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A Bayesian Mirror Architecture for Emergent Consciousness: Circular Hierarchies, Self-Manifolds, and Hybrid Event-Self Binding
Eduardo Righi Capanema de Almeida
TL;DR
The paper frames causal grammar identification as requiring the relationship between observation and execution, and formulates BMA as a circular generative architecture on belief space. It defines self-manifolds basin-wise under local Wasserstein contractivity and interprets basin continuation through ε-necks as choice.
Problem
Causal grammar identification is not determined from observation alone; the active grammar is revealed by the relationship between observation and execution.
Source: §10.4 · p. 12Method
BMA is formulated as a circular latent generative system on belief space, with self-state recursion and intent routing through a Wasserstein attention query over causal grammar prototypes.
Source: §11 · p. 14Results
Without assuming global strict contractivity, the paper establishes existence and uniqueness of a basin-restricted invariant measure under local Wasserstein contractivity and defines the self-manifold as its support.
Source: §11 · p. 14Takeaways & Limitations
Wasserstein ε-necks are transport bottlenecks between coherent belief basins, and realized basin continuation under vanishing conductance is interpreted as choice.
Source: §11 · p. 14Neural approximators need not generalize exact mathematical laws beyond their training domain; exact symbolic execution is presented as the mechanism for unbounded laws when extrapolation correctness is required.
Source: §10.8 · p. 13
Abstract
from arXiv · showhide
We present a foundational formulation of the Bayesian Mirror Architecture (BMA), a self-referential generative framework in which sensory abstractions, meta-abstractions, and a self-latent interact through circular recursion. The defining constraint is a closed update S_t <- H_{t-1}, where a hybrid event-self latent H_t binds self-representations to abstract world models and reinjects this coupling into the self-state. Consciousness, in a restricted sense, is not an optimization objective nor a semantic label, but an architectural property of systems possessing this circular structure. Because inference operates over posterior beliefs, BMA's intrinsic state space is a space of probability measures equipped with optimal-transport geometry. Stability and coherence are formulated in the 2-Wasserstein metric on P_2, yielding coordinate-free notions of self-stability and hybrid coherence along belief trajectories. We define a Causal Learning Regime (CLR) via bounds on Wasserstein belief drift together with an integration index capturing sustained coupling between self and world latents. CLR diagnoses whether the environment contains learnable causal structure; it is not a marker of consciousness. Global strict contractivity is not required: BMA may exhibit multiple coherent basins. We define self-manifolds basin-wise as supports of invariant measures under local Wasserstein contractivity. We identify Wasserstein epsilon-necks, transport bottlenecks where basins decouple, yielding a unique realized continuation in a vanishing-conductance limit. We interpret this selection as choice: internally determined yet externally unpredictable at finite resolution. Learning proceeds via variational free-energy minimization, with stability and agency emerging from what the environment affords to learn.
1 Introduction
The Bayesian Mirror Architecture (BMA) is a circular generative system whose hybrid event–self binding feeds back into the self-state. It frames selfhood and causal learning through belief dynamics and basin-wise geometry, while treating consciousness as an architectural property.
1 Introduction
- The architecture begins from perception-induced abstractions and iterated meaning-preserving maps that yield meta-abstractions and self-representations.Source: §1 · p. 2
- BMA’s defining constraint is compulsory self-recursion: a hybrid event–self binding feeds back into the self-state.The architecture is circular, not merely hierarchical.Source: §1 · p. 2
- Consciousness is defined as an architectural property of circular self-recursion, whereas the Causal Learning Regime diagnoses whether the environment affords learnable causal structure.The architecture also includes latent variables for events, abstractions, meta-abstractions, self-latents, hybrid bindings, and intent.Source: §1 · p. 2
- Because inference yields posterior beliefs, BMA uses 2-Wasserstein geometry to define self-stability and hybrid coherence along belief trajectories.Attractors are formulated through local Wasserstein contraction on basin-restricted dynamics.Source: §1 · p. 2
- The paper aims to define self-manifolds from basin-restricted invariant measures and interpret basin continuation through Wasserstein ε-necks as choice.It also outlines an agency trajectory from observation to goal-directed action, compositional planning, and model-based lookahead with replanning.Source: §1 · p. 2
2 Latent Hierarchy and Generative Structure
The generative hierarchy assigns functional roles to sensory events, abstractions, self-latents, hybrid bindings, and intent. Its minimal closure condition updates the self-state from the previous hybrid binding, with an agency extension incorporating predicted observations.
2 Latent Hierarchy and Generative Structure
- The model uses discrete time, with t = 1, 2, . . . , T.Source: §2 · p. 2
- The model’s latent variables represent sensory events, concept- and action-level abstractions, meta-abstractions, self-latents, hybrid event–self bindings, and intent.Source: §2 · p. 2
- The system observes o_t as an action or sensory event and predicts its execution or outcome e_t under the current causal grammar.Source: §2 · p. 3
- The active recursion S_t ← H_{t−1} is the minimal architectural condition that enforces self-referential closure.Source: §2 · p. 3
- In the agency extension, H_t incorporates world-model predictions and an expected next observation, allowing anticipated futures to shape S_{t+1}.Source: §2 · p. 3
3 Wasserstein Geometry of Mirror Beliefs
BMA treats inferred posterior marginals and joint beliefs as points in probability-measure space, using optimal-transport geometry to compare them. The 2-Wasserstein distance is defined for measures with finite second moments through couplings with the specified marginals.
3 Wasserstein Geometry of Mirror Beliefs
- P_2(X) denotes Borel probability measures on the Polish metric space X with finite second moment.Source: §3 · p. 3
- The 2-Wasserstein distance is introduced for μ, ν in P_2(X), with Π(μ, ν) denoting couplings whose marginals are μ and ν.Source: §3 · p. 3
- BMA applies W_2 to posterior marginals induced by inference and to joint beliefs over self, hybrid-binding, and meta-abstraction latents.Source: §3 · p. 3
4 Learning as Wasserstein Gradient Flow
Learning is framed as belief flow on P_2 rather than minimization of a static scalar functional. A reference Wasserstein gradient flow links data-fit and entropic smoothing, while its discrete-time JKO form makes each update an optimal-transport step.
4 Learning as Wasserstein Gradient Flow
- Learning is treated as a flow of beliefs on P_2 driven by an energy encoding posterior consistency, rather than as minimization of a static scalar functional.Source: §4 · p. 3
4.1 A reference form of the flow
- Under standard hypotheses on the energy, its W_2-gradient flow takes drift–diffusion (Fokker–Planck) form.The reference energy pairs a data-fit or negative log-likelihood potential with an entropic term for diffusion, regularization, exploration, and smoothing.Source: §4.1 · pp. 3–4
- When the data-fit potential is zero, the reference flow reduces to the heat equation, representing pure entropic smoothing.Source: §4.1 · p. 4
4.2 Discrete-time learning via JKO
- For discrete-time BMA, the JKO scheme discretizes the Wasserstein gradient flow.Source: §4.2 · p. 4
- Each JKO update trades off energy decrease against Wasserstein displacement, making learning a controlled deformation of belief.Source: §4.2 · p. 4
5 Variational Free Energy as the Sole Learning Objective
BMA learns by minimizing variational free energy over observation–execution pairs; stability, coherence, and integration are consequences or diagnostics, not additional objectives. Its circular self-update is architectural, while CLR diagnoses whether the environment’s causal structure has been learned.
5.1 The variational free energy functional
- The variational free energy combines prediction error and a complexity penalty, with β controlling the penalty’s strength.The recognition model maps observations to latent beliefs, while the generative model predicts executions or outcomes.Source: §5.1 · p. 4
5.2 Decomposition over the latent hierarchy
- BMA’s mirror interpretation is that the system observes an action or sensory event and learns to predict its execution or outcome.The self-latent’s circular constraint, S_t ← H_{t−1}, is enforced architecturally rather than by an additional KL term.Source: §5.1 · p. 4, §5.2 · p. 5
5.3 Learning dynamics
- Free energy is BMA’s sole learning objective; belief stability, hybrid coherence, and self–world integration emerge from its minimization rather than separate optimization targets.Learning metrics are diagnostic measurements applied to trained belief trajectories.Source: §5.3 · p. 5, §5.4 · p. 5
5.4 Relationship to learning metrics
- CLR thresholds on stability, coherence, and integration diagnose causal learning; predictable environments can yield high stability and coherence, while noisy ones may not.The metric differences reflect environmental causal learnability, not a system’s degree of consciousness.Source: §5.4 · p. 5
- The circular BMA architecture is the structural precondition for consciousness, whereas reaching CLR additionally depends on environmental causal structure and its discovery through free-energy minimization.A BMA system is architecturally conscious regardless of whether it reaches CLR.Source: §5.4 · p. 6
6 Learning Metrics and Definitions
The section defines stability, coherence, and integration as diagnostics of learned belief trajectories, then uses CLR to distinguish learnable causal structure from both chaotic fragmentation and rigid fixation.
6 Learning Metrics and Definitions
- Stability and coherence are bounded monotone transforms of average squared Wasserstein drift for self and hybrid beliefs, respectively.The transforms use calibration scales and a small stabilizer to avoid division by zero.Source: §6 · p. 6
- Event–self integration is defined using the joint belief over self and world latents.The passage identifies the joint belief over (S_t, M_t).Source: §6 · p. 6
- CLR is a diagnostic of causal learning, not consciousness; its metrics depend on whether the environment affords stable causal structure to learn.The recursion S_t ← H_{t−1} is active in the defined regime.Source: §6 · pp. 6–7
- CLR excludes both low-stability, low-coherence fragmentation and zero-drift fixation with vanishing integration.The latter is characterized as a collapsed or non-responsive model.Source: §6 · p. 7
- In active settings, stability concerns bounded self-beliefs under self-generated actions, coherence concerns multi-step prediction, and integration concerns world-model improvement over reactive policy.These measures remain diagnostic throughout the agency trajectory.Source: §6 · p. 7
7 Stochastic Mirror Dynamics and Basin-Wise Self-Manifolds
BMA’s stochastic dynamics induce a Markov evolution over belief measures, while self-manifolds are defined basin-wise rather than requiring global contractivity. Under local contractivity and bounded second moments, each basin has a unique invariant measure.
7 Stochastic Mirror Dynamics and Basin-Wise Self-Manifolds
- The stochastic update with environmental and inference noise induces a Markov kernel and an operator on probability measures.The state combines self, hybrid, and world latents.Source: §7 · p. 7
7.1 Coherent basins and local contractivity
- A coherent basin has positive probability along typical CLR belief trajectories and rare exits, making its restricted dynamics approximately closed.Multiple coherent basins may coexist, with persistent self-geometry defined relative to the realized basin continuation.Source: §7.1 · pp. 7–8
- Under local Wasserstein contractivity and second-moment boundedness, a basin has a unique invariant measure, and iterates from any initial belief converge to it.The basin-wise self-manifold is the support of that invariant measure.Source: §7.1 · p. 8
8 Differentiability of Subjective Time
Subjective time is represented as a trajectory of beliefs in Wasserstein space rather than a trajectory of latent-state points. With contractive evolution and bounded per-step displacement, interpolated belief trajectories are absolutely continuous and have bounded metric derivative.
8 Differentiability of Subjective Time
- In the Wasserstein framework, subjective time follows beliefs in P_2(Y), whose regularity is described by continuity and differentiability of the belief curve.It is not a trajectory of points in the latent state space.Source: §8 · p. 8
- With basin-wise contraction and uniformly bounded per-step Wasserstein displacement, piecewise-geodesic interpolation yields an absolutely continuous belief curve with bounded metric derivative.The metric derivative exists almost everywhere along the interpolated curve.Source: §8 · p. 8
- In CLR, bounded metric derivative means subjective experience does not jump discontinuously but deforms smoothly almost everywhere in Wasserstein geometry.The metric derivative quantifies the instantaneous rate of belief change.Source: §8 · p. 9
9 Wasserstein ε-Necks and Basin Selection
The section defines Wasserstein ε-necks as normalized transport corridors between coherent basins and treats their low-conductance limit as a locus where one basin continuation is realized despite multiple admissible limiting continuations.
9 Wasserstein ε-Necks and Basin Selection
- Wasserstein ε-necks model transport bottlenecks between coherent basins as regions close to a canonical cylindrical corridor after normalization, following Perelman’s ε-neck terminology [2].The paper’s notion is metric–measure: closeness is measured in Wasserstein distance rather than curvature.Source: §9 · p. 9, §9.2 · p. 10
9.1 A canonical neck model
- The canonical neck model is a corridor space C, such as Z × (−1, 1), with transport along its longitudinal coordinate dominating and a reference belief concentrated in a thin corridor.Source: §9.1 · p. 9
9.2 Conductance as a bottleneck functional
- A neck routes typical transitions between basins through N, where inter-basin conductance is small relative to intra-basin conductance and leakage per step is low.Source: §9.2 · pp. 9–10
9.3 Strong Wasserstein ε-necks
- A strong Wasserstein ε-neck requires normalized conditional beliefs to satisfy the neck condition throughout a preceding time window.The window extends from t0 − ∆ to t0 and uses time-indexed normalization maps.Source: §9.3 · p. 10
9.4 Critical slowing near ε-necks
- Low inter-basin conductance with high within-basin persistence makes switching rare and typical basin-switching times grow.Source: §9.4 · p. 10
9.5 Choice as realized continuation in a singular low-conductance limit
- As λ approaches zero, both basin-wise continuations remain globally admissible, yet finite-resolution dynamics realize exactly one continuation, selected by the system’s stabilized belief state and its microstructure.The authors interpret this internally determined, externally inscrutable selection as choice rather than an injected stochastic label.Source: §9.5 · pp. 10–11
10 Intent as Manifold Projection and Wasserstein Attention
The section develops intent as a basin-wise projection and a belief-based query for selecting causal grammars. It then extends the mechanism to symbolic execution and a staged progression from observation to planning.
10.1 Intent as Manifold Projection
- Under basin-wise concentration and low conditional variance, intent becomes a stable projection from the self-manifold to action space, preserving basin identity while compressing self-state into actionable form.Source: §10.1 · p. 11
10.2 Connection to Mirror Neurons; 10.3 From Euclidean to Wasserstein Attention
- Mirror-neuron observation–execution mirroring motivates mapping observed events to predicted outcomes, while BMA replaces Euclidean dot-product attention with Wasserstein proximity between beliefs.Source: §10.2 · p. 11, §10.3 · p. 11
10.4 The Transformation Signature
- The transformation signature is a belief encoding the causal relation between observation and execution, potentially using additive, multiplicative, and interaction views.Source: §10.4 · p. 12
10.5 Intent as the Attention Query
- Free-energy minimization shapes intent to encode the transformation signature, allowing Wasserstein attention to route to the active grammar for the current instance.The signature supplies routing-relevant information for predicting execution across multiple grammars.Source: §10.5 · p. 12
10.6 Grammar Keys as Belief Prototypes
- A grammar key may be the Fréchet mean of its transformation signatures, providing a belief prototype for that grammar.Source: §10.6 · p. 12
10.7 The Complete Wasserstein Attention Mechanism
- The complete Wasserstein attention mechanism predicts execution using the selected grammar, approaching hard grammar selection as τ → 0.Source: §10.7 · p. 13
10.8 Symbolic Grammars and the Generalization Theorem
- With correct routing, execution is exact; neural approximators may not extrapolate exact laws beyond training, motivating symbolic execution for unbounded laws when extrapolation correctness is required.The neuro-symbolic form pairs a neural router with a library of exact computable symbolic grammars.Source: §10.8 · p. 13
10.9 The Agency Trajectory: From Observer to Planner
- The agency trajectory progresses from observing and identifying transition grammars to acting, composing actions, and planning with a predictive world model integrated into the hybrid latent.Source: §10.9 · p. 14
11 Discussion and Conclusion
The conclusion frames BMA as a circular latent generative system whose active self-recursion closes the self-state update. It links belief-space learning and basin-wise dynamics to accounts of selfhood and choice.
11 Discussion and Conclusion
- BMA’s active recursion closes the self-state update, independently of any optimization interpretation.The conclusion presents BMA as a circular latent generative system.Source: §11 · p. 14
- Variational free energy is the sole learning objective; stability, coherence, and integration diagnose whether learning captured stable causal structure.The system learns by reducing surprise about observation–execution pairs, without consciousness-targeting losses.Source: §11 · p. 14
- Learning is treated as a Wasserstein gradient flow, reducing to heat-type diffusion in the simplest limit and otherwise drifting toward posterior consistency.Updates are described as drift–diffusion toward posterior consistency.Source: §11 · p. 14
- Local Wasserstein contractivity yields a unique basin-restricted invariant measure, whose support defines the self-manifold without requiring global strict contractivity.The conclusion defines the self-manifold basin-wise.Source: §11 · p. 14
- Wasserstein ε-necks separate coherent belief basins as transport bottlenecks; vanishing conductance yields realized basin continuation interpreted as choice.Intent is also formulated as a Wasserstein attention query over causal grammar prototypes, with routing separated from execution for exact mathematical laws.Source: §11 · p. 14