Alphabeta MathQuantitative Hyperbolic Geometry Toolkit
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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
Group Theory
Linear Algebra
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
- 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
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Hg toolkit slim triangles products and four point constants
Definition
Hg toolkit local geodesics and hausdorff control
Definition
Hg toolkit hyperbolic group and stable length
Lemma
Geodesic triangles in trees are tripods
Lemma
Slim triangles imply the gromov product inequality
Lemma
The gromov product inequality implies the four point condition
Lemma
The four point condition implies slim triangles
Lemma
Halfspace separation for the local-geodesic mesh
Lemma
Local geodesics in a hyperbolic space are uniform quasi geodesics
Lemma
Hg toolkit polygonal interpolation of quasi geodesics
Lemma
Exponential contraction of projection away from a quasiconvex set
Theorem
Morse stability with explicit parameter dependence
Lemma
A quasi isometry of geodesic spaces has a controlled coarse inverse
Lemma
Hyperbolicity is transported by a quasi isometry
Lemma
Short loop relators give a finite dehn presentation
Lemma
Linear isoperimetry implies uniformly thin geodesic bigons
Lemma
Infinite order elements have positive stable translation length
Lemma
Axis fellow travelling controls the centralizer
Definition
Hg toolkit gromov sequences and boundary product
Lemma
Asymptotic gromov sequences form an equivalence relation
Lemma
Boundary products have controlled representative and basepoint dependence
Lemma
Hg toolkit proper ray compactness and sequence comparison
Lemma
Quasi isometries extend to boundary homeomorphisms
Definition
Hg toolkit loxodromics and independent poles
Lemma
Hg toolkit finitely many cayley cone types
Lemma
Hg toolkit infinite hyperbolic groups have infinite order elements
Lemma
Hg toolkit non elementary groups have independent loxodromics
Lemma
Independent loxodromics have disjoint pole neighbourhoods
Lemma
Loxodromic elements have north south boundary dynamics
Lemma
Free Cayley trees from reduced-word normal form
Theorem
Quantitative hyperbolic geometry toolkit
Sources
- Druţu–Kapovich Chapter 9 §§9.1,9.5
- Druţu–Kapovich Theorem 9.38 and §9.6
- Hamann §5.1 definitions and Proposition 5.2.5
- Druţu–Kapovich §9.2 metric trees
- Druţu–Kapovich Lemmas 9.25 and 9.31
- Druţu–Kapovich §9.5 Gromov hyperbolicity
- Druţu–Kapovich Lemma 9.32; conservative two-inequality constant
- Drutu–Kapovich, revised Lemma 11.46, printed pp.375–377 (PDF indices 395–397)
- Druţu–Kapovich, revised Theorem 11.45 and Lemma 11.46, pp.375–378
- Gouëzel–Shchur, corrected quantitative Morse lemma, Lemma 2.1
- Gouëzel, AFP Gromov_Hyperbolicity, Isometries.thy, quasi_geodesic_made_lipschitz
- Gouëzel–Shchur, corrected quantitative Morse lemma, Lemmas 2.4–2.5
- Gouëzel, AFP Morse_Gromov_Theorem.thy, complete projection contraction proof
- Gouëzel–Shchur, Theorem 1.1 and corrected proof architecture
- Gouëzel, AFP Morse_Gromov_Theorem.thy, complete exact-constant proof
- Druţu–Kapovich Corollary 9.39; local published coarse-inverse theorem
- Druţu–Kapovich Corollary 9.39
- Druţu–Kapovich §9.13, Theorem 9.108, Proposition 9.109 and Lemma 9.112 pp.246–249
- Druţu–Kapovich, revised Proposition 9.104, Proposition 11.176 and Theorem 11.181
- Hamann Proposition 5.2.5 pp.87–89, complete proof read
- Hamann Theorem 5.2.6 pp.89–90, complete proof read
- Druţu–Kapovich §9.9 ray-boundary treatment; sequence comparison expanded locally
- Druţu–Kapovich §9.5 product inequality; §9.9 boundary conventions
- Druţu–Kapovich §9.9, ray topology; sequence-product comparison requires additional full treatment
- Druţu–Kapovich Lemma 9.61, Corollary 9.62 and Lemma 9.66; full comparison still to be sourced
- Druţu–Kapovich Theorem 9.83 (proper ray case); nonproper sequence proof supplied locally
- Hamann §5.3 after Proposition 5.2.5
- Hamann Proposition 5.2.3 pp.85–87; complete proof
- Hamann Proposition 5.2.2 and Theorem 5.2.4 pp.85–87; complete proofs
- Hamann Theorems 5.3.7–5.3.8 pp.91–94; complete torsion-allowed proof
- Canary Theorem 7.3 pp.32–33; four-pole separation
- Canary Proposition 5.1 pp.25–26 and Theorem 5.7 pp.28–29; complete convergence-action proof
- Reduced-word construction: local adaptation of the earlier proved free-group item; Drutu–Kapovich metric tree discussion
- Druţu–Kapovich Chapter 9; Hamann §§5.1–5.3; Canary §§5,7