Definition 1 (Base graph). A base graph* is a tuple \[G = (V, E, \mathcal{R}, \tau, w)\] where \(V\) is a finite set of nodes, \(E \subseteq V \times V\) is a set of directed edges, \(\mathcal{R}\) is a set of relation types, \(\tau : E \rightarrow \mathcal{R}\) assigns a type to each edge, and \(w : E \rightarrow \mathbb{R}\) assigns a real-valued weight to each edge. By default, non-evidential edge classes use non-negative weights; signed weights are reserved for edge types whose semantics require polarity, such as evidential support and contradiction.*

Definition 2 (Representational payload). A representational payload* is an internal structure associated with a node. For a node \(v\), its payload \(P_v\) may take one of the following forms:*

1. an empty payload \(\varnothing\), corresponding to an atomic node,

2. a linguistic payload, such as a word, phrase, sentence, or symbolic chunk,

3. a perceptual payload, such as a visual, auditory, tactile, or motor chunk,

4. a structured perceptual payload, i.e., a perceptual payload whose internal representation is a subgraph encoding a perceptual scene, feature bundle, or sensorimotor pattern,

5. a graph payload \(G_v\), where \(G_v\) is itself a base graph.

Definition 3 (Recursive node). A recursive node* is a tuple \[v = (c_v, P_v),\] where \(c_v\) is a node label or identifier and \(P_v\) is a representational payload. If \(P_v\) is a structured perceptual payload or a graph payload, then \(v\) is said to be internally structured, and its internal subgraph may encode the node’s perceptual scene, conceptual organization, episodic content, or procedural structure.*

Definition 4 (Edge type). Let \(G = (V, E, \mathcal{R}, \tau, w)\) be a base graph (Definition 1), where each node \(v \in V\) carries a representational payload \(P_v\) as in Definition 2. The relation-type set \(\mathcal{R}\) partitions into six pairwise-disjoint classes: \[\mathcal{R} = \mathcal{R}_{\mathrm{causal}} \cup \mathcal{R}_{\mathrm{contain}} \cup \mathcal{R}_{\mathrm{temp}} \cup \mathcal{R}_{\mathrm{assoc}} \cup \mathcal{R}_{\mathrm{epist}} \cup \mathcal{R}_{\mathrm{spatial}},\] each defined by an independent structural test below. For each edge \(e = (u, v) \in E\), the assigned type \(\tau(e) \in \mathcal{R}\) belongs to exactly one class. A particular graph may employ only a subset; belief-graph and working-memory instantiations specify which classes appear in \(\mathcal{R}_B\) and \(\mathcal{R}_{W_t}\). The six classes are:

1. Causal (\(\mathcal{R}_{\mathrm{causal}}\)). An edge \((u,v)\) with \(\tau(u,v) \in \mathcal{R}_{\mathrm{causal}}\) asserts that node \(u\) produces, enables, probabilistically raises, or inhibits the occurrence of node \(v\), independent of the payload types of \(u\) and \(v\): \[\mathcal{R}_{\mathrm{causal}} = \{r_{\mathrm{causes}},\;r_{\mathrm{enables}},\;r_{\mathrm{inhibits}},\;r_{\mathrm{counterfactual}}\}.\] Causal edges are antisymmetric: \((u,v) \in E_{\mathcal{R}_{\mathrm{causal}}} \Rightarrow (v,u) \notin E_{\mathcal{R}_{\mathrm{causal}}}\). Acyclicity over \(\mathcal{R}_{\mathrm{causal}}\) is a structural forbidden pattern in \(\mathcal{S}\). When \(u\) is a distinguished agent node, an optional attribution tag* \(a(u,v) \in V\) records whose causal model the edge belongs to—e.g., “agent \(a\) believes \(u\) causes \(v\)”.*

2. Containment (\(\mathcal{R}_{\mathrm{contain}}\)). An edge \((u,v)\) with \(\tau(u,v) \in \mathcal{R}_{\mathrm{contain}}\) asserts that \(v\)’s existence or identity is nested inside \(u\)’s scope, characterized by an existential-dependency test: \((u,v) \in E_{\mathcal{R}_{\mathrm{contain}}}\) iff removing \(u\) from \(G\) requires removing \(v\) as well. \[\mathcal{R}_{\mathrm{contain}} = \{r_{\mathrm{is\mathrm{-}a}},\;r_{\mathrm{part\mathrm{-}of}},\;r_{\mathrm{instance\mathrm{-}of}},\;r_{\mathrm{role\mathrm{-}of}}\}.\] When \(v\) carries a graph payload \(G_v\) (Definition 2, item 5) or a structured perceptual payload (item 4), its internal subgraph is linked to \(v\) via \(r_{\mathrm{part\mathrm{-}of}}\) edges. Containment is transitive and irreflexive; a directed cycle over \(\mathcal{R}_{\mathrm{contain}}\) is a forbidden pattern in \(\mathcal{S}\).

3. Temporal (\(\mathcal{R}_{\mathrm{temp}}\)). An edge \((u,v)\) with \(\tau(u,v) \in \mathcal{R}_{\mathrm{temp}}\) encodes ordering of nodes across time via a node timestamp function \(\theta_V : V \to \mathbb{R}\) (distinct from the working-memory time index \(t\)) such that \((u,v) \in E_{\mathcal{R}_{\mathrm{temp}}} \Rightarrow \theta_V(u) \leq \theta_V(v)\): \[\mathcal{R}_{\mathrm{temp}} = \{r_{\mathrm{before}},\;r_{\mathrm{after}},\;r_{\mathrm{during}},\;r_{\mathrm{next}}\}.\] Here \(r_{\mathrm{before}}\) and \(r_{\mathrm{after}}\) are mutually exclusive on the same ordered pair (a predicate incompatibility in \(\mathcal{F}\)); \(r_{\mathrm{next}}\) is the sequential successor used in action-sequence and planning subgraphs.

4. Associative (\(\mathcal{R}_{\mathrm{assoc}}\)). An edge \((u,v)\) with \(\tau(u,v) \in \mathcal{R}_{\mathrm{assoc}}\) encodes statistical co-occurrence or spreading-activation linkage with no stronger typed commitment [3], [4]: \[\mathcal{R}_{\mathrm{assoc}} = \{r_{\mathrm{related\mathrm{-}to}},\;r_{\mathrm{co\mathrm{-}occurs\mathrm{-}with}},\;r_{\mathrm{similar\mathrm{-}to}}\}.\] Associative edges are undirected, represented as symmetric pairs \((u,v),(v,u) \in E\) with equal weights \(w(u,v) = w(v,u)\). This is the default class: any relation not warranting a stronger typed commitment from classes 1–3 or 5–6 defaults here.

5. Evidential (\(\mathcal{R}_{\mathrm{epist}}\)). An edge \((u,v)\) with \(\tau(u,v) \in \mathcal{R}_{\mathrm{epist}}\) encodes inferential, evidential, or evaluative support between propositional or evaluative nodes: \[\mathcal{R}_{\mathrm{epist}} = \{r_{\mathrm{supports}},\;r_{\mathrm{contradicts}},\;r_{\mathrm{derived\mathrm{-}from}},\;r_{\mathrm{supersedes}}\}.\] Valence and deontic force are carried as signed edge weights* on \(r_{\mathrm{supports}}\) and \(r_{\mathrm{contradicts}}\) edges (extending \(w\) to \(\mathbb{R}\) on \(\mathcal{R}_{\mathrm{epist}}\) when needed); an obligation is an \(r_{\mathrm{supports}}\) edge with \(|w(u,v)| = 1\) directed at a norm node, and a prohibition is the corresponding \(r_{\mathrm{contradicts}}\) edge. Here \(r_{\mathrm{contradicts}}\) induces a conflict pair in \((\mathcal{F}, \mathcal{S})\); \(r_{\mathrm{supersedes}}\) is generated by belief-update operators when a node is revised rather than deleted.*

6. Spatial (\(\mathcal{R}_{\mathrm{spatial}}\)). An edge \((u,v)\) with \(\tau(u,v) \in \mathcal{R}_{\mathrm{spatial}}\) encodes geometric relations, restricted to node pairs where at least one endpoint carries a perceptual or structured perceptual payload (Definition 2, items 3–4): \[\mathcal{R}_{\mathrm{spatial}} = \{r_{\mathrm{above}},\;r_{\mathrm{below}},\;r_{\mathrm{left\mathrm{-}of}},\;r_{\mathrm{right\mathrm{-}of}},\;r_{\mathrm{near}},\;r_{\mathrm{affords}}\}.\] The sub-type \(r_{\mathrm{affords}}\) links a body-schema node \(u\) to a perceptual node \(v\), encoding a sensorimotor affordance relation.

For any edge class \(\mathcal{R}_k\), write \(E_{\mathcal{R}_k} = \{(u,v) \in E \mid \tau(u,v) \in \mathcal{R}_k\}\) for the induced edge set. For a subgraph \(Q \subseteq B\), the restriction \(E_{\mathcal{R}_k}(Q) = E_{\mathcal{R}_k} \cap (V_Q \times V_Q)\) denotes the edges of that class internal to \(Q\).

Definition 5 (Belief graph). A belief graph* is a base graph \[B = (V_B, E_B, \mathcal{R}_B, \tau_B, w_B),\] whose nodes are recursive nodes representing concepts, episodes, and other structured knowledge. \(B\) is intended to approximate long-term memory and background knowledge.*

Definition 6 (Mental model). A mental model* is a connected subgraph \[M = (V_M, E_M) \subseteq B\] such that the nodes and edges in \(M\) jointly represent a particular situation, domain, or system (e.g., a physical scenario, a social interaction, a mathematical problem).*

Definition 7 (Working-memory graph). A working-memory graph* at time \(t\), denoted \(W_t\), is a base graph \[W_t = (V_{W_t}, E_{W_t}, \mathcal{R}_{W_t}, \tau_{W_t}, w_t)\] subject to capacity constraints on nodes and edges (e.g., \(|V_{W_t}| \leq C_V\), \(|E_{W_t}| \leq C_E\), where \(C_V, C_E\) are architecture-specific constants). Most nodes in \(V_{W_t}\) are linked, via one or more edges, to nodes in the belief graph \(B\), but \(V_{W_t}\) may also contain transient nodes that encode sensory, hypothetical, or not-yet-integrated content. The weight function \(w_t\) may differ from \(w_B\) to reflect transient activation strengths.*

Definition 8 (Belief-anchored working memory). A working-memory state \(W_t\) is belief-anchored* if every node in \(V_{W_t}\) corresponds to some node in \(V_B\) and every edge in \(E_{W_t}\) corresponds to an edge in \(E_B\). In this case we write \(W_t \sqsubseteq B\) to emphasize that \(W_t\) is realized as a subgraph of the belief graph.*

Definition 9 (Perceptual input node). A perceptual input node* at time \(t\) is a node \[v^{\mathrm{perc}} \in V_{W_t} \setminus V_B\] such that its payload \(P_{v^{\mathrm{perc}}}\) is a perceptual payload, and either \[P_{v^{\mathrm{perc}}} \in \mathcal{P}_{\mathrm{atom}} \quad\text{or}\quad P_{v^{\mathrm{perc}}} = G_{\mathrm{perc}} = (V_{\mathrm{perc}}, E_{\mathrm{perc}}, \mathcal{R}_{\mathrm{perc}}, \tau_{\mathrm{perc}}, w_{\mathrm{perc}}),\] where \(\mathcal{P}_{\mathrm{atom}}\) is the set of atomic perceptual payloads (Definition 2, item 3) and \(G_{\mathrm{perc}}\) is a connected base graph encoding a perceptual scene, feature bundle, or sensorimotor pattern, whose edges draw on spatial, temporal, and property relations from the edge-type taxonomy (Definition 4, classes \(\mathcal{R}_{\mathrm{spatial}}\) and \(\mathcal{R}_{\mathrm{temp}}\)).*

Its activation \(\alpha_t^{W}(v^{\mathrm{perc}})\) is initialized by input strength and evolves under the working-memory transition operator \(U\) unless the node is integrated into \(B\).

The set of perceptual input nodes active at time \(t\) is \[V^{\mathrm{perc}}_t = \bigl\{ v \in V_{W_t} \setminus V_B : \bigl(P_v \in \mathcal{P}_{\mathrm{atom}} \lor (\exists\, G_{\mathrm{perc}} \text{ connected},\, P_v = G_{\mathrm{perc}})\bigr) \land \alpha_t^{W}(v) \geq \theta_W \bigr\}.\]

Definition 10 (Activation maps). An belief-node activation function* at time \(t\) is a mapping \[\alpha_t^B : V_B \rightarrow \mathbb{R}_{\ge 0}.\] For a subgraph \(X \subseteq B\), we write \[\alpha_t^B(X) = \sum_{v \in V_X} \alpha_t^B(v)\] for its total activation. When transient nodes are present in working memory, \(\alpha_t^B\) is extended to a working-memory activation map \[\alpha_t^{W} : V_{W_t} \rightarrow \mathbb{R}_{\ge 0},\] such that \(\alpha_t^{W}(v)=\alpha_t^{B}(v)\) for \(v \in V_B \cap V_{W_t}\), and \(\alpha_t^{W}(v)\) is initialized by input strength for \(v \in V_{W_t}\setminus V_B\), decaying under \(U\) unless the node is integrated into \(B\). For a subgraph \(X \subseteq W_t\), we write \(\alpha_t^{W}(X) = \sum_{v \in V_X} \alpha_t^{W}(v)\). Working-memory membership is defined by thresholding against an architecture-specific cutoff \(\theta_W \geq 0\): for \(v \in V_B\), we say that \(v\) is belief-active in working memory at \(t\), written \(\mathrm{active}_B(v,W_t)\), when \(\alpha_t^B(v) \geq \theta_W\). Accordingly, the belief-derived portion of working memory is \[V_{W_t}^{B}=\{v \in V_B \cap V_{W_t} : \mathrm{active}_B(v,W_t)\}.\] The transient portion is \(V_{W_t}^{\mathrm{tr}} = V_{W_t}\setminus V_B\); a transient node \(v\) is WM-active when \(\alpha_t^{W}(v) \geq \theta_W\). For \(X \subseteq V_B\), the active restriction is \[\mathrm{active}(X,W_t)=X\cap V_{W_t}^{B}.\]*

Definition 11 (Salience set). A salience set* at time \(t\) is any designated subset \(X_t \subseteq V_B\). Metrics over designated node-sets are written in terms of \(\mathrm{active}(X_t, W_t)\).*

Definition 12 (Node-set satisfaction). The node-set satisfaction of a finite nonempty salience set \(X_t\) is \[S_{\mathrm{sat}}(X_t, W_t) = \frac{|\mathrm{active}(X_t, W_t)|}{|X_t|}.\]

Definition 13 (Graph property functional). A graph property functional* is any mapping \[\Phi : \mathcal{W} \rightarrow \mathcal{D},\] where \(\mathcal{D}\) is a descriptor space of scalar or finite tuple outputs. In the present paper, \(\mathcal{D}\) is usually instantiated as \(\mathbb{R}^k\). Individual components are written \(\Phi_j(W_t)\) or with descriptive subscripts; domain-specific components are introduced where the corresponding domain is defined.*

Definition 14 (Incompatibility constraint). Let \(G = (V, E, \mathcal{R}, \tau, w)\) be a base graph. An incompatibility constraint* on \(G\) is a pair \((\mathcal{F}, \mathcal{S})\), where:*

1. Predicate incompatibility \(\mathcal{F}\). \(\mathcal{F}\) is a set of label pairs \(\{\ell_1,\ell_2\}\), where each \(\ell_i\) is either a node label or an edge type in \(\mathcal{R}\). Each pair \(\{\ell_1, \ell_2\}\) specifies that the two labels may not simultaneously apply to the same referent. If \(\ell_1\) and \(\ell_2\) are node labels, they may not both hold of the same node \(v \in V\); if they are edge types, they may not both hold of the same ordered pair \((u,v) \in V \times V\). A subgraph \(G' \subseteq G\) violates \(\mathcal{F}\) if there exists \(\{\ell_1, \ell_2\} \in \mathcal{F}\) such that a single referent in \(G'\) carries both labels.

2. Relational incompatibility \(\mathcal{S}\). \(\mathcal{S}\) is a set of forbidden pattern schemas \[P = (V_P, E_P, \mathcal{R}_P, \tau_P, \omega_P),\] where \(\omega_P\) is an optional predicate imposing constraints on the weights of matched edges. A subgraph \(G' \subseteq G\) matches* \(P\) if there exists a label-preserving isomorphism \(f : V_P \to V_{G'}\) such that the induced correspondence preserves edges and \(\tau_P\)-labels, and such that \(\omega_P\) is satisfied under the weight function \(w\).*

Two subgraphs \(X\) and \(Y\) of \(G\) are incompatible, written \(X \bowtie Y\), relative to \((\mathcal{F}, \mathcal{S})\), iff \(X \cup Y\) violates \(\mathcal{F}\) or matches some \(P \in \mathcal{S}\). A subgraph \(X\) is internally incompatible* if \(X \bowtie X\). The graph \(G\) is well-formed under \((\mathcal{F}, \mathcal{S})\) iff \(G\) is not internally incompatible. The sets \(\mathcal{F}\) and \(\mathcal{S}\) are assumed to be finite, or to belong to finitely parameterized families of forbidden patterns, so that incompatibility checking is decidable. For the belief graph \(B\), the relation-type set \(\mathcal{R}_B\) is equipped with a declared constraint \((\mathcal{F}_B, \mathcal{S}_B)\), which specifies the combinations and patterns that are ill-formed in \(B\) and in any working-memory graph derived from it.*

Definition 15 (WM–belief collation). Let \(W_t\) be a working-memory graph and \(B\) the belief graph. Write \(V^{\mathrm{tr}}_t = V_{W_t} \setminus V_B\) for transient WM nodes. A grounding correspondence* at time \(t\) is a finite relation \(\Gamma_t \subseteq V^{\mathrm{tr}}_t \times V_B\) with scores \(\sigma_t(v,u) \in [0,1]\) measuring similarity between transient node \(v\) and belief candidate \(u\). Let \[\Gamma_t^{\geq\theta_{\mathrm{gnd}}} = \bigl\{ (v,u) \in \Gamma_t : \sigma_t(v,u) \geq \theta_{\mathrm{gnd}} \bigr\}.\] Several transient nodes may ground to the same belief node; a transient node may have multiple qualifying candidates. The WM–belief collation graph is \[G^{\mathrm{col}}_t = \mathrm{Coll}(W_t, B, \Gamma_t),\] with node set \(V^{\mathrm{col}}_t = V_B \cup V^{\mathrm{tr}}_t\). It contains all edges of \(E_B\) and \(E_{W_t}\) whose endpoints lie in \(V^{\mathrm{col}}_t\), plus an associative grounding edge \((v,u)\) of type \(r_{\mathrm{related\mathrm{-}to}}\) for each \((v,u) \in \Gamma_t^{\geq\theta_{\mathrm{gnd}}}\). The graph \(G^{\mathrm{col}}_t\) is collation-well-formed when it is not internally incompatible under \((\mathcal{F}_B, \mathcal{S}_B)\) (Definition 14).*

Definition 16 (Cross-memory incompatibility). Let \((\mathcal{F}_B, \mathcal{S}_B)\) be the belief-graph incompatibility constraint. Regions \(X\) and \(Y\) are incompatible, written \(X \bowtie Y\), relative to \((W_t, B)\) and \((\mathcal{F}_B, \mathcal{S}_B)\), as follows. Single-graph regions are tested directly by Definition 14. WM–belief regions are tested in the collation graph \(G^{\mathrm{col}}_t\) (Definition 15), and incompatible whenever the induced WM–belief subgraph violates \((\mathcal{F}_B,\mathcal{S}_B)\) or a forbidden schema cross-matches a transient payload region. For a transient node \(v \in V^{\mathrm{tr}}_t\), the payload region* is \(V_{\mathrm{pl}}(v) = V_{G_v}\) when \(v\) carries a graph payload \(G_v\) and \(V_{\mathrm{pl}}(v) = \{v\}\) otherwise; a forbidden schema \(P \in \mathcal{S}_B\) cross-matches \((v, Y)\) when \(P\) matches the subgraph induced by \(V_{\mathrm{pl}}(v) \cup V_Y\) together with relevant grounding edges \((v,u) \in \Gamma_t^{\geq\theta_{\mathrm{gnd}}}\).*

Definition 17 (Conflict catalog constructor). Let \((\mathcal{F}_B, \mathcal{S}_B)\) be the belief-graph incompatibility constraint, let \(W_t\), \(B\), and \(\Gamma_t\) specify the current state, and let \(\mathrm{scope}\) be a predicate on region pairs \((X,Y)\). The conflict catalog constructor* is \[\mathrm{BuildConflictCatalog}(\mathcal{F}_B, \mathcal{S}_B, \mathrm{scope}, W_t, B, \Gamma_t) = \bigl\{ (X,Y) : X \bowtie Y \text{ under Definition~\ref{def:cross-memory-incompatibility}} \text{ and } \mathrm{scope}(X,Y) \bigr\}.\] Domain-specific catalogs such as \(\mathfrak{C}_{\mathrm{self}}\) and task-restricted confusion catalogs are scope-instantiations of \(\mathrm{BuildConflictCatalog}\), parametrized entirely by their \(\mathrm{scope}\) predicate.*

Definition 18 (Active conflict count). Let \(\mathfrak{C}_\kappa\) be a conflict catalog produced by Definition 17 or an equivalent finite enumeration of ordered pairs \((X, Y)\) such that \(X \bowtie Y\) under Definition 16. An entry \((X,Y)\) is actively conflicting* at time \(t\) when \[(V_X \cup V_Y)\cap V_{W_t} \neq \varnothing,\] and the incompatibility witness for \(X \bowtie Y\) involves at least one WM-active node or transient grounding currently present in \(W_t\). The active conflict count in \(W_t\) is \[\kappa(\mathfrak{C}_\kappa, W_t) = \bigl|\{(X, Y) \in \mathfrak{C}_\kappa : (X, Y) \text{ is actively conflicting at } t\}\bigr|.\] Domain-specific catalogs are written with descriptive subscripts; the WM–belief instance is \(\mathfrak{C}_{\mathrm{wb}}(W_t, B)\) (Proposition 8).*

Definition 19 (Subgraph coherence). Given a designated subgraph \(Q \subseteq B\) and a conflict catalog \(\mathfrak{C}_Q\) of region pairs whose entries lie in \(V_Q \times V_Q\), the coherence* of \(Q\) in \(W_t\) is \[\mathcal{H}(Q, \mathfrak{C}_Q, W_t) = \begin{cases} 1 & \text{if } \mathfrak{C}_Q = \varnothing, \\ 1 - \dfrac{\kappa(\mathfrak{C}_Q, W_t)}{|\mathfrak{C}_Q|} & \text{otherwise}. \end{cases}\] When \(\mathfrak{C}_Q = \varnothing\), coherence is \(1\) by convention (vacuous coherence: no catalogued conflicts to engage). Otherwise, \(\mathcal{H} = 1\) means no active conflicts within \(Q\); \(\mathcal{H} = 0\) means all catalogued conflicts are simultaneously active.*

Definition 20 (Subgraph involvement). Given a designated subgraph \(Q = (V_Q, E_Q) \subseteq B\) and a working-memory state \(W_t\), the involvement* of \(Q\) in \(W_t\) is \[\mathcal{I}(Q, W_t) = S_{\mathrm{sat}}(V_Q, W_t) = \frac{|\mathrm{active}(V_Q, W_t)|}{|V_Q|}.\] We say \(W_t\) is \(Q\)-involving when \(\mathcal{I}(Q, W_t) > 0\), and fully \(Q\)-involving when \(\mathcal{I}(Q, W_t) = 1\).*

Definition 21 (Signed subgraph evaluation). Given a designated subgraph \(Q \subseteq B\) and a signing function \(\sigma_Q : V_Q \rightarrow \mathbb{R}\) that assigns positive weights to favorable nodes and negative weights to unfavorable ones, the signed evaluation* of \(Q\) in \(W_t\) is \[\mathcal{E}(Q, \sigma_Q, W_t) = \sum_{v \in \mathrm{active}(V_Q, W_t)} \sigma_Q(v)\,\alpha_t^{W}(v).\] Grounding scores \(\sigma_t\) (Definition 15) use a separate symbol.*

Definition 22 (Working-memory transition operator). A working-memory transition operator* is any function \[U : \mathcal{W} \times \mathcal{B} \rightarrow \mathcal{W},\] that maps a working-memory state \(W_t\) and belief graph \(B\) to a successor state \(W_{t+1} = U(W_t, B)\), updating both graph structure and the activation maps \(\alpha_t^B\) and \(\alpha_t^{W}\). Input-driven, retrieval-driven, decay, and integration operations are instances of \(U\).*

Definition 23 (Subgraph salience bias). Given a designated subgraph \(Q \subseteq B\) and a salience coefficient \(\beta_Q \geq 1\), a subgraph salience bias* parametrizes the working-memory transition operator \(U\) (Definition 22) by supplying a salience map \(\beta^{\mathrm{sal}}_Q : V_B \rightarrow \mathbb{R}_{\ge 1}\) with \(\beta^{\mathrm{sal}}_Q(v) = \beta_Q\) for \(v \in V_Q\) and \(\beta^{\mathrm{sal}}_Q(v) = 1\) otherwise (distinct from the scalar coefficient \(\beta_Q\)). Under this parametrization, an activation update map \(U_{\mathrm{act}}(\alpha_t^{W},\beta^{\mathrm{sal}}_Q)\) applies multiplicative gain on activation update and reduced decay on nodes in \(V_Q\) (distinct from the WM transition operator \(U\), Definition 22): \[\alpha_{t+1}^{W}(v) = U_{\mathrm{act}}\bigl(\alpha_t^{W}, \beta^{\mathrm{sal}}_Q\bigr)(v) = \begin{cases} \beta_Q \cdot (\alpha_t^{W})^{\mathrm{base}}(v) & v \in V_Q, \\ (\alpha_t^{W})^{\mathrm{base}}(v) & v \notin V_Q, \end{cases}\] where \((\alpha_t^{W})^{\mathrm{base}}\) is the WM activation profile produced by the unbiased transition step, and decay on \(v \in V_Q\) is scaled by \(1/\beta_Q\) relative to nodes outside \(V_Q\) of equivalent base activation. Setting \(\beta_Q = 1\) recovers standard activation dynamics with no subgraph bias. Domain-specific biases are parametrizations of this definition by choice of \(Q\) and \(\beta_Q\), governed by the single multiplicative-gain and decay-scaling rule specified above.*

Definition 24 (Parameter bundle). An architecture parameter bundle* is a tuple \[\Theta = (\theta_W,\, \theta_{\mathrm{gnd}},\, \theta_{\mathrm{sim}},\, \theta_{\mathrm{dist}},\, \theta_{\mathrm{inv}},\, \theta_{\mathrm{coh}},\, \{\beta_Q\}_{Q \in \mathcal{Q}_{\mathrm{des}}}),\] where \(\theta_W\) governs WM membership (Definition 10), \(\theta_{\mathrm{gnd}}\) governs grounding correspondence in WM–belief collation (Definition 15), \(\theta_{\mathrm{sim}}\) governs graph-similarity predicates, \(\theta_{\mathrm{dist}}\) governs distraction thresholds on \(\Phi_{\mathrm{dist}}\), \(\theta_{\mathrm{inv}}\) governs involvement predicates on \(\mathcal{I}\), \(\theta_{\mathrm{coh}}\) governs coherence predicates on \(\mathcal{H}\), and \(\{\beta_Q\}_{Q \in \mathcal{Q}_{\mathrm{des}}}\) supplies salience coefficients for designated subgraphs \(Q \subseteq B\) under Definition 23 (\(\mathcal{Q}_{\mathrm{des}}\) is the family of designated subgraphs, distinct from any single subgraph symbol \(Q\)). Throughout the paper, threshold directions are fixed by quantity type: activation and similarity predicates use lower bounds; conflict and distraction predicates use upper bounds or nonzero tests; coherence predicates use lower bounds; and binary indicators are defined explicitly case by case. Application sections may assign threshold values, but they do not alter these comparison directions. Application sections instantiate \(\Theta\) by assigning values to these slots.*

Definition 25 (Fragmentation). Let \(W_t\) have connected components \(C_1, \dots, C_k\) (ignoring edge direction). The fragmentation component* of the graph property functional is \[\Phi_{\mathrm{frag}}(W_t) = k.\] We write \(F(W_t) = \Phi_{\mathrm{frag}}(W_t)\) and say that \(W_t\) is fragmented if \(F(W_t) > 1\).*

Definition 26 (Belief-update operator). A belief-update operator is any function \[\Delta_{\bullet} : \mathcal{B} \times \mathcal{W}^* \rightarrow \mathcal{B},\] that maps a belief graph \(B\) and a finite sequence of working-memory states \((W_{t-k}, \dots, W_t)\) to an updated belief graph \(B'\), using operations such as node and edge addition, deletion, retyping, and modification of representational payloads. Two canonical application interfaces are: \[\begin{align} \mathrm{ApplyMicroUpdate}(B, (W_{t-k},\dots,W_t), \mathfrak{C}_\kappa, \Theta) &\mapsto B' \\ \mathrm{ApplyMacroUpdate}(B, (W_{t-k},\dots,W_t), \mathfrak{C}_\kappa, \Theta, \tau_{\mathrm{trans}}) &\mapsto B', \end{align}\] where \(\mathfrak{C}_\kappa\) is a conflict catalog (Definition 17), \(\Theta\) is a parameter bundle (Definition 24), and \(\tau_{\mathrm{trans}} \in \{\mathrm{strict}, \mathrm{permit}\}\) controls whether ill-formed intermediate graphs are allowed during multi-step restructuring (distinct from the edge-type map \(\tau\) on base graphs, Definition 1). \(\mathrm{ApplyMicroUpdate}\) modifies a proper subgraph while preserving global topology and requires \(\kappa(\mathfrak{C}_{\mathrm{wb}}(W_t,B),W_t)=0\) at commit (Remark 12). \(\mathrm{ApplyMacroUpdate}\) may reorganize entire mental models or designated subgraphs \(Q \subseteq B\); when \(\tau_{\mathrm{trans}}=\mathrm{permit}\), transient violations of \((\mathcal{F}_B,\mathcal{S}_B)\) are allowed provided the committed output is well-formed. Specific instances are introduced where the corresponding domain is defined. A belief-update application is micro if it is realizable as \(\mathrm{ApplyMicroUpdate}\); it is macro* if it requires \(\mathrm{ApplyMacroUpdate}\) with nontrivial restructuring.*

Definition 27 (Trajectory functional). A trajectory functional* is any mapping \[\Psi : \mathcal{W}^* \rightarrow \mathcal{Z},\] from finite sequences of working-memory graphs to a representation space \(\mathcal{Z}\) (e.g., control signals, temporally annotated graphs, or scalar summaries). Concrete trajectory functionals are introduced in subsequent subsections.*

Definition 28 (Trajectory label sets). The canonical temporal and provenance label sets are \[\mathfrak{L}_{\mathrm{temp}}=\{\mathrm{static},\mathrm{dynamic},\mathrm{persisting},\mathrm{changing}\}, \qquad \mathfrak{L}_{\mathrm{prov}}=\{\mathrm{input},\mathrm{retrieval},\mathrm{integration},\mathrm{decay},\mathrm{chunk}\}.\] A temporal trajectory functional is a mapping \[\mathcal{T} : \mathcal{W}^* \times 2^{V_B} \rightarrow \mathfrak{L}_{\mathrm{temp}}\] that assigns each designated subgraph \(X\) a label in \(\mathfrak{L}_{\mathrm{temp}}\) relative to a finite trajectory \((W_{t-k},\dots,W_t)\), satisfying time-consistency: labels depend only on the restriction of the trajectory to nodes in \(X\) and must be invariant under padding with identical copies of \(W_t\). A provenance trajectory functional is a mapping \[\Pi : \mathcal{W}^* \times 2^{V_B} \rightarrow \mathfrak{L}_{\mathrm{prov}}\] that classifies the most recent transition-induced origin of subgraph content under the working-memory transition operator \(U\) (Definition 22). Both \(\mathcal{T}\) and \(\Pi\) are instances of the generic trajectory functional \(\Psi\) (Definition 27) with representation space \(\mathcal{Z} = \mathfrak{L}_{\mathrm{temp}} \times 2^{V_B}\) or \(\mathfrak{L}_{\mathrm{prov}} \times 2^{V_B}\). Architecture choices supply the decision rules that assign labels; the label sets and signatures are fixed by this definition.

Definition 29 (Self-image subgraph). Let \(B = (V_B, E_B, \mathcal{R}_B, \tau_B, w_B)\) be the belief graph. A self-image subgraph* is a connected subgraph \[S_{\mathrm{self}} = (V_{\mathrm{self}}, E_{\mathrm{self}}) \subseteq B\] distinguished by an anchor node \(v_{\mathrm{self}} \in V_{\mathrm{self}}\) whose payload denotes the agent as self.*

The node set \(V_{\mathrm{self}}\) contains nodes representing:

1. bodily attributes and interoceptive states,

2. autobiographical episodes and remembered events,

3. trait-like self-descriptors,

4. goals, intentions, commitments, and values,

5. social roles and identity-linked relational nodes,

6. meta-representations of the self, including beliefs about how the self is seen by others.

The edge set \(E_{\mathrm{self}}\) may include causal, containment, temporal, associative, evidential, and spatial relations, restricted to the relation classes already admitted by Definition 4. The self-image subgraph is well-formed when it is connected, internally compatible under \((\mathcal{F}_B,\mathcal{S}_B)\), and anchored by either one designated self node or one designated self anchor cluster treated as a single identity unit.

Definition 30 (Self-related salience). A subgraph \(Q \subseteq B\) is self-related* if \(V_Q \cap V_{\mathrm{self}} \neq \varnothing\) or if \(Q\) is linked to \(S_{\mathrm{self}}\) by one or more self-referential edges in \(E_B\). The self-salience of \(Q\) at time \(t\) is an activation-weighted measure of how much self-related content in \(Q\) is currently present in working memory: \[S_{\mathrm{self-sat}}(Q, W_t) = S_{\mathrm{sat}}(V_Q \cap V_{\mathrm{self}}, W_t)\] when \(V_Q \cap V_{\mathrm{self}} \neq \varnothing\) (Definition 12), and \(0\) otherwise. The score is activation-weighted through the underlying node-set satisfaction measure.*

Definition 31 (Self-involvement). The self-involvement* of the current working-memory state \(W_t\) is \[\mathcal{I}_{\mathrm{self}}(W_t) = \mathcal{I}(S_{\mathrm{self}}, W_t) = \frac{|\mathrm{active}(V_{\mathrm{self}}, W_t)|}{|V_{\mathrm{self}}|}\] (Definition 20). We say that \(W_t\) is self-involving when \(\mathcal{I}_{\mathrm{self}}(W_t) > 0\), and fully self-involving when \(\mathcal{I}_{\mathrm{self}}(W_t) = 1\).*

Definition 32 (Self-evaluation). Let \(\sigma_{\mathrm{eval}} : V_{\mathrm{self}} \rightarrow \mathbb{R}\) be a valuation assigning positive weight to desirable self-nodes and negative weight to undesirable self-nodes. The self-evaluation* of \(W_t\) is \[\mathcal{E}_{\mathrm{self}}(W_t) = \mathcal{E}(S_{\mathrm{self}}, \sigma_{\mathrm{eval}}, W_t) = \sum_{v \in \mathrm{active}(V_{\mathrm{self}}, W_t)} \sigma_{\mathrm{eval}}(v)\,\alpha_t^{W}(v).\] (Definition 21.) This quantity summarizes the affectively weighted activation of the self-image.*

Definition 33 (Self-consistency). Let \[\mathfrak{C}_{\mathrm{self}} = \mathrm{BuildConflictCatalog}(\mathcal{F}_B, \mathcal{S}_B, \mathrm{scope}_{\mathrm{self}}, W_t, B, \Gamma_t),\] where \(\mathrm{scope}_{\mathrm{self}}\) retains region pairs internal to \(S_{\mathrm{self}}\) (Definition 17). The self-consistency of \(S_{\mathrm{self}}\) in \(W_t\) is \[\mathcal{H}_{\mathrm{self}}(W_t) = \mathcal{H}(S_{\mathrm{self}}, \mathfrak{C}_{\mathrm{self}}, W_t).\] (Definition 19.) A value of \(1\) means that no catalogued self-conflicts are actively engaged, while \(0\) means that all catalogued self-conflicts are active.

Definition 34 (Self-update). A self-update* is an instance of \(\mathrm{ApplyMicroUpdate}\) or \(\mathrm{ApplyMacroUpdate}\) (Definition 26) whose application revises the self-image subgraph \(S_{\mathrm{self}} \subseteq B\) in response to self-relevant input. The update may:*

1. add a new self-node,

2. delete an obsolete self-node,

3. strengthen or weaken a self-edge,

4. retype a self-relation,

5. resolve a self-conflict,

6. compress a repeated pattern into a higher-level self-schema.

We write \(\Delta_{\mathrm{self}}\) for a self-update instance.

Definition 35 (Self-referential activation bias). A self-referential activation bias* instantiates subgraph salience bias (Definition 23) for \(Q = S_{\mathrm{self}}\) with coefficient \(\beta_{\mathrm{self}}\) from the parameter bundle \(\Theta\) (Definition 24), inducing salience map \(\beta^{\mathrm{sal}}_{S_{\mathrm{self}}}\): \[\alpha_{t+1}^{W}(v) = U_{\mathrm{act}}\bigl(\alpha_t^{W}, \beta^{\mathrm{sal}}_{S_{\mathrm{self}}}\bigr)(v),\] so that nodes \(v \in V_{\mathrm{self}}\) receive multiplicative gain and reduced decay under \(U\) as specified in Definition 23. This bias increases the likelihood that self-related nodes enter working memory and remain active across short time intervals.*

Definition 36 (Narrative self-coherence). For a finite working-memory trajectory \((W_{t_0}, W_{t_1}, \dots, W_{t_n}) \in \mathcal{W}^*\), narrative self-coherence instantiates involvement and coherence functionals (Definitions 20 and 19) over the self-image subgraph: \[N_{\mathrm{self}}(W_{t_0}, \dots, W_{t_n}) = \frac{1}{n+1} \sum_{i=0}^{n} \mathcal{I}_{\mathrm{self}}(W_{t_i}) \cdot \mathcal{H}_{\mathrm{self}}(W_{t_i}),\] where \(\mathcal{I}_{\mathrm{self}}(W_{t_i}) = \mathcal{I}(S_{\mathrm{self}}, W_{t_i})\) (Definition 31). This score is high when self-related content is both frequently active and internally consistent across time.

Definition 37 (Awareness functional). An awareness functional* is a mapping \[\mathcal{A} : \mathcal{W} \rightarrow 2^{\mathcal{W}},\] where \(2^{\mathcal{W}}\) denotes the power set of subgraphs of working-memory graphs. For a working-memory state \(W_t\), the value \(\mathcal{A}(W_t)\) is a subgraph \(A_t \subseteq W_t\)—the portion of working memory that the system currently has access to. A node or subgraph \(X \subseteq W_t\) is in awareness at \(t\) if \(X \subseteq \mathcal{A}(W_t)\).*

The system is said to be aware* at \(t\) if \(\mathcal{A}(W_t) \neq \varnothing\), i.e., at least some content of \(W_t\) is accessed. No connectivity constraint is imposed: \(\mathcal{A}(W_t)\) may be a single node, a fragment, or any subgraph of \(W_t\).*

Unlike trajectory functionals \(\Psi\) (Definition 27), which map finite WM sequences to summary representations, \(\mathcal{A}\) is a state-level access selector: its input is a single \(W_t\), not a trajectory \((W_{t-k},\dots,W_t)\). Temporal and regulatory summaries over trajectories—including control via \(\mathcal{K}\) (Section 2.8) and transformed-awareness annotations via \(\mathcal{T}\) and \(\Pi\) (Section 2.7)—are supplied by the \(\Psi\)-family; \(\mathcal{A}\) specifies which subgraph of \(W_t\) those summaries annotate at time \(t\).

Definition 38 (Total awareness). A working-memory state \(W_t\) is in total awareness* at \(t\) if \[\mathcal{A}(W_t) = W_t,\] i.e., the awareness functional covers the entirety of the current working-memory graph, including all nodes and edges in the current working-memory graph. Total awareness is the special case of awareness in which no content of \(W_t\) is excluded from access.*

Definition 39 (Transformed awareness). Let \(W_t\) be a working-memory graph at time \(t\), and let \(\mathcal{A}(W_t) \subseteq W_t\) be the subgraph in awareness. The state \(W_t\) exhibits transformed awareness* if there exists \(k \geq 0\), a finite trajectory \((W_{t-k},\dots,W_t)\) on the interval \([t-k,t]\), a subgraph \(X \subseteq \mathcal{A}(W_t)\), and a temporal trajectory functional \(\mathcal{T}\) (Definition 28) such that \(\mathcal{T}(W_{t-k},\dots,W_t, X)\) assigns to \(X\) a label in \(\mathfrak{L}_{\mathrm{temp}}\).*

Definition 40 (Temporal objectification). Let \(W_t\) be a working-memory state, let \(k \geq 0\), and let \((W_{t-k},\dots,W_t)\) be a finite trajectory on \([t-k,t]\). A subgraph \(X \subseteq W_t\) is temporally objectified* at time \(t\) if \[X \subseteq \mathcal{A}(W_t)\] and a temporal trajectory functional \(\mathcal{T}\) (Definition 28) assigns to \(X\) a label in \(\mathfrak{L}_{\mathrm{temp}}\) relative to \((W_{t-k},\dots,W_t)\). Thus, temporal objectification is a specialized form of transformed awareness.*

Definition 41 (Psychological time). Let \(W_t\) be a working-memory state and let \(k \geq 0\). A state \(W_t\) exhibits psychological time* on \([t-k,t]\) if there exists a subgraph \(X \subseteq \mathcal{A}(W_t)\) and a temporal trajectory functional \(\mathcal{T}\) (Definition 28) such that \(\mathcal{T}(W_{t-k},\dots,W_t, X)\) assigns to \(X\) a label in \(\mathfrak{L}_{\mathrm{temp}}\). Equivalently, psychological time is present exactly when at least one subgraph in awareness is temporally objectified.*

Definition 42 (Provenance objectification). Let \(W_t\) be a working-memory state, let \(k \geq 0\), and let \((W_{t-k},\dots,W_t)\) be a finite trajectory on \([t-k,t]\). A subgraph \(X \subseteq W_t\) is provenance-objectified* at time \(t\) if \[X \subseteq \mathcal{A}(W_t)\] and a provenance trajectory functional \(\Pi\) (Definition 28) assigns to \(X\) a label in \(\mathfrak{L}_{\mathrm{prov}}\) relative to \((W_{t-k},\dots,W_t)\). Thus, provenance objectification is a specialized form of transformed awareness.*

Definition 43 (Conflict objectification). Let \(W_t\) be a working-memory state and let \(\mathfrak{C}_\kappa\) be a domain-specific conflict catalog (Definition 18) whose entries are defined under Definition 16. A subgraph \(X \subseteq W_t\) is conflict-objectified* at time \(t\) if \(X \subseteq \mathcal{A}(W_t)\) and \(X\) is counted as actively conflicting by \(\kappa(\mathfrak{C}_\kappa,W_t)\).*

Definition 44 (Conflict-aware transformed awareness). The state \(W_t\) exhibits conflict-aware transformed awareness* if there exists a subgraph \(X \subseteq W_t\) that is conflict-objectified at \(t\) under Definition 43.*

Definition 45 (Insightful perception). Let \(W_t\) be a working-memory state at time \(t\), let \(k \geq 0\), and let \((W_{t-k},\dots,W_t)\) be a finite trajectory on \([t-k,t]\). The state \(W_t\) exhibits insightful perception* at \(t\) if there exists a WM-active subgraph \(X \subseteq W_t\) (Definition 10) such that \(X \subseteq \mathcal{A}(W_t)\) and at least one of the following predicates holds:*

1. \(X\) is temporally objectified at \(t\) under Definition 40,

2. \(X\) is conflict-objectified at \(t\) under Definition 43,

3. there exist a designated subgraph \(Q \subseteq B\) and an architecturally designated signing function \(\sigma_Q : V_Q \to \mathbb{R}\) (e.g., \(\sigma_{\mathrm{eval}}\) when \(Q = S_{\mathrm{self}}\), Definition 32) such that \(X \subseteq Q\) and \(\mathcal{E}(Q,\sigma_Q,W_t) \neq 0\) with nonzero contribution from \(\mathrm{active}(V_X,W_t)\) (Definition 21),

4. \(X\) is provenance-objectified at \(t\) under Definition 42.

Definition 46 (Total insight). Let \(W_t\) be a working-memory state at time \(t\), let \(k \geq 0\), and let \((W_{t-k},\dots,W_t)\) be a finite trajectory on \([t-k,t]\). The state \(W_t\) exhibits total insight* at \(t\) if \[\mathcal{A}(W_t)=W_t\] and at least one predicate in Definition 45 holds with \(X=W_t\). Equivalently, total insight is insightful perception on the whole working-memory graph under total awareness (Definition 38).*

Definition 47 (Control functional). A control functional* is a trajectory functional \[\mathcal{K} : \mathcal{W}^* \rightarrow \mathfrak{O},\] where \(\mathfrak{O}\) is a space of control operations on working memory and retrieval dynamics.*

Definition 48 (Control operation). A control operation* is any operation emitted by \(\mathcal{K}\) that modulates subsequent working-memory evolution under the transition operator \(U\) (Definition 22). Typical instances include:*

1. attentional focusing on a designated subgraph \(Q\),

2. retrieval amplification from belief graph regions linked to \(Q\),

3. suppression or pruning of distractor subgraphs,

4. persistence bias on designated active nodes,

5. conflict-resolution routing when \(\kappa(\mathfrak{C}_\kappa,W_t)>0\) for a catalog \(\mathfrak{C}_\kappa\) from Definition 17,

6. self-directed regulation when self-related quantities exceed thresholds in the parameter bundle \(\Theta\) (Definition 24).

The set of all admissible control operations is denoted \(\mathfrak{O}\), the codomain of \(\mathcal{K}\). A control operation \(c \in \mathfrak{O}\) is local* if it modifies a proper subgraph of \(W_t\) or a single mental model in \(B\); it is global if it reorganizes the large-scale structure of \(W_t\) or \(B\).*

Definition 49 (Control-sensitive trajectory). A finite trajectory \((W_{t-k},\dots,W_t)\) is control-sensitive* if \(\mathcal{K}(W_{t-k},\dots,W_t)\) depends on at least one graph-property quantity defined over the trajectory, including but not limited to fragmentation \(F(W_t)\) (Definition 25), self-involvement \(\mathcal{I}(S_{\mathrm{self}},W_t)\) (Definition 31), self-consistency \(\mathcal{H}(S_{\mathrm{self}},\mathfrak{C}_{\mathrm{self}},W_t)\) (Definition 33), signed evaluation \(\mathcal{E}(S_{\mathrm{self}},\sigma_{\mathrm{eval}},W_t)\) (Definition 32), or active conflict counts \(\kappa(\mathfrak{C}_\kappa,W_t)\) (Definition 18).*

Definition 50 (Awareness-conditioned control). A control functional \(\mathcal{K}\) is awareness-conditioned* if its output at time \(t\) depends only on subgraphs contained in \(\mathcal{A}(W_t)\) together with trajectory-derived annotations available within transformed awareness.*

Definition 51 (Graph similarity). Let \(G_{\mathrm{q}} = (V_{\mathrm{q}}, E_{\mathrm{q}})\) be a query graph and \(M = (V_M, E_M)\) a candidate graph with typed edges. A partial match from \(G_{\mathrm{q}}\) to \(M\) is an injective mapping \(f_{\mathrm{match}} : V_{\mathrm{q}} \rightarrow V_M\) preserving a specified subset of edge types from the edge-type taxonomy (Definition 4). Let \(m_V = |f_{\mathrm{match}}(V_{\mathrm{q}})|\) and let \(m_E\) be the number of edges in \(E_{\mathrm{q}}\) whose images are present in \(E_M\) with matching types under \(f_{\mathrm{match}}\). The similarity measure* of \(G_{\mathrm{q}}\) to \(M\) under \(f_{\mathrm{match}}\) is \[\mathrm{sim}_{\mathrm{match}}(G_{\mathrm{q}}, M; f_{\mathrm{match}}) = \lambda_{\mathrm{match}} \frac{m_V}{|V_{\mathrm{q}}|} + (1-\lambda_{\mathrm{match}})\frac{m_E}{|E_{\mathrm{q}}|},\] for some \(\lambda_{\mathrm{match}} \in [0,1]\) in the parameter bundle \(\Theta\) (Definition 24).*

Definition 52 (Recognition event). Recognition is a derived binary predicate over graph similarity (Definition 51) with threshold \(\theta_{\mathrm{sim}}\) from \(\Theta\). Let \(W_t\) be the current working-memory graph and \(\mathcal{M}_{\mathrm{lib}}\) a library of candidate mental models in \(B\). A recognition event* occurs when there exists \(M \in \mathcal{M}_{\mathrm{lib}}\) and a partial match \(f_{\mathrm{match}}\) such that \[\mathrm{sim}_{\mathrm{match}}(W_t, M; f_{\mathrm{match}}) \geq \theta_{\mathrm{sim}}.\] A complementary failed recognition case holds when \(X \subseteq W_t\) partially matches \(M \in \mathcal{M}_{\mathrm{lib}}\) under \(f_{\mathrm{match}}\) with \(\mathrm{sim}_{\mathrm{match}}(X, M; f_{\mathrm{match}}) \geq \theta_{\mathrm{sim}}\) but \(X \bowtie M\) in \(G^{\mathrm{col}}_t = \mathrm{Coll}(W_t, B, \Gamma_t)\) (Definitions 15 and 16): the episode matches structurally but is rejected by WM–belief incompatibility and contributes to \(\kappa(\mathfrak{C}_{\mathrm{wb}}(W_t, B), W_t)\) (Proposition 8).*

Definition 53 (Distraction measure). Distraction is a derived scalar measure over \(\Phi\) (Definition 13). Let \(W_t\) have connected components \(C_1, \dots, C_k\) with \(k = F(W_t)\) (Definition 25), and let \(p_i = \alpha_t^{W}(C_i) / \alpha_t^{W}(V_{W_t})\) be the normalized activation mass on component \(C_i\). The distraction measure* is \[\Phi_{\mathrm{dist}}(W_t) = H_{\mathrm{ent}}(\boldsymbol{p}) = -\sum_{i=1}^k p_i \log p_i,\] where \(H_{\mathrm{ent}}\) denotes Shannon entropy over the component mass vector \(\boldsymbol{p}=(p_1,\dots,p_k)\).*

Definition 54 (Distracted predicate). The distracted* predicate holds when \(F(W_t) > 1\) and \(\Phi_{\mathrm{dist}}(W_t) > \theta_{\mathrm{dist}}\) for the distraction threshold \(\theta_{\mathrm{dist}}\) in the parameter bundle \(\Theta\) (Definition 24). Distraction is distinct from partial awareness: when \(W_t\) is fragmented, \(\mathcal{A}(W_t)\) may cover one component, several components, or any other subgraph of \(W_t\) without requiring a single connected access region.*

Definition 55 (Confusion). The confused* predicate is built from conflict catalogs (Definition 17) and subgraph coherence \(H_{\mathrm{coh}}\). Let \(R^{\mathrm{wm}} \subseteq V_{W_t}\) be the task-relevant WM node set for a given task representation, and let \(E_{R^{\mathrm{wm}}}\) be the edges among them. Let \(G^{\mathrm{conf}}_{R^{\mathrm{wm}}} = (R^{\mathrm{wm}}, E_{R^{\mathrm{wm}}})\) denote the induced task-relevant subgraph (salience set \(R_t \subseteq V_B\) is a separate symbol). Let \[\mathfrak{C}_{\mathrm{conf}}^{\mathrm{intra}}(R^{\mathrm{wm}}) = \mathrm{BuildConflictCatalog}(\mathcal{F}_B, \mathcal{S}_B, \mathrm{scope}_{\mathrm{intra}}(R^{\mathrm{wm}}), W_t, B, \Gamma_t),\] where \(\mathrm{scope}_{\mathrm{intra}}(R^{\mathrm{wm}})\) retains intra-WM pairs \((X,Y)\) with \(V_X \cup V_Y \subseteq R^{\mathrm{wm}}\). Let \[\mathfrak{C}_{\mathrm{conf}}^{\mathrm{wb}}(R^{\mathrm{wm}}) = \mathrm{BuildConflictCatalog}(\mathcal{F}_B, \mathcal{S}_B, \mathrm{scope}_{\mathrm{wb}}(R^{\mathrm{wm}}), W_t, B, \Gamma_t),\] where \(\mathrm{scope}_{\mathrm{wb}}(R^{\mathrm{wm}})\) retains WM–belief pairs \((X,Y)\) with \(V_X \cap R^{\mathrm{wm}} \neq \varnothing\). The confused predicate relative to \(R^{\mathrm{wm}}\) is \[\mathrm{Confused}(W_t; R^{\mathrm{wm}}) \iff \mathrm{Disconnected}(R^{\mathrm{wm}}) \lor \mathrm{Incoherent}(R^{\mathrm{wm}}) \lor \mathrm{BeliefConflict}(R^{\mathrm{wm}}),\] where \[\mathrm{Disconnected}(R^{\mathrm{wm}}) \iff (R^{\mathrm{wm}}, E_{R^{\mathrm{wm}}}) \text{ is disconnected},\] \[\mathrm{Incoherent}(R^{\mathrm{wm}}) \iff H_{\mathrm{coh}}(G^{\mathrm{conf}}_{R^{\mathrm{wm}}}, \mathfrak{C}_{\mathrm{conf}}^{\mathrm{intra}}(R^{\mathrm{wm}}), W_t) < \theta_{\mathrm{coh}},\] \[\mathrm{BeliefConflict}(R^{\mathrm{wm}}) \iff \kappa(\mathfrak{C}_{\mathrm{conf}}^{\mathrm{wb}}(R^{\mathrm{wm}}), W_t) > 0,\] and \(\theta_{\mathrm{coh}}\) is the coherence threshold in \(\Theta\) (Definition 24). The corresponding confusion indicator is the \(0/1\) quantity \(\Phi_{\mathrm{conf}}(W_t; R^{\mathrm{wm}}) = 1\) if and only if \(\mathrm{Confused}(W_t; R^{\mathrm{wm}})\). The three disjuncts are under-connection (\(\mathrm{Disconnected}\)), intra-WM incoherence (\(\mathrm{Incoherent}\)), and cross-memory conflict (\(\mathrm{BeliefConflict}\)).*

Definition 56 (Cognitive load measures). The three load measures are scalar measures defined as follows.

1. Intrinsic load. Given a task that requires representing a minimal schema \(S^\ast = (V^\ast, E^\ast)\), \[\Phi_{\mathrm{load\mathrm{-}int}}(S^\ast) = L_{\mathrm{int}} = |V^\ast| + \lambda |E^\ast|,\] where \(\lambda \geq 0\) weights edge complexity.

2. Extraneous load. For a working-memory state \(W_t\) during task execution, with task-relevant belief nodes \(R_t \subseteq V_B\) (Definition 11) and \(E^{\mathrm{sal}}_t \subseteq E_{W_t}\) the edges with both endpoints in \(\mathrm{active}(R_t, W_t)\), \[\Phi_{\mathrm{load\mathrm{-}ext}}(W_t) = L_{\mathrm{ext}}(W_t) = |V_{W_t} \setminus R_t| + \lambda |E_{W_t} \setminus E^{\mathrm{sal}}_t|,\] counting unnecessary nodes and edges, where \(E^{\mathrm{sal}}_t\) is distinct from confusion edge sets \(E_{R^{\mathrm{wm}}}\) (Definition 55).

3. Germane load. The germane load measure \(L_{\mathrm{ger}}\) quantifies coherence-preserving restructuring across an update \(B \to B'\), for example via \(\delta_{\mathrm{coh}}(B,B')\). For belief graphs \(B\) and \(B'\) produced during a sequence \((W_t, \dots, W_{t+n})\), \[\Phi_{\mathrm{load\mathrm{-}ger}}(B, B') = L_{\mathrm{ger}} = \delta_{\mathrm{coh}}(B, B'),\] where \(\delta_{\mathrm{coh}}\) quantifies the increase in structural coherence, e.g., via reductions in contradictions or increases in alignment between schemas and mental models.

Definition 57 (Effective element interactivity). Let \(R_t \subseteq V_B\) be the task-relevant salience set (Definition 11) for a given task at time \(t\). For a learner with belief graph \(B\), the effective element interactivity scalar is \[E_{\mathrm{eff}}(R_t, B) = \min \bigl\{ k : R_t \subseteq \bigcup_{i=1}^{k} S^{\mathrm{ch}}_i,\;S^{\mathrm{ch}}_i \subseteq V_B\;\text{is a stored schema or chunk in } B \bigr\},\] i.e., the minimal number of stored chunks needed to cover the task-relevant elements.

Definition 58 (Conceptual change). **Conceptual change* is a mapping-level construct: an instance of \(\mathrm{ApplyMacroUpdate}\) from the generic belief-update operator (Definition 26) acting on the belief graph \(B\) in response to a sequence of working-memory states \((W_{t}, \dots, W_{t+n})\). It restructures theory-bearing subgraphs of \(B\), potentially changing node labels, edge types, and compatibility relations. In this application—written \(\Delta\) when the interpretation is conceptual change—\(\Delta\) may realize node addition or deletion, edge addition, deletion, or retyping, and reconfiguration of representational payloads, as already permitted under the generic belief-update framework. The specialization lies in targeting theory-bearing subgraphs of \(B\) (including mental models and nested internal graphs \(G_v\)) during learning and theory revision [9], [10]. Conceptual change is distinguished from other belief-update applications such as self-update \(\Delta_{\mathrm{self}}\) (Section 2.5) by target subgraph; psychologically, it corresponds to theory revision and restructuring during learning. Applications may be micro—modifying a proper subgraph while preserving global topology—or macro—reorganizing entire mental models or substantially reconfiguring a designated subgraph (Definition 26). Productive belief-graph restructuring tracked by germane load \(L_{\mathrm{ger}}\) (Section 3.5) is realized under this interpretation as conceptual change.*

Definition 59 (Field and expert-accessible subgraph). Let \(B\) be the belief graph. A field* (or domain) is a designated connected mental model \[M = (V_M, E_M) \subseteq B\] (Definition 6). A particular expert’s accessible knowledge is represented by a possibly fragmented subgraph \[M_{\mathrm{e}} = (V_{\mathrm{e}}, E_{\mathrm{e}}) \subseteq M,\] where \(M_{\mathrm{e}}\) captures the expert’s internally available portion of the field. When \(M_{\mathrm{e}}\) is fragmented, write \(M_{\mathrm{e},1},\ldots,M_{\mathrm{e},k}\) for its connected components (ignoring edge direction, as in Definition 25).*

Definition 60 (Question as query subgraph). A question* is a query subgraph \[G_{\mathrm{q}} = (V_{\mathrm{q}}, E_{\mathrm{q}}) \subseteq W_t\] active in working memory at time \(t\) (Definition 51). The subgraph \(G_{\mathrm{q}}\) may encode an incomplete relation, a candidate proposition, a required successor state, a predicted outcome, or a true/false alternative whose status is to be evaluated relative to the expert-accessible portion of the domain. Its resolution is evaluated through grounding correspondence \(\Gamma_t\) and \(G^{\mathrm{col}}_t\), partial matches \(f_{\mathrm{match}} : V_{\mathrm{q}} \to V_{M_{\mathrm{e}}}\), similarity scores \(\mathrm{sim}_{\mathrm{match}}(G_{\mathrm{q}}, M_{\mathrm{e}}; f_{\mathrm{match}})\), cross-memory incompatibility \(X \bowtie Y\) in \(G^{\mathrm{col}}_t\), active conflict count \(\kappa(\mathfrak{C}_{\mathrm{wb}}(W_t, B), W_t)\), and control outputs from \(\mathcal{K}\) that stabilize or revise the active query neighborhood in \(W_t\).*

Definition 61 (Context graph). Let \(W_t\) be a working-memory graph at time \(t\), let \(R_t\) be a task-relevant salience set, let \(\mathcal{A}(W_t)\) be the subgraph currently in awareness, and let \(\mathcal{K}\) emit control operations \(c \in \mathfrak{O}\) that select and stabilize working-memory structure. A context graph* at time \(t\), denoted \(C_t\), is the active task-relevant subgraph of \(W_t\) selected under salience, awareness, and control: \[C_t = (V_{C_t}, E_{C_t}) \subseteq W_t,\] such that \[V_{C_t} \subseteq R_t \cup V_{\mathcal{A}(W_t)}.\] When useful, \(C_t\) may be defined as the maximal subgraph of \(W_t\) satisfying a task context predicate \(\Phi^{\mathrm{ctx}}_T\) (distinct from graph-property functional \(\Phi\), Definition 13): \[C_t = \max\{X \subseteq W_t : \Phi^{\mathrm{ctx}}_T(X,t)=1\}.\] The evolution of \(C_t\) is governed by the working-memory transition operator \(U\), modulated by salience, awareness, and control; its relation to stored knowledge is mediated through \(\Gamma_t\) and \(G^{\mathrm{col}}_t\).*

Definition 62 (Answerability). Relative to expert-accessible subgraph \(M_{\mathrm{e}}\) and query subgraph \(G_{\mathrm{q}} \subseteq W_t\), the answer status* of \(G_{\mathrm{q}}\) is determined as follows:*

1. Supported when partial matches \(f_{\mathrm{match}} : V_{\mathrm{q}} \to V_{M_{\mathrm{e}}}\) yield high \(\mathrm{sim}_{\mathrm{match}}(G_{\mathrm{q}}, M_{\mathrm{e}}; f_{\mathrm{match}})\) (Definition 51), grounding scores \(\sigma_t(v,u)\) exceed \(\theta_{\mathrm{gnd}}\) in \(G^{\mathrm{col}}_t\) (Definition 15), and collation remains well-formed under \((\mathcal{F}_B, \mathcal{S}_B)\).

2. Contradicted when \(X \subseteq G_{\mathrm{q}}\) and \(Y \subseteq M_{\mathrm{e}}\) satisfy \(X \bowtie Y\) under Definition 16 in \(G^{\mathrm{col}}_t\) (cf.failed recognition in Definition 52), contributing to \(\kappa(\mathfrak{C}_{\mathrm{wb}}(W_t, B), W_t)\) (Proposition 8).

3. Unresolved when the solution path requires nodes or edges in \(M \setminus M_{\mathrm{e}}\), spans a gap between disconnected components \(M_{\mathrm{e},i}\) and \(M_{\mathrm{e},j}\) of \(M_{\mathrm{e}}\), or remains undecided because grounding, coverage, or active conflict \(\kappa(\mathfrak{C}_\kappa, W_t)>0\) leaves the query open despite retrieval and control operations from \(\mathcal{K}\).

Definition 63 (Task specification). A task specification is a tuple \[T = \langle R_T,\Gamma_T,\Omega_T,\Theta_T\rangle,\] where:

1. \(R_T \subseteq V_B\) is the specification-level task-relevant salience set, and \(R_t\) is its runtime instantiation (Definition 11);

2. \(\Gamma_T\) is a set of admissibility constraints;

3. \(\Omega_T\) is a set of candidate operations or action options;

4. \(\Theta_T\) is a set of task predicates, including success and failure conditions \(\Psi^{\mathrm{task}}_{\mathrm{succ}}\) and \(\Psi^{\mathrm{task}}_{\mathrm{fail}}\) (distinct from trajectory functional \(\Psi\), Definition 27) and from the global threshold bundle \(\Theta\) (Definition 24).

Definition 64 (Goal-designated subgraph). A motivational or task goal state is represented by a designated subgraph \[G_t=(V_{G_t},E_{G_t}) \subseteq B,\] whose nodes encode desired outcomes, maintained constraints, or preferred target states. This is not a new primitive object class: it is a designated belief subgraph whose task-layer runtime realization may appear inside the context graph \(C_t\).

Definition 65 (Goal satisfaction). Let \(G_t=(V_{G_t},E_{G_t}) \subseteq B\) be a designated goal subgraph. Given a realization predicate \(\mathrm{realized}(g,W_t)\) on goal nodes \(g \in V_{G_t}\), the goal-satisfaction measure* is the involvement-style quantity \[S_{\mathrm{goal}}(G_t,W_t) = \frac{|\{g \in V_{G_t} : \mathrm{realized}(g,W_t)\}|}{|V_{G_t}|}.\] When realization is identified with working-memory activation or explicit attainment inside a context graph, this is an application-specific specialization of Definition 20.*

Definition 66 (Failure state). A working-memory state \(W_t\) is a failure state* relative to task \(T\) iff \[\mathrm{Fail}^{\mathrm{task}}(W_t,T) \iff \neg \Psi^{\mathrm{task}}_{\mathrm{succ}}(W_t,T)\wedge \Psi^{\mathrm{task}}_{\mathrm{fail}}(W_t,T).\] where \(\mathrm{Fail}^{\mathrm{task}}\) is distinct from the fragmentation index \(F(W_t)\) (Definition 25).*

Definition 67 (Chunking operator). A chunking operator* is a two-stage operator family consisting of a working-memory compression step \(U_\chi\) and, when committed, a belief-update step \(\Delta_\chi\). The first stage compresses a set of co-active nodes in \(W_t\) into a single new recursive node; the second stage stores the resulting chunk in \(B\) when update conditions of the architecture are satisfied. Let \(\{v_1, \dots, v_k\} \subseteq V_{W_t}\) be co-active nodes—including perceptual input nodes—compressed into \[v^\ast = (c^\ast, G^\ast),\] where \(G^\ast\) is a graph payload encoding the internal structure of \(\{v_1, \dots, v_k\}\) and their mutual edges. The WM compression stage is \[U_\chi(W_t, B) = W_{t+1},\] where \(V_{W_{t+1}} = (V_{W_t} \setminus \{v_1, \dots, v_k\}) \cup \{v^\ast\}\), edges incident to any \(v_i\) are redirected to \(v^\ast\), and \(\alpha_{t+1}^{W}(v^\ast) = \sum_{i=1}^{k} \alpha_t^{W}(v_i)\). The operator reduces \(|V_{W_t}|\) by \(k-1\), freeing capacity. A node \(v^\ast\) formed this way is called a chunk; if a matching chunk already exists in \(B\), the operation is a recognition event rather than a novel chunk-creation event (Definition 52). When update conditions are satisfied, \(\Delta_\chi\) commits the chunk structure to \(B\).*