  Oct 2026

• Separating example: the two theories are not Morita equivalentProved

  Oct 2026

• Separating example: the two theories are categorically equivalentProved

  Oct 2026

• Two kinds of things: one sort with p, q vs two sortsProved

  Oct 2026

• Translation of Σ⁺-formulas with old-sorted free variablesProved

  Oct 2026

• Morita extensions are conservativeProved

  Oct 2026

• The reduct functor Π is fullProved

  Oct 2026

• The reduct functor Π is faithfulProved

  Oct 2026

• Every model of T expands uniquely to a model of T⁺Proved

  Oct 2026

• Many-sorted first-order logic, Morita extensions and Morita equivalenceDefinition

  Oct 2026

• The holonomy functor is an equivalence of categoriesProved

  Oct 2026

• Aharonov-Bohm: curvature forgets structure on a circleProved

  Oct 2026

• The gauge-orbit functor forgets only stuffProved

  Oct 2026

• Gauge stabilizer equals the centralizer of the holonomiesProved

  Oct 2026