NOTATION.md

slug	notation
type	reference

AAD Notation and Conventions

All symbols used in Adaptation and Actuation Dynamics, serving as a single authoritative reference. When a symbol appears in any AAD document, its meaning is as defined here unless explicitly noted otherwise.

Notation conventions are adopted from TFT (_obs/old-tf-00-notation-conventions.md) with extensions for AAD's purposeful-agent machinery.

The Adaptive Cycle

One complete traversal of the agent-environment feedback loop — the unit of adaptive work. The five phases are named from Greek philosophical vocabulary; see LEXICON.md for the full prose discussion (why these terms, what they mean beyond the formalism, and what the cycle is not).

Phase	Greek	Formalism
Prolepsis (πρόληψις)	Anticipation	$\hat o_t = \mathbb{E}[o_t \mid M_{t-1}, a_{t-1}]$
Aisthesis (αἴσθησις)	Perception	$o_t$ arrives
Aporia (ἀπορία)	Perplexity	$\delta_t = o_t - \hat o_t$
Epistrophe (ἐπιστροφή)	Turning-toward	$M_t = M_{t-1} + \eta^\ast \cdot g(\delta_t)$
Praxis (πρᾶξις)	Informed action	$a_t = \pi(M_t)$

The cycle is: Prolepsis → Aisthesis → Aporia → Epistrophe → Praxis → (Prolepsis).

Primitives ( #agent-environment, #observation-function, #action-transition)

Symbol	Type	Meaning
$\Omega$	State space	Environment state (unobservable totality)
$\Omega_t$	State	Environment state at time $t$
$\mathcal{O}$	Set	Observation space
$\mathcal{A}$	Set	Action space
$o_t$	$\in \mathcal{O}$	Observation at time $t$
$a_t$	$\in \mathcal{A}$	Action at time $t$
$h$	Function	Observation function: $o_t = h(\Omega_t, a_{t-1}, \varepsilon_t)$
$T$	Distribution	Transition: $\Omega_{t+1} \sim T(\cdot \mid \Omega_t, a_t)$
$\varepsilon_t$	Random variable	Observation noise
$H(\cdot)$	Functional	Shannon entropy
$I(\cdot; \cdot)$	Functional	Mutual information

Temporal Structure ( #causal-structure, #chronica)

Symbol	Type	Meaning
$\mathcal C_t$	Sequence	Interaction history (chronica): $(o_1, a_1, \ldots, a_{t-1}, o_t)$
$do(\cdot)$	Operator	Pearl's intervention operator (Level 2)
$\text{CIY}(a)$	Scalar $\geq 0$	Causal information yield of action $a$ ( #causal-information-yield)

The Model ( #agent-model, #information-bottleneck)

Symbol	Type	Meaning
$M_t$	$\in \mathcal{M}$	Model state (epistemic substate) at time $t$
$\mathcal{M}$	Set	Model space (the class of representable models)
$\phi$	Function	Compression: $M_t = \phi(\mathcal C_t)$
$f$	Function	Recursive update: $M_{\tau^+} = f(M_{\tau^-}, e_\tau)$
$S(M_t)$	$\in [0, 1]$	Model sufficiency ( #model-sufficiency)
$\mathcal{F}(\mathcal{M})$	$\in [0, 1]$	Model class fitness ( #model-class-fitness)
$\beta$	Scalar $\gt 0$	Information bottleneck trade-off parameter

Event-Driven Dynamics ( #event-driven-dynamics)

Symbol	Type	Meaning
$e$	Event	An atomic observation or action-completion event
$\tau$	$\in \mathbb R_{\geq 0}$	Continuous timestamp of an event
$\mathcal{E}$	Sequence	Event stream: ${(e_i, \tau_i)}$
$\nu^{(k)}$	Rate (Hz)	Event rate on channel $k$
$M_{\tau^-}$, $M_{\tau^+}$	$\in \mathcal{M}$	Model state just before / after event at $\tau$
$\mathcal{I}(e_\tau)$	Scalar $\geq 0$	Event information content: $I(e_\tau;, \Omega_\tau \mid M_{\tau^-})$
$U_o^{(k)}$	Scalar $\gt 0$	Observation uncertainty on channel $k$

Mismatch Signal ( #mismatch-signal)

Symbol	Type	Meaning
$\hat o_t$	$\in \mathcal{O}$	Predicted observation: $\mathbb{E}[o_t \mid M_{t-1}, a_{t-1}]$
$\delta_t$	$\in \mathcal{O}$	Mismatch signal (prediction error): $o_t - \hat o_t$
$\tilde{\delta}_t$	$\in T_M\mathcal{M}$	Score-function mismatch: $-\nabla_M \log P(o_t \mid M_{t-1}, a_{t-1})$

Update Gain ( #update-gain)

Symbol	Type	Meaning
$\eta$	Scalar, vector, or matrix	Update gain (general)
$\eta^\ast$	Same	Optimal update gain
$\eta^{(k)\ast}$	Same	Optimal gain on channel $k$
$U_M$	Scalar $\gt 0$	Model uncertainty: $\text{Var}{M{t-1}}[\hat o_t \mid a_{t-1}]$
$U_o$	Scalar $\gt 0$	Observation uncertainty: $\text{Var}[\varepsilon_t]$
$g$	Function	Mismatch transform: maps mismatch to update direction

Action Selection ( #action-selection)

Symbol	Type	Meaning
$\pi$	Function or distribution	Policy: $a_t = \pi(M_t)$ or $a_t \sim \pi(\cdot \mid M_t)$
$\lambda(M_t)$	Scalar $\geq 0$	Exploration-exploitation balance weight ( #causal-information-yield)

Adaptive Tempo and Dynamics ( #adaptive-tempo, #mismatch-dynamics)

Symbol	Type	Meaning
$\mathcal{T}$	Rate ($t^{-1}$)	Adaptive tempo (epistemic): $\sum_k \nu^{(k)} \cdot \eta^{(k)\ast}$
$\mathcal{T}_\Sigma$	Rate ($t^{-1}$)	Strategic tempo: $\sum_{(i,j) \in E} \nu_{ij} \cdot \eta_{\text{edge},ij}$ ( #strategic-tempo)
$\rho(t)$	Rate (surprise/time)	Environment change rate (mismatch injection rate)
$\rho_\Sigma$	Rate	Strategic disturbance rate (rate of causal-link invalidation)
$\lVert\delta\rVert_{ss}$	Scalar $\geq 0$	Steady-state mismatch: $\rho / \mathcal{T}$ (linear approximation)

Lyapunov / Sector-Condition Analysis ( #sector-condition-stability)

Symbol	Type	Meaning
$F(\mathcal{T}, \delta)$	Function	Correction function (general nonlinear)
$w(t)$	Vector	Disturbance (new mismatch), $\lVert w(t)\rVert \leq \rho$
$\alpha$	Scalar $\gt 0$	Lower sector bound of correction function
$R$	Scalar $\gt 0$	Radius of sector-condition region (model class capacity)
$R^\ast$	Scalar $\gt 0$	Ultimately bounded mismatch radius: $\rho/\alpha$
$\Delta\rho^\ast$	Scalar $\geq 0$	Adaptive reserve: $\alpha R - \rho$
$\gamma_A$	Scalar $\gt 0$	Coupling effectiveness of $A$'s actions on $B$'s disturbance
$V(\delta)$	Scalar $\geq 0$	Lyapunov function: $\frac{1}{2}\lVert\delta\rVert^2$

Deliberation ( #deliberation-cost)

Symbol	Type	Meaning
$\Delta\tau$	Duration $\geq 0$	Deliberation duration
$\Delta\eta^\ast(\Delta\tau)$	Scalar $\geq 0$	Gain improvement from deliberation
$\rho_{\text{delib}}$	Rate	Local mismatch drift rate during deliberation pauses

Purposeful Agent State (Section II)

Symbol	Type	Meaning
$X_t$	State	Complete agent state: $(M_t, G_t)$
$G_t$	State	Purposeful substate: $(O_t, \Sigma_t)$
$O_t$	Objective	What the agent wants — induces value functional $V_{O_t}$
$\Sigma_t$	Strategy	How the agent plans to get it — a probabilistic causal DAG
$V_O(M_t, \pi; N_h)$	Scalar	Value object: horizon- and policy-conditioned
$Q_O(M_t, a; \pi_{\text{cont}}, N_h)$	Scalar	Action-value object
$\delta_{\text{sat}}$	Scalar	Satisfaction gap: $V_{O_t}^{\min} - A_O(M_t; \Pi, N_h)$
$\delta_{\text{regret}}$	Scalar $\geq 0$	Control regret: $A_O - V_O(\pi_{\text{current}})$
$\delta_{\text{strategic}}$	Scalar $\geq 0$	Strategic calibration: edge residual aggregate
$p_{ij}$	$\in [0, 1]$	Edge confidence weight in strategy DAG
$\gamma(v)$	$\in {\text{AND}, \text{OR}}$	Node combination rule in strategy DAG

Multi-Agent (Section III)

Symbol	Type	Meaning
$U_M$	Scalar $\in [-1, 1]$	Epistemic unity (shared model)
$U_O$	Scalar $\in [-1, 1]$	Teleological unity (shared objective)
$U_\Sigma$	Scalar $\in [-1, 1]$	Strategic unity (coordinated action)
$\eta_{ji}^\ast$	Scalar	Communication gain: $U_{M_i}/(U_{M_i} + U_{o,ji} + U_{\text{src},j} + U_{\text{align},ji})$
$C_{\text{coord}}$	Rate ($t^{-1}$)	Coordination overhead (tempo-equivalent)

Conventions

Subscript $t$: Discrete time index or macroscopic continuous time. Context disambiguates: $M_t = f(M_{t-1}, o_t, a_{t-1})$ is discrete; $d\lVert\delta\rVert/dt$ is continuous.

Subscript $\tau$: Continuous timestamp of an individual event. Used in the event-driven formulation for microscopic update dynamics.

Superscript $(k)$: Channel index. Distinguishes observation or action channels with distinct rates and noise.

Calligraphic letters ($\mathcal{M}$, $\mathcal{O}$, $\mathcal{A}$, $\mathcal{E}$, $\mathcal{C}$, $\mathcal{T}$, $\mathcal{I}$): Sets, spaces, or aggregate quantities. $\mathcal{C}$ for chronica (not $\mathcal{H}$, to avoid collision with entropy). Exceptions: $\mathcal{T}$ is a scalar rate (calligraphic distinguishes from temperature); $\mathcal{I}(e_\tau)$ is a scalar (distinguishes from mutual information $I$).

$\lVert\cdot\rVert$: Norm (Euclidean or information-theoretic). Used for mismatch magnitude.

$\lvert\cdot\rvert$: Cardinality of a set (e.g., $\lvert\mathcal{A}\rvert$). Not for mismatch — use $\lVert\cdot\rVert$.

Scalar reduction of gain and tempo. When $\eta^\ast$ appears as scalar in mismatch dynamics, it represents the effective correction fraction along the current mismatch direction: $\eta^\ast_{\text{eff}} = \delta^T K \delta / \lVert\delta\rVert^2$. Scalar tempo $\mathcal{T} = \nu \cdot \eta^\ast_{\text{eff}}$ is the correction rate along this direction. The sector condition parameter $\alpha$ corresponds to the worst-case scalar projection. The full anisotropic treatment requires a tempo tensor; the scalar reduction is valid when correction dynamics are approximately isotropic.

Units

The theory uses natural (dimensionless information-theoretic) units where possible:

Quantity	Units	Notes
$\eta^\ast$	Dimensionless $\in [0, 1]$	Ratio
$\nu^{(k)}$	Events per unit time (Hz)
$\mathcal{T}$	Inverse time ($t^{-1}$)	Effective correction rate
$\rho$	Surprise $\cdot t^{-1}$	Mismatch injection rate
$S(M_t)$	Dimensionless $\in [0, 1]$	Ratio

Dimensional analysis of the mismatch ODE. In $d\lVert\delta\rVert/dt = -\mathcal{T}\lVert\delta\rVert + \rho$: LHS has units [surprise $\cdot t^{-1}$]; $\mathcal{T}\lVert\delta\rVert$ has $[t^{-1}] \cdot [\text{surprise}]$; $\rho$ has [surprise $\cdot t^{-1}$]. All consistent. Note $\mathcal{T}$ and $\rho$ have different units — the persistence condition $\mathcal{T} \gt \rho/\lVert\delta_{\text{critical}}\rVert$ is dimensionally consistent. The shorthand "$\mathcal{T} \gt \rho$" is valid only when $\lVert\delta_{\text{critical}}\rVert$ is normalized to 1.

Global Assumptions

Load-bearing assumptions that appear locally but are referenced by multiple results:

ID	Assumption	Used by
GA-1	Fresh noise. $\varepsilon_t$ is conditionally independent of $\mathcal C_{t-1}$ given $(\Omega_t, a_{t-1})$.	#mismatch-decomposition
GA-2	Bounded disturbance (Model D). $\lVert w(t)\rVert \leq \rho$ for finite $\rho$. Deterministic worst-case bound; no distributional assumption.	#sector-condition-stability, #persistence-condition
GA-2S	Stochastic disturbance (Model S). $w(t)$ is zero-mean with $\mathbb{E}[\lVert w(t)\rVert^2] = \sigma_w^2$. Alternative to GA-2 for environments with noise rather than drift.	#sector-condition-stability (Prop A.1S), #persistence-condition, #adversarial-exponent-regimes
GA-3	Sector condition (continuous). $\delta^T F(\mathcal{T}, \delta) \geq \alpha\lVert\delta\rVert^2$ for $\lVert\delta\rVert \leq R$. Derived from the gain principle when the update rule has directional fidelity (B1); for gradient-based agents, equivalent to local strong convexity of the loss. Remains an independent assumption for non-gradient agents. The discrete-time analog (DA2') adds a Lipschitz upper bound $\lVert F_d(\delta)\rVert \leq c_{\max}\lVert\delta\rVert$ — strictly stronger than an inner-product upper bound, required because the discrete contraction involves $\lVert F_d\rVert^2$. See #gain-sector-bridge, #discrete-sector-condition.	#sector-condition-stability, #discrete-sector-condition
GA-4	Local deliberation drift. Mismatch accumulates at rate $\rho_{\text{delib}}$ during inaction.	#deliberation-cost
GA-5	Fluid limit. Event rate high relative to dynamics timescale ($\eta^\ast \ll 1$).	#mismatch-dynamics