涌现意识的贝叶斯镜像架构：循环层级、自流形与混合事件–自我绑定

# A Bayesian Mirror Architecture for Emergent Consciousness: Circular Hierarchies, Self-Manifolds, and Hybrid Event–Self Binding
Source: [https://arxiv.org/html/2610.08792](https://arxiv.org/html/2610.08792)
###### Abstract

We present a foundational formulation of the*Bayesian Mirror Architecture*\(BMA\), a self\-referential generative framework in which sensory\-driven abstractions, meta\-abstractions, and a self\-latent interact through recursive and circular dynamics\. The defining architectural constraint is a closed self\-recursionSt←Ht−1S\_\{t\}\\leftarrow H\_\{t\-1\}, where a hybrid event–self binding latentHtH\_\{t\}couples self\-representations to abstract world models and re\-injects that coupling into the self\-state\. Consciousness, in a restricted sense, is not treated as an optimization objective nor as a semantic label; it is an*architectural property*of systems possessing this circular recursion\.

Since the fundamental objects of inference are posterior beliefs, the intrinsic state space of BMA is taken to be a space of probability measures, equipped with optimal\-transport geometry\. We formulate stability and contractivity statements in the22\-Wasserstein metric on𝒫2\\mathcal\{P\}\_\{2\}, yielding coordinate\-free notions of self\-stability and hybrid coherence as properties of belief trajectories\. We define a*Causal Learning Regime*\(CLR\) by quantitative thresholds on Wasserstein belief drift together with an integration index measuring sustained coupling between self and world latents\. Critically, CLR is a*diagnostic*of successful causal learning—it indicates whether the environment contains learnable structure—not a marker of consciousness itself\.

A key structural point is that global strict contractivity is neither required nor expected: BMA may exhibit multiple coherent basins \(ergodic components\)\. Accordingly, we define*self\-manifolds*basin\-wise as supports of basin\-restricted invariant measures under local Wasserstein contractivity within each coherent basin\. We further identify*Wassersteinε\\varepsilon\-necks*—transport bottlenecks in belief geometry—as the canonical locus where two coherent belief basins become effectively decoupled, resulting in a unique realized basin continuation in a singular \(vanishing\-conductance\) limit\. We interpret this realized basin selection as*choice*: internally determined by the system’s stabilized belief state, yet externally inscrutable because it is not predictable from locally observable dynamics at finite resolution\. Learning proceeds via variational free\-energy minimization; the system observes actions and predicts executions, with stability, coherence, and integration emerging as observable consequences of what the world affords to learn\. We further develop an*agency trajectory*showing how the circular architecture enables progression from passive observation through action selection, goal composition, to model\-based planning with lookahead and replanning under uncertainty\.

## 1Introduction

The mirror\-based view of intelligence asserts that perception induces internal abstraction and that abstractions may be mirrored across multiple representational carriers\. When a system learns transformations between carriers encoding the same underlying content, it constructs meaning\-preserving maps; iterating such maps yields higher\-order representations, including meta\-abstractions and, ultimately, self\-representations\.

The Bayesian Mirror Architecture \(BMA\) imposes a structural closure: the architecture is not merely hierarchical but*circular*, with a compulsory self\-recursion that continuously feeds a hybrid event–self binding back into the self\-state:

St←Ht−1\.S\_\{t\}\\;\\leftarrow\\;H\_\{t\-1\}\.The mathematical emphasis of this work is that the relevant dynamics are dynamics of*beliefs*\. The architecture is specified as a generative model with latent variables for events, abstractions, meta\-abstractions, self\-latents, hybrid bindings, and intent\. However, consciousness is not identified with minimization of a scalar training loss\. Instead, consciousness is defined as an architectural property of systems possessing circular self\-recursion\. The*Causal Learning Regime*\(CLR\), motivated by the mirror\-neuron literature, is introduced as a diagnostic regime indicating when the environment affords learnable causal structure\.

Because the state variables of interest are posterior measures \(marginals and joints induced by inference\), we equip belief space with optimal\-transport geometry\. The22–Wasserstein metric provides a coordinate\-free distance between beliefs, sensitive to multimodality and independent of latent parameterization\. In this geometry, we define self\-stability and hybrid coherence as Wasserstein continuity properties of belief trajectories, and we formulate attractor statements by imposing*local*Wasserstein contraction conditions on basin\-restricted dynamics\.

The paper has four goals\. First, to define BMA as a precise stochastic generative system with circular self\-recursion and hybrid event–self binding\. Second, to give a metric and dynamical framework in which persistent selfhood can be defined and proved to exist under*basin\-wise*contractive assumptions: the*self\-manifold*is the support of a basin\-restricted invariant measure of the induced belief dynamics\. Third, to describe how*Wassersteinε\\varepsilon\-necks*yield a mathematically precise decision locus in a singular low\-conductance limit, resulting in realized basin continuation; we interpret this as*choice*, governed only by the system’s own belief state\. Fourth, to establish an*agency trajectory*showing how the circular architecture enables increasingly sophisticated action: from passive observation, through goal\-directed action and compositional planning, to model\-based lookahead with replanning under uncertainty\.

## 2Latent Hierarchy and Generative Structure

Time is discrete:t=1,2,…,Tt=1,2,\\dots,T\.

We consider latent variables

Et,Ct,At,Mt,St,Ht,It,E\_\{t\},\\;C\_\{t\},\\;A\_\{t\},\\;M\_\{t\},\\;S\_\{t\},\\;H\_\{t\},\\;I\_\{t\},representing sensory events, concept\-level abstractions, action abstractions, meta\-abstractions, self\-latents, hybrid event–self bindings, and intent\. These variables encode functional roles in the generative hierarchy\.

#### Observed variables \(observation/execution\)\.

To avoid notational drift, we use a single observed pair throughout:

ot:=Et\(observation\),et:=Xt\(execution/outcome\)\.o\_\{t\}:=E\_\{t\}\\quad\\text\{\(observation\)\},\\qquad e\_\{t\}:=X\_\{t\}\\quad\\text\{\(execution/outcome\)\}\.Intuitively, the system observesoto\_\{t\}\(an observed action/sensory event\) and must predict its execution/outcomeete\_\{t\}under the current causal grammar\.

The generative model factors as:

pθ​\(St∣St−1,Ht−1\),\\displaystyle p\_\{\\theta\}\(S\_\{t\}\\mid S\_\{t\-1\},H\_\{t\-1\}\),pθ​\(Mt∣Mt−1,St\),\\displaystyle\\qquad p\_\{\\theta\}\(M\_\{t\}\\mid M\_\{t\-1\},S\_\{t\}\),\(1\)pθ​\(At∣Mt\),\\displaystyle p\_\{\\theta\}\(A\_\{t\}\\mid M\_\{t\}\),pθ​\(Ct∣At\),pθ​\(ot∣Ct\),\\displaystyle\\qquad p\_\{\\theta\}\(C\_\{t\}\\mid A\_\{t\}\),\\qquad p\_\{\\theta\}\(o\_\{t\}\\mid C\_\{t\}\),\(2\)pθ​\(Ht∣Mt,St−1\),\\displaystyle p\_\{\\theta\}\(H\_\{t\}\\mid M\_\{t\},S\_\{t\-1\}\),pθ​\(It∣St,Mt,Ht\),\\displaystyle\\qquad p\_\{\\theta\}\(I\_\{t\}\\mid S\_\{t\},M\_\{t\},H\_\{t\}\),\(3\)pθ​\(et∣It\)\.\\displaystyle p\_\{\\theta\}\(e\_\{t\}\\mid I\_\{t\}\)\.\(4\)

## 3Wasserstein Geometry of Mirror Beliefs

Let\(𝒳,d\)\(\\mathcal\{X\},d\)be a Polish metric space\. Denote by𝒫2​\(𝒳\)\\mathcal\{P\}\_\{2\}\(\\mathcal\{X\}\)the set of Borel probability measures on𝒳\\mathcal\{X\}with finite second moment\.

###### Definition 3\.1\(Wasserstein distance\)\.

Forμ,ν∈𝒫2​\(𝒳\)\\mu,\\nu\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{X\}\), the22–Wasserstein distance is

W2​\(μ,ν\)=\(infπ∈Π​\(μ,ν\)∫𝒳×𝒳d​\(x,y\)2​𝑑π​\(x,y\)\)1/2,W\_\{2\}\(\\mu,\\nu\)=\\left\(\\inf\_\{\\pi\\in\\Pi\(\\mu,\\nu\)\}\\int\_\{\\mathcal\{X\}\\times\\mathcal\{X\}\}d\(x,y\)^\{2\}\\,d\\pi\(x,y\)\\right\)^\{1/2\},whereΠ​\(μ,ν\)\\Pi\(\\mu,\\nu\)denotes the set of couplings with marginalsμ,ν\\mu,\\nu\.

## 4Learning as Wasserstein Gradient Flow

We do not treat learning as the mere minimization of a static scalar functional, but as a*flow*of beliefs on𝒫2\\mathcal\{P\}\_\{2\}driven by an energy that encodes posterior consistency\.

### 4\.1A reference form of the flow

Letρt\\rho\_\{t\}denote a time\-dependent density on a latent space𝒴\\mathcal\{Y\}\(with respect to a reference volume\), representing a belief overY=\(S,H,M\)Y=\(S,H,M\)\. Consider an energy functional of the form

ℰ​\(ρ\)=∫𝒴V​\(y\)​ρ​\(y\)​𝑑y\+λent​∫𝒴ρ​\(y\)​log⁡ρ​\(y\)​𝑑y,\\mathcal\{E\}\(\\rho\)=\\int\_\{\\mathcal\{Y\}\}V\(y\)\\,\\rho\(y\)\\,dy\+\\lambda\_\{\\mathrm\{ent\}\}\\int\_\{\\mathcal\{Y\}\}\\rho\(y\)\\log\\rho\(y\)\\,dy,\(5\)whereVVplays the role of a data\-fit / negative log\-likelihood potential and the entropic term imposes diffusion \(regularization, exploration, smoothing\)\.

###### Proposition 4\.1\(Wasserstein gradient flow form\)\.

Under standard hypotheses onℰ\\mathcal\{E\}\(e\.g\. proper, lower semicontinuous, and geodesicallyλ\\lambda\-convex on\(𝒫2​\(𝒴\),W2\)\(\\mathcal\{P\}\_\{2\}\(\\mathcal\{Y\}\),W\_\{2\}\)\), theW2W\_\{2\}\-gradient flow ofℰ\\mathcal\{E\}is the drift–diffusion \(Fokker–Planck\) equation

∂tρt=∇⋅\(ρt​∇V\)\+λent​Δ​ρt,\\partial\_\{t\}\\rho\_\{t\}=\\nabla\\cdot\\bigl\(\\rho\_\{t\}\\nabla V\\bigr\)\+\\lambda\_\{\\mathrm\{ent\}\}\\,\\Delta\\rho\_\{t\},\(6\)equivalently the continuity equation∂tρt\+∇⋅\(ρt​ut\)=0\\partial\_\{t\}\\rho\_\{t\}\+\\nabla\\cdot\(\\rho\_\{t\}u\_\{t\}\)=0with velocity fieldut=−∇\(V\+λent​log⁡ρt\)u\_\{t\}=\-\\nabla\\bigl\(V\+\\lambda\_\{\\mathrm\{ent\}\}\\log\\rho\_\{t\}\\bigr\)\.

### 4\.2Discrete\-time learning via JKO

In practice BMA operates in discrete time\. The canonical discretization of Wasserstein gradient flows is the Jordan–Kinderlehrer–Otto \(JKO\) scheme\.

###### Definition 4\.3\(JKO step\)\.

Given step sizeτ\>0\\tau\>0, defineρk\+1\\rho^\{k\+1\}fromρk\\rho^\{k\}by

ρk\+1∈arg⁡minρ∈𝒫2​\(𝒴\)⁡\{ℰ​\(ρ\)\+12​τ​W2​\(ρ,ρk\)2\}\.\\rho^\{k\+1\}\\in\\arg\\min\_\{\\rho\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{Y\}\)\}\\left\\\{\\mathcal\{E\}\(\\rho\)\+\\frac\{1\}\{2\\tau\}W\_\{2\}\(\\rho,\\rho^\{k\}\)^\{2\}\\right\\\}\.\(7\)

## 5Variational Free Energy as the Sole Learning Objective

Having established the geometric framework for learning as Wasserstein gradient flow, we now specify the concrete energy functional that BMA minimizes\. The central principle is that*variational free energy minimization is the only learning objective*\. All other quantities—including the learning metrics defined in subsequent sections—are observational diagnostics of the learned system, not optimization targets\.

### 5\.1The variational free energy functional

Letqϕ​\(Zt∣ot\)q\_\{\\phi\}\(Z\_\{t\}\\mid o\_\{t\}\)denote the recognition model \(encoder\) that maps observationsoto\_\{t\}to posterior beliefs over latent variablesZt=\(St,Ht,Mt,It\)Z\_\{t\}=\(S\_\{t\},H\_\{t\},M\_\{t\},I\_\{t\}\), and letpθ​\(et∣Zt\)p\_\{\\theta\}\(e\_\{t\}\\mid Z\_\{t\}\)denote the generative model for executions/outcomes\. The variational free energy for a single observation–execution pair\(ot,et\)\(o\_\{t\},e\_\{t\}\)is

ℱt=−𝔼qϕ​\(Zt∣ot\)​\[log⁡pθ​\(et∣Zt\)\]⏟reconstruction / prediction error\+β​KL​\(qϕ​\(Zt∣ot\)∥p​\(Zt\)\)⏟complexity penalty,\\mathcal\{F\}\_\{t\}=\\underbrace\{\-\\mathbb\{E\}\_\{q\_\{\\phi\}\(Z\_\{t\}\\mid o\_\{t\}\)\}\\bigl\[\\log p\_\{\\theta\}\(e\_\{t\}\\mid Z\_\{t\}\)\\bigr\]\}\_\{\\text\{reconstruction / prediction error\}\}\+\\underbrace\{\\beta\\,\\mathrm\{KL\}\\bigl\(q\_\{\\phi\}\(Z\_\{t\}\\mid o\_\{t\}\)\\,\\\|\\,p\(Z\_\{t\}\)\\bigr\)\}\_\{\\text\{complexity penalty\}\},\(8\)wherep​\(Zt\)p\(Z\_\{t\}\)is the prior over latents andβ\>0\\beta\>0controls the strength of the complexity term\.

### 5\.2Decomposition over the latent hierarchy

Using the factorization \([1](https://arxiv.org/html/2610.08792#S2.E1)\)–\([4](https://arxiv.org/html/2610.08792#S2.E4)\), one may express the expected negative log\-likelihood term in \([8](https://arxiv.org/html/2610.08792#S5.E8)\) via conditional independences \(e\.g\.ete\_\{t\}depends onItI\_\{t\}, andItI\_\{t\}depends on\(St,Mt,Ht\)\(S\_\{t\},M\_\{t\},H\_\{t\}\)\)\. A compact reference form is:

ℱ=𝔼​\[−log⁡pθ​\(et∣It\)\]\+β​\(KL​\(q​\(Mt\)∥p​\(Mt\)\)\+KL​\(q​\(Ht\)∥p​\(Ht\)\)\+KL​\(q​\(It\)∥p​\(It\)\)\),\\mathcal\{F\}=\\mathbb\{E\}\\Bigl\[\-\\log p\_\{\\theta\}\(e\_\{t\}\\mid I\_\{t\}\)\\Bigr\]\+\\beta\\Bigl\(\\mathrm\{KL\}\(q\(M\_\{t\}\)\\\|p\(M\_\{t\}\)\)\+\\mathrm\{KL\}\(q\(H\_\{t\}\)\\\|p\(H\_\{t\}\)\)\+\\mathrm\{KL\}\(q\(I\_\{t\}\)\\\|p\(I\_\{t\}\)\)\\Bigr\),\(9\)where expectations are taken over recognition distributions and we have suppressed conditioning for clarity\. The self\-latentStS\_\{t\}enters through the circular constraintSt←Ht−1S\_\{t\}\\leftarrow H\_\{t\-1\}, which is enforced architecturally rather than through an additional KL term\.

### 5\.3Learning dynamics

Learning proceeds by gradient descent on the free energy with respect to parameters\(θ,ϕ\)\(\\theta,\\phi\):

\(θ,ϕ\)←\(θ,ϕ\)−η​∇\(θ,ϕ\)ℱ\.\(\\theta,\\phi\)\\;\\leftarrow\\;\(\\theta,\\phi\)\-\\eta\\,\\nabla\_\{\(\\theta,\\phi\)\}\\mathcal\{F\}\.\(10\)This is the*only*update rule in the foundational definition\. No auxiliary objectives \(stability targets, coherence penalties, integration bonuses\) are added to the loss\. The system learns solely by minimizing its surprise about executions/outcomes given its observation\-conditioned beliefs\.

###### Proposition 5\.2\(Free energy as the unique objective\)\.

Under the BMA framework, the parameter update \([10](https://arxiv.org/html/2610.08792#S5.E10)\) is the complete specification of learning\. All emergent properties of the learned system—including belief stability, hybrid coherence, and self–world integration—arise as consequences of free energy minimization on structured observation–execution pairs, not as independently optimized quantities\.

### 5\.4Relationship to learning metrics

The learning metrics defined in Section 6 \(Stability, Coherence, Integration\) are diagnostic measurements applied to the belief trajectories of a trained system\.

###### Definition 5\.3\(Causal learning diagnostics\)\.

The Causal Learning Regime \(CLR\) thresholds on Stability, Coherence, and Integration serve as diagnostics for*successful causal learning*:

- •High Stability indicates that the system has learned predictable self\-dynamics: future self\-beliefs are predictable from past hybrid beliefs\.
- •High Coherence indicates that hybrid event–self bindings evolve smoothly: the system has learned coherent temporal structure\.
- •High Integration indicates sustained statistical dependence between self\-latents and world\-latents: the system has learned that self and world are causally coupled\.

## 6Learning Metrics and Definitions

The metrics in this section are*diagnostic observations*of a trained system, not optimization objectives\. They quantify whether the system has successfully learned stable causal structure from its environment\.

For eachtt, let the self and hybrid belief marginals be

μtS,μtH∈𝒫2​\(𝒳\)\.\\mu^\{S\}\_\{t\},\\;\\mu^\{H\}\_\{t\}\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{X\}\)\.LetΔS¯\\overline\{\\Delta\_\{S\}\}andΔH¯\\overline\{\\Delta\_\{H\}\}denote average squared Wasserstein drift:

ΔS¯:=1T−1​∑t=2TW2​\(μtS,μt−1S\)2,ΔH¯:=1T−1​∑t=2TW2​\(μtH,μt−1H\)2\.\\overline\{\\Delta\_\{S\}\}:=\\frac\{1\}\{T\-1\}\\sum\_\{t=2\}^\{T\}W\_\{2\}\(\\mu^\{S\}\_\{t\},\\mu^\{S\}\_\{t\-1\}\)^\{2\},\\qquad\\overline\{\\Delta\_\{H\}\}:=\\frac\{1\}\{T\-1\}\\sum\_\{t=2\}^\{T\}W\_\{2\}\(\\mu^\{H\}\_\{t\},\\mu^\{H\}\_\{t\-1\}\)^\{2\}\.LetΣS2,ΣH2\>0\\Sigma\_\{S\}^\{2\},\\Sigma\_\{H\}^\{2\}\>0be calibration scales \(e\.g\. baseline drift levels estimated on a reference regime\), and letε0\>0\\varepsilon\_\{0\}\>0be a small numerical stabilizer to avoid division by zero\.

#### Self\-Stability\.

We use a bounded monotone transform of drift:

Stability=11\+ΔS¯/\(ΣS2\+ε0\)\.\\mathrm\{Stability\}=\\frac\{1\}\{1\+\\overline\{\\Delta\_\{S\}\}/\(\\Sigma\_\{S\}^\{2\}\+\\varepsilon\_\{0\}\)\}\.

#### Hybrid Coherence\.

Coherence=11\+ΔH¯/\(ΣH2\+ε0\)\.\\mathrm\{Coherence\}=\\frac\{1\}\{1\+\\overline\{\\Delta\_\{H\}\}/\(\\Sigma\_\{H\}^\{2\}\+\\varepsilon\_\{0\}\)\}\.

#### Event–Self Integration\.

LetμtS​M∈𝒫2​\(𝒮×ℳ\)\\mu^\{SM\}\_\{t\}\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{S\}\\times\\mathcal\{M\}\)denote the joint belief over\(St,Mt\)\(S\_\{t\},M\_\{t\}\), with marginalsμtS\\mu^\{S\}\_\{t\}andμtM\\mu^\{M\}\_\{t\}\. We define the Wasserstein integration index by

Integration=1T​∑t=1TW2​\(μtS​M,μtS⊗μtM\)\.\\mathrm\{Integration\}=\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}W\_\{2\}\\\!\\bigl\(\\mu^\{SM\}\_\{t\},\\;\\mu^\{S\}\_\{t\}\\otimes\\mu^\{M\}\_\{t\}\\bigr\)\.
###### Definition 6\.1\(Causal Learning Regime\)\.

Given thresholdsαS\>0\\alpha\_\{S\}\>0,αH\>0\\alpha\_\{H\}\>0,αI\>0\\alpha\_\{I\}\>0, we say the system has achieved a*Causal Learning Regime*\(CLR\) if

Stability≥αS,Coherence≥αH,Integration≥αI,\\mathrm\{Stability\}\\geq\\alpha\_\{S\},\\quad\\mathrm\{Coherence\}\\geq\\alpha\_\{H\},\\quad\\mathrm\{Integration\}\\geq\\alpha\_\{I\},and the recursionSt←Ht−1S\_\{t\}\\leftarrow H\_\{t\-1\}is active\.

## 7Stochastic Mirror Dynamics and Basin\-Wise Self\-Manifolds

LetYt:=\(St,Ht,Mt\)∈𝒴Y\_\{t\}:=\(S\_\{t\},H\_\{t\},M\_\{t\}\)\\in\\mathcal\{Y\}and write the induced update as

Yt=Gθ​\(Yt−1\)\+ζt,Y\_\{t\}=G\_\{\\theta\}\(Y\_\{t\-1\}\)\+\\zeta\_\{t\},\(11\)with noiseζt\\zeta\_\{t\}\(representing both environmental variability and algorithmic/inference variability\)\.

This induces a Markov kernelKθ​\(y,⋅\)K\_\{\\theta\}\(y,\\cdot\)on𝒴\\mathcal\{Y\}and hence a Markov operator𝒯θ\\mathcal\{T\}\_\{\\theta\}on𝒫2​\(𝒴\)\\mathcal\{P\}\_\{2\}\(\\mathcal\{Y\}\)by𝒯θ​\(μ\)=∫Kθ​\(y,⋅\)​𝑑μ​\(y\)\\mathcal\{T\}\_\{\\theta\}\(\\mu\)=\\int K\_\{\\theta\}\(y,\\cdot\)\\,d\\mu\(y\)\.

### 7\.1Coherent basins and local contractivity

We allow the possibility of multiple coherent basins \(metastable regions or distinct ergodic components\) within which dynamics are contractive\.

###### Definition 7\.1\(Coherent basin\)\.

A measurable setℬ⊂𝒴\\mathcal\{B\}\\subset\\mathcal\{Y\}is a*coherent basin*if \(i\) it has positive probability under typical belief trajectories in CLR, and \(ii\) transitions leavingℬ\\mathcal\{B\}are rare on the time scales of interest, so that the dynamics restricted toℬ\\mathcal\{B\}are approximately closed\.

For a basinℬ\\mathcal\{B\}, define the*basin\-restricted \(renormalized\) operator*on measures supported inℬ\\mathcal\{B\}by

𝒯θ,ℬ​\(μ\):=\(𝒯θ​\(μ\)\)\|ℬ\(𝒯θ​\(μ\)\)​\(ℬ\),μ∈𝒫2​\(ℬ\),\\mathcal\{T\}\_\{\\theta,\\mathcal\{B\}\}\(\\mu\):=\\frac\{\\left\(\\mathcal\{T\}\_\{\\theta\}\(\\mu\)\\right\)\\big\|\_\{\\mathcal\{B\}\}\}\{\\left\(\\mathcal\{T\}\_\{\\theta\}\(\\mu\)\\right\)\(\\mathcal\{B\}\)\},\\qquad\\mu\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{B\}\),whenever the denominator is positive\. This captures internal evolution conditioned on remaining in the basin\.

\(AB\) Local Wasserstein contractivity\.There exists0<Lℬ<10<L\_\{\\mathcal\{B\}\}<1such that for allμ,ν∈𝒫2​\(ℬ\)\\mu,\\nu\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{B\}\),

W2​\(𝒯θ,ℬ​\(μ\),𝒯θ,ℬ​\(ν\)\)≤Lℬ​W2​\(μ,ν\)\.W\_\{2\}\(\\mathcal\{T\}\_\{\\theta,\\mathcal\{B\}\}\(\\mu\),\\mathcal\{T\}\_\{\\theta,\\mathcal\{B\}\}\(\\nu\)\)\\leq L\_\{\\mathcal\{B\}\}\\,W\_\{2\}\(\\mu,\\nu\)\.
\(BB\) Second\-moment boundedness\.Along the basin\-restricted evolution, second moments stay finite\.

###### Definition 7\.2\(Self\-manifold, basin\-wise\)\.

Letπθ,ℬ∈𝒫2​\(ℬ\)\\pi\_\{\\theta,\\mathcal\{B\}\}\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{B\}\)be an invariant measure of𝒯θ,ℬ\\mathcal\{T\}\_\{\\theta,\\mathcal\{B\}\}\(when it exists\)\. The*self\-manifold in basinℬ\\mathcal\{B\}*is

ℳS​\(ℬ\):=supp​\(πθ,ℬ\)\.\\mathcal\{M\}\_\{S\}\(\\mathcal\{B\}\):=\\mathrm\{supp\}\(\\pi\_\{\\theta,\\mathcal\{B\}\}\)\.

###### Theorem 7\.3\(Existence and uniqueness of basin\-wise self\-manifold\)\.

Under \(AB\) and \(BB\), there exists a unique invariant measureπθ,ℬ\\pi\_\{\\theta,\\mathcal\{B\}\}of𝒯θ,ℬ\\mathcal\{T\}\_\{\\theta,\\mathcal\{B\}\}and for any initialμ0∈𝒫2​\(ℬ\)\\mu\_\{0\}\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{B\}\)one has

W2​\(𝒯θ,ℬt​\(μ0\),πθ,ℬ\)→0\.W\_\{2\}\(\\mathcal\{T\}\_\{\\theta,\\mathcal\{B\}\}^\{\\,t\}\(\\mu\_\{0\}\),\\pi\_\{\\theta,\\mathcal\{B\}\}\)\\to 0\.

###### Proof\.

By \(AB\),𝒯θ,ℬ\\mathcal\{T\}\_\{\\theta,\\mathcal\{B\}\}is a strict contraction on the complete metric space\(𝒫2​\(ℬ\),W2\)\(\\mathcal\{P\}\_\{2\}\(\\mathcal\{B\}\),W\_\{2\}\)\. Banach’s fixed point theorem yields a unique fixed pointπθ,ℬ\\pi\_\{\\theta,\\mathcal\{B\}\}and convergence of iterates\. Assumption \(BB\) ensures the fixed point lies in𝒫2​\(ℬ\)\\mathcal\{P\}\_\{2\}\(\\mathcal\{B\}\)\. ∎

## 8Differentiability of Subjective Time

In the Wasserstein framework, subjective time is not a trajectory of pointsYt∈𝒴Y\_\{t\}\\in\\mathcal\{Y\}but a trajectory of beliefsμt∈𝒫2​\(𝒴\)\\mu\_\{t\}\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{Y\}\)\. The natural notion of regularity is therefore Wasserstein continuity and differentiability of the belief curvet↦μtt\\mapsto\\mu\_\{t\}\.

###### Definition 8\.1\(Absolutely continuous curve in𝒫2\\mathcal\{P\}\_\{2\}\)\.

A curve\(μt\)t∈\[0,T\]⊂𝒫2​\(𝒴\)\(\\mu\_\{t\}\)\_\{t\\in\[0,T\]\}\\subset\\mathcal\{P\}\_\{2\}\(\\mathcal\{Y\}\)is*absolutely continuous*if there existsm∈L1​\(\[0,T\]\)m\\in L^\{1\}\(\[0,T\]\)such that

W2​\(μs,μt\)≤∫stm​\(r\)​𝑑rfor all​0≤s≤t≤T\.W\_\{2\}\(\\mu\_\{s\},\\mu\_\{t\}\)\\leq\\int\_\{s\}^\{t\}m\(r\)\\,dr\\qquad\\text\{for all \}0\\leq s\\leq t\\leq T\.The*metric derivative*at timettis

\|μt′\|:=limh→0W2​\(μt\+h,μt\)\|h\|\|\\mu^\{\\prime\}\_\{t\}\|:=\\lim\_\{h\\to 0\}\\frac\{W\_\{2\}\(\\mu\_\{t\+h\},\\mu\_\{t\}\)\}\{\|h\|\}whenever this limit exists\.

###### Lemma 8\.2\(Wasserstein regularity of belief trajectories\)\.

Assume the setting of Theorem[7\.3](https://arxiv.org/html/2610.08792#S7.Thmtheorem3)on a basinℬ\\mathcal\{B\}\. If the Markov operator𝒯θ,ℬ\\mathcal\{T\}\_\{\\theta,\\mathcal\{B\}\}satisfies

W2​\(𝒯θ,ℬ​\(μ\),𝒯θ,ℬ​\(ν\)\)≤Lℬ​W2​\(μ,ν\)W\_\{2\}\(\\mathcal\{T\}\_\{\\theta,\\mathcal\{B\}\}\(\\mu\),\\mathcal\{T\}\_\{\\theta,\\mathcal\{B\}\}\(\\nu\)\)\\leq L\_\{\\mathcal\{B\}\}\\,W\_\{2\}\(\\mu,\\nu\)withLℬ<1L\_\{\\mathcal\{B\}\}<1, and the per\-step Wasserstein displacement is uniformly bounded,

W2​\(𝒯θ,ℬ​\(μ\),μ\)≤Cfor all​μ∈𝒫2​\(ℬ\),W\_\{2\}\(\\mathcal\{T\}\_\{\\theta,\\mathcal\{B\}\}\(\\mu\),\\mu\)\\leq C\\qquad\\text\{for all \}\\mu\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{B\}\),then any piecewise\-geodesic interpolation of the discrete belief sequence\(μt\)t=0,1,2,…\(\\mu\_\{t\}\)\_\{t=0,1,2,\\ldots\}is absolutely continuous with uniformly bounded metric derivative\. In particular, the metric derivative\|μt′\|\|\\mu^\{\\prime\}\_\{t\}\|exists almost everywhere along the interpolated curve\.

###### Proof\.

By the bounded displacement assumption, consecutive beliefs satisfyW2​\(μt\+1,μt\)≤CW\_\{2\}\(\\mu\_\{t\+1\},\\mu\_\{t\}\)\\leq C\. A piecewise\-geodesic \(constant\-speed\) interpolation on each unit interval\[t,t\+1\]\[t,t\+1\]has metric derivative at mostCCalmost everywhere\. The curve is therefore Lipschitz in theW2W\_\{2\}metric with constantCC, and by standard results for absolutely continuous curves in\(𝒫2,W2\)\(\\mathcal\{P\}\_\{2\},W\_\{2\}\)\(cf\. Ambrosio–Gigli–Savaré\), the metric derivative exists for almost everytt\. ∎

## 9Wassersteinε\\varepsilon\-Necks and Basin Selection

Even within CLR, belief dynamics may admit two coherent basins separated by a transport bottleneck\. This is the analogue of neck formation, but in metric–measure geometry rather than Riemannian curvature\. Following Perelman’s use ofε\\varepsilon\-necks \(and strongε\\varepsilon\-necks\) as the canonical surgery locus in Ricci flow\[[2](https://arxiv.org/html/2610.08792#bib.bib2)\], we formalize*Wassersteinε\\varepsilon\-necks*as regions which, after suitable normalization, areε\\varepsilon\-close \(in Wasserstein metric\) to a canonical “cylindrical” transport corridor\.

Letℬ1,ℬ2⊂𝒴\\mathcal\{B\}\_\{1\},\\mathcal\{B\}\_\{2\}\\subset\\mathcal\{Y\}be disjoint coherent basins, and letμt∈𝒫2​\(𝒴\)\\mu\_\{t\}\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{Y\}\)denote the belief trajectory\.

### 9\.1A canonical neck model

Fix a reference neck model\(𝒞,ν\)\(\\mathcal\{C\},\\nu\), where𝒞\\mathcal\{C\}is a canonical corridor space \(the “belief cylinder”\) andν∈𝒫2​\(𝒞\)\\nu\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{C\}\)is a reference neck belief\. Concretely,𝒞\\mathcal\{C\}may be taken as a product𝒵×\(−1,1\)\\mathcal\{Z\}\\times\(\-1,1\)equipped with a ground cost for which transport along the longitudinal coordinate dominates, andν\\nua measure concentrated in a thin corridor in𝒞\\mathcal\{C\}\.

To compare a candidate region to the model, we allow a normalization map \(rescaling\)Ψ:𝒩→𝒞\\Psi:\\mathcal\{N\}\\to\\mathcal\{C\}and measure closeness by Wasserstein distance on𝒫2​\(𝒞\)\\mathcal\{P\}\_\{2\}\(\\mathcal\{C\}\)\.

### 9\.2Conductance as a bottleneck functional

LetKθK\_\{\\theta\}be the Markov kernel on𝒴\\mathcal\{Y\}induced by \([11](https://arxiv.org/html/2610.08792#S7.E11)\)\. For a probability measureπ\\pion𝒴\\mathcal\{Y\}\(e\.g\. a long\-run empirical measure on the relevant time window\), define the*one\-step conductance*fromℬ1\\mathcal\{B\}\_\{1\}toℬ2\\mathcal\{B\}\_\{2\}by

Φπ​\(ℬ1→ℬ2\):=∫ℬ1Kθ​\(y,ℬ2\)​𝑑π​\(y\)π​\(ℬ1\)\.\\Phi\_\{\\pi\}\(\\mathcal\{B\}\_\{1\}\\\!\\to\\\!\\mathcal\{B\}\_\{2\}\):=\\frac\{\\int\_\{\\mathcal\{B\}\_\{1\}\}K\_\{\\theta\}\(y,\\mathcal\{B\}\_\{2\}\)\\,d\\pi\(y\)\}\{\\pi\(\\mathcal\{B\}\_\{1\}\)\}\.Intuitively,Φπ\\Phi\_\{\\pi\}measures the effective leakage probability per step\.

###### Definition 9\.1\(Wassersteinε\\varepsilon\-neck\)\.

Fixε\>0\\varepsilon\>0\. A measurable subset𝒩⊂𝒴\\mathcal\{N\}\\subset\\mathcal\{Y\}is a*Wassersteinε\\varepsilon\-neck*betweenℬ1\\mathcal\{B\}\_\{1\}andℬ2\\mathcal\{B\}\_\{2\}if there exist a timett, a normalization mapΨ:𝒩→𝒞\\Psi:\\mathcal\{N\}\\to\\mathcal\{C\}, and the conditional neck belief

μt𝒩:=μt\|𝒩μt​\(𝒩\)∈𝒫2​\(𝒩\),μ^t𝒩:=Ψ\#​μt𝒩∈𝒫2​\(𝒞\),\\mu\_\{t\}^\{\\mathcal\{N\}\}:=\\frac\{\\mu\_\{t\}\|\_\{\\mathcal\{N\}\}\}\{\\mu\_\{t\}\(\\mathcal\{N\}\)\}\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{N\}\),\\qquad\\widehat\{\\mu\}\_\{t\}^\{\\mathcal\{N\}\}:=\\Psi\_\{\\\#\}\\mu\_\{t\}^\{\\mathcal\{N\}\}\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{C\}\),such that:

1. \(i\)\(ε\\varepsilon\-cylindricity\) the normalized neck belief isε\\varepsilon\-close to the model, W2​\(μ^t𝒩,ν\)≤ε;W\_\{2\}\\\!\\bigl\(\\widehat\{\\mu\}\_\{t\}^\{\\mathcal\{N\}\},\\,\\nu\\bigr\)\\leq\\varepsilon;
2. \(ii\)\(routing\) transport between basin\-concentrated measures is effectively routed through𝒩\\mathcal\{N\}in the sense that typical transition paths contributing toKθ​\(⋅,ℬ2\)K\_\{\\theta\}\(\\cdot,\\mathcal\{B\}\_\{2\}\)fromℬ1\\mathcal\{B\}\_\{1\}place non\-negligible intermediate mass in𝒩\\mathcal\{N\};
3. \(iii\)\(bottleneck\) the inter\-basin conductance is small relative to intra\-basin conductance on the relevant window: Φπ​\(ℬ1→ℬ2\)≪1,Φπ​\(ℬ2→ℬ1\)≪1,\\Phi\_\{\\pi\}\(\\mathcal\{B\}\_\{1\}\\\!\\to\\\!\\mathcal\{B\}\_\{2\}\)\\ll 1,\\qquad\\Phi\_\{\\pi\}\(\\mathcal\{B\}\_\{2\}\\\!\\to\\\!\\mathcal\{B\}\_\{1\}\)\\ll 1,while intra\-basin persistence remains high\.

### 9\.3Strong Wassersteinε\\varepsilon\-necks

###### Definition 9\.3\(Strong Wassersteinε\\varepsilon\-neck\)\.

A Wassersteinε\\varepsilon\-neck𝒩\\mathcal\{N\}at timet0t\_\{0\}is*strong*if there existΔ\>0\\Delta\>0and normalization mapsΨt:𝒩→𝒞\\Psi\_\{t\}:\\mathcal\{N\}\\to\\mathcal\{C\}\(fort∈\[t0−Δ,t0\]t\\in\[t\_\{0\}\-\\Delta,t\_\{0\}\]\) such that for allttin the window the normalized conditional beliefs satisfy

W2​\(μ^t𝒩,ν\)≤ε\.W\_\{2\}\\\!\\bigl\(\\widehat\{\\mu\}\_\{t\}^\{\\mathcal\{N\}\},\\,\\nu\\bigr\)\\leq\\varepsilon\.

### 9\.4Critical slowing nearε\\varepsilon\-necks

### 9\.5Choice as realized continuation in a singular low\-conductance limit

Letλ\>0\\lambda\>0be a parameter that controls inter\-basin communication \(e\.g\. a temperature/noise scale, an entropy weight in inference, or a numerical regularization strength\)\. Suppose that asλ↓0\\lambda\\downarrow 0, the dynamics approach a limiting evolution for which the basins become effectively invariant:

Φπ\(λ\)​\(ℬ1→ℬ2\)→0,Φπ\(λ\)​\(ℬ2→ℬ1\)→0\.\\Phi\_\{\\pi\}^\{\(\\lambda\)\}\(\\mathcal\{B\}\_\{1\}\\\!\\to\\\!\\mathcal\{B\}\_\{2\}\)\\to 0,\\qquad\\Phi\_\{\\pi\}^\{\(\\lambda\)\}\(\\mathcal\{B\}\_\{2\}\\\!\\to\\\!\\mathcal\{B\}\_\{1\}\)\\to 0\.In this regime, the limiting evolution admits multiple globally admissible basin\-wise continuations \(remaining inℬ1\\mathcal\{B\}\_\{1\}or remaining inℬ2\\mathcal\{B\}\_\{2\}\), while finite\-λ\\lambdadynamics realize exactly one of them\.

###### Definition 9\.5\(Choice as basin selection at a strong Wassersteinε\\varepsilon\-neck\)\.

A*choice*is the realized basin continuation event occurring at a*strong Wassersteinε\\varepsilon\-neck*in a regime where inter\-basin conductance is asymptotically negligible\. Concretely, at some timet0t\_\{0\}there exist disjoint coherent basinsℬ1,ℬ2\\mathcal\{B\}\_\{1\},\\mathcal\{B\}\_\{2\}and a strongε\\varepsilon\-neck𝒩\\mathcal\{N\}between them such that:

1. \(i\)\(global underdetermination in the limit\) in the singular limit \(vanishing conductance\), both basin continuations are globally admissible \(i\.e\. the limiting evolution can remain in either basin indefinitely\);
2. \(ii\)\(unique realized continuation\) for finite resolution \(λ\>0\\lambda\>0\), the realized belief trajectory becomes concentrated on exactly one basinℬi\\mathcal\{B\}\_\{i\},i∈\{1,2\}i\\in\\\{1,2\\\}, and subsequently evolves according to the basin\-wise dynamics associated withℬi\\mathcal\{B\}\_\{i\}\.

The selection criterion is*internal*: it depends on the system’s total stabilized belief state \(including microscopic details not captured by coarse observation\), and is therefore*externally inscrutable*\.

## 10Intent as Manifold Projection and Wasserstein Attention

The intent latentItI\_\{t\}occupies a distinguished position in the BMA hierarchy: it is generated bypθ​\(It∣St,Mt,Ht\)p\_\{\\theta\}\(I\_\{t\}\\mid S\_\{t\},M\_\{t\},H\_\{t\}\)and then drives the execution/outcome distributionpθ​\(et∣It\)p\_\{\\theta\}\(e\_\{t\}\\mid I\_\{t\}\)\. In this sense,ItI\_\{t\}mediates between self\-identity \(StS\_\{t\}\), meta\-cognition \(MtM\_\{t\}\), hybrid binding \(HtH\_\{t\}\), and the predicted executionete\_\{t\}\.

### 10\.1Intent as Manifold Projection

###### Proposition 10\.1\(Intent as emergent projection\)\.

Assume Theorem[7\.3](https://arxiv.org/html/2610.08792#S7.Thmtheorem3)holds in a basinℬ\\mathcal\{B\}and that onℳS​\(ℬ\)\\mathcal\{M\}\_\{S\}\(\\mathcal\{B\}\)the hybrid latent is an effectively deterministic functionHt≈hθ​\(St,Mt\)H\_\{t\}\\approx h\_\{\\theta\}\(S\_\{t\},M\_\{t\}\)while the conditional variance ofpθ​\(It∣St,Mt,Ht\)p\_\{\\theta\}\(I\_\{t\}\\mid S\_\{t\},M\_\{t\},H\_\{t\}\)becomes small along learning\. ThenItI\_\{t\}becomes an effectively deterministic map of\(St,Mt\)\(S\_\{t\},M\_\{t\}\)onℳS​\(ℬ\)\\mathcal\{M\}\_\{S\}\(\\mathcal\{B\}\):

It≈πI​\(St,Mt\)on​ℳS​\(ℬ\),I\_\{t\}\\approx\\pi\_\{I\}\(S\_\{t\},M\_\{t\}\)\\quad\\text\{on \}\\mathcal\{M\}\_\{S\}\(\\mathcal\{B\}\),i\.e\., intent approaches a stable projection from the basin\-wise self\-manifold to the space of possible actions\.

###### Proof\.

OnℳS​\(ℬ\)\\mathcal\{M\}\_\{S\}\(\\mathcal\{B\}\), the belief overYt=\(St,Ht,Mt\)Y\_\{t\}=\(S\_\{t\},H\_\{t\},M\_\{t\}\)concentrates on the invariant measureπθ,ℬ\\pi\_\{\\theta,\\mathcal\{B\}\}\. IfHt≈hθ​\(St,Mt\)H\_\{t\}\\approx h\_\{\\theta\}\(S\_\{t\},M\_\{t\}\)is effectively deterministic andVar​\[It∣St,Mt,Ht\]→0\\mathrm\{Var\}\[I\_\{t\}\\mid S\_\{t\},M\_\{t\},H\_\{t\}\]\\to 0, then the compositionIt=𝔼​\[It∣St,Mt,hθ​\(St,Mt\)\]I\_\{t\}=\\mathbb\{E\}\[I\_\{t\}\\mid S\_\{t\},M\_\{t\},h\_\{\\theta\}\(S\_\{t\},M\_\{t\}\)\]becomes a deterministic functionπI​\(St,Mt\)\\pi\_\{I\}\(S\_\{t\},M\_\{t\}\)on the support of the invariant measure\. ∎

### 10\.2Connection to Mirror Neurons

Di Pellegrino et al\.\[[5](https://arxiv.org/html/2610.08792#bib.bib5)\]reported neurons in macaque premotor cortex \(area F5\) that discharge both during action execution and during observation of similar goal\-directed actions\. This observation–execution mirroring provides biological motivation for the BMA core: learning a mapping from observed events to predicted executions/outcomes\.

### 10\.3From Euclidean to Wasserstein Attention

Transformer attention uses dot\-product similarity inℝd\\mathbb\{R\}^\{d\}\[[4](https://arxiv.org/html/2610.08792#bib.bib4)\]\. In BMA, the native objects are*beliefs*; the principled analogue replaces Euclidean similarity with Wasserstein proximity\.

###### Definition 10\.3\(Wasserstein Attention\)\.

Letq=𝒩​\(μq,Σq\)q=\\mathcal\{N\}\(\\mu\_\{q\},\\Sigma\_\{q\}\)be a query belief and\{kg=𝒩​\(μg,Σg\)\}g=1G\\\{k\_\{g\}=\\mathcal\{N\}\(\\mu\_\{g\},\\Sigma\_\{g\}\)\\\}\_\{g=1\}^\{G\}be grammar key beliefs\. The Wasserstein attention weights are

αg=exp⁡\(−W22​\(q,kg\)/τ\)∑g′exp⁡\(−W22​\(q,kg′\)/τ\),\\alpha\_\{g\}=\\frac\{\\exp\(\-W\_\{2\}^\{2\}\(q,k\_\{g\}\)/\\tau\)\}\{\\sum\_\{g^\{\\prime\}\}\\exp\(\-W\_\{2\}^\{2\}\(q,k\_\{g^\{\\prime\}\}\)/\\tau\)\},whereτ\>0\\tau\>0is a temperature parameter andW22W\_\{2\}^\{2\}is the squared22\-Wasserstein distance\.

### 10\.4The Transformation Signature

A critical insight is that causal grammar identification is not determined from observation alone: the active grammar is revealed by the*relationship*between observation and execution\.

###### Definition 10\.5\(Transformation Signature\)\.

Given observationoto\_\{t\}and executionete\_\{t\}, the transformation signature is a belief encoding the causal relationship:

σt=ℰϕ​\(ot,et\)∈𝒫2​\(ℝd\),\\sigma\_\{t\}=\\mathcal\{E\}\_\{\\phi\}\(o\_\{t\},e\_\{t\}\)\\in\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{d\}\),whereℰϕ\\mathcal\{E\}\_\{\\phi\}may extract multiple views of the transformation, e\.g\. additive \(et−ote\_\{t\}\-o\_\{t\}\), multiplicative \(et⊘ote\_\{t\}\\oslash o\_\{t\}\), and interaction \(ot⊙eto\_\{t\}\\odot e\_\{t\}\) components\.

### 10\.5Intent as the Attention Query

###### Proposition 10\.6\(Intent as emergent query\)\.

In a BMA agent with multiple causal grammars\{𝒢g\}g=1G\\\{\\mathcal\{G\}\_\{g\}\\\}\_\{g=1\}^\{G\}, the intent beliefItI\_\{t\}is shaped by free\-energy minimization to encode \(a representation of\) the transformation signatureσt\\sigma\_\{t\}, so that Wasserstein attention withItI\_\{t\}as query routes to the active grammar:

g∗​\(t\)≈arg⁡ming⁡W22​\(It,kg\),g^\{\*\}\(t\)\\approx\\arg\\min\_\{g\}W\_\{2\}^\{2\}\(I\_\{t\},k\_\{g\}\),wherekgk\_\{g\}is a learned prototype belief for grammar𝒢g\\mathcal\{G\}\_\{g\}\.

###### Proof\.

Minimizing the prediction term in \([8](https://arxiv.org/html/2610.08792#S5.E8)\) requires accurate prediction ofete\_\{t\}fromoto\_\{t\}\. When multiple grammars exist, this requires identifying which grammar mapsoto\_\{t\}toete\_\{t\}on the current instance\. The transformation signatureσt\\sigma\_\{t\}is a sufficient statistic for this routing in the grammar library\. SinceItI\_\{t\}mediates prediction ofete\_\{t\}viapθ​\(et∣It\)p\_\{\\theta\}\(e\_\{t\}\\mid I\_\{t\}\), gradient descent shapesItI\_\{t\}toward carrying the routing\-relevant information contained inσt\\sigma\_\{t\}, while keyskgk\_\{g\}learn to represent grammar\-typical signatures\. ∎

### 10\.6Grammar Keys as Belief Prototypes

###### Definition 10\.7\(Grammar key as Fréchet mean\)\.

For grammar𝒢g\\mathcal\{G\}\_\{g\}, the grammar keykg∈𝒫2​\(ℝd\)k\_\{g\}\\in\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{d\}\)may be defined as the Fréchet mean of transformation signatures:

kg=arg⁡mink⁡𝔼\(o,e\)∼𝒢g​\[W22​\(σ​\(o,e\),k\)\]\.k\_\{g\}=\\arg\\min\_\{k\}\\mathbb\{E\}\_\{\(o,e\)\\sim\\mathcal\{G\}\_\{g\}\}\\\!\\left\[W\_\{2\}^\{2\}\(\\sigma\(o,e\),k\)\\right\]\.

### 10\.7The Complete Wasserstein Attention Mechanism

###### Theorem 10\.9\(Wasserstein grammar attention\)\.

Let\{\(𝒢g,Wg,kg\)\}g=1G\\\{\(\\mathcal\{G\}\_\{g\},W\_\{g\},k\_\{g\}\)\\\}\_\{g=1\}^\{G\}be a grammar library with transformationsWgW\_\{g\}and key beliefskgk\_\{g\}\. Given observationoto\_\{t\}and a query belief \(encoded byItI\_\{t\}\), the predicted execution is

e^t=∑g=1Gαg​\(It\)​Wg​\(ot\),αg​\(It\)=exp⁡\(−W22​\(It,kg\)/τ\)∑g′exp⁡\(−W22​\(It,kg′\)/τ\)\.\\hat\{e\}\_\{t\}=\\sum\_\{g=1\}^\{G\}\\alpha\_\{g\}\(I\_\{t\}\)\\,W\_\{g\}\(o\_\{t\}\),\\qquad\\alpha\_\{g\}\(I\_\{t\}\)=\\frac\{\\exp\(\-W\_\{2\}^\{2\}\(I\_\{t\},k\_\{g\}\)/\\tau\)\}\{\\sum\_\{g^\{\\prime\}\}\\exp\(\-W\_\{2\}^\{2\}\(I\_\{t\},k\_\{g^\{\\prime\}\}\)/\\tau\)\}\.Asτ→0\\tau\\to 0, the mechanism approaches hard grammar selection:e^t→Wg∗​\(ot\)\\hat\{e\}\_\{t\}\\to W\_\{g^\{\*\}\}\(o\_\{t\}\)withg∗=arg⁡ming⁡W22​\(It,kg\)g^\{\*\}=\\arg\\min\_\{g\}W\_\{2\}^\{2\}\(I\_\{t\},k\_\{g\}\)\.

### 10\.8Symbolic Grammars and the Generalization Theorem

The preceding development allows grammar transformationsWgW\_\{g\}to be neural networks\. A limitation emerges: neural approximators need not generalize exact mathematical laws beyond their training domain\. This motivates a distinction between*neural grammars*\(flexible approximation\) and*symbolic grammars*\(exact computation\)\.

###### Theorem 10\.10\(A basic extrapolation gap for nonlinear growth\)\.

Letf:ℝn→ℝf:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}be a function with superlinear growth, i\.e\. there existk\>1k\>1andc\>0c\>0such that\|f​\(x\)\|≥c​‖x‖k\|f\(x\)\|\\geq c\\\|x\\\|^\{k\}for large‖x‖\\\|x\\\|\. Letf^θ\\hat\{f\}\_\{\\theta\}be any globally Lipschitz function with constantLθL\_\{\\theta\}\(e\.g\. a bounded\-parameter neural network\)\. Then

lim‖x‖→∞\|f^θ​\(x\)−f​\(x\)\|=∞\.\\lim\_\{\\\|x\\\|\\to\\infty\}\|\\hat\{f\}\_\{\\theta\}\(x\)\-f\(x\)\|=\\infty\.

###### Proof\.

By Lipschitzness,\|f^θ​\(x\)\|≤Lθ​‖x‖\+\|f^θ​\(0\)\|\|\\hat\{f\}\_\{\\theta\}\(x\)\|\\leq L\_\{\\theta\}\\\|x\\\|\+\|\\hat\{f\}\_\{\\theta\}\(0\)\|grows at most linearly, while\|f​\(x\)\|\|f\(x\)\|grows at least like‖x‖k\\\|x\\\|^\{k\}withk\>1k\>1\. Therefore the difference diverges as‖x‖→∞\\\|x\\\|\\to\\infty\. ∎

###### Definition 10\.12\(Neuro\-symbolic grammar\)\.

A*neuro\-symbolic grammar*is a pair\(ℛϕ,𝒢sym\)\(\\mathcal\{R\}\_\{\\phi\},\\mathcal\{G\}\_\{\\mathrm\{sym\}\}\)where:

- •ℛϕ:𝒫2​\(ℝd\)→\{1,…,G\}\\mathcal\{R\}\_\{\\phi\}:\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{d\}\)\\to\\\{1,\\ldots,G\\\}is a neural router \(e\.g\. Wasserstein attention followed by hard selection\),
- •𝒢sym=\{g1,…,gG\}\\mathcal\{G\}\_\{\\mathrm\{sym\}\}=\\\{g\_\{1\},\\ldots,g\_\{G\}\\\}is a library of symbolic grammars \(exact computable functions\)\.

The predicted execution ise^t=gℛϕ​\(It\)​\(ot\)\\hat\{e\}\_\{t\}=g\_\{\\mathcal\{R\}\_\{\\phi\}\(I\_\{t\}\)\}\(o\_\{t\}\)\.

###### Theorem 10\.13\(Perfect generalization given correct routing\)\.

If the routerℛϕ\\mathcal\{R\}\_\{\\phi\}correctly identifies the active grammarg∗g^\{\*\}for an instance, then the executed prediction is exact:

e^t=g∗​\(ot\)=et\.\\hat\{e\}\_\{t\}=g^\{\*\}\(o\_\{t\}\)=e\_\{t\}\.

###### Proof\.

Ifℛϕ​\(It\)=g∗\\mathcal\{R\}\_\{\\phi\}\(I\_\{t\}\)=g^\{\*\}, thene^t=g∗​\(ot\)\\hat\{e\}\_\{t\}=g^\{\*\}\(o\_\{t\}\)\. Since the environment execution iset=g∗​\(ot\)e\_\{t\}=g^\{\*\}\(o\_\{t\}\)under the active grammar, the equality holds\. ∎

### 10\.9The Agency Trajectory: From Observer to Planner

The preceding sections established BMA as a passive inference system: given observations, it infers latent states and learns grammars\. But intelligence requires*agency*—the capacity to act on the world, not merely observe it\. We now formalize a developmental trajectory from passive observation to active planning, showing how the circular architectureSt←Ht−1S\_\{t\}\\leftarrow H\_\{t\-1\}provides the foundation for increasingly sophisticated agency\.

###### Definition 10\.14\(Agency trajectory\)\.

The*agency trajectory*is a sequence of increasingly capable architectures:

Observer→Agent→Composer→Planner,\\text\{Observer\}\\to\\text\{Agent\}\\to\\text\{Composer\}\\to\\text\{Planner\},where each stage subsumes the previous and adds new capabilities\.

#### Stage 1: Observer\.

The system observes world transitions\(ot,ot\+1\)\(o\_\{t\},o\_\{t\+1\}\)without agency\. It learns to recognize which grammar generated the transition\.

#### Stage 2: Agent\.

The system gains an action head that maps intent to action selection and closes the causal loop

St→It→at→ot\+1→Ht→St\+1\.S\_\{t\}\\to I\_\{t\}\\to a\_\{t\}\\to o\_\{t\+1\}\\to H\_\{t\}\\to S\_\{t\+1\}\.

#### Stage 3: Composer\.

The system learns to compose multiple actions to achieve goals unreachable in one step, chunking frequent compositions into higher\-level grammars\.

#### Stage 4: Planner\.

The system adds a world model for lookahead and replanning under uncertainty, and integrates predictions into the hybrid latent:

Ht=g​\(Mt,St,o^t\+1\)\.H\_\{t\}=g\(M\_\{t\},S\_\{t\},\\hat\{o\}\_\{t\+1\}\)\.

## 11Discussion and Conclusion

We introduced a mathematically explicit formulation of the Bayesian Mirror Architecture as a circular latent generative system on belief space\. The core architectural condition is the active recursionSt←Ht−1S\_\{t\}\\leftarrow H\_\{t\-1\}, enforcing self\-referential closure independently of any optimization interpretation\.

We established that variational free energy minimization is the*sole*learning objective\. The system learns by reducing its surprise about observation–execution pairs; no auxiliary consciousness\-targeting losses are imposed\. The learning metrics \(Stability, Coherence, Integration\) are diagnostic measurements that indicate whether the learned model has captured stable causal structure\.

We treated learning as a Wasserstein gradient flow on belief space\. In the simplest limit the learning dynamics reduce to heat\-type diffusion; in general they realize drift–diffusion towards posterior consistency\.

Crucially, we did*not*assume global strict contractivity\. Instead, we introduced basin\-wise dynamics and proved existence and uniqueness of a basin\-restricted invariant measure under local Wasserstein contractivity, defining the self\-manifold as the support of that basin\-wise invariant measure\.

We introduced Wassersteinε\\varepsilon\-necks as transport bottlenecks separating coherent belief basins and interpreted realized basin continuation under vanishing conductance as*choice*\. Finally, we formulated intent as a Wasserstein attention query over causal grammar prototypes and motivated a neuro\-symbolic separation between routing and execution for exact mathematical laws\.

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