1 Introduction↩︎

We describe a system that is inspired by the cybernetic decision theory of G. E. Pugh [1], as well as the concept of reentry [2].

This system consists of modules that correspond to perception, valuation, memory, control, and action. These modules are linked by the continuous exchange of latent representations. They can change their own state and share selected hidden content. A module also receives translated content from other components and updates its future dynamics. This repeated exchange leads to the emergence of stable patterns. These patters can influence what is noticed and avoided, as well as what is remembered and forgotten.

The full state vector takes the form \[Z(t) = (H_L, X_R, \Theta, Y, P, Q_L, W_R, M, A, \mathcal{W}, \mathcal{S}, \mathcal{U})(t),\]

where the symbols stand for symbolic state (\(H_L\)), geometric state (\(X_R\)), and controller state (\(\Theta\)). In addition, we have the valuative state (\(Y\)), executive state (\(P\)), attention (\(Q_L\)), awareness (\(W_R\)) and memory (\(M\)). Additional elements are the external action (\(A\)), world model (\(\mathcal{W}\)), self-model (\(\mathcal{S}\)) and meta-control modules (\(\mathcal{U}\)), which are further specified in the appendix. The state vector used in the main body of the paper concerns more restricted principal subsystems and is specified in the main Stability section of the paper.

Each of the modules maintains a history buffer from which it reads delayed coupling signals. The interconnector contains the propagation delays \(\tau_{R \to L}\) and \(\tau_{L \to R}\). The resulting structure is that of a retarded functional differential equation (RFDE), which is a differential equation whose derivative of the state depends on the current state and on state values at fixed past times, see also [3]–[5].

The symbolic and geometric field have the following additional structure: \(H_L \in \mathbb{R}^{T \times d_L}\) is indexed by sequence positions \(\{1, \cdots, T\}\), and \(X_R\) is indexed by the nodes of a graph \(G_R\). The symbolic field is modeled on the graph \(G_L = (V_L, E_L)\), where \(V_L = \{1, \cdots, T\}\) are the token-position nodes and the edge weights \(Q_L \colon E_L \to \mathbb{R}_{> 0}\) are the precision weights. The geometric field similarly is modeled on a graph \(G_R = (V_R, E_R)\) with awareness weights \(W_R\). In the RFDE formulation, both fields diffuse via weighted Laplacians on finite graphs. They receive delayed coupling signals from each other, and contribute to the principal small-gain margin. The main equation consists of a combination of two graph Laplacians \(\Delta_{G_L}(Q_L)\) and \(\Delta_{G_R}(W_R)\) plus the bipartite coupling kernel \(\mathcal{K}\).

2 Reaction-Diffusion Model↩︎

The latent representations of the model are treated as fields over representational spaces which are the finite base graphs. The coarse graining that we describe in the appendix preserves the main control variables of the system.

Base graphs↩︎

We define a base space for each of the two primary fields. The idea is to view both fields in a unified manner, by having both live and evolve on finite graphs with possible additional structure that is specific to an architecture.

Sequence graph \(G_L\)↩︎

Let \(G_L = (V_L, E_L)\) be a finite connected graph with node set \(V_L = \{1, \cdots, T\}\) (token positions) and edge set \(E_L\) containing sequential edges \((s, s+1)\) and \((s+1, s)\) for adjacent positions, together with skip edges that encode positional structure. Each edge carries a positional encoding label \(p(s, s') \in \mathbb{R}^{d_{\mathrm{pos}}}\) which is an architecture constant.

Scene graph \(G_R\)↩︎

Let \(G_R = (V_R, E_R)\) be a finite connected graph with node set \(V_R\) (encoding geometric entities) and edge set \(E_R\). We endow each edge \((i, j)\) with a fixed rigid-frame label \(g_{ij} = (R_{ij}, t_{ij}) \in SE(d)\) which is viewed as an architecture constant. These edge labels encode the geometric relationships among nodes.

Primary fields↩︎

Symbolic field↩︎

\(H_L \colon V_L \to \mathbb{R}^{d_L}\) is a section of the trivial vertex bundle over \(G_L\), equivalently \(H_L \in \mathbb{R}^{T \times d_L}\). The precision weights \(Q_L \colon E_L \to \mathbb{R}_{> 0}\) define the weighted graph Laplacian \[\bigl(\Delta_{G_L}(Q_L)\,H_L\bigr)_s =\sum_{s'\colon(s,s')\in E_L}Q_L(s,s')(h_s - h_{s'}),\] which acts as precision-weighted diffusion on \(H_L\). In the formal symmetric regime, connectivity of \(G_L\) and \(Q_L\geq\varepsilon_Q>0\) give the spectral gap \(\lambda_2(\Delta_{G_L}(Q_L))\geq\varepsilon_Q\lambda_2(G_L)>0\).

Geometric field.↩︎

\(X_R\colon V\to\mathbb{R}^{d_R}\) is a section of the trivial vertex bundle over \(G_R\), equivalently \(X_R\in\mathbb{R}^{|V|\times d_R}\). The rigid-frame labels \(g_{ij}\) provide scalar invariant edge features used by the reaction term, such as \(\|t_{ij}\|^2\), traces of rotation features, and prescribed inner products. Because the node embeddings \(e_i\) do not transform under \(\mathrm{SE}(d)\), the geometric feature field is \(\mathrm{SE}(d)\)-invariant. The awareness weights \(W_R\in\prod_j\Delta^{k_j}\) define the weighted graph Laplacian \[\bigl(\Delta_{G_R}(W_R)\,X_R\bigr)_j =\sum_{i\colon(i,j)\in E}W_{R,ij}(e_j - e_i),\] For the formal dissipativity estimates, the associated symmetric conductance field is \(\overline{W}_{ij}=\tfrac12(W_{R,ij}+W_{R,ji})\). The spectral gap assumption is imposed on the Laplacian defined by \(\overline{W}\): if \(\overline{W}_{ij}\geq\varepsilon_W\) on a connected support, then \(\lambda_2(\Delta_{G_R}(\overline{W}))\geq\varepsilon_W\lambda_2(G_R)>0\).

Coupling kernel↩︎

A coupling kernel \(\mathcal{K}\colon V_L\times V_R\to\mathbb{R}^{d_L\times d_R}\) induces two Hilbert-Schmidt operators: \[\begin{align} \mathcal{K}&\;\colon\; L^2(G_R,\mathbb{R}^{d_R})\to L^2(G_L,\mathbb{R}^{d_L}), &(\mathcal{K}X_R)_\ell&=\sum_{i\in V}\mathcal{K}(\ell,i)\,e_i,\\ \mathcal{K}^* &\;\colon\; L^2(G_L,\mathbb{R}^{d_L})\to L^2(G_R,\mathbb{R}^{d_R}), &(\mathcal{K}^*H_L)_i&=\sum_{\ell\in V_L}\mathcal{K}(\ell,i)^\top h_\ell, \end{align}\] with HS norm \(\|\mathcal{K}\|_{\mathrm{HS}}^2=\sum_{\ell,i}\|\mathcal{K}(\ell,i)\|_F^2\leq C_\mathcal{K}^2\). In practice the coupling uses state-dependent attention weights \(\alpha_{\ell i}(Z)\in \Delta^{|V|-1}\) and \(\beta_{i\ell}(Z)\in\Delta^{T-1}\): \[\begin{align} C_{R\to L,\ell}(t) &= \sum_{i\in V}\alpha_{\ell i}(Z(t))\,\mathcal{K}(\ell,i)\,e_i(t-\tau_{R\to L}),\\ C_{L\to R,i}(t) &= \sum_{\ell\in V_L}\beta_{i\ell}(Z(t))\,\mathcal{K}(\ell,i)^\top h_\ell(t-\tau_{L\to R}). \end{align}\] Since \(\alpha_{\ell i}\in[0,1]\), the state-dependent operators satisfy \(\|\mathcal{K}_\alpha(Z)\|_{\mathrm{HS}}\leq C_\mathcal{K}\) uniformly in \(Z\), and likewise for \(\mathcal{K}^*_\beta(Z)\). The Lipschitz bounds \[\|\mathcal{K}_\alpha(Z)-\mathcal{K}_\alpha(Z')\|_{\mathrm{op}}\leq L_\alpha\|Z-Z'\|, \qquad \|\mathcal{K}_\beta^*(Z)-\mathcal{K}_\beta^*(Z')\|_{\mathrm{op}}\leq L_\beta\|Z-Z'\| .\] are needed for the well-posedness and attractor compactness. The global stability theorem below is stated for the closed principal regime in which the interfield operators are fixed bounded Hilbert-Schmidt operators \(\mathcal{K}\) and \(\mathcal{K}^*\) with norm at most \(C_\mathcal{K}\).

Master equation↩︎

The full state is \(Z=(H_L,X_R,Q_L,W_R,\mathcal{R}_\Theta,Y,P,M, \{\rho_i\},\{z_i\},\{\theta_i\})\in\mathcal{Z}\). Let \(\mathcal{E}\) be the finite-dimensional Euclidean product space containing all components of \(Z\), and let \(\mathcal{Z}\subset\mathcal{E}\) denote the compact admissible state set. The ambient history space and the admissible phase space are \[\mathcal{X}_{\mathcal{E}}:=C([-\tau_{\max},0],\mathcal{E}), \qquad \mathcal{X}:=C([-\tau_{\max},0],\mathcal{Z})\subset \mathcal{X}_{\mathcal{E}},\] both equipped with the sup norm on histories. The ambient space \(\mathcal{X}_{\mathcal{E}}\) is a Banach space, while \(\mathcal{X}\) is a closed complete metric subspace. We set \(\tau_{\max}=\max(\tau_{R\to L},\tau_{L\to R})\). We consider the RFDE \[\label{eq:RFDE} \dot{Z}(t)=\mathcal{D}(Z(t))+\mathcal{N}(Z(t),u(t))+\mathcal{C}(Z(t),Z(t-\tau_1),Z(t-\tau_2)),\tag{1}\] with \(\tau_1=\tau_{R\to L}\), \(\tau_2=\tau_{L\to R}\). The equation is composed of the following blocks.

Diffusion block \(\mathcal{D}(Z)\): \[\mathcal{D}(Z)=\bigl(-\Delta_{G_L}(Q_L)\,H_L,\;-\Delta_{G_R}(W_R)\,X_R,\;-\kappa_YY,\;0,\ldots\bigr).\]

Reaction block \(\mathcal{N}(Z,u)\): the instantaneous nonlinear updates \(F_L,F_R,G_Y\) and the auxiliary field updates \(\Psi_{Q_L},\Psi_{W_R},\Psi_{\mathcal{R}}, \mathcal{P},\Phi_M\), and the learning dynamics \(-\lambda_z z_i+\nabla_{\theta_i}\log\pi_i\), \(-\lambda_{\mathrm{reg}}\theta_i+\eta_i\delta z_i\). All components are evaluated at the current time \(t\). The continuous-time reaction terms are defined as residuals of the transformer maps: \(F_L(H_L,\ldots):=\mathcal{T}_L(H_L,\ldots)-H_L\) for the symbolic field, and \(F_R(X_R,\ldots):=\mathcal{G}_R(X_R,\ldots)-X_R\) for the geometric field. The remaining auxiliary fields are assigned continuous-time vector fields \[\Psi_{Q_L},\quad \Psi_{W_R},\quad \Psi_{\mathcal{R}},\quad \Phi_M, \quad \Psi_{\rho_i},\quad \Psi_{z_i},\quad \Psi_{\theta_i},\] which are globally Lipschitz on \(\mathcal{Z}\) and point into the tangent cone of the relevant compact domain. The displayed stagewise rules are explicit Euler or operator-splitting discretizations of these vector fields.

Delayed coupling block \(\mathcal{C}\): the only terms depending on past state, \[\mathcal{C}(Z(t),Z(t-\tau_1),Z(t-\tau_2)) =\bigl(\mathcal{K}_\alpha(Z(t))\,X_R(t-\tau_1),\;\;\mathcal{K}^*_\beta(Z(t))\,H_L(t-\tau_2),\;\;0,\ldots\bigr).\] The equation is retarded (as opposed to neutral), i.e. the right-hand side depends on \(Z(t+\theta)\) for \(\theta\leq 0\) as opposed to \(\dot{Z}(t+\theta)\) for \(\theta<0\). The initial data is given by a history segment \(Z_0\in\mathcal{X}\).

For constant input the RFDE defines a continuous semiflow on \(\mathcal{X}\). In the standard history representation the shift derivative acts on \([-\tau_{\max},0)\) and the RFDE boundary condition is imposed at zero, cf. [3], Chapter 2, and [5], Chapter I.

Dissipativity constants and stability condition↩︎

Let \(\mu_L,\mu_R,\mu_P>0\) denote the one-sided dissipativity constants of the principal instantaneous \(H_L\), \(X_R\), and \(P\) vector fields. The constants \(\mu_L\) and \(\mu_R\) include the corresponding diffusion and reaction contributions and exclude the delayed interfield coupling. We have fixed interfield operators whose operator norms are bounded by \(C_\mathcal{K}\) and we have the small-gain condition \[\label{eq:SC} C_\mathcal{K}^2 \;<\; \mu_L\,\mu_R\tag{2}\] which is the positivity condition for the Schur margin of the coupled principal fields. The executive constant \(\mu_P\) contributes separately to convergence of the \(P\) component.

Radial margin conditions↩︎

The stability condition 2 controls convergence near equilibrium. A separate condition controls viability at the compact boundary of the principal balls. To wit, at \(\|H_L\|_F=R_L\) the instantaneous reaction term must overcome the maximum outward contribution of the delayed coupling. A sufficient condition is \[\label{eq:radialmargin} \eta_L\ge R_L\,C_\mathcal{K}\,R_R,\qquad \eta_R\ge R_R\,C_\mathcal{K}\,R_L,\tag{3}\] where \(\eta_L,\eta_R>0\) are the radial damping margins of the reaction terms \(F_L\) and \(F_R\) at the respective ball boundaries (i.e.\(\langle H_L,-\Delta_{G_L}(Q_L)H_L +F_L\rangle\le-\eta_L\) when \(\|H_L\|_F=R_L\), and analogously for \(F_R\)). The delayed coupling contributes at most \(R_L C_\mathcal{K}R_R\) to the outward radial velocity, so 3 guarantees that the combined inward radial velocity is nonpositive on the boundary. When the principal fields are implemented as projected relaxation flows, the margin condition is satisfied automatically by the projection. In the case where they are implemented as explicit Laplacian-plus-reaction systems, the reaction term \(F_L\) must be sufficiently dissipative to enforce 3 .

Standing assumption (fixed dimensions and finite delay horizon)↩︎

Throughout this paper, the sequence length \(T\), the graph node count \(|V|\), the edge count \(|E|\), the subsystem count \(n_s\), and the delay scalars \(\tau_{R\to L}\), \(\tau_{L\to R}\geq 0\) are fixed finite architecture constants. Both graphs \(G_L\) and \(G_R\) are assume to be connected. The maximum delay is set as \(\tau_{\max}=\max(\tau_{R\to L},\tau_{L\to R})\). We note that architectures with dynamic token length, variable graph size, or online-adapted delays require a separate analysis.

Uniformized master equation↩︎

The two graph fields \(H_L\) on \(G_L\) and \(X_R\) on \(G_R\), together with the bipartite coupling kernel \(\mathcal{K}\), can be assembled into a single field on the bipartite graph \(G_{LR}=G_L\cup G_R\) whose bipartite edges are indexed by the support of \(\mathcal{K}\). The joint primary field \(Z_{\mathrm{field}}=(H_L,X_R)\) lives on \(G_{LR}\) and the auxiliary field \(Z_{\mathrm{aux}}=(Q_L,W_R,\mathcal{R}_\Theta,Y,P,M,\{\rho_i\},\{z_i\},\{\theta_i\})\) contains the remaining components. The RFDE 1 then has the block form \[\begin{align} \partial_t Z_{\mathrm{field}}(t) &= -\Delta_{G_{LR}}(w)\,Z_{\mathrm{field}}(t) +F(Z_{\mathrm{field}}(t),Z_{\mathrm{aux}}(t),u(t)) +\mathcal{C}_\tau[Z_{\mathrm{field},t}],\label{eq:uniformized}\\ \partial_t Z_{\mathrm{aux}}(t) &= B(Z_{\mathrm{field}}(t),Z_{\mathrm{aux}}(t),u(t)),\notag \end{align}\tag{4}\] where the block Laplacian \(\Delta_{G_{LR}}(w)\) has diagonal blocks \(\Delta_{G_L}(Q_L)\) and \(\Delta_{G_R}(W_R)\) and zero off-diagonal blocks and the coupling enters through \(\mathcal{C}_\tau\). The combined edge-weight tuple is \(w=(Q_L,W_R)\), and \(\mathcal{C}_\tau\) groups the delayed cross-coupling terms. This form shows that \(H_L\) and \(X_R\) are instances of a single object which is given by a dissipative graph field driven by a weighted Laplacian and a bipartite delayed coupling, see also Section 13. As \(T\to\infty\) and \(|V|\to\infty\), this bipartite-graph RFDE can be shown to approach a partial FDE on \(L^2([0,1],\mathbb{R}^{d_L})\times L^2(\Omega,\mathbb{R}^{d_R})\); see [6] for well-posedness of the continuum limit.

Role of the design specifications↩︎

Standard known transformer or graph-neural modules fail to fit the required hypotheses of the theory. We therefore include design specifications in the appendix which describe the design choices needed in order to have a stabilized architecture. One instance of such a specification involves the residual backbones which are needed for connectivity and mixing constants for the two primary (symbolic / geometric) fields. These have corresponding overlays that give us the task-dependent selectivity. The viability conditions stated in the main body of the paper are fulfilled by the use of projection onto Frobenius balls. These projections preserve the direction of an overlarge matrix-valued state and rescale its amplitude to the admissible radius. This procedure prevents unbounded amplification and yields a continuous self-map of the compact domain. It does however disregard the magnitude of the state. In practice, the radii are chosen so that projection is normally inactive or weakly active. As an alternative, hard projection may be replaced by smooth norm saturation. In the table below we summarize the other mechanisms that might be employed to turn a given neural network based architecture into a stabilized candidate that accords to the hypotheses.

3 Stability Criteria↩︎

This section assembles the specification conditions from the appendix into formal hypotheses which then yield the main results. On the other hand we also describe the discrete-time implementation which is recorded as a numerical discretization of the RFDE. The distinction between both viewpoints (fields vs. discrete time system) should be kept in mind.

Whenever a spectral gap, \(\lambda_2\), or graph-Laplacian dissipativity estimate is used, the relevant Laplacian is the symmetric conductance Laplacian. Thus either the graph weights are symmetric, or the estimate is applied to the symmetrized conductance field. For the geometric awareness field this means \[\overline{W}_{R,ij}:=\tfrac12(W_{R,ij}+W_{R,ji}),\] and the gap is imposed on \(\Delta_{G_R}(\overline{W}_R)\). The stability constants \(\mu_L,\mu_R\) are derived from symmetric or symmetrized dissipative operators, or else assumed directly as one-sided dissipativity constants.

Assumptions. The main results rely on the following assumptions.

1. Fixed finite architecture and finite delays. The sequence length \(T\), graph node count \(|V|\), edge count \(|E|\), subsystem count \(n_s\), and delay scalars \(\tau_{R\to L},\tau_{L\to R}\geq0\) are fixed finite constants.

2. Compact closed state domain. The state domain \(\mathcal{Z}\) is the compact product domain displayed below, with bounded Euclidean components, floored positive weights, and simplex-valued routing components.

3. Global Lipschitz RFDE vector field on histories. The RFDE right-hand side \(\mathcal{F}:\mathcal{X}\to\mathcal{E}\) is globally Lipschitz on the admissible history space \(\mathcal{X}=C([-\tau_{\max},0],\mathcal{Z})\) and admits a locally Lipschitz extension to an open neighbourhood of \(\mathcal{X}\) in the ambient Banach history space \(\mathcal{X}_{\mathcal{E}}=C([-\tau_{\max},0],\mathcal{E})\). Equivalently, after componentwise Whitney-McShane extension in the finite-dimensional target \(\mathcal{E}\), the Picard theorem is applied in \(\mathcal{X}_{\mathcal{E}}\) and the tangent-cone condition below selects the admissible subset \(\mathcal{X}\).

4. Positive invariance. The continuous-time vector field points into the tangent cone of \(\mathcal{Z}\) at boundary points.

5. Joint non-emptiness. The core class is nonempty by the minimal construction below. For richer architectures, we impose joint admissibility as a modular condition: each added module must preserve compactness, Lipschitz continuity, tangent-cone viability, and the relevant dissipativity or bounded-coupling estimates.

6. Equilibrium existence for constant input. For \(u\equiv u^*\) the RFDE has at least one equilibrium \(Z^*\in\mathcal{Z}\).

 

1. Closed principal stability regime. For Theorem 4, the delayed interfield operators are fixed bounded Hilbert-Schmidt operators \(\mathcal{K}\) and \(\mathcal{K}^*\) with norms at most \(C_\mathcal{K}\). The instantaneous principal dynamics are one-sided dissipative with constants \(\mu_L, \mu_R\), and \(\mu_P\).

Remark. a) Equilibrium existence. When the full RFDE is written in relaxation form \(\dot{Z}=\Gamma(Z_t)(T(Z_t)-Z)\) on compact convex \(\mathcal{Z}\), every fixed point \(Z^*=T(Z^*)\) of the stationary target map \(T:\mathcal{Z}\to\mathcal{Z}\) is an equilibrium. If \(T\) is continuous, Brouwer’s fixed-point theorem gives at least one such fixed point.

b) Viability condition. Condition (A4) is realistic when the continuous-time dynamics are implemented as projected or relaxation-type flows, or through softmax/logit parameterizations for simplex variables. We note that, unless additional damping, projection, or radius-margin conditions are imposed, the delayed coupling terms and unprojected policy or memory drifts can point outward at the boundary. The condition means that for every admissible history \(\varphi \in C([-\tau_{\mathrm{max}}, 0], \mathcal{Z})\), the RFDE vector field must satisfy \(\mathcal{F}(\varphi, u^{\ast}) \in T_{\mathcal{Z}}(\varphi(0))\), where the tangent cone is taken at the present state \(\varphi(0)\), and \(\mathcal{F}\) may depend on the delayed history. Componentwise, this means that ball-valued variables have nonpositive outward radial velocity at their boundary, box-valued variables point inward at their lower and upper faces, and simplex-valued variables preserve total mass while assigning nonnegative velocity to zero coordinates.

3.1 Formal closure↩︎

We verify that the discretized system according to the specifications is well-defined and closes in the right manner.

The formal map treats \(A^t\in\mathcal{A}\) as an exogenous input supplied by the environment or by a separate sampling procedure. Given \(({\tilde{Z}^t},u^t,A^t)\), the map \(\tilde{\mathcal{F}}\) is deterministic. The architecture contains action policy parameters \(\theta_{\rm act}\) and a policy \(\pi_{\rm act}\) governing the sampling of \(A^t\). The analysis of these parameters depends on the externally supplied action. In that sense the agent is an open system, because perceptual inputs \(u^t\) arrive from outside and actions \(A^t\) exit to perturb the environment. A fully closed agent-environment loop lies outside the scope of the present analysis.

The full state is \[Z = \bigl(H_L,\;X_R,\;Q_L,\;W_R,\;\mathcal{R}_\Theta,\;Y,\; P,\;M,\;\{\rho_i\},\;\{z_i\},\;\{\theta_i\}\bigr)\in\mathcal{Z}.\]

Three classes of auxiliary variable require explicit classification. State-augmented variables: the reliability variables \(\rho_i\), eligibility traces \(z_i\), and policy parameters \(\theta_i\) are dynamic state variables with their own update rules and bounds and are included in \(Z\) above. Stagewise-derived scalars: the homeostatic deviation \(h\), prediction error \(\varepsilon_{\mathrm{pred}}\), novelty signal \(n\), and outcome feedback \(r\) are functions of \((u^t,A^t,Z^t)\) computed at each step. Delay scalars and history buffer: the delay scalars \(\tau_{R\to L}\) and \(\tau_{L\to R}\) are fixed architecture constants; the history buffer is the finite discretized tail used by the implementation. Controller broadcast: the broadcast \(B_\Theta^t\) is computed from \(\mathcal{R}_\Theta^t\in Z^t\) and subsystem export vectors at the opening of the update step. Memory gate: the gate \(g_M\) is computed as a deterministic Lipschitz function of \(Z^t\) at Stage 8.5.

The variable types are summarized in the following table.

The state domain \(\mathcal{Z}\) is defined by the explicit constraints: \[H_L\in B_{R_L},\quad e_i\in B_{R_R},\quad Q_L\in[\varepsilon_Q,R_Q]^{T\times T}, \quad W_R\in\textstyle\prod_j\Delta^{k_j},\quad \mathcal{R}_\Theta\in\textstyle\prod_{i=1}^{n_s}\Delta^{n_s-1},\] \[Y\in B_{R_Y},\quad P\in B_{R_P},\quad M\in B_{R_M},\quad \rho_i\in[0,1],\quad z_i\in B_{R_{z,i}},\quad \theta_i\in B_{R_{\theta,i}}.\] The constants \(\varepsilon_Q>0\), \(R_Q\), \(R_L\), \(R_R\), \(R_Y\), \(R_P\), \(R_M\), \(R_{z,i}\), and \(R_{\theta,i}\) are finite parameters of the architecture. The radius \(R_M\) satisfies \(R_M\geq\max(\|M^0\|,C_M)\).

The continuous-time vector field assigns to every component of \(Z\) a Lipschitz vector field on this domain: \[\begin{array}{lll} H_L& -\Delta_{G_L}(Q_L)H_L+F_L(Z,u)+\mathcal{K}_\alpha(Z)X_R(t-\tau_{R\to L}) & \text{principal field},\\ X_R& -\Delta_{G_R}(W_R)X_R+F_R(Z,u)+\mathcal{K}_\beta^*(Z)H_L(t-\tau_{L\to R}) & \text{principal field},\\ Q_L & \Psi_{Q_L}(Z) & \text{compact positive box},\\ W_R & \Psi_{W_R}(Z) & \text{product of simplices},\\ \mathcal{R}_\Theta& \Psi_{\mathcal{R}}(Z) & \text{routing simplex product},\\ Y& -\kappa_YY+G_Y(H_L,X_R,u) & \text{valuative field},\\ P& \mathcal{P}(Z) & \text{executive field},\\ M& \Phi_M(Z) & \text{memory field},\\ \rho_i & \Psi_{\rho_i}(Z) & \text{reliability variable},\\ z_i & \Psi_{z_i}(Z) & \text{eligibility trace},\\ \theta_i & \Psi_{\theta_i}(Z) & \text{projected smooth policy drift}. \end{array}\] Each auxiliary vector field is globally Lipschitz on \(\mathcal{Z}\) and satisfies the tangent cone condition in (A4). The stagewise algorithm that follows below is the discretization of the continuous-time equation.

Each theorem below appeals to the conditions of Specifications 1–8 and the admissibility class \(\mathfrak{C}_P\).

Remark (Discrete-time implementation). The following stages define an explicit Euler or operator-splitting implementation of the continuous-time RFDE 1 . Define the augmented state \(\tilde{Z}^t:=(Z^t,Z^{t-1},\ldots,Z^{t-K})\) and the shift map \(\tilde{\mathcal{F}}(\tilde{Z}^t,u^t,A^t):=(Z^{t+1},Z^t,\ldots,Z^{t-K+1})\). Under (A5), \(Z^{t+1}\) is uniquely and acyclically determined; \(\tilde{\mathcal{F}}\) is a well-defined deterministic map on \(\tilde{\mathcal{Z}}:=\mathcal{Z}^{K+1}\).

The update evaluates the components of \(Z^{t+1}\) in eight stages. At each stage the inputs are either elements of \(Z^t\), elements of \((u^t,A^t)\), or outputs of earlier stages within the same step.

Stage 1 (Neuromodulatory signals). Compute \(\mu=\mathcal{V}_\mu(Y^t)\in(0,1)^5\) from \(Y^t\) using the neuromodulatory readout component of \(\mathcal{V}\). Well-definedness follows from Specification 5(ii)-(iii). The single input \(Y^t\) is an element of \(Z^t\).

Stage 2 (Precision and awareness fields). The base logit field \(a_L\in\mathbb{R}^{T\times T}\) and the modulation weight \(b_L\in\mathbb{R}^{T\times T}\) (with \(b_L\geq 0\)) are fixed architecture constants. Define \(q_L^t:=a_L+\mu_{\mathrm{ACh}}^t\,b_L\) as the modulated logit field at step \(t\) which furnishes a deterministic function of the architecture constants and \(\mu_{\mathrm{ACh}}^t\) from Stage 1. Compute \(Q_L^{t+1}=\varepsilon_Q+(R_Q-\varepsilon_Q)\,\sigma(q_L^t)\) from Stage 1 and Specification 1(v)-(viii). Similarly, let \(\omega^t:=\omega(Z^t)\) be the awareness logit field which is a deterministic function of \(Z^t\). Compute \(W_R^{t+1}\) from \(\omega^t\) and \(\mu_{\mathrm{NE}}^t\) (Stage 1) using Specification 2(v)-(vii). Both are well-defined by Lipschitz continuity of the respective maps.

Stage 3 (Interconnector signals). Let \(n=\lfloor\tau/\Delta t\rfloor\geq 1\) be the integer delay index from the delay discretization and define \(\operatorname{Delay}_\tau(X_R^t,\ldots,X_R^{t-n}):=X_R^{t-n}\) (or the weighted combination). Compute \(C_{R\to L}=g_{R\to L}^t\,\Phi_{R\to L}(\operatorname{Delay}_\tau(X_R^t,\ldots,X_R^{t-n}))\) from the history buffer and \(\mathcal{R}_\Theta^t\), and \(C_{L\to R}=g_{L\to R}^t\,\Phi_{L\to R}(\operatorname{Delay}_\tau(H_L^t,\ldots,H_L^{t-n}))\) from the history buffer and \(\mathcal{R}_\Theta^t\). Gate values are entries of \(\mathcal{R}_\Theta^t\in Z^t\). Boundedness follows from Specification 3(i)-(iii).

Stage 4 (Symbolic state). Compute \(H_L^{t+1}=\mathcal{T}_L(H_L^t, B_\Theta^t,C_{R\to L},Q_L^{t+1},P^t,M^t)\) using Stages 2-3 and \(Z^t\). Well-definedness follows from Specification 1(i).

Stage 5 (Geometric state). Compute \(X_R^{t+1}=\mathcal{G}_R(X_R^t, B_\Theta^t,C_{L\to R},W_R^{t+1},Y^t,M^t)\) using Stages 2-3 and \(Z^t\). Well-definedness follows from Specification 2(i).

Stage 6 (Valuative state). Compute \(Y^{t+1}=\mathcal{V}_Y(Y^t, h^t,\varepsilon_{\mathrm{pred}}^t,n^t,r^t,H_L^{t+1},X_R^{t+1},M^t,P^t)\) using the outputs of Stages 4-5, where \(h^t,\varepsilon_{\mathrm{pred}}^t,n^t,r^t\) are stagewise-derived scalars computed from \((u^t,A^t,Z^t)\). Well-definedness follows from Specification 5(iii)-(iv).

Stage 7 (Routing matrix). Compute \(\mathcal{R}_\Theta^{t+1}=K(H_L^{t+1},X_R^{t+1},Y^{t+1},\rho^t,Z^t_{\mathrm{rest}})\) where \(Z^t_{\mathrm{rest}}\) denotes the remaining components of \(Z^t\) not yet updated. The domain of \(K\) at this stage is the partial-update product space \(B_{R_L}\times\prod_i B_{R_R}\times B_{R_Y}\times[0,1]^{n_s}\times\mathcal{Z}_{\mathrm{rest}}\); this is consistent with Specification 4(iii) provided that specification is interpreted as a condition on the routing function’s inputs at Stage 7. Reliability scores \(\rho^t\in Z^t\) require no further computation at this stage. Well-definedness follows from Specification 4(iii).

Stage 8 (Reliability, parameters, executive, memory). The sub-order within Stage 8 is fixed as follows; each sub-step relies solely on quantities already available:

1. Compute \(\delta\) from \(Y^{t+1}\) (Stage 6) via Specification 6(iii); \(\delta\in[-D,D]\).

2. Compute \(\rho_i^{t+1}=(1-\alpha)\rho_i^t+\alpha f(\varepsilon_i^t)\) using Specification 7(i)-(iii); well-defined since \(\rho_i^t\in Z^t\) and \(\varepsilon_i^t\) is a stagewise-derived scalar available at the start of the step.

3. Compute \(z_i^{t+1}=\lambda_i z_i^t+\nabla_{\theta_i}\log\pi_i^\varepsilon\) and \(\Delta\theta_i^t=\eta_i(\delta z_i^t-\lambda_{\mathrm{reg}}\theta_i^t)\); then project \(\theta_i^{t+1}\leftarrow\Pi_{B_{R_{\theta,i}}}(\theta_i^t+\Delta\theta_i^t)\). Uses \(z_i^t\) (as opposed to the just-computed \(z_i^{t+1}\)); one-step-delayed REINFORCE convention.

4. Compute \(P^{t+1}=\mathcal{P}(P^t,\{\Delta\theta_i^t\},Y^{t+1})\) using Specification 9(i)-(iii). Uses \(\Delta\theta_i^t\) from sub-step 8.3 and \(Y^{t+1}\) from Stage 6; does not use \(M^{t+1}\).

5. Compute \(M^{t+1}=(1-g_M)M^t+g_M\Phi_M(H_L^{t+1}, Y^{t+1})\) using Specification 8(i)-(iii). Uses \(H_L^{t+1}\) (Stage 4) and \(Y^{t+1}\) (Stage 6); does not use \(P^{t+1}\) or \(z_i^{t+1}\).

At each stage and sub-stage the required inputs are in \(\tilde{Z}^t\), \((u^t,A^t)\), or outputs of earlier stages. No component of \(Z^{t+1}\) appears as an input to its own computation. The within-step dependency graph is acyclic.

Remark (Covariant closure). The RFDE establishes that \(Z(t)\) evolves continuously from \(Z_0\in\mathcal{X}\). A stronger condition, covariant closure, would require the dynamics to be consistent under the symmetries of the physical environment. Specifically, if \(\phi\in\mathrm{Aut}(G)\times\mathrm{SE}(d)\) is applied to the physical scene, the field trajectories of the transformed and untransformed systems should be related by \(\phi\) throughout. Since the rigid-frame transforms \(g_{ij}\) are fixed structural labels and the node embeddings \(e_i\) are scalar (\(\mathrm{SE}(d)\)-invariant) features, the relevant symmetry property is permutation-equivariance under \(\mathrm{Aut}(G)\) combined with \(\mathrm{SE}(d)\)-invariance in the feature sector. A stronger condition, full covariant closure, would additionally require the interconnector to intertwine the symmetry action. By Specification 2(ii), the geometric update \(\mathcal{G}_R\) satisfies: \(\mathcal{G}_R(\sigma\cdot\{e_i\},\{g_{\sigma(i)\sigma(j)}\})=\sigma\cdot\mathcal{G}_R(\{e_i\},\{g_{ij}\})\) for any \(\sigma\in\mathrm{Aut}(G)\), and the outputs are invariant to \(\mathrm{SE}(d)\) relabelings of \(g_{ij}\), cf. [7]. Covariant closure of the full system would additionally require that the interconnector intertwines this action with a corresponding representation \(\rho_L\) of \(\mathrm{Aut}(G)\times\mathrm{SE}(d)\) on the symbolic state space, i.e. \(\Phi_{R\to L}(\phi\cdot\{e_i\})=\rho_L(\phi)\cdot\Phi_{R\to L}(\{e_i\})\), and that the symbolic update operator \(\mathcal{T}_L\) commutes with \(\rho_L\). If this requirement is not fulfilled, the geometric module is equivariant in isolation but the equivariance is not transmitted through the interconnector, and the agent’s symbolic dynamics may represent different tokens for physically equivalent scenes in different frames. Whether a consolidated architecture satisfying all of these conditions simultaneously exists is an open question. The conditions above identify what full covariant closure requires and are left for further development.

Standing joint-satisfiability assumptions (B5). We record the following cross-specification requirements.

1. NE-modulation vs.sparsity. \(\mu_{\mathrm{NE}}\) modulation of the awareness logit must not push all in-degree edges into the active support, or the sparsity bound of Specification 2(vii) is violated.

2. Layer-normalization variance. The Lipschitz constant bound for layer normalization in the Specification 2 existence argument requires a positive lower bound on the pre-normalization variance that is not stated as a specification condition.

3. Interconnector gate gradient. The gate values \(g_{R\to L},g_{L\to R}\in[0,1]\) must have non-vanishing gradients for training; no smooth function is simultaneously bounded and gradient-nonvanishing everywhere.

4. SE\((d)\)-invariant routing logits. The salience logits of Specification 4 must depend on \(Z\) only through \(\mathrm{SE}(d)\)-invariant functions of \(X_R\) for routing to be consistent with the symmetry condition of Specification 2(ii).

5. Existence arguments non-conflicting. The individual specification existence arguments must be simultaneously satisfiable; this is the joint non-emptiness assumption (A5).

The coupling kernel formulation of Specification 3 provides a natural framework in which several of these conditions can be recast at the level of the interaction vertex; this is developed further in the final section.

The following table summarizes the dependency structure.

3.2 Well-posedness and stability↩︎

Remark. i) Under the hypotheses of Theorem 4, allow \(\mathcal{K}_\alpha(Z)\), \(\mathcal{K}_\beta^*(Z)\), \(\Delta_{G_L}(Q_L)\), and \(\Delta_{G_R}(W_R)\) to depend Lipschitz-continuously on \(Z\), with Lipschitz constants \(L_\alpha\), \(L_\beta\), \(L_Q\), and \(L_W\), respectively. Centering at an equilibrium \(Z^*\) produces, in addition to the fixed-coupling delayed terms, the instantaneous perturbations \[\begin{align} & \bigl(\mathcal{K}_\alpha(Z)-\mathcal{K}_\alpha(Z^*)\bigr)X_R^*,\qquad \bigl(\mathcal{K}_\beta^*(Z)-\mathcal{K}_\beta^*(Z^*)\bigr)H_L^*,\\ & \bigl(\Delta_{G_L}(Q_L)-\Delta_{G_L}(Q_L^*)\bigr)H_L^*,\qquad \bigl(\Delta_{G_R}(W_R)-\Delta_{G_R}(W_R^*)\bigr)X_R^* . \end{align}\] Consequently, the centered \(\tilde{H}_L\) equation receives an instantaneous perturbation bounded by \[m_L\|\tilde{Z}\|, \qquad m_L:=L_\alpha\|X_R^*\|+L_Q\|H_L^*\|,\] and the centered \(\tilde{X}_R\) equation receives an instantaneous perturbation bounded by \[m_R\|\tilde{Z}\|, \qquad m_R:=L_\beta\|H_L^*\|+L_W\|X_R^*\|.\] A global stability conclusion for the fully state-dependent adaptive system requires an auxiliary Lyapunov or input-to-state stability (ISS) estimate for the remaining components of \(\tilde{Z}\) and positivity of the resulting coupled dissipation matrix.

ii) The underlying assumptions of Theorem 4 entail zero cross-gain. That means the auxiliary variables, including \(Y\), policy deltas, routing, and memory, are frozen at equilibrium, so the centered executive block contributes only its intrinsic dissipativity \[\frac{d}{dt}\frac{1}{2}\|p\|^2 \le -\mu_P\|p\|^2 .\] If active executive forcing is retained, write \[\|R_P(h,x,y)\| \le c_{PH}\|h\|+c_{PX}\|x\|+c_{PY}\|y\|,\] where all \(c_{\cdot}\) are nonnegative gain bounds. With the valuative relaxation \[\dot{Y} = \kappa_Y\bigl(\Phi_Y(H_L,X_R,u)-Y\bigr),\] define \[R_\Phi(h,x) := \Phi_Y(H_L^*+h,X_R^*+x,u^*) - \Phi_Y(H_L^*,X_R^*,u^*),\] and assume \[\|R_\Phi(h,x)\| \le L_{\Phi H}\|h\|+L_{\Phi X}\|x\|.\] Thus valuation may react to symbolic and geometric perturbations, but only with bounded gain. Since the fast valuative coordinate tracks \(y\approx R_\Phi(h,x)\), the valuation-mediated gains propagated into the executive block are \[c_{PY}L_{\Phi H}, \qquad c_{PY}L_{\Phi X}.\] Consequently define \[c_{PH}^{\mathrm{eff}} := c_{PH}+c_{PY}L_{\Phi H}, \qquad c_{PX}^{\mathrm{eff}} := c_{PX}+c_{PY}L_{\Phi X}.\] After the delayed symbolic-geometric terms have been absorbed, the available margins are \[\omega_L := \frac{\mu_L}{2}-\frac{C_\mathcal{K}^2}{2\mu_R}, \qquad \omega_R := \frac{\mu_R}{2}-\frac{C_\mathcal{K}^2}{2\mu_L}.\] The executive cross-gain condition is then \[M_P^{\mathrm{eff}} := \begin{pmatrix} \omega_L & 0 & -\frac{1}{2} c_{PH}^{\mathrm{eff}}\\[2mm] 0 & \omega_R & -\frac{1}{2} c_{PX}^{\mathrm{eff}}\\[2mm] -\frac{1}{2} c_{PH}^{\mathrm{eff}} & -\frac{1}{2} c_{PX}^{\mathrm{eff}} & \mu_P \end{pmatrix} >0 .\]

For finite \(\kappa_Y\), this condition is understood together with the usual \(O(\kappa_Y^{-1})\) tracking-error allowance for the fast valuative relaxation.

By Specification 3(ii), \(C_{R\to L,\ell}=\sum_{i\in V}\alpha_{\ell i}(Z)\mathcal{K}(\ell,i)e_i\) with \(\mathcal{K}(\ell,i)\in\mathbb{R}^{d_L\times d_R}\) and \(e_i\in\mathbb{R}^{d_R}\), so \(C_{R\to L,\ell}\in\mathbb{R}^{d_L}\) and \(C_{R\to L}\in\mathbb{R}^{T\times d_L}\), matching the token dimension of \(H_L\). By Specification 3(iii), \(C_{L\to R,i}=\sum_{\ell=1}^T\beta_{i\ell}(Z)\mathcal{K}(\ell,i)^\top H_{L,\ell}\) with \(\mathcal{K}(\ell,i)^\top\in\mathbb{R}^{d_R\times d_L}\) and \(H_{L,\ell}\in\mathbb{R}^{d_L}\), so \(C_{L\to R,i}\in\mathbb{R}^{d_R}\) and \(C_{L\to R}\in\mathbb{R}^{|V|\times d_R}\), matching the node embedding space. The routing compatibility score used in Specification 4 is a scalar for \(Z_i,Z_j\in\mathbb{R}^{d_Z}\).

Remark. i) Admissible auxiliary modules may be appended by product extension. If an auxiliary state \(A\in K_A\) evolves according to a globally Lipschitz tangent-cone-compatible vector field and enters the core equations through bounded Lipschitz maps with gains within the small-gain budget, then the enlarged system remains admissible.

ii) The \(W^{1,\infty}\) hypothesis of Theorem 5 is automatic in the smooth subregime when \(G_Y\) has bounded first derivative on the compact state domain and the RFDE vector field is uniformly bounded. In the case of Lipschitz or nonsmooth architectural choices this is viewed as a regularity assumption along the considered trajectory.

4 Extensions and Interpretations↩︎

Continuum limit. A continuum limit would require a specified graph-convergence regime as \(T\to\infty\) and \(|V|\to\infty\), a scaling under which \(\Delta_{G_L}(Q_L)\) and \(\Delta_{G_R}(W_R)\) converge to limiting differential or integral operators, convergence of the reentrant kernels \(\mathcal{K}\), and dissipativity estimates uniform in the finite approximants. Under such additional hypotheses one may seek a limiting PFDE on a space such as \[L^2([0,1],\mathbb{R}^{d_L})\times L^2(\Omega,\mathbb{R}^{d_R}),\] together with upper semicontinuity of attractors. See also [6].

Stochastic extension. A stochastic RFDE extension would require a noise model compatible with the constrained state domain \(\mathcal{Z}\). That requires for instance reflecting, projected, or tangent noise. Or alternatively, a reformulation on an unconstrained dissipative state space. In infinite-dimensional versions, the covariance of the noise must also be chosen so that the stochastic convolution is state-valued. The existence of invariant measures, small-noise concentration, and corrections to the Lyapunov-Krasovskii functional are additional problems to be resolved.

Specification compliance and constraints. The description in the appendix of this paper specifies the admissibility conditions an architecture must satisfy, and establishes consistency properties from those conditions alone. The central challenge is reentrant instability: when the output of a module feeds back into its own input through a chain of translations, delays, and gain modulations, the closed loop may amplify instead of dampen perturbations, particularly when the controller routing matrix places large weight on a single feedback path and the delay \(\tau\) is small relative to the characteristic relaxation time of the receiving module. Bounded-range conditions such as Specification 1(ii) and Specification 2(iv) constrain module outputs unconditionally, which provide one line of defense. However, the gain of the full closed-loop system is not directly constrained by any individual specification, and stability of the coupled field equations must be analyzed separately for each implementation. Discrete-time integration introduces numerical stiffness wherever diffusion coefficients are large relative to the integration step, requiring either implicit solvers or carefully tuned step sizes. Each specification class contains both smooth and non-smooth members. The non-smooth implementations within a class, such as piecewise-affine awareness projections or sparse routing operators with support boundaries, satisfy all specification conditions and the formal results but require additional care in gradient-based learning to avoid support instability.

Neurogeometry and the geometric module. The equivariant message-passing formulation in the Geometric Module section is a high-level abstraction over a specific cortical substrate. A potentially more anatomically faithful treatment of the dorsal visual processing stream is available within the framework of neurogeometry [8]. In this framework the primary visual cortex is modeled as a fiber bundle over the retinal surface. At each point in the retinotopic map the fiber encodes the local orientation preference. The bundle carries a contact structure which is inspired by the physiological observation that cortical neurons sensitive to the same orientation and adjacent retinal positions are connected preferentially through short-range lateral interactions. Cortical columns constitute the local fibers of this bundle. Here, a hypercolumn spans the complete cycle of orientation preferences at a given retinotopic location. The long-range horizontal connections between hypercolumns implement the parallel transport of orientation signals across the retinotopic plane. The rigid-frame transforms \(g_{ij}\in\mathrm{SE}(d)\) correspond to this fixed anatomical contact structure. The frame relation between two adjacent hypercolumns is determined by the cortical wiring and does not change from moment to moment, which is consistent with their treatment as fixed edge labels in the formal architecture. The sparse awareness field \(W_R\) then corresponds to the dynamic selectivity of long-range horizontal projections, i.e. which connections are currently active is a function of the ongoing computation. A full cortical interpretation of the architecture using the classical terms of retinotopy, orientation columns, hypercolumns, and long-range horizontal connectivity is plausible but is outside the scope of the present text [9], [10].

Tradeoff between valuative and intellectual complexity. [1] points out a systematic inverse relationship between the resolution of the value system and the effective scope of intellectual deliberation. A value organization with high dimensionality and fine differentiation, i.e.many drives, many homeostatic variables, many neuromodulatory channels, directs intellectual resources onto a narrow set of options and resolves behavioral ambiguity rapidly, at the cost of reduced flexibility. A sparsely specified value organization admits a wider range of deliberated options but introduces indeterminacy when competing courses of action generate similar valuation signals. In the present architecture this relationship is expressed by the coupling between the dimensionality and precision of the neuromodulatory vector \(\mu=(\mu_{\mathrm{DA}},\mu_{\mathrm{ACh}},\mu_{\mathrm{NE}},\mu_{5HT},\mu_{\mathrm{OP}})\) and the effective support of the routing matrix \(\mathcal{R}_\Theta\). This entails that a finer-grained valuative signal modulates salience scores \(s_i\) with higher specificity, concentrating routing mass on fewer target subsystems and leaving less of the symbolic and geometric processing capacity open to unconstrained deliberation. The tradeoff is therefore a structural property of any closed system in which valuation and deliberation share the same routing infrastructure.

Metalearning and the evolutionary timescale. The metacognitive layer described in the Metacognition section operates on the timescale of individual learning, adjusting routing parameters, reliability estimates, and norm weights in response to within-lifetime prediction errors. A complete account of the value system would include a third temporal tier in which the architecture of the neuromodulatory pathways, the functional form of the homeostatic drive \(f_h\), the prior distribution over world models, and the initial norm weights \(w_k^{(0)}\) are themselves shaped by selection pressure across evolutionary time. This is the biological origin of primary values [1]: the drives and affective responses that present themselves as given within an individual lifetime are the product of a long optimization over reproductive fitness across generations. In a synthetic system this corresponds to a hyperprior over the parametric structure of the valuative module, with the structural parameters of Specifications 5-7 determined by a process analogous to evolutionary search.

Latent World Model. It is instructive to take into consideration other natural choices for architectures that fit as the latent world-model mechanism. As a particular instance, the JEPA is admissible in the current framework when its encoders and predictor are treated as bounded Lipschitz latent maps on compact finite-dimensional state spaces, and its prediction error is routed into the reliability, valuative, memory, or interconnector dynamics. The JEPA prediction error becomes one of the system’s reliability errors \(\epsilon_i\), feeding the metacognitive reliability update, valuative modulation, and possibly attention/awareness precision fields.

Effective field theory interpretation and future directions. We describe an analogy with effective field theory. The symbolic field \(H_L\) and the geometric field \(X_R\) are two dynamical systems defined on different base spaces: sequence positions \(\{1,\ldots,T\}\) and graph nodes \(V\) respectively. In isolation each evolves according to its own field equation (Specifications 1 and 2). By analogy, these play the role of the free-field terms of the effective action. The coupling kernel \(\mathcal{K}\) of Specification 3 plays the role of the interaction vertex: it assigns to each (sequence position, graph node) pair a linear map through which the two fields exchange information. The Hilbert-Schmidt norm bound \(\|\mathcal{K}\|_{\mathrm{HS}}\leq C_\mathcal{K}\) plays a role loosely reminiscent of a finiteness condition: it means that the total coupling strength is finite regardless of the number of interacting modes. This should be viewed as a finite-dimensional norm bound, as opposed to a renormalizability result in the technical sense of quantum field theory. The coupling bottleneck \(d_R\) plays a role loosely reminiscent of a scale cutoff, i.e. all information exchanged between the two fields is compressed to dimension \(d_R\). The remaining specifications correspond to modulating fields that adjust effective coupling strengths and field parameters in response to the system’s state. The Lipschitz conditions bound the rate at which fields and couplings may vary. These conditions thereby play a role analogous to regularity conditions. The symmetry condition of Specification 2(ii) is a kind of symmetry consistency requirement. The inter-specification consistency conditions identified in the Formal Closure section ensure that the effective description of each module is compatible with the adjacent modules. A natural direction for further development is the minimal coupling. By that we mean the determination of the smallest kernel \(\mathcal{K}\) (in the sense of smallest \(C_\mathcal{K}\)) that reproduces the cognitive phenomena attributed to the architecture. This is the analogy for the minimal coupling principle in gauge theories, and it points toward a more axiomatic reformulation in which the field equations are derived from symmetry and minimality conditions rather than assembled from independently motivated modules. Furthermore, the most promising extensions are to replace the current compact-Lipschitz-dissipative admissibility regime with structural stability principles that use the architecture’s own mechanisms. Concretely, the aim should be to prove stability for the full adaptive system with state-dependent attention, routing, valuation, memory, reliability, and policy dynamics. Then the strong residual damping should be replaced by incremental passivity or compositional small-gain conditions. Another aim is to show that prediction-error-driven reliability or valuation can self-regulate delayed reentrant coupling and enforce stability margins. One should analyze delay-dependent regimes where reentry produces oscillations, synchronization, or Hopf bifurcations. Moreover, develop continuum and large-graph limits with convergence of attractors and incorporate online learning as well as stochasticity through parameter dynamics or stochastic RFDEs. Finally, one should remove artificial compact projections by proving absorbing-set and attractor results under coercive dissipativity.

5 Symbolic Module↩︎

The symbolic module should fulfill the following requirements. It must maintain a dense sequential workspace adequate for linguistic, rule-based, counterfactual, and justificatory representations. At the same time, it must remain open to grounding constraints arriving from non-symbolic systems. We verify this via the conditions below.

The symbolic module maintains the field \(H_L\in\mathbb{R}^{T\times d_L}\). This field represents utterances, plans, hypotheses, rules, obligations, and narrative summaries. In addition, it provides serial abstraction. It converts distributed system state into propositions that can be compared, revised, explained, and remembered. The module can receive four classes of input: broadcast from the controller, geometric grounding packets from the interconnector, precision modulation from the valuative system, and norm constraints from the executive state. The update therefore has the form \[H_L^+ = \mathcal{T}_L\bigl(H_L,\;B_\Theta,\;C_{R\to L},\;Q_L,\; P,\;M\bigr).\] The precision field \(Q_L\in\mathbb{R}^{T\times T}_{>0}\) modulates attention over symbolic positions. This yields a positive field over token pairs, increased by precision demand and decreased by uncertainty.

Existence argument. A concrete element of \(\mathfrak{C}_L\) is the stack of \(L_{\mathrm{sym}}\) transformer blocks [11] with additional normalization. One step is layer normalization [12], applied after each attention sublayer and feed-forward sublayer. Condition (i): each sublayer operation, scaled dot-product attention, feed-forward network, layer normalization, is Lipschitz on bounded inputs. Their composition over \(L_{\mathrm{sym}}\) layers is Lipschitz by the chain rule, with constant \(\mathrm{Lip}_L\) depending on depth, embedding dimension, and the spectral norms of all weight matrices (query, key, value, and feed-forward projections). The Lipschitz claim requires bounded spectral norms as a standing assumption; in practice, spectral normalization [13] or the radial projection already appended in condition (ii) serves this purpose. Layer normalization with damping parameter \(\varepsilon>0\), \(\mathrm{LN}_\varepsilon(x)=\gamma\odot(x-\bar x)/\sqrt{\mathrm{Var}(x)+\varepsilon}+\beta\), has Lipschitz constant bounded by \(2\|\gamma\|_\infty/\sqrt{\varepsilon}\). For \(\varepsilon>0\) fixed this constant is finite, and the overall Lipschitz constant \(\mathrm{Lip}_L\) of the composed network is bounded accordingly. Condition (ii): residual connections accumulate across layers and layer normalization alone fails to bound the output norm of a standard residual transformer. To verify condition (ii) unconditionally, the architecture appends a final projection \(H_L\leftarrow\Pi_{B_{R_L}}(H_L)\) (clipping to the Frobenius-norm ball of radius \(R_L\)) after the last transformer block; this satisfies Specification 1(ii) by construction and is Lipschitz with constant one. Condition (iii): \(Q_L\) enters the attention logit as a positive additive term. Increasing \(Q_L(\ell,s)\) raises the pre-softmax logit of the corresponding position pair, increasing the attention weight within that row. Condition (iv): \(C_{R\to L}\) is an additive contribution to the initial hidden state \(H_L^{(0)}\); any non-zero grounding packet shifts \(H_L^{(0)}\) and propagates through all subsequent layers. Conditions (v)-(viii): use the logistic construction from condition (viii), \(Q_L(\ell,s)=\varepsilon_Q+(R_Q-\varepsilon_Q)\,\sigma(\tilde{q}_L(\ell,s))\), with \(\tilde{q}_L(\ell,s)=a_L(\ell,s)+\mu_{\mathrm{prec}}\,b_L(\ell,s)\) and \(b_L\ge0\) pointwise. Bounded range: \(\sigma\) maps \(\mathbb{R}\) into \((0,1)\), so \(Q_L\in(\varepsilon_Q,R_Q)\subset[\varepsilon_Q,R_Q]\) by construction. Positivity: \(Q_L>\varepsilon_Q>0\). Monotonicity in \(\mu_{\mathrm{prec}}\): \(\partial Q_L/\partial\mu_{\mathrm{prec}}=(R_Q-\varepsilon_Q)\,\sigma'(\tilde{q}_L)\,b_L\ge0\), with strict increase whenever \(b_L>0\). Lipschitz continuity: \(\sigma\) is \(\tfrac{1}{4}\)-Lipschitz, \(\tilde{q}_L\) is linear in \((\mu_{\mathrm{prec}},a_L,b_L)\), and the composition is Lipschitz on any compact parameter domain.

Dissipativity construction. A graph Laplacian has a zero eigenvalue whose eigenvector is the constant field. Since diffusion alone cannot contract the constant mode, strict dissipativity of the symbolic block must come from the reaction term. A sufficient construction is the residual damped transformer: \[F_L(H_L,\ldots)\;=\;-\alpha H_L+\Phi_L(H_L,\ldots), \qquad\mathrm{Lip}(\Phi_L)<\alpha,\] which gives \(\langle F_L(H)-F_L(H'),H-H'\rangle_F \leq(-\alpha+\mathrm{Lip}(\Phi_L))\|H-H'\|_F^2\leq-\mu_L\|H-H'\|_F^2\) with \(\mu_L=\alpha-\mathrm{Lip}(\Phi_L)>0\). A standard transformer block with spectral-norm-bounded weights and a residual linear damping term \(-\alpha H\) satisfies this when \(\mathrm{Lip}(\Phi_L)\leq\alpha-\mu_L\) for some \(\mu_L>0\).

Spectral gap of the precision field. Under the standing assumption that \(G_L\) is connected and \(Q_L\geq\varepsilon_Q>0\) (condition (viii)), the symmetric part of \(\Delta_{G_L}(Q_L)\) has spectral gap at least \(\varepsilon_Q\lambda_2(G_L)>0\), where \(\lambda_2(G_L)\) is the second eigenvalue of the symmetrized Laplacian (valid for undirected graphs, compare the description in Section 3). Higher \(Q_L\) pointwise yields faster mixing of \(H_L\) toward its quasi-stationary configuration. The monotonicity condition (vi) expresses this at the level of the modulating signal. We need to explain next how to ensure the lower bound \(Q_L\geq\varepsilon_Q>0\). Note that Softmax attention has strictly positive entries, but fails to supply an input-uniform lower bound: attention weights can be exponentially small, and for long sequences even uniform attention scales like \(1/T\). Sparse attention can set entries exactly to zero. Moreover, transformer attention is generally directed, whereas the spectral-gap estimate above is a statement about a symmetric conductance Laplacian. In the present framework, \(Q_L\) is therefore a formal precision/conductance field. A realistic way to obtain the required gap is by adding a connected residual backbone or floor, \[Q_L(Z)=Q_{\mathrm{base}}+\widetilde{Q}_L(Z), \qquad Q_{\mathrm{base}}\geq\varepsilon_Q \;\text{on a fixed connected graph }G_0, \qquad \widetilde{Q}_L(Z)\geq0,\] or equivalently by using a floored symmetric conductance field derived from attention. Then \[\lambda_2(\Delta_{G_L}(Q_L)) \geq \lambda_2(\Delta_{G_0}(Q_{\mathrm{base}}))>0,\] independently of whether the attention component becomes sharp, sparse, or directed.

Coarse-graining. The coarse-graining map \(\Phi_L\) extracts the abstract classes \(\mathcal{A}_L\) (dissipative operator on \(L^2(G_L,\mathbb{R}^{d_L})\) with \(\mathrm{Aut}(G_L)\)-equivariance, dissipativity constant \(\mu_L>0\)) and \(\mathcal{A}_{Q_L}\) (edge weight function \(w_L\colon E_L\to\mathbb{R}_{>0}\) with \(\inf w_L\geq\varepsilon_Q\)). Microscopic details (layer count, head count, feed-forward dimension) are irrelevant to \(\mathcal{A}_L\); see Section 13.

6 Geometric Module↩︎

The rationale for the geometric module is that it should supply the system with a structured world. This module detects that a plan is impossible, that a path is blocked, that an action threatens another agent, or that a safe route exists. The representations are local and relational. They preserve structure under transformation. The geometric module therefore functions as the system’s spatial and affordance based contact with its environment. The geometric module should represent the world as a sparse equivariant relational structure and the awareness field selects objects, agents, affordances, and transformations relevant to action.

The geometric module maintains the field \(X_R=\{e_i\in\mathbb{R}^{d_R},\,g_{ij}\in\mathrm{SE}(d)\}_{(i,j)\in E}\). Its state is a sparse structure over different kinds of vertices. These can include objects, body parts, regions, agents, paths, hazards, and affordances [14], [15]. The edges carry relations such as proximity, containment, occlusion, reachability, support, collision risk, and causal influence [16]. A convenient way to represent transformations between local frames is via groupoid morphisms [17]. In our setting here, the rigid-frame transforms \(g_{ij}\in\mathrm{SE}(d)\) are treated as fixed structural edge labels set at initialization that are not updated by the field equations. The node embeddings \(e_i\) are dynamic state variables. The sparse awareness field \(W_R\in[0,1]^{|E|}\) selects which relations should be propagated, exported, or used for action. The module’s update has the form \[X_R^+ = \mathcal{G}_R\bigl(X_R,\;B_\Theta,\;C_{L\to R},\;Y,\; P,\;W_R,\;M\bigr).\] The field \(C_{L\to R}\) injects symbolic goals, labels, hypotheses, and norms as constraints on graph propagation.

Existence argument. A concrete element of \(\mathfrak{C}_R\) is the scalar-feature sector of an EGNN-style architecture in the sense of [7]. Each node \(i\in V\) carries an embedding \(e_i\in\mathbb{R}^{d_R}\) and each directed edge \((i,j)\in E\) carries a rigid-frame transform \(g_{ij}=(R_{ij},t_{ij})\in\mathrm{SE}(d)\) used as a fixed edge attribute, with relative position vector \(r_{ij}:=t_{ij}\in\mathbb{R}^d\) extracted from its translational component. Under a global rigid motion \((R,t)\in\mathrm{SE}(d)\) applied to node positions, \(r_{ij}\mapsto R\,r_{ij}\), so \(\|r_{ij}\|^2\) is invariant. In this instantiation the position-coordinate dependence is captured entirely by \(r_{ij}\), which enters only through the \(\mathrm{SE}(d)\)-invariant distance \(\|r_{ij}\|^2\). The scalar features \(e_i\) are therefore \(\mathrm{SE}(d)\)-invariant. The messages are computed as \(m_{ij}=\phi_e(e_i,e_j,\|r_{ij}\|^2,a_{ij})\) where \(a_{ij}\) encodes additional edge attributes and \(\mathrm{SE}(d)\)-invariance holds because \(m_{ij}\) depends on \(g_{ij}\) only through \(\mathrm{SE}(d)\)-invariant quantities (distances and inner products). The permutation-equivariance holds because the node update \(e_j^+=\phi_h(e_j,\sum_{i\in S^*}W_{R,ij}m_{ij})\) reindexes correctly under any \(\sigma\in\mathrm{Aut}(G)\). This architecture satisfies the scalar-feature part of condition (ii), following the invariant message construction, cf. [7]. Condition (i): the message and update functions \(\phi_e,\phi_h\) are MLPs with layer normalization (\(\varepsilon>0\), Lipschitz constant \(2\|\gamma\|_\infty/\sqrt{\varepsilon}\)) and bounded range, giving a finite Lipschitz constant \(\mathrm{Lip}_R\) since each factor is Lipschitz. Condition (iii) holds by definition of the awareness-weighted message \(m_j=\sum_{i\in S^*}W_{R,ij}m_{ij}\), where only active-support edges contribute. Condition (iv): layer normalization by itself is insufficient, since it fails to imply that \(\|e_j^+\|\leq R_R\) when residual connections or subsequent affine maps are present. To enforce bounded range unconditionally, the architecture appends a final projection \(e_j\leftarrow\Pi_{B_{R_R}}(e_j)\) (clipping to the Euclidean ball of radius \(R_R\)) after the last EGNN layer and this satisfies Specification 2(iv) by construction, i.e. is Lipschitz with constant one. Conditions (v)-(vii) are satisfied by the sparsemax projection [18] \(W_{R,ij}=\max(\omega_{ij}-\tau,0)\), where \(\tau\) is the threshold enforcing \(\sum_{i\in S^*}W_{R,ij}=1\): simplex membership holds by construction. The Lipschitz continuity holds because the sparsemax map is the Euclidean projection onto the simplex, which is non-expansive. The sparsity bound holds because \(|S^*|\) is bounded by the in-degree \(k_j\leq K_{\max}\), and summing over all nodes gives \(|\mathrm{supp}(W_R)|\leq |V|\cdot K_{\max}=:K_{\mathrm{sp}}\). Softmax has full support over all edges, so it satisfies the sparsity bound of condition (vii) only if \(K_{\mathrm{sp}}=|E|\) (i.e.the bound is vacuous) or the total edge count is bounded. Strict sparsity requires a different approach. Top-\(K\) normalization satisfies conditions (v) and (vii) but is discontinuous at ties, placing it outside the admissible class for the formal Lipschitz results. Other Lipschitz projections onto the simplex with bounded support (e.g.sparsemax with bounded in-degree) are admissible within \(\mathfrak{C}_R\) provided they satisfy conditions (v)-(vii).

Dissipativity construction. As with the symbolic block, the graph-Laplacian diffusion does not contract constant-field modes. The strict dissipativity of the geometric block must come from the reaction term. A sufficient construction is: \[F_R(X_R,\ldots)\;=\;-\alpha_RX_R+\Phi_R(X_R,\ldots), \qquad\mathrm{Lip}(\Phi_R)<\alpha_R,\] giving \(\langle F_R(e)-F_R(e'),e-e'\rangle \leq-\mu_R\|e-e'\|^2\) with \(\mu_R=\alpha_R-\mathrm{Lip}(\Phi_R)>0\). The EGNN existence argument establishes Lipschitz and bounded range. Augmenting with a residual linear damping \(-\alpha_R e\) satisfies dissipativity condition (viii).

Spectral gap of the awareness field. The simplex condition on \(W_R\) gives nonnegative normalized awareness weights. In the formal undirected regime,the symmetric awareness conductance is obtained from \(W_R\), for example by \[\overline{W}_{R,ij}=\tfrac12(W_{R,ij}+W_{R,ji}),\] and the relevant Laplacian is \(\Delta_{G_R}(\overline{W}_R)\). A positive gap holds whenever the symmetrized active support \[E_R(Z)=\{\{i,j\}:\overline{W}_{R,ij}(Z)>0\}\] is connected. Equivalently, \[\lambda_2(\Delta_{G_R}(\overline{W}_R))>0\] is an additional connectivity condition on the awareness graph. For sparse awareness fields this is the intended interpretation: \(W_R\) may be highly selective, provided its symmetrized active support remains connected, or provided a fixed sparse residual backbone supplies the gap. In the latter case one may write \[\begin{align} & \overline{W}_R(Z)=W_{\mathrm{base}}+\widetilde{W}_R(Z), \; W_{\mathrm{base}}\geq \varepsilon_R>0 \\ &\text{on a fixed connected sparse graph }G_{\mathrm{base}}, \; \widetilde{W}_R(Z)\geq0, \end{align}\] which gives the uniform lower bound \[\lambda_2(\Delta_{G_R}(\overline{W}_R(Z))) \geq \lambda_2(\Delta_{G_{\mathrm{base}}}(W_{\mathrm{base}}))>0.\] More generally, for a sparse connected active graph, Cheeger-type estimates relate the spectral gap to the conductance \(h(G_R,\overline{W}_R)\), for instance \[\lambda_2(\Delta_{G_R}(\overline{W}_R)) \gtrsim h(G_R,\overline{W}_R)^2,\] up to the normalization convention for the graph Laplacian. The gap is supplied by connected conductance support or by an explicitly assumed residual backbone.

Coarse-graining. The coarse-graining map \(\Phi_R\) extracts the abstract classes \(\mathcal{A}_R\) (dissipative operator on \(L^2(G_R,\mathbb{R}^{d_R})\) with \(\mathrm{SE}(d)\)-invariance in the scalar feature sector, \(\mathrm{Aut}(G_R)\)-equivariance, dissipativity constant \(\mu_R>0\)) and \(\mathcal{A}_{W_R}\) (edge weight function \(w_R\colon E_R\to\mathbb{R}_{\geq 0}\) on simplices with connected support). Microscopic details (e.g. EGNN layer count, message-passing readout, MLP width) are not relevant to \(\mathcal{A}_R\); see Section 13.

7 Interconnector↩︎

The interconnector should act as a delayed, gated translation operator between heterogeneous latent spaces. It is supposed to preserve specialization while allowing coordinated behavior. The function of the interconnector can be described as disciplined exchange. The left side sends hypotheses such as "test whether this path is safe." The right side sends constraints such as "this action would collide with another agent." Its healthy regime is partial synchronization: enough coupling to coordinate, enough separation to preserve distinct computations.

This module is supposed to play the role of the analogue of the corpus callosum. It is defined as a pair of translation maps with routing gains and delays. The dense symbolic content is turned into into graph constraints, labels, goals, and queries by the interconnector. The sparse geometric content is transformed into symbolic packets, contradictions, affordance summaries, and risk statements. We write the interconnector as \[C_{R\to L}=g_{R\to L}\,\Phi_{R\to L}\!\bigl(X_R(t-\tau_{R\to L})\bigr), \qquad C_{L\to R}=g_{L\to R}\,\Phi_{L\to R}\!\bigl(H_L(t-\tau_{L\to R})\bigr),\] where \(\tau_{R\to L}\) and \(\tau_{L\to R}\) are non-negative delay parameters and \(g_{R\to L},g_{L\to R}\in[0,1]\) are scalar gates controlled by the valuative and executive state. The delays allow phase, anticipation, and synchronization effects. These translation maps are calibrated by experience.

Existence argument. The simplest element of \(\mathfrak{C}_\mathcal{I}\) is the state-independent constant kernel \(\mathcal{K}(\ell,i)=W_{RL}\in\mathbb{R}^{d_L\times d_R}\) for all \((\ell,i)\). Its Hilbert-Schmidt norm is \(\sqrt{T|V|}\,\|W_{RL}\|_F\), bounded by the architectural constraint \(\|W_{RL}\|_F\leq C_{RL}:=C_\mathcal{K}/\sqrt{T|V|}\) which is enforced by projecting \(W_{RL}\) onto the Frobenius-norm ball after each update, because weight decay alone does not give a hard bound. With this scaling the HS norm satisfies \(\|\mathcal{K}\|_{\mathrm{HS}}=\sqrt{T|V|}\,\|W_{RL}\|_F\leq C_\mathcal{K}\), and the coupling strength per interaction pair scales as \(1/\sqrt{T|V|}\), so the total coupling remains bounded as the number of interacting modes grows. With simplex weights \(\alpha_{\ell i}\) computed by any Lipschitz function of \(Z\) (e.g.a sparsemax over dot products between token and node features), the right-to-left signal becomes \(C_{R\to L,\ell}=W_{RL}\,\bar{e}_\ell\) where \(\bar{e}_\ell=\sum_i\alpha_{\ell i}\,e_i\in\mathbb{R}^{d_R}\) which recovers the bounded linear map as a special case. A more informative instance is the cross-attention kernel: \(\mathcal{K}(\ell,i)=W_V\) (a learned value projection, still constant in \((\ell,i)\)), with \(\alpha_{\ell i}=\mathrm{sparsemax}(q_\ell^\top k_i/\sqrt{d_\mathcal{K}})_i\) where \(q_\ell=W_Q H_{L,\ell}\) and \(k_i=W_K e_i\) are Lipschitz functions of \(Z\). The Lipschitz constants depend on the spectral norms \(\|W_Q\|\), \(\|W_K\|\), \(\|W_V\|\), which are assumed to be bounded: \(\|W_Q\|,\|W_K\|,\|W_V\|\leq C_W\) for some fixed architecture constant \(C_W\). The coupling weights are now sequence- and node-dependent while the kernel matrix remains shared. A fully \((\ell,i)\)-dependent kernel \(\mathcal{K}(\ell,i,Z)=f(H_{L,\ell},e_i)\) for some Lipschitz \(f\) is the most general admissible instance. The HS norm of that kernel is bounded when \(f\) has bounded Lipschitz constant and the state domain is compact. We include an overview of these different kernel types below.

The delay parameters \(\tau_{R\to L}\) and \(\tau_{L\to R}\) are treated here as externally specified schedule constants which means that they are initialized from structural priors and may be adjusted between episodes, but are fixed during any single rollout. They are not components of \(Z\) and have no within-step update rule. If online adaptation of delays is desired, the update rule and bounds must be included in an augmented Markov state, which is outside the scope of the present analysis. The interconnector gates are specific entries of the routing matrix \(\mathcal{R}_\Theta\): \(g_{R\to L}=\mathcal{R}_{\Theta,R\to L}\) and \(g_{L\to R}=\mathcal{R}_{\Theta,L\to R}\), where the entries are already in \([0,1]\) by the routing specification below.

Coarse-graining. The coarse-graining map \(\Phi_\mathcal{K}\) extracts the abstract class \(\mathcal{A}_\mathcal{K}\): Hilbert-Schmidt bipartite operators \(\mathcal{K}\colon L^2(G_R,\mathbb{R}^{d_R})\to L^2(G_L,\mathbb{R}^{d_L})\) with \(\|\mathcal{K}\|_{\mathrm{HS}}\leq C_\mathcal{K}\). The attention weights \(\alpha_{\ell i}\), gating functions, and concrete cross-attention implementation enter the Lipschitz RFDE class through the maps \(Z\mapsto \mathcal{K}_\alpha(Z)\) and \(Z\mapsto \mathcal{K}_\beta^*(Z)\); the closed stability class freezes them to fixed bounded operators; see Section 13.

Kernel families and Hilbert-Schmidt scaling↩︎

The Hilbert-Schmidt constraint \(\|\mathcal{K}(Z)\|_{\mathrm{HS}}\le C_\mathcal{K}\) is an energy budget for symbolic-geometric reentry. The budget interacts with system size in different ways depending on the kernel family.

Constant shared kernel. \(\mathcal{K}(\ell,i)=W_{RL}\) for all \((\ell,i)\). Then \(\|\mathcal{K}\|_{\mathrm{HS}}=\sqrt{T|V|}\,\|W_{RL}\|_F\), so the budget requires \(\|W_{RL}\|_F\le C_\mathcal{K}/\sqrt{T|V|}\). Dense all-to-all coupling must scale inversely with the square root of the number of token-node pairs.

Attention-weighted translation kernel. \(\mathcal{K}_\alpha(\ell,i;Z)=\alpha_{\ell i}(Z)W_V\) with \(\alpha_\ell(\cdot,Z)\in\Delta^{|V|-1}\). Since attention weights are simplex-valued, \(\sum_i\alpha_{\ell i}^2\le1\) and \(\|\mathcal{K}_\alpha(Z)\|_{\mathrm{HS}}^2\le T\|W_V\|_F^2\), so the budget requires only \(\|W_V\|_F\le C_\mathcal{K}/\sqrt{T}\), removing the \(\sqrt{|V|}\) penalty of the constant kernel.

Low-rank structured kernel. \(\mathcal{K}(\ell,i)=\sum_{r=1}^m a_r(\ell)b_r(i)A_r\) with \(\sum_\ell a_r^2\le1\) and \(\sum_i b_r^2\le1\). Then \(\|\mathcal{K}\|_{\mathrm{HS}}\le\sum_r\|A_r\|_F\), independent of \(T\) and \(|V|\). Each rank channel \(r\) can encode a mode of cross-domain alignment (object identity, affordance, threat, task relevance).

Gated mixture kernel. \(\mathcal{K}(\ell,i;Z)=\sum_{r=1}^m g_r(Z)\mathcal{K}_r(\ell,i)\) with \(0\le g_r(Z)\le\bar g_r\). Then \(\|\mathcal{K}(Z)\|_{\mathrm{HS}}\le\sum_r\bar g_r\|\mathcal{K}_r\|_{\mathrm{HS}}\). Executive and valuative variables can open and close coupling channels without violating the energy budget. State-dependent gates contribute additional Lipschitz terms to the RFDE and must be accounted for in the state-dependent stability margin.

Low rank gated attention. The following variant combines gating, low-rank structure, and attention: \[\label{eq:bestkernel} \mathcal{K}(\ell,i;Z)=\sum_{r=1}^m g_r(Z)\,a_r(\ell,Z)\,b_r(i,Z)\,A_r, \qquad \|\mathcal{K}(Z)\|_{\mathrm{HS}}\le C_\mathcal{K}.\tag{5}\] This type of kernel can represent multiple reentrant channels and gives executive and valuative variables a precise role as channel gates. For the global stability theorem, the gates are frozen to their equilibrium values, making \(\mathcal{K}\) a fixed Hilbert-Schmidt operator. This recommendation presupposes a compute budget adequate for dynamic gating. The constant shared kernel is adequate for small \(T\) and \(|V|\).

High dimensional scaling. Since dense unnormalized kernels have HS-norm scaling like \(\sqrt{d_Ld_R}\), or like \(\sqrt{T|V|d_Ld_R}\) for dense token-node coupling, they shrink the admissible small-gain margin for a reentry gain \(k\). Therefore, one should select low-rank normalized channels \(K_0=USV^\top\) with controlled singular values, attention-weighted token-node selection, and bounded state-dependent gates.

8 Controller↩︎

Routing determines which content is transmitted between modules and in what proportion. The controller implements this as a routing field over a product of simplices; the admissibility conditions require the simplex to be positively invariant and the routing map to be Lipschitz-regular. The controller is also the site of arbitration. If the symbolic module proposes a plan and the geometric module detects infeasibility, the controller increases right-to-left routing, suppresses direct action, and requests replanning. If the valuative system signals urgent harm, the controller broadens broadcast and shortens action latency. If the executive system detects norm conflict, the controller routes other-agent geometry and relevant memories into the symbolic module.

This module is an analog of the thalamus. In the present framework it serves as a dynamic routing system. Concretely, it receives exports from the symbolic module, geometric module, valuative system, executive system, and memory. It selects contents for broadcast, opens and closes channels, changes gain, coordinates timing, and determines which subsystem currently constrains action.

Let \(Z_i\) be exported latent packets from subsystem \(i\). The controller computes a sparse routing matrix \(\mathcal{R}_\Theta\) and a broadcast \(B_\Theta\): \[B_\Theta=\mathrm{TopS}_{s_i}\{Z_i\}, \qquad M_{i\to j}=\mathcal{R}_{\Theta,ij}\,T_{i\to j}(Z_i),\] where \(\mathrm{TopS}\) selects the subset of subsystems with highest salience scores \(s_i\), \(s_i\) is a Lipschitz function of \(Z_i\) and the routing state, and \(T_{i\to j}\) is a subsystem-specific learned translation. The routing matrix is state-dependent and changes with uncertainty, conflict, danger, novelty, and goal relevance.

Existence argument (formal case). For the formal Lipschitz theorem, set \(K_s=n_s\). Row-wise sparsemax [18] applied to a score matrix \(S\in\mathbb{R}^{n_s\times n_s}\) satisfies all three conditions: row-stochasticity holds by construction; sparsity condition (ii) holds trivially with \(K_s=n_s\); Lipschitz continuity holds because sparsemax is a Euclidean projection onto the simplex and therefore non-expansive with Lipschitz constant one. Row-wise softmax also satisfies conditions (i) and (iii) with \(K_s=n_s\). The formal case therefore covers dense routing only: with \(K_s=n_s\) every row of \(\mathcal{R}_\Theta\) may have full support, making the sparsity condition vacuous. The sparse-routing design principle is realized by the hard top-\(K_s\) construction and approximately by Gumbel-Softmax, both described below. However, these fall outside the Lipschitz conditions.

Caveat (outside the admissible class). Hard top-\(K_s\) normalization with \(K_s<n_s\) enforces strict sparsity but is discontinuous at score ties and therefore does not satisfy condition (iii); it is outside the admissible class \(\mathfrak{C}_K\) used for the formal results. Gumbel-Softmax with any fixed positive temperature has full support over \(n_s\) entries and therefore does not satisfy condition (ii) exactly; it too is outside \(\mathfrak{C}_K\). Both are viable implementations, but neither is covered by the formal propositions below. In the utilization of these constructions one must therefore independently verify any consistency properties claimed for the architecture. Sparsemax is a Euclidean projection onto the simplex. Hence it is globally Lipschitz and piecewise affine; its active support can change across faces of the simplex, while the map itself remains continuous. The interconnector gate values \(g_{R\to L}\) and \(g_{L\to R}\) are specific entries of \(\mathcal{R}_\Theta\) and therefore lie in \([0,1]\) by condition (i).

Stability and the routing matrix. The row-stochastic condition (i) gives \(\|\mathcal{R}_\Theta\|_\infty=1\), the natural operator norm for routing. A spectral norm bound for routing is a separate architectural condition. The closed principal stability theorem below absorbs only the fixed interfield operators \(\mathcal{K}\) and \(\mathcal{K}^*\) through the margin \(C_\mathcal{K}^2<\mu_L\mu_R\). If routing is added directly to the principal stability block, then its operator norm must be included in the same block small-gain matrix. An optional strengthening, achieved by making each row strictly sub-stochastic (\(\sum_j\mathcal{R}_{ij}<1\) for each \(i\)), gives \(\|\mathcal{R}_\Theta\|_{\mathrm{op}}<1\) as an independent contractive routing condition. Perturbative attractor comparison then requires the standard hypotheses of upper semicontinuity for the corresponding semiflows.

The discussion above results in three corresponding regimes:

1. Use softmax or another smooth positive-temperature map. The routing map is smooth, the RFDE vector field is classical, and derivative-based Lyapunov arguments apply directly.

2. Use sparsemax. The RFDE vector field remains Lipschitz and single-valued, so Picard well-posedness holds. Variational stability arguments at support boundaries use the Clarke generalized Jacobian \(\partial_C\mathcal{R}_\Theta\).

3. Use hard top-\(K_s\). This requires a separate tie-margin, hybrid, or differential inclusion analysis and is outside the Lipschitz conditions stated here.

The formal derivative-based stability theorem uses regime (1). Regime (2) is admissible for well-posedness and requires nonsmooth analysis for derivative-based stability.

Coarse-graining. The coarse-graining map \(\Phi_{R_\Theta}\) extracts the abstract class \(\mathcal{A}_{R_\Theta}\): weighted adjacency matrices on the subsystem graph \(G_C\) with \(\|\mathcal{R}_\Theta\|_{\mathrm{op}}<1\). Broadcast implementation, phase-control mechanism, and row-stochastic parameterization are irrelevant to \(\mathcal{A}_{R_\Theta}\); see Section 13.

9 Valuative System↩︎

The valuative system should regulate gain, priority, plasticity, and action pressure by converting homeostatic deviation, prediction error, novelty, relief, and norm relevance into modulating signals.

This module is an analogue of the hypothalamus which tracks viability variables and produces drive. The analogue of the frontal system regulates inhibition, planning, and norms. Both modules together then define the valuative organization of the system. The valuative system is supposed to signal the states that matter, the errors that deserve learning, memories which deserve consolidation, and actions which should be suppressed.

Let \(h\) denote homeostatic deviation, \(\varepsilon_{\mathrm{pred}}\) prediction error, \(n\) novelty, \(r\) outcome feedback, and \(\mu\) the neuromodulatory context. The map \(\mathcal{V}\) is decomposed as \(\mathcal{V}=(\mathcal{V}_Y,\mathcal{V}_\mu)\), where \(\mathcal{V}_Y\) produces the updated valuative state and \(\mathcal{V}_\mu\) produces the neuromodulatory readout. We can write the update as \[Y^+ = \mathcal{V}_Y(Y,h,\varepsilon_{\mathrm{pred}},n,r,H_L,X_R,M,P), \qquad \mu=\mathcal{V}_\mu(Y)=(\mu_{\mathrm{DA}},\mu_{\mathrm{ACh}},\mu_{\mathrm{NE}},\mu_{5HT},\mu_{\mathrm{OP}}).\] Here \(\mu_{\mathrm{DA}}\) is a dopamine-like value error, \(\mu_{\mathrm{ACh}}\) an acetylcholine-like precision demand, \(\mu_{\mathrm{NE}}\) a norepinephrine-like unexpected uncertainty, \(\mu_{5HT}\) a serotonin-like temporal stabilization, and \(\mu_{\mathrm{OP}}\) an opioid-like relief.

Existence argument. Condition (i) is satisfied by the discrete leaky integrator \(h^{t+1}=(1-\kappa_h\Delta t)h^t+\Delta t\,f_h(u^t)\). Under the condition \(0<\Delta t<1/\kappa_h\), we have that \(0<1-\kappa_h\Delta t<1\), so the recursion \(\|h^{t+1}\|\leq(1-\kappa_h\Delta t)\|h^t\|+\Delta t B_u\) is contractive; its fixed point satisfies \(\|h^*\|=B_u/\kappa_h\), giving \(\|h^t\|\leq\max(\|h^0\|,B_u/\kappa_h)\) for all \(t\geq 0\). (The continuous formula is \(h(t)=e^{-\kappa_h t}h(0)+\int_0^t e^{-\kappa_h(t-s)}f_h(u(s))\,ds\) and the discrete estimate above is the operative bound.) Condition (ii) is satisfied by any strictly monotone map into the open interval \((0,1)\). A sigmoid applied to a linear function of the state vector, \(\mu_k=\sigma(\mathbf{w}_k^\top z)\), satisfies conditions (ii) and (iii) for any bounded weight vector \(\mathbf{w}_k\) and bounded input domain. Other maps such as softsign, \(\tanh\) shifted to \((0,1)\), or any other bounded smooth activation are equally admissible. Condition (iii) holds for all smooth bounded activations by standard Lipschitz estimates. Condition (iv) is satisfied by any bounded activation applied to the output layer of \(\mathcal{V}_Y\), for instance a \(\tanh\) or clipped linear map with range \([-R_Y,R_Y]^{n_Y}\); this is an additional constraint analogous to Specification 1(ii).

The valuative system modulates left attention \(Q_L\), right awareness \(W_R\), controller routing \(\mathcal{R}_\Theta\), memory consolidation, and action selection through the five neuromodulatory signals. Delayed credit assignment is implemented by eligibility traces, cf. [19], [20]. For a parameter family \(\theta_i\) belonging to action, attention, awareness, routing, or memory, \[z_i^{t+1} = \lambda_i z_i^t + \nabla_{\theta_i}\log\pi_i^t, \qquad \Delta\theta_i^t = \eta_i\,\delta^t\,z_i^t.\] Here \(\nabla_{\theta_i}\log\pi_i^t\) denotes the REINFORCE score function evaluated at the action \(A^t\) actually taken: for the action policy this is \(\nabla_{\theta_{\mathrm{act}}}\log p(A^t\mid\tilde{Z}^t;\theta_{\mathrm{act}})\), and analogously for each parameter family at the sample drawn at step \(t\). Both the trace update and the parameter update use \(z_i^t\), the trace from step \(t\), as opposed to the just-computed \(z_i^{t+1}\); this is the one-step-delayed convention used throughout. The traces \(z_i\) retain recent causes. The signal \(\delta\) supplies delayed evaluation. This allows the system to learn which attentional choices, awareness choices, routes, memories, and actions contributed to later success, error, harm, or relief.

The five parameter families with their associated distributions \(\pi_i\) and downstream effects are:

• \(\theta_{\mathrm{att}}\): attention weights in the symbolic module; \(\pi_{\mathrm{att}}\) is the attention distribution over token positions; updates shift the precision-modulated attention distribution and hence \(Q_L\).

• \(\theta_{\mathrm{aw}}\): awareness scoring weights in the geometric module; \(\pi_{\mathrm{aw}}\) is the awareness distribution over edges; updates shift \(W_R\).

• \(\theta_{\mathrm{rt}}\): routing weights in the controller; \(\pi_{\mathrm{rt}}\) is the routing distribution over subsystems; updates shift \(\mathcal{R}_\Theta\).

• \(\theta_{\mathrm{act}}\): action policy weights; \(\pi_{\mathrm{act}}=p(A\mid\cdot)\); updates shift the action distribution directly.

• \(\theta_{\mathrm{mem}}\): memory read and write weights; \(\pi_{\mathrm{mem}}=p(\mathrm{read}\mid\cdot)\); updates shift which memories are consolidated or retrieved.

Existence argument. Condition (i): the softmax distribution \(\pi_i=\mathrm{softmax}(\ell(\theta_i))\) is real-analytic and strictly positive everywhere, so it satisfies (i) as a special case. The sparsemax distribution is piecewise linear and hence piecewise smooth; at support boundaries the Clarke subdifferential exists and is non-empty because sparsemax is Lipschitz (it is a Euclidean projection onto the simplex), cf. [18], [21]. Any other piecewise-smooth projection onto the simplex also satisfies (i). When sparsemax is used in the eligibility trace update \(z_i^+=\lambda_i z_i+\nabla_{\theta_i}\log\pi_i\), the log-policy is undefined at actions assigned zero probability. The canonical remedy is \(\varepsilon\)-flooring: \(\pi_i^\varepsilon=(1-\varepsilon)\,\mathrm{sparsemax}(\ell(\theta_i))+\varepsilon\,\mathrm{Unif}\), which is strictly positive so \(\log\pi_i^\varepsilon\) is finite everywhere. The gradient is piecewise \(C^\infty\) on each smooth piece of sparsemax but remains undefined at support boundaries even after flooring which preserves positivity of \(\pi_i^\varepsilon\). The floored policy lies in \(\mathfrak{C}_\pi\) and converges to the unmodified sparsemax as \(\varepsilon\to 0\). Condition (ii) is a parameter specification: \(\lambda_i\in(0,1)\) is set by hand or by meta-learning. Condition (iii) is satisfied by \(\delta=\mu_{\mathrm{DA}}\), which lies in \((0,1)\) by Specification 5(ii) and is Lipschitz in \(Y\) by Specification 5(iii). More generally, any reward prediction error \(\delta=r-\hat{r}\) clipped to a bounded range is admissible.

Each update \(\Delta\theta_i=\eta_i\,\delta\,z_i\) modifies the weights used in the next forward pass, closing the feedback loop from valuative error into the field variables \(Q_L\), \(W_R\), \(\mathcal{R}_\Theta\), \(A\), and \(M\). The three precision and routing fields are modulated by the neuromodulatory signals: \(\mu_{\mathrm{ACh}}\) sharpens symbolic attention as specified in Specification 1(vi); \(\mu_{\mathrm{NE}}\) shifts geometric awareness logits; \(\mu_{\mathrm{DA}}\) shifts routing salience globally, identifying \(\delta=\mu_{\mathrm{DA}}\) as the common credit signal across parameter families, weighted by individual learning rates \(\eta_i\) and eligibility decays \(\lambda_i\).

Viability for traces and policy parameters. The continuous-time eligibility trace equation is \(\dot{z}_i=-\lambda_z z_i+s_i(Z_t)\), where \(s_i\) is the mean-field score function bounded by \(\|s_i(Z_t)\|\le S_i\). At \(\|z_i\|=R_{z,i}\), the inner product satisfies \(\langle z_i,\dot{z}_i\rangle\le-\lambda_z R_{z,i}^2+R_{z,i}S_i\le0\) whenever \(\lambda_z R_{z,i}\ge S_i\). This is the trace radial margin condition: the leakage rate \(\lambda_z\) must dominate the maximum score magnitude normalized by the trace radius. The policy parameter equation is \(\dot{\theta}_i=-\lambda_\theta\theta_i+\eta_i\delta z_i\), viable in \(B_{R_{\theta,i}}\) when \(\lambda_\theta R_{\theta,i}\ge\eta_i D R_{z,i}\). If these margin conditions are not imposed by design, the projected form \(\dot{z}_i=\Pi_{T_{B_{R_{z,i}}}(z_i)}(-\lambda_z z_i+s_i)\) establishes viability unconditionally.

Convergence characterization. The RFDE analysis treats the score-function estimator \(\nabla_{\theta_i}\log\pi_i\) as a deterministic gradient field, corresponding to the mean-field approximation in which eligibility traces track the expected gradient. A stochastic treatment would model \(\nabla_{\theta_i}\log\pi_i(A^t)\) as a random variable and apply stochastic RFDE theory, cf. [22]. The deterministic attractor result then applies to the expected dynamics. Under the specification conditions and the mean-field approximation, the eligibility trace update \(z_i^+=\lambda_i z_i+\nabla_{\theta_i}\log\pi_i^\varepsilon\) is contractive in \(z_i\) with rate \(\lambda_i\in(0,1)\), and the parameter update with weight decay is contractive in \(\theta_i\) under \(\eta_i\lambda_{\mathrm{reg}}<1\). The combined \((Z,\theta)\) system is an RFDE on an extended state space, and the global attractor of Theorem 3 applies to this extended system. The attractor projects onto the set \(\{\theta:\mathbb{E}[\delta z_i]=0\}\), which is the REINFORCE stationarity condition; convergence to this set holds in the mean-field (expected-gradient) sense.

Coarse-graining. The coarse-graining maps \(\Phi_Y\) and \(\Phi_\theta\) extract the abstract classes \(\mathcal{A}_Y\) (scalar forcing \(G_Y\) with \(\mathrm{Lip}(G_Y)<\kappa_Y\)) and \(\mathcal{A}_\theta\) (contractive gradient flow on \(\Theta\) with \(\eta_i\lambda_{\mathrm{reg}}<1\)). Neuromodulator identities (\(\mu_{\mathrm{ACh}}\), \(\mu_{\mathrm{NE}}\), \(\mu_{\mathrm{DA}}\)), homeostatic targets, and score-function estimator are irrelevant to the abstract classes; see Section 13.

10 Metacognition↩︎

Metacognition, as developed here, arises from self and world models that predict internal and external dynamics. These models provide second-order variable such as reliability, expected value of computation and internal conflict that are used to meta-learn the control of routing, learning, attention, and action. Meta-control chooses internal actions: attend, scan, query memory, simulate, or consolidate - as well as inhibit, replan, or request additional perception. It also adjusts learning rates and routing sparsity. The system thereby learns how to learn and how to regulate its own computation. Metacognition is, at its core, the controlled use of a self-world model to alter future cognition and behavior.

A world model predicts the effects of action on the environment, see e.g. [23], [24]. A self model predicts the effects of internal and external action on the system’s own future state. These models introduce the second-order variables: reliability, confidence, calibration, and internal conflict, among other second-order variables such as bias and the expected value of further computation.

Let \(\mathcal{W}\) predict external change and \(\mathcal{S}\) predict internal change. Let \(\rho_i\) denote the context-dependent reliability of subsystem \(i\). The metacognitive layer is summarized by \[\widehat{S}_{\mathrm{ext}}^+ = \mathcal{W}(S_{\mathrm{ext}},A,Z), \qquad \widehat{Z}_{\mathrm{int}}^+ = \mathcal{S}(Z_{\mathrm{int}},A,B_\Theta), \qquad \rho_i^+ = (1-\alpha)\rho_i+\alpha\,f(\varepsilon_i),\] where \(\varepsilon_i\) is the prediction error of subsystem \(i\) and \(f\) is a reliability kernel specified in the specification below. The world model applies the geometric and symbolic update rules under a hypothetical action \(A\): \[\widehat{X_R}^+ = \mathcal{W}_{\mathrm{geom}}(X_R,A,B_\Theta), \qquad \widehat{H_L}^+ = \mathcal{W}_{\mathrm{sym}}(H_L,A,B_\Theta).\] The world-model prediction error is \(\varepsilon_{\mathcal{W}}^t=\widehat{X_R}^+ -X_R^{t+1}\), where \(\widehat{X_R}^+\) was predicted at step \(t\) using \(Z^t\) and \(A^t\), and \(X_R^{t+1}\) is the observation available at the end of step \(t\). This error is therefore a stagewise-derived scalar at step \(t\), available for computing \(\rho_i^{t+1}\) in Stage 8.2. The normalized magnitude enters the neuromodulatory signals. The self model predicts the next internal regulation state: \[\widehat{Z}_{\mathrm{int}}^+ = \mathcal{S}\!\bigl((Q_L,W_R,\mathcal{R}_\Theta,Y,P,M), \,A,\,B_\Theta\bigr),\] with self-model error \(\varepsilon_{\mathcal{S}}^t=\widehat{Z}_{\mathrm{int}}^+ -Z_{\mathrm{int}}^{t+1}\); the same timing convention applies. The index \(i\) runs over \(\{\mathcal{W}_{\mathrm{geom}}, \mathcal{W}_{\mathrm{sym}},\mathcal{S},\Phi_{R\to L},\Phi_{L\to R}\}\).

Existence argument. Condition (i) follows from condition (ii): the convex combination \((1-\alpha)\rho_i+\alpha f(\varepsilon_i)\) of values in \([0,1]\) lies in \([0,1]\) by the specification requirement \(\alpha\in(0,1)\) and \(f:\mathbb{R}^{d_i}\to[0,1]\). Condition (iii) restricts to radially non-increasing kernels; examples include \(f(\varepsilon)=\exp(-\|\varepsilon\|^2/d_i)\), \(f(\varepsilon)=(1+\|\varepsilon\|/\sqrt{d_i})^{-1}\), and \(f(\varepsilon)=\max(0,\,1-\|\varepsilon\|/(\sqrt{d_i}\,\sigma))\) for a threshold \(\sigma>0\). The normalization by \(\sqrt{d_i}\) is essential: without it the exponential kernel \(\exp(-\|\varepsilon\|^2)\) collapses toward zero in high-dimensional spaces for any fixed per-dimension error, causing reliability to be insensitive to prediction quality. With normalization, \(f(\varepsilon)\) depends on the mean squared error per component, which is the statistically appropriate measure of model accuracy and is bounded away from zero for any accurate predictor regardless of \(d_i\).

Viability. The continuous-time reliability field is the relaxation flow \[\dot{\rho}_i=\alpha_i\bigl(f(\varepsilon_i)-\rho_i\bigr),\qquad \alpha_i>0.\] Since \(f(\varepsilon_i)\in[0,1]\), at \(\rho_i=0\) one has \(\dot{\rho}_i=\alpha_i f(\varepsilon_i)\ge0\) and at \(\rho_i=1\) one has \(\dot{\rho}_i=\alpha_i(f(\varepsilon_i)-1)\le0\). Both box-cone conditions (Lemma 1(ii)) are satisfied, verifying (A4) for \(\rho_i\in[0,1]\).

Reliability feeds back into routing and translation: the salience of subsystem \(i\) is augmented by \(\beta_\rho\rho_i\), and the translation gates are attenuated in proportion to translation reliability, so that unreliable channels are progressively closed and reliable subsystems receive stronger routing priority.

11 Memory↩︎

Memory consolidation occurs selectively: the gate \(g_M\) determines which symbolic and valuative content is written to the persistent field \(M\), and the consolidation rate is calibrated by the executive state. The admissibility conditions formalize this selectivity.

The memory field \(M\) accumulates content from the symbolic and valuative systems and makes it available to all modules. Its update has the form \[M^{t+1} = (1-g_M)\,M^t + g_M\,\Phi_M(H_L^{t+1}, Y^{t+1}),\] where \(g_M\in(0,1)\) is a consolidation gate and \(\Phi_M\) is a learned write projection.

Existence argument. The gated consolidation form already satisfies condition (i) by definition because it is a parameterized family that includes the GRU gated update and the write gate of differentiable neural computers as instances of this family. Any such form satisfies (i) by construction. Condition (ii) is satisfied whenever the gate is computed by a bounded differentiable function of \(Z\), such as a sigmoid applied to a linear form. The source of the gate signal, i.e. valuative state, content novelty, executive instruction, or any combination, is an implementation choice within \(\mathfrak{C}_M\). Condition (iii) is satisfied by any bounded linear write projection \(\Phi_M\). This includes a learned matrix applied to the concatenation \([H_L^{t+1};Y^{t+1}]\) after mean pooling.

Viability. The continuous-time memory field is the relaxation flow \[\dot{M}= g_M(Z_t)\bigl(\Phi_M(Z_t)-M\bigr),\qquad g_M\ge0.\] At \(\|M\|=R_M\), the inner product \(\langle M,\dot{M}\rangle= g_M(\langle M,\Phi_M\rangle-\|M\|^2)\le g_M(R_M^2-R_M^2)=0\) by condition (iii) and the Cauchy-Schwarz inequality. Hence the ball \(B_{R_M}\) is positively invariant, verifying (A4) for the memory component.

12 Executive System↩︎

The executive module implements a feedback loop that overrides stimulus-driven impulses in favor of planned, value-driven behaviors. Its admissibility conditions require dissipativity and bounded range, which together ensure the executive state converges independently of the interfield coupling.

Existence argument. A simple sufficient construction consists of the application of a linear projection to the concatenation of the inputs, followed with a bounded activation such as \(\tanh\) (with output clipped to \([-R_P,R_P]^{n_P}\)). This satisfies conditions (i)–(iii) under bounded input domains and weight norms bounded as in the Standing Assumption. Dissipativity condition (iv) is satisfied by the implementation \(\mathcal{P}(x)=-\mu_P x+\phi(W_P x)\), where \(\phi\) is \(1\)-Lipschitz (e.g.\(\tanh\) applied coordinate-wise) and \(\|W_P\|_{\mathrm{op}}<\mu_P\). Then: \[\begin{align} \langle\mathcal{P}(u)-\mathcal{P}(v),u-v\rangle &= -\mu_P\|u-v\|^2+\langle\phi(W_P u)-\phi(W_P v),u-v\rangle\\ &\leq -\mu_P\|u-v\|^2+\|W_P\|_{\mathrm{op}}\|u-v\|^2 \;\leq\; -(\mu_P-\|W_P\|_{\mathrm{op}})\|u-v\|^2\;<\;0. \end{align}\] The condition \(\|W_P\|_{\mathrm{op}}<\mu_P\) is achievable by spectral normalization of \(W_P\).

13 Coarse Graining↩︎

A coarse-graining replaces each of the specification conditions with an abstract operator class defined by dissipativity constants, symmetry, compact viability, and spectral conditions.

Coarse-graining maps↩︎

For each specification \(i\), the coarse-graining map \(\Phi_i\colon\mathcal{C}_i\to\mathcal{A}_i\) extracts the field-theoretic content of the admissibility class \(\mathcal{C}_i\) as an abstract operator class \(\mathcal{A}_i\). Each \(\mathcal{A}_i\) specifies: (a) the function space the operator acts on; (b) its dissipativity or spectral constant; (c) its symmetry group. Architectural or microscopic details (layer count, head count, edge feature dimension) are not visible in \(\mathcal{A}_i\).

Abstract operator classes↩︎

The nine specification operator classes are:

• \(\mathcal{A}_L\): dissipative operators on \(L^2(G_L,\mathbb{R}^{d_L})\), with \(\mathrm{Aut}(G_L)\)-equivariance and dissipativity constant \(\mu_L>0\).

• \(\mathcal{A}_R\): dissipative operators on \(L^2(G_R,\mathbb{R}^{d_R})\) with \(\mathrm{SE}(d)\)-invariance in the scalar feature sector, \(\mathrm{Aut}(G_R)\)-equivariance, and dissipativity constant \(\mu_R>0\).

• \(\mathcal{A}_\mathcal{K}\): Hilbert-Schmidt bipartite operators \[\mathcal{K}\colon L^2(G_R,\mathbb{R}^{d_R})\to L^2(G_L,\mathbb{R}^{d_L}),\quad \|\mathcal{K}\|_{\mathrm{HS}}\leq C_\mathcal{K}.\]

• \(\mathcal{A}_{Q_L}\): edge weight functions \(w_L\colon E_L\to\mathbb{R}_{>0}\) with \(\inf w_L\geq\varepsilon_Q\).

• \(\mathcal{A}_{W_R}\): edge weight functions \(w_R\colon E_R\to\mathbb{R}_{\geq 0}\) on simplices with connected support.

• \(\mathcal{A}_{R_\Theta}\): weighted adjacency matrices on the subsystem graph \(G_C\) with \(\|R_\Theta\|_{\mathrm{op}}<1\).

• \(\mathcal{A}_Y\): scalar forcing functions \(G_Y\) satisfying \(\mathrm{Lip}(G_Y)<\kappa_Y\).

• \(\mathcal{A}_\theta\): contractive gradient flows on the parameter manifold \(\Theta\) with weight decay satisfying \(\eta\lambda_{\mathrm{reg}}<1\).

• \(\mathcal{A}_P\): dissipative operators on \(\mathbb{R}^{n_P}\) with dissipativity constant \(\mu_P>0\).

Any tuple \((T_L,G_R,\mathcal{K},Q_L,W_R,R_\Theta,G_Y,\pi_\theta,P)\in\prod_i\mathcal{A}_i\) satisfying the RFDE assumptions defines a well-posed RFDE with a compact attractor. The closed principal stability theorem applies to the corresponding reduced fixed-coupling regime under the stability condition 2 . These are compatibility classes for assembling the master RFDE. The classes \(\mathcal{A}_L\) and \(\mathcal{A}_R\) provide the internally dissipative symbolic and geometric field dynamics, while \(\mathcal{A}_{Q_L}\) and \(\mathcal{A}_{W_R}\) supply the stabilized conductance fields whose residual backbones yield the spectral gaps entering the constants \(\mu_L\) and \(\mu_R\). The class \(\mathcal{A}_{\mathcal{K}}\) supplies Hilbert-Schmidt symbolic-geometric reentrant kernels; its norm bound \(C_{\mathcal{K}}\) is compatible with the principal dissipativity through the small-gain requirement \(C_{\mathcal{K}}^2<\mu_L\mu_R\). The remaining classes \(\mathcal{A}_{R_\Theta},\mathcal{A}_Y,\mathcal{A}_\theta,\mathcal{A}_P\) govern routing, valuation, policy adaptation, and executive modulation; their contraction, Lipschitz, or dissipativity hypotheses ensure that these auxiliary feedback loops enrich the dynamics without destroying boundedness or stability. Selecting operators from all nine classes therefore defines an admissible stabilized attention-awareness-diffusion architecture, i.e. a block vector field \[\dot{Z}(t)=\mathcal{F}(Z_t,u^*)\] on the compact phase domain \(\mathcal{Z}\) with the required properties.

References↩︎