Alphabeta MathDiscrete masked diffusion and finite-state reverse generation
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
- 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
Bayesian Computational Statistics
Information Theory
Linear Algebra
Mathematical Statistics
Measure Theory
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue Integral and the Convergence Theorems
- The Radon Nikodym Theorem and Lebesgue Decomposition
Probability
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convex and Semicontinuous Functions on Rⁿ
- Convexity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- 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
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Integral Logarithm and the Equivalence of Its Characterisations
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Statistical Learning and Classical Models
2 · Logical flowchart
3 · Items
Definition
Finite masking diffusion chain
Lemma
Uniform row error bounds mixture-kernel error
Lemma
Finite product KL is the sum of coordinate KLs
Theorem
Finite-chain reverse-kernel error telescope
Lemma
Finite masking-chain transitions define a probability law
Algorithm
Finite-grid masked reverse sampler
Lemma
Finite-grid masked sampler defines a normalized law
Theorem
Posterior of a finite masking chain
Lemma
Finite masking path law factors backward by its posterior rows
Theorem
Masked-diffusion KL reduces to weighted cross-entropy
Definition
Finite masked-diffusion generative model
Lemma
Finite masked-diffusion joint model is normalized
Lemma
Forward path law is admissible for the finite masked model
Corollary
Negative expected finite masked-diffusion ELBO is weighted cross-entropy
Remark
Factorization and finite-endpoint warnings
Sources
- Subham Sekhar Sahoo et al., Simple and Effective Masked Diffusion Language Models, arXiv v1
- Jacob Austin et al., Structured Denoising Diffusion Models in Discrete State-Spaces
- David A. Levin and Yuval Peres, Markov Chains and Mixing Times, second edition
- Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning
- David M. Blei, Alp Kucukelbir, and Jon D. McAuliffe, Variational Inference: A Review for Statisticians
- Ronghuo Zheng, Zhiting Hu, and Yuntian Deng, CMU 10-708 Lecture 13: Variational Inference II