Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Chern–Simons boundary geometry and the three-dimensional gravity interface — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Binary Operations, Monoids, Groups and Subgroups
- Chern–Simons boundary geometry and the three-dimensional gravity interface
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Exterior Powers, Orientation and Hodge Duality
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Hereditary and Productive Behaviour of the Separation Axioms
- Integration of Forms and the General Stokes Theorem
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Manifolds with Boundary Collars and Orientations
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Order, Zorn's Lemma, and the Axiom of Choice
- Partitions of Unity and Paracompactness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rank Theorems and Embedded Submanifolds
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Exterior Derivative and Cartan Calculus
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Vector Fields Flows and Lie Derivatives
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Summary
3 · Logical flowchart
4 · Definitions, postulates, theorems and proofs
None yet.
5 · Examples, experiments and counterexamples
A degenerate BF triad defines no Lorentz metric
Pure mathematics; an explicit classical configuration, with no empirical or quantum claim.
Statement refuted
Every smooth solution of the zero-cosmological-constant three-dimensional BF equations has an invertible triad and determines a nondegenerate Lorentz metric satisfying the vacuum Einstein equation.
Statement references
Facts & Assumptions
Given: , a nonempty oriented three-manifold with a global trivialization, and the zero-cosmological-constant BF field equations.
The Axiom of Countable Choice (): countable choice, as required by the conditional gravity proposition used for comparison.
Chern–Simons and BF action data: the BF curvature is and the covariant derivative of the BF field is .
Invertible triads identify zero-cosmological-constant three-dimensional BF gravity: the BF equations are and , and only the invertible-coframe sector defines a nondegenerate Lorentz metric equivalent to vacuum gravity.
Counterexample
Take to be the closed unit three-ball, choose the trivial bundle, and set and . Direct substitution gives and , so [F2] says this is a BF solution.
At every , the map has rank zero, so it is not invertible. The induced symmetric tensor is identically zero and therefore degenerate; it is not a Lorentz metric. This solution lies outside the metric-gravity sector, refuting the statement.