Alphabeta MathLandscape Statistics And Selection Examples
About this page
Pipeline-generated
How 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.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Category Theory
Complex Analysis
Differential Geometry
- Connections Levi Civita and Parallel Transport
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Integration of Forms and the General Stokes Theorem
- Jacobi Fields, Conjugate Points, and the Cut Locus
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Riemann Curvature and Riemannian Submanifolds
- Riemannian Comparison Theorems
- Riemannian Metrics Length Distance and Volume
- Singular Cochains Mayer Vietoris and Smooth Singular Comparison
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The De Rham Complex Homotopy and Mayer Vietoris
- The de Rham Theorem and Degree
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
Functional Analysis
General Relativity
Homological Algebra
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Constant Rank, Submersions, Immersions and Regular Level Sets
- 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
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Further Trigonometric Identities and Inverse Functions
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
- Volumes of Elementary Solids and Solids of Revolution
Set Theory
Special Relativity
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Experiment
DESI DR2 BAO measurements and qualified dark-energy inference
Open problem
Regulator-independent landscape probability and selection problem
Experiment
Planck PR4 and BICEP/Keck 2018 tensor-to-scalar upper limit
Sources
- DESI Collaboration, DESI DR2 Results II: Measurements of baryon acoustic oscillations and cosmological constraints, arXiv:2503.14738v3
- S. D. Landy and A. S. Szalay, Bias and variance of angular correlation functions, Astrophysical Journal 412(1), 64–71 (1993)
- D. J. Spiegelhalter, N. G. Best, B. P. Carlin, and A. van der Linde, Bayesian measures of model complexity and fit, Journal of the Royal Statistical Society: Series B 64(4), 583–639 (2002)
- DESI Collaboration, DESI DR2 Results I: Baryon Acoustic Oscillations from the Lyman Alpha Forest, arXiv:2503.14739v2
- N. Padmanabhan et al., A 2% Distance to z=0.35 by Reconstructing Baryon Acoustic Oscillations — I: Methods and Application to the Sloan Digital Sky Survey, arXiv:1202.0090v1
- Frederik Denef and Michael R. Douglas, Distributions of Flux Vacua, arXiv:hep-th/0404116v2
- Raphael Bousso, TASI Lectures on the Cosmological Constant, arXiv:0708.4231v2
- Michael R. Douglas, The String Theory Landscape, Universe 2019, 5(7), 176, doi:10.3390/universe5070176
- Michele Cicoli, Joseph P. Conlon, Anshuman Maharana, Susha Parameswaran, Fernando Quevedo, and Ivonne Zavala, String Cosmology: from the Early Universe to Today, arXiv:2303.04819v2
- Bjoern Hassfeld, Arthur Hebecker, Manfred Salmhofer, Jonah Cedric Strauss, and Johannes Walcher, A Local Wheeler–DeWitt Measure for the String Landscape, arXiv:2205.09772v4, Nuclear Physics B 992 (2023) 116230
- Justin Khoury and Sam S. C. Wong, Bayesian Reasoning in Eternal Inflation: A Solution to the Measure Problem, arXiv:2205.11524v2, Physical Review D 108 (2023) 023506
- M. Tristram et al., Improved limits on the tensor-to-scalar ratio using BICEP and Planck, arXiv:2112.07961v2
- M. Cicoli et al., String Cosmology: from the Early Universe to Today, arXiv:2303.04819v2
- Planck Collaboration, Planck intermediate results LVII: Joint Planck LFI and HFI data processing, arXiv:2007.04997v1
- BICEP2 Collaboration, BICEP2 II: Experiment and three-year data set, arXiv:1403.4302v3
- BICEP/Keck Collaboration, BICEP/Keck XV: The BICEP3 CMB polarimeter and the first three-year data set, arXiv:2110.00482v1
- BICEP/Keck Collaboration, BICEP/Keck XIII: Improved constraints on primordial gravitational waves using observations through 2018, arXiv:2110.00483v1