# Canonical Notation and Mathematical Conventions

This file is the source of truth for the notation used throughout the manuscript. The philosophical language is intentionally extravagant; the notation is not allowed to be.

The formalism below is a model specification, not a theorem that the world literally has these coordinates. Whenever a real system is represented in a Hilbert space, the vector is the representation used by the model, not the external object itself.

Unless a section says otherwise, every declared input, state, trace, action, and outcome domain is equipped with a sigma-algebra; every deterministic encoder, update, interaction, aggregation, and utility map is measurable; and every object written as a conditional distribution is a probability kernel. Finite-dimensional Euclidean spaces carry their Borel sigma-algebras. When regular conditional laws or measurable state realizations are invoked, the relevant spaces are assumed standard Borel. Equalities between conditional laws are understood up to the usual almost-sure qualification for the chosen versions.

## 1. Typographic conventions

- Calligraphic capitals such as \(\mathcal G\) denote spaces, sets, or sigma-algebraic domains.
- Ordinary capitals such as \(T_{i,t}\), \(Q_{i,t}\), \(S_{C,t}\), and \(D_{ab,t}\) denote structured mathematical states that need not be vectors. Ordinary uppercase Roman symbols may also denote declared maps, kernels, or matrices.
- Bold lowercase symbols such as \(\mathbf z_{i,t}\) denote finite-dimensional vectors.
- Nonbold lowercase symbols may denote scalars, realized values, or generic elements of nonvector spaces; their type is declared where they appear.
- A hat, as in \(\widehat D_{ab,t}\), denotes a filtered or learned estimate constructed from information available before the current decision.
- A tilde, as in \(\widetilde D_{ab,t+1}\), denotes a random quantity simulated from the fitted model.
- A superscript \(\star\) denotes an unknown data-generating object when it is useful to distinguish reality from the fitted model.
- The subscript \(\theta\) denotes learned parameters.
- \(\mathcal L(X\mid Z)\) denotes the conditional law of random variable \(X\) given \(Z\). The symbol \(\mathcal L\) is not used for the training loss; the training loss is written \(\mathscr J\).

Where the random-versus-realized distinction matters for propositions, traces, and outcomes, capitals denote random variables and lowercase symbols denote realized values. Thus \(X_t\) is the proposition selected at decision time \(t\), \(x_t\) is the proposition actually selected, \(O_{t+1}\) is a trace-valued random variable, and \(o_{t+1}\) is an observed trace. Structured state symbols retain the capitalization conventions above, and bold lowercase symbols may still denote random latent vectors.

## 2. Indices and clocks

| Symbol | Meaning |
|---|---|
| \(i,j\) | generic individual human actors |
| \(a\) | salesperson in the first application |
| \(b\) | executive or buyer in the first application |
| \(C_a\) | salesperson’s company |
| \(C_b\) | executive’s company |
| \(C\) | a generic corporation |
| \(t\) | a decision epoch |
| \(r,n\) | chronological record indices |
| \(v_n\) | timestamp of chronological record \(\mathsf h_n\) |
| \(\nu\) | evolutionary stage in the toy evolutionary construction |
| \(\tau\) | prediction task |
| \(\Delta\) | elapsed-time outcome horizon measured from decision epoch \(t\) |
| \(H\) | number of future decision steps in a planning or rollout problem |
| \(f\) | categorical feature family |
| \(\kappa\) | category index when a category itself needs a label |
| \(\sigma\) | categorical source channel |
| \(\rho\) | role or regime |
| \(\ell\) | primary outcome-head index |
| \(m\) | auxiliary probe-head index |

The symbol \(\pi\) is reserved for a decision policy. It is not used for a task projection, salience map, or dimensionality-reduction operator.

The symbol \(H\) is reserved for a finite planning length. Histories are written with calligraphic \(\mathcal H\), for example \(\mathcal H_{i,<t}\).

## 3. External conditions and GIDS

Let \(\mathcal N\) denote the external or noumenal domain. It is deliberately not assumed to be a vector space. Let \(\mathcal N_{\mathrm{loc}}\subseteq\mathcal N\) denote local external configurations that a model attempts to register.

Let

\[
\mathcal G
\]

be **God’s Infinite Dimensional Space**: an idealized real separable Hilbert representation arena rich enough to encode distinctions and combinations of distinctions that could enter the experience or response of actors. The construction does not require one orthogonal basis coordinate for every named distinction. For a concrete model one may take \(\mathcal G\cong\ell^2\), but the manuscript does not claim that physical reality is literally \(\ell^2\).

A registration map

\[
\operatorname{Reg}:\mathcal N_{\mathrm{loc}}\to\mathcal G
\]

assigns a model-side representation to a local external configuration. If \(\omega_t\in\mathcal N_{\mathrm{loc}}\), then

\[
\mathbf g_t^{\mathrm{reg}}=\operatorname{Reg}(\omega_t)\in\mathcal G.
\]

The vector \(\mathbf g_t^{\mathrm{reg}}\) is an encoding selected by the framework. It is not the external condition itself, and \(\operatorname{Reg}\) need not be injective, linear, or information-preserving.

## 4. Lineage access and inherited structure

Let

\[
\mathcal M^{\mathrm{spec}}
=
\operatorname{span}\{\mathbf v_1,\ldots,\mathbf v_d\}
\subset\mathcal G,
\qquad d<\infty,
\]

be a finite-dimensional idealization of the distinctions available to a lineage. When \((\mathbf v_j)_{j=1}^d\) is orthonormal, the orthogonal projection is

\[
P^{\mathrm{spec}}\mathbf g
=
\sum_{j=1}^{d}
\langle\mathbf v_j,\mathbf g\rangle\mathbf v_j.
\]

“Best approximation” here means closest in the norm chosen on \(\mathcal G\). It does not establish that biological perception literally performs an orthogonal projection.

Write the lineage-level template as

\[
G^{\mathrm{spec}}
=
\bigl(
\mathcal M^{\mathrm{spec}},
P^{\mathrm{spec}},
\mathfrak I^{\mathrm{spec}}
\bigr),
\]

where \(\mathfrak I^{\mathrm{spec}}\) denotes a species-compatible family of developmental and interpretive organizations.

The inherited seed of individual \(i\) is written

\[
G_i
=
\bigl(
\mathcal M_i^{0},
A_i^{0},
\mathcal I_i^{0}
\bigr),
\]

where \(\mathcal M_i^{0}\) is an individual starting repertoire compatible with the lineage,

\[
A_i^{0}:\mathcal G\to\mathcal M_i^{0}
\]

is the inherited access map at the chosen resolution, and \(\mathcal I_i^{0}\) is the inherited initial interpretive organization. Unlike \(P^{\mathrm{spec}}\), the access map is not assumed to be an orthogonal projection or even linear.

## 5. Objects, categories, and propositions

Objects of experience, categories, people, propositions, and corporations are not assumed to be primitive basis vectors in \(\mathcal G\).

For actor \(i\), let \(A_{i,t}:\mathcal G\to\mathcal V_{i,t}^{\mathrm{acc}}\) be an actor-specific access map and let

\[
\mathcal I_{i,t}:
\mathcal V_{i,t}^{\mathrm{acc}}\times\Phi_i\times\mathcal C_i^{\mathrm{ctx}}
\to
\mathcal V_{i,t}^{\mathrm{obj}}
\]

be the actor’s nonlinear organization of accessible distinctions, phenomenal state, and context. An external configuration \(\omega_t\in\mathcal N_{\mathrm{loc}}\) can induce the actor-relative representation

\[
\zeta_{i,t}^{\mathrm{obj}}(\omega_t)
=
\mathcal I_{i,t}
\!\left(
A_{i,t}\operatorname{Reg}(\omega_t),
\phi_{i,t},
 c_{i,t}
\right).
\]

The access codomain \(\mathcal V_{i,t}^{\mathrm{acc}}\) and object-representation codomain \(\mathcal V_{i,t}^{\mathrm{obj}}\) are actor-relative spaces; neither needs to be a linear subspace of \(\mathcal G\).

For a category indexed by \(\kappa\), first declare a measurable representation domain

\[
\mathcal V_{\kappa}^{\mathrm{cat}}.
\]

The domain may be \(\mathcal G\), an actor-relative object space such as \(\mathcal V_{i,t}^{\mathrm{obj}}\), or a finite learned feature space. The category may then be represented by any mathematically appropriate object, including:

- a measurable region \(\mathcal C_\kappa\subseteq\mathcal V_{\kappa}^{\mathrm{cat}}\);
- a prototype point \(\operatorname{proto}_\kappa\in\mathcal V_{\kappa}^{\mathrm{cat}}\);
- a probability measure \(\mathbb P_\kappa\) on \(\mathcal V_{\kappa}^{\mathrm{cat}}\);
- a scoring function \(\operatorname{cat}_{\kappa}:\mathcal V_{\kappa}^{\mathrm{cat}}\to[0,1]\);
- or a more complicated learned object over activations and relations.

Only the prototype is literally a point of the declared representation domain. A region is a subset; a distribution or scoring function is defined **on** the domain. A category is not required to be a vector or a primitive coordinate in \(\mathcal G\).

Let \((\mathcal X,\mathscr X)\) be the measurable master proposition space. Let \(\mathcal X_t^{\mathrm{adm}}\in\mathscr X\) be the admissible proposition set and let the nonempty measurable set \(\mathcal X_t^{\mathrm{cand}}\subseteq\mathcal X_t^{\mathrm{adm}}\) contain the propositions actually available at decision \(t\). The realized proposition satisfies \(x_t\in\mathcal X_t^{\mathrm{cand}}\). Write \(\widehat{\mathcal S}\) for the operational individual-state space defined in Section 8. For finite dimensions \(d_x,d_s,d_p^{(s)},d_h^{(s)}\), use the measurable maps

\[
E_{x,\theta}:\mathcal X\to\mathbb R^{d_x},
\qquad
E_{s,\theta}:\widehat{\mathcal S}\to\mathbb R^{d_s},
\]

\[
\mathcal P_{s,\theta}:\mathbb R^{d_x}\times\widehat{\mathcal S}\to\mathbb R^{d_p^{(s)}},
\qquad
\Psi_{s,\theta}:\mathbb R^{d_s}\times\mathbb R^{d_p^{(s)}}\to\mathbb R^{d_h^{(s)}}.
\]

The context-free proposition encoding is

\[
\mathbf e_t^x=E_{x,\theta}(x_t).
\]

For an individual actor-state model, the actor-relative proposition representation is

\[
\mathbf p_{i,t}^{(s)}(x_t)
=
\mathcal P_{s,\theta}
\!\left(
\mathbf e_t^x,
 \widehat s_{i,t}
\right),
\]

and the corresponding interaction representation is

\[
\mathbf h_{i,t}^{(s,\mathrm{int})}(x_t)
=
\Psi_{s,\theta}
\!\left(
E_{s,\theta}(\widehat s_{i,t}),
\mathbf p_{i,t}^{(s)}(x_t)
\right).
\]

The dyadic maps \(\mathcal P_{D,\theta}\) and \(\Psi_{D,\theta}\) are separately typed in Section 11. The same external proposition can therefore yield different actor-relative representations without one map being asked to accept two incompatible state types.

## 6. Realized person, phenomenal state, and Chimera

The slowly changing realized organization of person \(i\) at time \(t\) is

\[
T_{i,t}
=
\mathcal E_{\mathrm{ind}}
\!\left(
G_i,
\mathbf u_{i,<t}^{\mathrm{lang}},
\mathcal H_{i,<t}^{\mathrm{life}}
\right),
\]

where \(\mathbf u_{i,<t}^{\mathrm{lang}}\) summarizes language, culture, and socialization available before decision \(t\), and \(\mathcal H_{i,<t}^{\mathrm{life}}\) denotes life history recorded before that decision. The object \(T_{i,t}\) is not assumed to be one finite vector.

The full phenomenal state is

\[
\phi_{i,t}\in\Phi_i.
\]

The person-in-role object, called the **Chimera**, is

\[
\chi_{i,t}=(T_{i,t},c_{i,t}),
\]

where \(c_{i,t}\) is the active role-and-institution context.

For statements about complete one-step dynamics, define the ideal actor–world state

\[
\Sigma_{i,t}^{\star}
:=
\bigl(T_{i,t},\phi_{i,t},c_{i,t},w_t\bigr).
\]

Let \((\mathcal S_i^{\star},\mathscr S_i^{\star})\) be its measurable state space and let \((\mathcal E_{\Xi,i}^{\star},\mathscr E_{\Xi,i}^{\star})\) be the ideal one-step exogenous-innovation space. The complete transition is a Markov kernel

\[
K_i^{\star}:
\mathcal S_i^{\star}\times\mathcal X\times\mathcal E_{\Xi,i}^{\star}
\rightsquigarrow
\mathcal S_i^{\star}.
\]

The familiar equation for \(\phi_{i,t+1}\) is the phenomenal component of a transition on \(\Sigma_{i,t}^{\star}\); it treats the slow and exogenous components as fixed over the explanatory step unless their updates are written explicitly. The output state lies in the same complete state space as the next input.

## 7. General and task-conditioned predictive states

Let

\[
\mathsf I_{i,t}
:=
\bigl(
\mathcal H_{i,<t},
T_{i,t},
 c_{i,t},
 w_t
\bigr)
\]

be the ideal pre-proposition information state relative to which the predictive problem is defined. It is not the complete actor–world state \(\Sigma_{i,t}^{\star}\), because the predictor is not granted direct access to \(\phi_{i,t}\). Assume the relevant state and outcome spaces are standard Borel spaces so that regular conditional laws exist.

Let \(\boldsymbol\Xi_{t+1:t+H}\) denote a random exogenous path and \(\boldsymbol\xi_{t+1:t+H}\) a realized or supplied scenario value. For a finite decision horizon \(H\), a proposition sequence \(\mathbf x_{t:t+H-1}\), and a scenario path \(\boldsymbol\xi_{t+1:t+H}\), choose a version of the observational regular conditional response kernel

\[
\mathscr R_{i,t}^{(H),\mathrm{obs}}
\!\left(
\cdot
\mid
\mathsf I_{i,t},
\mathbf x_{t:t+H-1},
\boldsymbol\xi_{t+1:t+H}
\right)
:=
\mathcal L
\!\left(
O_{i,t+1:t+H}
\mid
\mathsf I_{i,t},
\mathbf X_{t:t+H-1}=\mathbf x_{t:t+H-1},
\boldsymbol\Xi_{t+1:t+H}=\boldsymbol\xi_{t+1:t+H}
\right).
\]

This kernel is observationally identified only for proposition and scenario values in the support of the data-generating regime; regular conditional laws are unique only almost surely. Off-support response laws require an explicit structural model, intervention, or extrapolation assumption. A conditional law indexed by a fixed historical action sequence does not, by itself, identify the law under a new adaptive policy, even when every candidate action has support. For a factual policy, define the corresponding joint observational law as part of the data-generating regime. For a counterfactual policy, use controlled kernels together with an identification argument, or state the required extrapolation assumptions explicitly. Two ideal information states are predictively equivalent for the declared modeled family when they induce the same supported response law for every sequence or policy, scenario, horizon, and measurable future event in that family. The equivalence relation defines an abstract predictive information object, but the quotient is not automatically a standard-Borel measurable space. Whenever a usable measurable realization exists, denote it by \(Q_{i,t}\); otherwise, use the indexed response-kernel family itself as the predictive object. Either object may be infinite-dimensional. An interventional analogue replaces \(\mathscr R^{\mathrm{obs}}\) with a controlled or \(\operatorname{do}\)-indexed response kernel.

For task \(\tau\) and elapsed-time outcome horizon \(\Delta\), write

\[
q_{i,t}^{(\tau,\Delta)}
=
\Pi_{\tau,\Delta}(Q_{i,t})
\]

when a task summary with the required sufficiency property exists. The map \(\Pi_{\tau,\Delta}\) need not be linear or orthogonal.

The decision-associated outcome is

\[
Y_{i,t}^{(\tau,\Delta)}.
\]

It is indexed by the decision that generated the prediction, while \(\Delta\) records the horizon at which the outcome is evaluated. For every measurable \(B\) in the outcome space, predictive sufficiency means

\[
\mathbb P
\!\left(
Y_{i,t}^{(\tau,\Delta)}\in B
\mid
\mathsf I_{i,t},
X_t=x
\right)
=
\mathbb P
\!\left(
Y_{i,t}^{(\tau,\Delta)}\in B
\mid
q_{i,t}^{(\tau,\Delta)},
X_t=x
\right)
\]

for admissible \(x\) in the support of the predictive regime. This is an observational predictive statement. A causal predictive state is obtained only if the equality is defined using interventional laws \(\operatorname{do}(X_t=x)\).

## 8. Learned person-state and approximation error

The slow learned person vector is

\[
\widehat{\mathbf t}_{i,t}\in\mathbb R^{d_T},
\]

and the fast latent vector is

\[
\mathbf z_{i,t}\in\mathbb R^{d_Z}.
\]

Let \(\mathbf c_{i,t}\) and \(\mathbf w_t\) be finite encodings of context and world state. The operational person-state is

\[
\widehat s_{i,t}
=
\bigl(
\widehat{\mathbf t}_{i,t},
\mathbf z_{i,t},
\mathbf c_{i,t},
\mathbf w_t
\bigr).
\]

Write the measurable space \((\widehat{\mathcal S},\widehat{\mathscr S})\) for the operational individual state containing \(\widehat s_{i,t}\). Inside a dyadic model, the shared world vector appears once, so

\[
\widehat s_{i,t}^{\mathrm{person}}
=
\bigl(
\widehat{\mathbf t}_{i,t},
\mathbf z_{i,t},
\mathbf c_{i,t}
\bigr).
\]

Write

\[
(\widehat{\mathcal S}_{\mathrm{person}},
 \widehat{\mathscr S}_{\mathrm{person}})
=
\left(
\mathbb R^{d_T}\times\mathbb R^{d_Z}\times\mathbb R^{d_c},
\mathcal B(\mathbb R^{d_T})\otimes
\mathcal B(\mathbb R^{d_Z})\otimes
\mathcal B(\mathbb R^{d_c})
\right)
\]

for the measurable person-substate space, assuming the context encoder has dimension \(d_c\). The full individual-state space \(\widehat{\mathcal S}\) additionally carries the world coordinate. This declaration makes the person inputs to relationship and corporation operators well typed.

Let \((\mathcal E_{\Xi,s},\mathscr E_{\Xi,s})\) be the individual one-step exogenous-innovation space. The individual interaction kernel has the type

\[
K_{s,\theta}^{\mathrm{int}}
:
\mathbb R^{d_h^{(s)}}\times\mathcal E_{\Xi,s}
\rightsquigarrow
\widehat{\mathcal S}.
\]

For every measurable next-state set \(B\in\widehat{\mathscr S}\), define the shorthand

\[
K_{s,\theta}
\!\left(
B\mid\widehat s_{i,t},X_t=x_t,
\boldsymbol\Xi_{t+1}=\boldsymbol\xi
\right)
:=
K_{s,\theta}^{\mathrm{int}}
\!\left(
B\mid
\mathbf h_{i,t}^{(s,\mathrm{int})}(x_t),
\boldsymbol\xi
\right).
\]

Thus a simulated individual next state is a random element of the same operational state space used at the following step. The notation \(K(\cdot\mid\ldots,\boldsymbol\Xi=\boldsymbol\xi)\) evaluates a parameterized kernel at the supplied scenario value and does not require the singleton event to have positive probability. An unconditional forecast integrates this kernel against a declared exogenous law.

If \(\widehat s_{i,t}\) is a measurable function of \(\mathsf I_{i,t}\), define the state-compression gap under the observational law by

\[
\epsilon_{\tau,\Delta}^{\mathrm{state}}(\widehat s)
:=
I
\!\left(
Y_{i,t}^{(\tau,\Delta)};
\mathsf I_{i,t}
\mid
\widehat s_{i,t},X_t
\right).
\]

Equivalently,

\[
\epsilon_{\tau,\Delta}^{\mathrm{state}}(\widehat s)
=
\mathbb E
\!\left[
D_{\mathrm{KL}}
\!\left(
\mathbb P(Y\in\cdot\mid\mathsf I_{i,t},X_t)
\,\middle\|\,
\mathbb P(Y\in\cdot\mid \widehat s_{i,t},X_t)
\right)
\right],
\]

where \(Y=Y_{i,t}^{(\tau,\Delta)}\). It is zero exactly when the operational state is sufficient under the joint observational law and its supported propositions, up to null sets. Uniform or causal sufficiency across interventions requires the corresponding family of interventional laws.

For a fitted conditional law \(P_{Y,\theta,\tau,\Delta}(\cdot\mid\widehat s,X)\), define the model-estimation gap

\[
\epsilon_{\theta,\tau,\Delta}^{\mathrm{model}}(\widehat s)
:=
\mathbb E
\!\left[
D_{\mathrm{KL}}
\!\left(
\mathbb P(Y\in\cdot\mid \widehat s_{i,t},X_t)
\,\middle\|\,
P_{Y,\theta,\tau,\Delta}(\cdot\mid \widehat s_{i,t},X_t)
\right)
\right].
\]

To state the log-score decomposition without assuming a discrete outcome, suppose the relevant conditional laws are dominated by one fixed reference measure. Let \(p^{\star}_{\tau,\Delta}(y\mid\mathsf I_{i,t},X_t)\) be a full-information conditional density or mass function, let \(p_{\widehat s,\tau,\Delta}(y\mid\widehat s_{i,t},X_t)\) be the true conditional density or mass function after compression, and let \(p_{Y,\theta,\tau,\Delta}(y\mid\widehat s_{i,t},X_t)\) be the fitted density or mass function. Define

\[
\mathscr R_{\log}^{\star}
:=
\mathbb E\!\left[
-\log p^{\star}_{\tau,\Delta}
\!\left(
Y_{i,t}^{(\tau,\Delta)}
\mid
\mathsf I_{i,t},X_t
\right)
\right].
\]

Whenever the displayed expectations and divergences are finite, the exact decomposition is

\[
\begin{aligned}
&\mathbb E\!\left[
-\log p_{Y,\theta,\tau,\Delta}
\!\left(
Y_{i,t}^{(\tau,\Delta)}
\mid
\widehat s_{i,t},X_t
\right)
\right]
-
\mathscr R_{\log}^{\star}
\\[4pt]
&\qquad=
\epsilon_{\tau,\Delta}^{\mathrm{state}}(\widehat s)
+
\epsilon_{\theta,\tau,\Delta}^{\mathrm{model}}(\widehat s).
\end{aligned}
\]

For a discrete outcome with counting measure, \(\mathscr R_{\log}^{\star}=H(Y_{i,t}^{(\tau,\Delta)}\mid\mathsf I_{i,t},X_t)\). For continuous outcomes it is an expected negative log density, not an invariant conditional entropy. The first gap is information discarded by state compression; the second is error in the fitted law conditional on that state. This is the precise replacement for an unexplained \(s\approx q\).

## 9. Relevance, salience, memory, and fast-state updates

The task relevance map is written \(\Lambda_\tau\), not \(\pi_\tau\):

\[
\Lambda_\tau
\!\left(
T_{i,t},
\phi_{i,t},
 c_{i,t},
 w_t,
 x_t
\right)
\in\mathbb R^{d_\tau}.
\]

Salience is

\[
\boldsymbol\lambda_{i,t}^{(\tau)}(x_t)
\in[0,1]^{d_\tau},
\]

and the ideal active slice is

\[
\mathbf r_{i,t}^{(\tau)}(x_t)
=
\boldsymbol\lambda_{i,t}^{(\tau)}(x_t)
\odot
\Lambda_\tau
\!\left(
T_{i,t},
\phi_{i,t},
 c_{i,t},
 w_t,
 x_t
\right).
\]

The symbol \(\mathbf z_{i,t}\) is reserved exclusively for the fast latent state.

A tractable memory field can be written

\[
\mathbf m_{i,t}^{\mathrm{mem}}
=
\sum_{j=1}^{N_i}
\varpi_{ij,t}\mathbf h_{ij}^{\mathrm{mem}},
\]

where \(\mathbf h_{ij}^{\mathrm{mem}}\) is a trace representation and \(\varpi_{ij,t}\) its current weight. A proposition-conditioned retrieval rule may produce weights \(\widetilde\varpi_{ij,t}(x_t)\) and the retrieved vector

\[
\widetilde{\mathbf m}_{i,t}^{\mathrm{mem}}(x_t)
=
\sum_{j=1}^{N_i}
\widetilde\varpi_{ij,t}(x_t)\mathbf h_{ij}^{\mathrm{mem}}.
\]

Let \((\mathsf h_n)_{n\ge1}\) be the chronological record sequence and let

\[
v_n:=\operatorname{time}(\mathsf h_n).
\]

When the chronological record stream is shared across actors, let

\[
\mathbf d_{i,n}^{\mathrm{rec}}
\]

be the record-applicability vector: it contains a binary applicability flag indicating whether record \(n\) pertains to actor \(i\), together with indicators for the actor-specific fields available. It is neither a memory vector nor a label-observation mask. When the applicability flag is zero, \(U_{z,\theta}\) is required to act as the identity on \(\mathbf z_{i,n-1}\). For an ordered dyad \((a,b)\), let

\[
d_{ab,n}^{\mathrm{rel}}\in\{0,1\}
\]

indicate whether record \(n\) pertains to that relationship. A relationship-update map receiving \(d_{ab,n}^{\mathrm{rel}}=0\) must act as the identity on the existing relationship state. Starting from \(\mathbf z_{i,0}\), process record \(n\) by

\[
\mathbf z_{i,n}
=
U_{z,\theta}
\!\left(
\mathbf z_{i,n-1},
\widehat{\mathbf t}_{i}(v_n^-),
\mathbf c_i(v_n^-),
\mathbf w(v_n^-),
\mathsf h_n,
\mathbf d_{i,n}^{\mathrm{rec}}
\right),
\qquad n\ge1.
\]

For decision epoch \(t\), define

\[
N(t)
:=
\#\{n:v_n<\operatorname{time}(\mathsf d_t)\},
\qquad
\mathbf z_{i,t}:=\mathbf z_{i,N(t)}.
\]

Thus a prediction at decision \(t\) uses records strictly before that decision. The record created by proposition \(x_t\), and every response to it, cannot enter \(\mathbf z_{i,t}\). If timestamps tie, the logging system must supply an ordering key that preserves the actual decision-before-response order.

For categorical traces, \(f\) indexes feature family, \(\sigma\) source channel, and \(\rho\) role or regime. Raw token bags \(\mathcal B_{i,r}^{(f,\sigma)}\) are contextually typed by

\[
\widetilde{\mathcal B}_{i,r}^{(f,\sigma)}
=
\operatorname{Lift}_{\mathrm{ctx}}
\!\left(
\mathcal B_{i,r}^{(f,\sigma)},c_{i,r}
\right)
\]

before pooling. Learned empty-cell representations use \(\mathbf e_{\varnothing}\), binary availability masks use \(M\), slow aggregation weights use \(\beta^{\mathrm{slow}}\), and fast retrieval weights use \(\alpha^{(\tau)}\). These pooled categorical objects are trace estimators; they are not identified with the phenomenal state or with the fast state \(\mathbf z_{i,t}\).

## 10. Composite corporations

Let \(\mathcal J_C(t)\) be the set of people relevant to a corporation’s decision at time \(t\). Define the abstract member state

\[
\varsigma_{j,t}^{\mathrm{person}}
:=
\bigl(T_{j,t},\phi_{j,t},c_{j,t}\bigr).
\]

A conceptual composite corporation-state can then be written

\[
S_{C,t}
=
\mathcal A_{\mathrm{corp}}
\!\left(
\bigl(\varsigma_{j,t}^{\mathrm{person}}\bigr)_{j\in\mathcal J_C(t)},
\mathsf{Facts}_{C,t},
\mathsf{Org}_{C,t},
\mathsf{Mem}_{C,t},
\mathsf{Inc}_{C,t},
\mathsf{EnvHist}_{C,t}
\right).
\]

The indexed family preserves member identity and multiplicity; an ordinary set could collapse distinct members with identical represented states. The arguments respectively denote relevant member states, facts and statistics, organizational structure, institutional memory, incentives and constraints, and environmental history already absorbed into the organization. They are structured objects and need not be vectors. The aggregation operator is not an arithmetic average; it must be capable of representing authority, veto rights, communication paths, coalitions, and unequal participation.

For dataset construction, the measurable company-side bundle is

\[
\mathbf u_{C,t}^{\mathrm{corp}}
=
\left[
\mathbf f_{C,t}^{\mathrm{corp}},
\mathbf g_{C,t}^{\mathrm{org}},
\mathbf m_{C,t}^{\mathrm{inst}},
\boldsymbol\iota_{C,t}^{\mathrm{inc}},
\mathbf h_{C,t}^{\mathrm{env}}
\right].
\]

Let \(\mathcal J_C^{\mathrm{obs}}(t)\subseteq\mathcal J_C(t)\) be the people whose usable states are observed, and let \(\mathbf m_{C,t}^{\mathrm{miss}}\) encode missing membership, authority, and company fields. Write \((\widehat{\mathcal C},\widehat{\mathscr C})\) for the measurable learned corporation-state space. A learned corporation-state is the random element

\[
\widehat S_{C,t}
=
\mathcal A_{\mathrm{corp},\theta}
\!\left(
\bigl(\widehat s_{j,t}^{\mathrm{person}}\bigr)_{j\in\mathcal J_C^{\mathrm{obs}}(t)},
\mathbf u_{C,t}^{\mathrm{corp}},
\mathbf m_{C,t}^{\mathrm{miss}}
\right)
\in\widehat{\mathcal C},
\]

where \(\mathcal A_{\mathrm{corp},\theta}\) is measurable on the structured, variable-cardinality input domain used by the implementation. The measurable bundle and observed member states are evidence supplied to this operator; neither is, by itself, the complete corporation-state. When a focal person is also carried separately in a dyadic state, either exclude that person from the company-context summary or estimate the coupled person/company representation jointly so duplicate paths are not treated as independent evidence.

This construction supports treating a corporation as a higher-order actor for prediction. It does not, by itself, claim that a corporation has human phenomenal consciousness. Let \((\mathcal E_{\Xi,C},\mathscr E_{\Xi,C})\) be the corporate one-step exogenous-innovation space. A shared learned corporate transition kernel has type

\[
K_{\mathrm{corp},\theta}
:
\widehat{\mathcal C}
\times\mathcal X
\times\mathbb R^{d_w}
\times\mathcal E_{\Xi,C}
\rightsquigarrow
\widehat{\mathcal C}.
\]

Thus

\[
\widetilde S_{C,t+1}
\sim
K_{\mathrm{corp},\theta}
\!\left(
\cdot
\mid
\widehat S_{C,t},X_t=x_t,\mathbf w_t,
\boldsymbol\Xi_{C,t+1}=\boldsymbol\xi_{C,t+1}
\right),
\]

with company identity and structure carried by the state and covariates rather than by a separate parameter set for every corporation. The next state lives in the same measurable space as the current learned corporation-state, so the corporate transition is recursively closed at its declared level of abstraction.

## 11. The sales dyad and recursively closed dynamics

The first application uses salesperson \(a\), executive \(b\), salesperson’s company \(C_a\), and executive’s company \(C_b\). Let \((\widehat{\mathcal R},\widehat{\mathscr R})\) be the measurable relationship-state space and let

\[
\widehat\Gamma_{ab,n}\in\widehat{\mathcal R}
\]

be the relationship state after chronological record \(n\). With a measurable record space \((\mathcal H_{\mathrm{rec}},\mathscr H_{\mathrm{rec}})\), the relationship update has type

\[
U_{\Gamma,\theta}:
\widehat{\mathcal R}
\times\mathcal H_{\mathrm{rec}}
\times\widehat{\mathcal S}_{\mathrm{person}}
\times\widehat{\mathcal S}_{\mathrm{person}}
\times\{0,1\}
\to
\widehat{\mathcal R},
\]

where \(\widehat{\mathcal S}_{\mathrm{person}}\) is the person-substate space. It satisfies

\[
U_{\Gamma,\theta}(\gamma,h,s_a,s_b,0)=\gamma.
\]

Thus only records applicable to ordered dyad \((a,b)\) can change \(\widehat\Gamma_{ab,n}\), and \(\widehat\Gamma_{ab,t}:=\widehat\Gamma_{ab,N(t)}\) at decision epoch \(t\). The filtered dyadic state is

\[
\widehat D_{ab,t}
=
\left(
\widehat s_{a,t}^{\mathrm{person}},
\widehat S_{C_a,t},
\widehat s_{b,t}^{\mathrm{person}},
\widehat S_{C_b,t},
\widehat\Gamma_{ab,t},
\mathbf w_t
\right).
\]

Write the measurable space \((\widehat{\mathcal D},\widehat{\mathscr D})\) for the dyadic state containing \(\widehat D_{ab,t}\). For finite dimensions \(d_D,d_p^{(D)},d_h^{(D)}\), the dyadic proposition and interaction maps have types distinct from the individual-state maps:

\[
E_{D,\theta}:\widehat{\mathcal D}\to\mathbb R^{d_D},
\]

\[
\mathcal P_{D,\theta}:\mathbb R^{d_x}\times\widehat{\mathcal D}\to\mathbb R^{d_p^{(D)}},
\qquad
\Psi_{D,\theta}:\mathbb R^{d_D}\times\mathbb R^{d_p^{(D)}}\to\mathbb R^{d_h^{(D)}}.
\]

Thus

\[
\mathbf p_{b,t}^{(D)}(x_t)
=
\mathcal P_{D,\theta}
\!\left(
\mathbf e_t^x,
\widehat D_{ab,t}
\right),
\]

\[
\mathbf h_{ab,t}^{(D,\mathrm{int})}(x_t)
=
\Psi_{D,\theta}
\!\left(
E_{D,\theta}(\widehat D_{ab,t}),
\mathbf p_{b,t}^{(D)}(x_t)
\right).
\]

Let \(\boldsymbol\Xi_{t+1}\) denote a random exogenous innovation and \(\boldsymbol\xi_{t+1}\) a realized or supplied scenario value. At the ideal level,

\[
\Sigma_{i,t+1}^{\star}
\sim
K_i^{\star}
\!\left(
\cdot
\mid
\Sigma_{i,t}^{\star},
X_t=x_t,
\boldsymbol\Xi_{t+1}=\boldsymbol\xi_{t+1}
\right).
\]

Let \((\mathcal E_{\Xi,D},\mathscr E_{\Xi,D})\) be the dyadic one-step exogenous-innovation space. The learned dyadic interaction kernel

\[
K_{D,\theta}^{\mathrm{int}}
:
\mathbb R^{d_h^{(D)}}\times\mathcal E_{\Xi,D}
\rightsquigarrow
\widehat{\mathcal D}
\]

is a Markov kernel into the next dyadic-state space. The shorthand conditional kernel is defined, for every next-state set \(B\in\widehat{\mathscr D}\), by

\[
K_{D,\theta}
\!\left(
B\mid\widehat D_{ab,t},X_t=x_t,
\boldsymbol\Xi_{t+1}=\boldsymbol\xi
\right)
:=
K_{D,\theta}^{\mathrm{int}}
\!\left(
B\mid
\mathbf h_{ab,t}^{(D,\mathrm{int})}(x_t),
\boldsymbol\xi
\right).
\]

The notation \(K(\cdot\mid\ldots,\boldsymbol\Xi=\boldsymbol\xi)\) is shorthand for evaluating a parameterized Markov kernel at the supplied scenario value \(\boldsymbol\xi\); it does not require the singleton event \(\{\boldsymbol\Xi=\boldsymbol\xi\}\) to have positive probability. Conditioning on \(\boldsymbol\Xi_{t+1}=\boldsymbol\xi_{t+1}\) therefore defines a scenario forecast through this kernel. Given a declared one-step exogenous law \(\mathbb Q_{\Xi,\mathfrak e}(\mathrm d\boldsymbol\xi\mid\widehat D_{ab,t})\), common across the candidate propositions being compared under evaluation regime \(\mathfrak e\), the corresponding marginal transition is

\[
K_{D,\theta,\mathfrak e}^{\mathrm{marg}}
\!\left(
B\mid\widehat D_{ab,t},X_t=x_t
\right)
=
\int
K_{D,\theta}^{\mathrm{int}}
\!\left(
B\mid
\mathbf h_{ab,t}^{(D,\mathrm{int})}(x_t),
\boldsymbol\xi
\right)
\mathbb Q_{\Xi,\mathfrak e}
\!\left(
\mathrm d\boldsymbol\xi
\mid\widehat D_{ab,t}
\right).
\]

If a modeled future variable is causally affected by the proposition, it belongs in the transition state rather than in the action-invariant exogenous law. An observational forecast may instead condition an empirical exogenous distribution on the factual action, but that action-dependent distribution must not be silently reused for counterfactual candidate ranking. A simulated next operational state under a supplied scenario is

\[
\widetilde D_{ab,t+1}
\sim
K_{D,\theta}^{\mathrm{int}}
\!\left(
\cdot
\mid
\mathbf h_{ab,t}^{(D,\mathrm{int})}(x_t),
\boldsymbol\xi_{t+1}
\right).
\]

The immediate observable trace bundle is

\[
\widetilde O_{t+1}
\sim
R_{O,\theta}
\!\left(
\cdot
\mid
\widetilde D_{ab,t+1}
\right).
\]

This emission law asserts that the next operational state is sufficient for the immediate trace. If that Markov-style assumption is not adopted, use the more general kernel \(R_{O,\theta}(\cdot\mid\widetilde D_{ab,t+1},\widehat D_{ab,t},X_t=x_t)\).

Here \(O_{t+1}\) denotes the trace bundle assigned to the inter-decision interval \((t,t+1]\); several timestamped observation records may contribute to that bundle.

After actual records arrive, filtering produces

\[
\widehat D_{ab,t+1}
=
\mathcal F_\theta
\!\left(
\widehat D_{ab,t},
 x_t,
\mathcal H_{(t,t+1]}
\right).
\]

Here \(\mathcal H_{(t,t+1]}\) is the record bundle that arrived after decision \(t\) and no later than the next decision epoch. It includes the exogenous company and world changes actually observed in the interval, so they are not passed to the filter a second time. The transition is recursively closed because \(\widetilde D_{ab,t+1}\) has the same state type as the next input. Simulation and filtering remain distinct operations.

For a proposition sequence \(x_{t:t+H-1}\) and scenario path \(\boldsymbol\xi_{t+1:t+H}\), repeated application of \(K_{D,\theta}\) and \(R_{O,\theta}\) induces the trajectory law

\[
\mathbb P_\theta
\!\left(
\widetilde D_{ab,t+1:t+H},\widetilde O_{t+1:t+H}
\mid
\widehat D_{ab,t},
\mathbf X_{t:t+H-1}=\mathbf x_{t:t+H-1},
\boldsymbol\Xi_{t+1:t+H}=\boldsymbol\xi_{t+1:t+H}
\right).
\]

A delayed outcome \(Y_t^{(\tau,\Delta)}\) is a measurable functional of records and states over the elapsed-time window \((t,t+\Delta]\), not generally an immediate emission from \(D_{t+1}\). A direct outcome head

\[
P_{Y,\theta,\tau,\Delta}
\!\left(
\cdot
\mid
\widehat D_{ab,t},X_t=x
\right)
\]

is a fitted conditional probability kernel relative to a declared continuation policy, future candidate-set process, censoring convention, and exogenous regime. When those regime components are suppressed from the notation, they remain part of the estimand. Under a proposed continuation policy \(\pi\), exogenous-path law \(\mathbb Q_{\Xi}\), and future candidate-set law \(\mathbb Q_{\mathcal X}\), the corresponding outcome law should instead be induced by the rollout and written

\[
\mathbb P_{\theta,\pi,\mathbb Q_{\Xi},\mathbb Q_{\mathcal X}}
\!\left(
Y_t^{(\tau,\Delta)}\in\cdot
\mid
\widehat D_{ab,t},X_t=x
\right).
\]

When several primary outcomes and probes enter one utility, their coherent direct-head representation is a joint kernel

\[
(\widetilde{\mathbf Y}_t,\widetilde{\mathbf Z}_t)
\sim
P_{YZ,\theta,\mathfrak e}
\!\left(
\cdot
\mid
\widehat D_{ab,t},X_t=x
\right).
\]

The head-specific kernels \(P_{Y,\theta,\tau_\ell,\Delta_\ell}\) and \(P_{Z,\theta,m,\Delta_m}\) are marginals or conditionals of that joint law when such a law is modeled. Training separate marginal heads does not, by itself, define their dependence. A joint utility may use only marginal expectations, a declared coupling, or a jointly modeled law; it must not silently manufacture independence.

## 12. Event time, records, outcomes, and censoring

The pre-decision history is

\[
\mathcal H_{<t}
=
\bigl(
\mathsf h_n:
\operatorname{time}(\mathsf h_n)
<
\operatorname{time}(\mathsf d_t)
\bigr).
\]

A decision record is

\[
\mathsf d_t
=
\left(
\operatorname{id}_t,
\mathcal X_t^{\mathrm{cand}},
 x_t,
\delta_t,
\mathbf e_t^{\mathrm{cat}},
\eta_t,
\mathsf I_t^{\mu}
\right),
\]

where \(\mathcal X_t^{\mathrm{cand}}\) is the candidate set actually available, \(\mathsf I_t^{\mu}\) is the information actually available to the behavior policy, and

\[
\eta_t
:=
\mu_t
\!\left(
 x_t
\mid
\mathsf I_t^{\mu},
\mathcal X_t^{\mathrm{cand}}
\right)
\]

is the logged assignment probability for a discrete action, or the logged assignment density for a continuous action. The denominator in off-policy evaluation is \(\eta_t\), not a propensity retroactively conditioned on information the logging policy never had.

For primary outcomes, define

\[
\mathcal J_Y
=
\{(\tau_\ell,\Delta_\ell):\ell=1,\ldots,L_Y\},
\qquad
\mathbf Y_t
=
\left(
Y_t^{(\tau_\ell,\Delta_\ell)}
\right)_{\ell=1}^{L_Y}.
\]

Auxiliary probes are

\[
\mathbf Z_t
=
\left(
Z_t^{(m,\Delta_m)}
\right)_{m=1}^{M_Z}.
\]

Their availability indicators are

\[
R_{t,m}^{Z,\mathrm{obs}}
=
\mathbf 1\{Z_t^{(m,\Delta_m)}\text{ is observed by the analysis cutoff}\},
\qquad
\mathbf R_t^{Z,\mathrm{obs}}
=
\left(R_{t,m}^{Z,\mathrm{obs}}\right)_{m=1}^{M_Z}.
\]

The primary label-availability indicator is

\[
R_{t,\ell}^{\mathrm{obs}}
=
\mathbf 1\{Y_t^{(\tau_\ell,\Delta_\ell)}\text{ is observed by the analysis cutoff}\}.
\]

The primary availability bundle is

\[
\mathbf R_t^{\mathrm{obs}}
=
\left(
R_{t,\ell}^{\mathrm{obs}}
\right)_{\ell=1}^{L_Y}.
\]

An unobserved long-horizon label is censored or missing; it is not automatically a negative. Informative censoring requires survival methods, inverse-probability-of-censoring weights, or an explicit joint model.

## 13. Forecasting, ranking, and causal value

A value over more than one future decision is undefined until the continuation regime is declared. Let \(\mathfrak e\) denote a **predictive evaluation regime** containing at least:

- a continuation policy \(\pi^{\mathrm{cont}}\) after the candidate proposition;
- a future candidate-set process, either included in the state dynamics or declared explicitly;
- an exogenous-path law \(\mathbb Q_{\Xi}\);
- and the outcome, censoring, and terminal-utility convention used by the score.

For a genuinely one-step utility that terminates before another decision can affect it, the continuation-policy component is irrelevant. If a single candidate proposition is scored by a delayed outcome that can be changed by later decisions, the continuation regime is still part of the estimand. For a multi-step score, it is never optional. Assume the utility below is measurable and integrable under every model and evaluation regime being compared. The predictive/model-based value of a candidate proposition is

\[
V_{\theta,\mathfrak e}^{\mathrm{pred}}(x\mid\widehat D_{ab,t})
=
\mathbb E_{\theta,\mathfrak e}
\!\left[
U_\tau
\!\left(
\widetilde D_{ab,t+1:t+H},
\widetilde O_{t+1:t+H},
\widetilde{\mathbf Y}_t,
\widetilde{\mathbf Z}_t
\right)
\mid
\widehat D_{ab,t},
X_t=x
\right].
\]

The tilded outcome and probe bundles are either computed as measurable functionals of the same rollout or sampled from the coherent joint kernel \(P_{YZ,\theta,\mathfrak e}\). If only marginal heads exist, the utility must use marginal expectations or a declared coupling. The bundles are not treated as independent duplicate futures. When the maximum is attained, candidate ranking over the nonempty available set chooses

\[
x_t^\star
\in
\arg\max_{x\in\mathcal X_t^{\mathrm{cand}}}
V_{\theta,\mathfrak e}^{\mathrm{pred}}(x\mid\widehat D_{ab,t}).
\]

A nonempty finite candidate set guarantees attainment. For an infinite candidate set, compactness together with upper semicontinuity is one sufficient condition; otherwise the formal objective is a supremum and the implementation returns an approximate optimizer. This is a forecasting-based score. It is not automatically a causal effect.

The corresponding causal value under the same declared continuation and exogenous regime is

\[
V_{\mathfrak e}^{\mathrm{causal}}(x\mid\widehat D_{ab,t})
=
\mathbb E_{\mathfrak e}
\!\left[
U_\tau
\mid
\widehat D_{ab,t},
\operatorname{do}(X_t=x)
\right],
\]

or equivalently an expectation over potential outcomes. Equality between \(V_{\theta,\mathfrak e}^{\mathrm{pred}}\) and \(V_{\mathfrak e}^{\mathrm{causal}}\) requires an identification argument and a fitted model that correctly estimates the identified law.

A policy is

\[
\pi_t
\!\left(
 x
\mid
\widehat D_{ab,t},
\mathcal X_t^{\mathrm{cand}}
\right).
\]

Let \(\mathfrak P_{\mathrm{adm}}\) be the nonempty class of admissible policies, let \(\gamma\in[0,1]\) be the discount factor, and let \(\mathbb Q_{\Xi}\) be the declared exogenous-path law and \(\mathbb Q_{\mathcal X}\) the future candidate-set law. Assume the step utilities and terminal value are integrable under every admissible policy being compared. For a finite decision horizon \(H\), a standard model-based sequential objective is

\[
J_{\theta,\mathbb Q_{\Xi},\mathbb Q_{\mathcal X}}(\pi\mid\widehat D_{ab,t})
=
\mathbb E_{\theta,\pi,\mathbb Q_{\Xi},\mathbb Q_{\mathcal X}}
\!\left[
\sum_{k=0}^{H-1}
\gamma^k
u_\tau^{\mathrm{step}}
\!\left(
\widetilde D_{ab,t+k+1},
X_{t+k},
\widetilde O_{t+k+1}
\right)
+
\gamma^H V_\tau^{\mathrm{term}}(\widetilde D_{ab,t+H})
\mid
\widehat D_{ab,t}
\right].
\]

The policy search problem is

\[
\pi^\star
\in
\arg\max_{\pi\in\mathfrak P_{\mathrm{adm}}}
J_{\theta,\mathbb Q_{\Xi},\mathbb Q_{\mathcal X}}(\pi\mid\widehat D_{ab,t}),
\]

provided the maximum is attained; otherwise \(\sup\) is the mathematically correct objective. This construction avoids assigning the full eventual transaction value independently to every earlier message. It remains model-based until causal identification and policy evaluation are supplied.

## 14. Training and off-policy evaluation

For \(L_Y\) primary heads and \(M_Z\) probe heads, the minimization objective is

\[
\mathscr J(\theta)
=
\sum_{\ell=1}^{L_Y}
\lambda_\ell^Y
\mathscr J_\ell^Y(\theta)
+
\sum_{m=1}^{M_Z}
\lambda_m^Z
\mathscr J_m^Z(\theta)
+
\lambda_{\mathrm{reg}}\Omega(\theta),
\]

with all weights nonnegative and \(\Omega(\theta)\ge0\). The positive signs are required because every term is minimized. Masks or censoring weights must be applied inside the corresponding head loss.

A slow-state arithmetic refresh is written

\[
\widehat{\mathbf t}_{i,t+1}
=
(1-\alpha_i)\widehat{\mathbf t}_{i,t}
+
\alpha_i\widehat{\mathbf t}_{i,t}^{\mathrm{new}},
\qquad 0<\alpha_i\le1,
\]

only when the old and refreshed vectors are expressed in the same latent coordinate chart. This holds when the encoder is fixed, the refreshed representation is explicitly aligned to the old chart, or the relevant histories are re-encoded into one common chart after an encoder change.

For one-step off-policy evaluation, let \(\mathcal T_{\mathrm{ope}}\) be the predeclared set of eligible decisions whose scalar utility \(u_t^{\mathrm{obs}}\) is mature and usable under the chosen censoring rule, and let \(N_{\mathrm{ope}}=|\mathcal T_{\mathrm{ope}}|>0\). If the utility extends beyond the immediate response, this estimator targets an intervention on the current proposition followed by the declared or logged continuation regime; evaluating an adaptive sequence policy requires a sequential estimator. The basic inverse-propensity estimator is

\[
\widehat V_{\mathrm{IPS}}(\pi)
=
\frac{1}{N_{\mathrm{ope}}}
\sum_{t\in\mathcal T_{\mathrm{ope}}}
\frac{
\pi_t(x_t\mid\widehat D_{ab,t},\mathcal X_t^{\mathrm{cand}})
}{
\eta_t
}
 u_t^{\mathrm{obs}}.
\]

The target-policy numerator must be a function only of information available before the evaluated decision. For continuous actions, the numerator and denominator are policy densities with respect to the same dominating measure. Report the importance-weight distribution and effective sample size. Clipping and self-normalization reduce variance at the cost of bias or a changed finite-sample estimand and should be reported as sensitivity analyses; doubly robust estimators are preferable when their nuisance models are credible. This expression is valid only when:

- the logged mass or density \(\eta_t\) is correct;
- the target policy acts on the logged candidate support;
- consistency holds;
- overlap holds;
- the assignment mechanism is randomized or sequentially ignorable conditional on the recorded information;
- interference is absent or explicitly modeled;
- and censoring and delayed outcomes are handled appropriately.

If eligibility or censoring is informative, the estimator requires an additional censoring model or weighting term rather than complete-case deletion. A learned target policy should be frozen and evaluated on held-out data, or estimated with appropriate sample splitting or cross-fitting. Sequential policies require sequential estimators with products or per-decision products of importance ratios, or doubly robust sequential alternatives. The one-step expression is not silently extended to whole trajectories.

## 15. Symbols intentionally not reused

The following separations are enforced throughout the series:

- \(\nu\) indexes evolutionary stage; \(\tau\) indexes a prediction task; learned empty-cell vectors use \(\mathbf e_{\varnothing}\), not \(\nu\).
- \(\Pi_{\tau,\Delta}\) is a task-summary map; \(\pi\) is a policy.
- \(\mathbf z_{i,t}\) is only the fast latent state.
- \(\mathbf r_{i,t}^{(\tau)}(x)\) is the proposition-conditioned active slice.
- \(\mathcal A_{\mathrm{corp}}\) aggregates a corporation; \(\mathbf Z_t\) denotes auxiliary probes.
- \(\mathscr R^{\mathrm{obs}}\) is a response-law family; \(\mathcal R_{\mathrm{ret},\theta}\) retrieves memory; \(R_{O,\theta}\) emits immediate traces.
- \(P_{Y,\theta,\tau,\Delta}\) denotes a delayed-outcome predictive law.
- \(U_{z,\theta}\) updates fast latent state; \(\mathcal F_\theta\) filters the full dyadic state.
- \(\sigma\) indexes a categorical source channel; \(\widehat s_{i,t}\) is an operational state.
- \(\kappa\) indexes categories; \(C\) denotes a corporation.
- \(v_n\) is a record timestamp; \(u_t^{\mathrm{obs}}\) is realized scalar utility.
- \(H\) is a planning length; \(\mathcal H\) is a history; \(\mathsf{EnvHist}_{C,t}\) is corporate environmental history.
- \(\gamma\) is a discount factor; \(c_{\mathrm{drift}}\) is a drift threshold.
- \(\mathfrak e\) is a predictive evaluation regime; \(\mathbb Q_{\mathcal X}\) is a future candidate-set law; \(\mathfrak P_{\mathrm{adm}}\) is a policy class.
- \(\mathcal P_{s,\theta}\) and \(\Psi_{s,\theta}\) act on individual operational states; \(\mathcal P_{D,\theta}\) and \(\Psi_{D,\theta}\) act on dyadic states.
- \(\mathbf d_{i,n}^{\mathrm{rec}}\) is actor-record applicability; \(d_{ab,n}^{\mathrm{rel}}\) is relationship-record applicability; \(\mathbf m_{i,t}^{\mathrm{mem}}\) is memory; \(\mathbf R_t^{\mathrm{obs}}\) is label availability.
- \(a,b\) are people; \(C_a,C_b\) are corporations.