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NOTATION REFERENCE INDEX

Subscript Plus Notation (₊): - Denotes next stage relative to current stage - Used for states (xₐ₊), periods (𝕊₊), and value functions (𝒜₊) - In branching: Subscript j (e.g., 𝒜₊ⱼ) indicates branch-j-specific

Stage and Period Structure: - 𝕊: A period containing an ordered sequence of stages - 𝕊[i]: Stage at position i in period (e.g., 𝕊[0] first, 𝕊[-1] last) - κ(s): The kind of stage s - N₊: Set of stage kinds that can follow current stage - t: Time period index (shown only when emphasizing inter-period relationships)

State Spaces: - 𝓧ₐ: Arrival states - 𝓧ᵥ: Decision states - 𝓧ₑ: Continuation states - 𝒵ₐᵥ: Shock space for arvl→dcsn transition, with probability measure Pₐᵥ(ζₐᵥ) - 𝒵ᵥₑ: Shock space for dcsn→cntn transition, with probability measure Pᵥₑ(ζᵥₑ) - Either or both shock spaces may be trivial (𝒵 = {∅}) - Π(xᵥ): Choice set at decision state xᵥ - 𝓧ₐʳᵉᵠ: Required components of arrival state space - 𝓧ₐᵒᵖᵗ: Optional components of arrival state space (before extension) - 𝓧ₐᵒᵖᵗ⊥: Optional components extended with ⊥ (= 𝓧ₐᵒᵖᵗ ∪ {⊥}) - ⊥: Sentinel value indicating an optional state component is absent

Value Functions: - 𝒜(xₐ): Value at arrival state - 𝒱(xᵥ): Value at decision state - ℰ(xₑ): Value at continuation state - 𝒜₊ⱼ(xₐ): Arrival value for branch j ∈ N₊ in branching stages - ℰ₀(xₑ): Terminal value function (special case)

Within-Stage Perch Transitions: - gₐᵥ: 𝓧ₐ × 𝒵ₐᵥ → 𝓧ᵥ (arrival to decision, with pre-decision shocks ζₐᵥ) - gᵥₑ: 𝓧ᵥ × Π × 𝒵ᵥₑ → 𝓧ₑ (decision to continuation, with post-decision shocks ζᵥₑ) - When a shock space is trivial (𝒵 = {∅}), the corresponding argument is omitted

Connector Functions: - gₑₐ₊: 𝓧ₑ → 𝓧ₐ₊ (sequential stage connector) - gᵥₐ₊: 𝓧ᵥ × Π → 𝓧ₐ₊ (branching stage connector) - "Connector" is used for both within-period (stage-to-stage) and between-period (period-to-period) mappings, qualified by scope. (The term "twister" used in spec_0.1h and the builder code corresponds to what this document calls a "between-period connector.")

Population Measures: - μₐ: Measure over arrival states - μᵥ: Measure over decision states - μₑ: Measure over continuation states - μₐ₊ⱼ: Branch-specific arrival measure for j ∈ N₊

Branching: - p(j|xᵥ,𝜋): Probability of transitioning to branch j ∈ N₊ - j ∈ N₊: Index over possible next stages in branching case

Stage Identifiers: Each stage is indexed by the variable that characterizes its dcsn perch: - c: cons stage (consumption decision) - ς: port-able stage (portfolio share decision) - β: disc stage (time discounting "decision")

Discounting Stage (disc): - The disc stage appears as the last stage in every period - 𝒱_disc(xᵥ) = β ℰ(xₑ): value at dcsn perch = β × continuation value - Identity transitions: gₐᵥ = identity, gᵥₑ = identity - Singleton choice set: the "decision" is applying β - No shocks, no reward - All "economic" stages (cons, port-able) have β = 1; the disc stage is the sole bearer of β < 1