Notation And Dependencies
Symbol Table
The “Introduced” column lists the chapter that owns the general definition of each symbol. Many symbols first appear concretely in the Chapter 0 worked example before they are defined in general; the column points to the latter.
Introduced In Chapter 0
: finite state space with 
states (general definition in Chapter 2).
: finite action space with 
actions (general definition in Chapter 2).
: actor parameter in 
(general definition in Chapter 2).
: critic parameter in 
(general definition in Chapter 2).
: dynamic state distribution in 
(general definition in Chapter 2).
Introduced In Chapter 1
: semiflow on 
(constructed for the coupled system in Chapter 3).
Introduced In Chapter 2
: softmax policy.
: actor feature vectors.
: critic feature vectors.
: product occupancy 
.
: generator family for the controlled chain.
, 
: critic matrix and forcing vector.
: boundary-confinement damping matrix 
.
: prescribed Lipschitz closure map 
.
- : phase space
.
: critic coercivity constant (coercivity role in Chapter 4).
: fast-law timescale parameter (central in Chapter 7).
Introduced In Chapter 4
: critic drift bound.
: absorbing critic radius 
.
- : compact absorbing set
.
Introduced In Chapter 5
: omega-limit set of .
: compact global attractor.
Introduced In Chapter 6
: frozen invariant law of .
: invariant-law closure map 
.
: stationary occupancy after the bridge theorem.
Introduced In Chapter 7
: exact attractor at timescale .
: reduced (lifted) attractor.
Chapter Dependency Diagram
The chapters are designed for linear reading:
- Ch 0: Worked Example
- Ch 1: Prerequisite Bridge
- Ch 2: General Model
- Ch 3: Well-Posedness
- Ch 4: A Priori Estimates
- Ch 5: Global Attractor
- Ch 6: Controlled Chain Closure
- Ch 7: Fast-Slow Reduction
- Ch 8: Outlook and Open Problems
Chapter 6 also draws on the generator family introduced in Chapter 2. Chapter 8 contains no new theorems and can be read independently after Chapter 7.
Chapters 0—8 above form the theoretical core. The application chapters are read after Chapter 8 in the order Ch 9 → Ch 10 → Ch 11 (Chapter 9 defines the Model Specification Protocol that Chapters 10 and 11 execute, and Chapter 11 builds on Chapter 10); they introduce model-specific symbols local to each chapter (the Y-graph generators 
of Chapter 10, the hub-and-spoke construction of Chapter 11) rather than new core notation. Appendix A maps the core theorems to their Lean formalization, and Appendix B collects computational methods and a phase-portrait blueprint.
Conventions
- Indices. States are indexed by
, actions by 
. State-action pairs are 
.
- Probability. denotes the probability simplex on . Elements
are column vectors with 
and 
; the adjoint generator 
acts on distributions, while acts on observables.
- Norms.
is the Euclidean norm unless otherwise specified. The simplex uses the 
distance 
, which equals twice the total-variation distance.
- Overloaded letters. The letter names both the number of actions in the model data and the compact absorbing set constructed in Chapter 4; context always disambiguates.
- Closure maps. always denotes a prescribed Lipschitz closure map. denotes the invariant-law map identified by the bridge theorem (Chapter 6). The two coincide once the bridge theorem is in place, but the distinction matters in Chapters 2—5.