Alphabeta MathVerifiable generation, group-relative updates, and test-time sampling
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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.
✓ 4 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 4 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
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
Combinatorics
Linear Algebra
Machine Learning Algorithms
Measure Theory
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Outer Measure and the Caratheodory Extension Theorem
- 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 Lᵖ Spaces Holder Minkowski and Riesz Fischer
- 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
- Countability and Uncountability
- 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 Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Finite truncated response spaces, correctness predicates, and verifiers
Algorithm
Budgeted verifier-guided generation and best-of-N sampling
Lemma
Budgeted verifier generation terminates with bounded response and selection calls
Theorem
Independent generation success and verifier false-acceptance bounds
Lemma
Finite response likelihood ratios change measure by cancellation
Algorithm
Finite-budget group-relative policy update
Lemma
Finite group-relative updates preserve admissibility and have bounded oracle calls
Lemma
Finite-response score mean and reward-gradient identities
Theorem
The group-centered score estimator has factor one minus one over G
Remark
Reasoning-training evidence and evaluation targets
Sources
- DeepSeek-AI, DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning, original arXiv v1
- Richard S. Sutton and Andrew G. Barto, Reinforcement Learning: An Introduction, second-edition draft
- DeepSeek-AI, DeepSeek-R1, §2.3.3 (rejection-sampling background; finite call bounds proved locally)
- John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov, Proximal Policy Optimization Algorithms, original arXiv v1
- DeepSeek-AI, DeepSeek-R1, §2.2.1 (group-relative procedure background; finite oracle bounds proved locally)
- Schulman et al., Proximal Policy Optimization Algorithms, §2.2 and §3 (clipped-surrogate background)