Geometry of Qualia: Isomorphic Mapping between Entanglement Manifold Curvature in Hilbert Space and Subjective Experience

Geometry of Qualia: Isomorphic Mapping between Entanglement Manifold Curvature in Hilbert Space and Subjective Experience Anonymous Author November 26, 2025 Abstract The so-called “qualia” in consciousness is one of the most stubborn problems at the experiential level: there seems to be a chasm from physical processes to “what it feels like to see red”. Developments in information theory and neuroscience suggest that brain activity can be viewed as trajectories on a high-dimensional state space, while information geometry and quantum information geometry provide rigorous characterization of “intrinsic structure of state space”. Based on the discrete ontology framework of quantum cellular automaton (QCA) and optical path conservation, this paper proposes a unified route: viewing subjective experience as the intrinsic geometric and topological structure of entangled state manifolds in high-dimensional Hilbert space. First, we describe the physical state of a conscious system (such as the human brain) as a family of density operators {ρ(θ)}on its Hilbert space, and introduce the quantum Fisher information metric (QFIM) on the parameter manifold. Using quantum information geometry and monotone metric classification theorems, we prove: under natural psychophysical postulates, the subjective experience space can constitute an isometric isomorphism with the “psychological manifold” endowed with QFIM, i.e., psychological distance is equivalent to QFIM geodesic distance in a one-to-one correspondence sense. Second, we introduce Berry connection and curvature on pure state submanifolds, proposing a geometric proposition of “qualia as curvature”: typical qualia (such as color, taste) correspond to submanifolds with non-trivial Berry curvature on the entanglement manifold, with their intensity and type characterized respectively by the magnitude and eigendirection of curvature. Furthermore, within Friston’s free energy principle framework, we define emotional valence (pleasure and pain) as the directional derivative and acceleration of the free energy potential function along the conscious trajectory on the QFIM manifold, characterizing “rapid flow toward low free energy valleys” as positive valence experience and “being trapped in high curvature high free energy basins” as negative valence experience. At the global structure level, we view the set of conscious states as a high-dimensional entanglement manifold, using homology theory and persistent homology tools to define its Betti numbers and related topological invariants, arguing that their values provide lower bounds for the system’s realizable irreducible experiential complexity. This paper finally presents a series of testable engineering proposals, including: constructing approximate QFIM manifolds through neural population activity, using Berry curvature and persistent homology to analyze curvature and hole structure of “psychological space”, and constructing geometric criteria for “whether consciousness exists” scaled by quantum or classical information geometry in artificial intelligence systems. Keywords: Qualia; Consciousness; Quantum information geometry; Quantum Fisher information; Berry curvature; Entanglement manifold; Free energy principle; Neural manifolds; Homology theory 1 1 Introduction & Historical Context 1.1 The “Hard Problem” of Consciousness and Structural Isomorphism Approach In consciousness research, the commonly distinguished “easy problems” and “difficult problems”, the former involves functional mechanisms such as attention, memory, and behavioral control, while the latter points to why specific experiences themselves “feel like something”. Traditional physical theories mainly deal with input-output, causal relationships and dynamics, but find it difficult to provide geometric or operator-level characterization of “what red looks like” or “what pain itself is”. Existing leading theories mostly start from information or computational structure. Integrated Information Theory (IIT) advocates that consciousness is equivalent to a computable integrated information quantity Φ in the causal structure of systems, attempting to construct the “shape” of experiential space from physical causal networks, but there remains controversy about testability and formalization details. Global workspace theory and related neurodynamic models focus more on reportability and whole-brain broadcasting mechanisms of consciousness. The free energy principle and active inference framework characterize how the brain minimizes surprise and maintains self-organizing structure in long-term statistical sense from Bayesian inference and variational free energy perspectives. These theories have made substantial progress in explaining “why consciousness is sensitive to the physical world” and “how consciousness participates in cognition and behavior”, but still lack unified characterization of “geometric structure of experiential space itself”. On the other hand, Max Tegmark and others propose “consciousness as a phase of matter” (perceptronium), attempting to provide phase diagrams of conscious matter through principles such as information integration and independence, partially absorbing IIT’s ideas and relating Hilbert space decomposition and entanglement structure to “observer-selected decomposition methods”. These works collectively point to a direction: consciousness may be some kind of “geometric-information” structure rather than an isolated variable. This paper adopts a more thorough geometric standpoint: conscious experience is not the “content” of information, but the “shape” of information in high-dimensional state space. We propose a psychophysical isomorphism principle: if rigorously characterizing the brain’s states and evolutionary trajectories in Hilbert space, then the geometric and topological structure of subjective experience space has a one-to-one correspondence with upper-layer quantum information geometry. The hard problem is restated as: finding precise geometric-topological isomorphisms, rather than chasing “where does color come from” in classical three-dimensional space. 1.2 Neural Manifolds and Neurogeometry Over the past two decades, developments in neural population recording technology have gradually shaped the concept of “neural manifolds”. Extensive work shows that in high-dimensional neural activity space, neural activity corresponding to specific tasks or perceptual variables often distributes on submanifolds with dimensions far lower than full dimension, and these manifolds often have smoothness and non-trivial geometric structure. Research in “neurogeometry” has depicted representations of features such as color, orientation, and binocular disparity in visual cortex as convolution and integral structures on certain Lie groups or sub-Riemannian manifolds. Koenderink and others even directly measured the “intrinsic curvature” of visual space in psychophysical experiments, showing that subjective space itself is not simply Euclidean. Neural manifolds and neurogeometry suggest: there is indeed some geometric structure behind experience, but existing work mostly adopts classical probability spaces and Euclidean embeddings, not yet fully utilizing tools of quantum information geometry and entanglement 2 manifolds. 1.3 Information Geometry and Quantum Information Geometry Information geometry views families of probability distributions as manifolds with Riemannian metrics, with classical Fisher information providing the unique metric satisfying natural monotonicity and covariance requirements (under reasonable postulates). Amari and collaborators systematically developed this theory and applied it to neural networks, learning algorithms, and statistical model analysis. In the quantum case, density operator space can be endowed with multiple “monotone metrics”, among which the metric corresponding to Bures distance or Helstrom quantum Fisher information plays a central role in quantum estimation and quantum statistics. Braunstein and Caves proved that quantum Fisher information gives the maximal metric satisfying natural requirements among quantum statistical distances and is closely related to quantum Cram´er– Rao bounds. Petz systematically classified all quantum monotone metrics. Bengtsson and ˙ Zyczkowski systematically summarized density matrix space, Fubini–Study metric, Bures metric and entanglement structure from geometric perspective. Recent reviews on “from classical to quantum information geometry” point out that quantum Fisher information and Berry curvature together constitute the natural Riemannian-symplectic geometric structure on quantum state space, with important applications in many-body system phase transitions and topological phases. These achievements provide mathematical foundations for the proposal to “characterize conscious state space with quantum information geometry”. 1.4 Geometric Consciousness Conception and Contributions of This Paper A few works have attempted to describe consciousness or experiential space from geometric angles, such as discussing “geometry of consciousness” from perspectives of reference frame selection, intrinsic geometry of perceptual space, and shape of information manifolds. However, these attempts often remain at macroscopic or conceptual levels, lacking systematic connection with quantum information geometry and entanglement manifolds. This paper is based on the following principles: 1. Psychophysical isomorphism principle: There exists a structural isomorphism between experiential space Qand some equivalence class manifold Mpsy describing brain physical states, preserving operable distinguishability relations and continuity structure; 2. Information geometry priority principle: Metric and curvature are not additional structures, but geometric objects uniquely selected from minimal distinguishability and statistical decision theory; 3. Entanglement topology principle: Structures manifested as qualia differences at macroscopic scales correspond microscopically to topological and symplectic geometric invariants of entangled state manifolds; 4. Free energy dynamics principle: Emotion and value are not externally added labels, but intrinsic properties of free energy gradient flow and trajectory geometry on this manifold. Based on this, the main contributions of this paper can be summarized as:  Within the QCA and optical path conservation framework, construct the conscious system Hilbert space state manifold Mpsy, and prove that under natural postulates its QFIM metric is isometrically isomorphic to subjective experience space metric; 3  Introduce Berry connection and curvature on pure state submanifolds, proposing “qualia as curvature”: typical qualia correspond to local structures of non-trivial Berry curvature on entanglement manifolds;  Give geometric definition of emotional valence within free energy principle framework, viewing pleasure and pain as directional derivatives and second derivatives of free energy along geodesics on QFIM manifolds;  Define Betti numbers of conscious manifolds through homology theory and persistent homology, arguing they provide lower bounds for irreducible experiential complexity, and provide topological indicators comparing different systems (human brain, simple neural networks, classical automata);  Propose a series of feasible experimental and engineering proposals for estimating QFIM, Berry curvature and topological invariants in neural data and artificial systems, thus providing testable paths for the “geometric consciousness” framework. 2 Model & Assumptions 2.1 QCA Universe and Conscious Subsystem In QCA ontology, the universe is modeled as local unitary evolution Udefined on discrete spatial grid Λ, with Hilbert space Huniv =O x∈Λ Hx, where each grid point carries finite-dimensional local degrees of freedom (such as qubits or finite-dimensional local multi-level systems). Global time evolution is realized by finite-depth local quantum circuits, with optical path conservation principle requiring signal transmission speed limited by finite neighborhood propagation. A conscious system (such as the human brain) in this framework is a subsystem on some finite region B⊂Λ, with Hilbert space HB=O x∈B Hx, environment as HE=Nx∈Λ\BHx. Under appropriate coarse-graining and decoherence conditions, consciousness-related states can be approximately described as mixed states ρB(t) on HB, satisfying ρB(t) = TrEUtρunivU†t. This definition does not depend on specific QCA details, only on local unitarity and finite information capacity. We introduce the concept of information quality MIto characterize the maximum ordered information capacity the subsystem can carry (such as related to its Hilbert space dimension or effective entanglement entropy upper bound), to distinguish systems with complex experiential potential from simple systems. 2.2 Psychophysical Equivalence Classes and Conscious Manifold The physical state ρBof conscious systems contains far more information than subjects can access. Subjects can only perceive their own states through finite sets of measurements and internal readout channels within finite time windows. We define an equivalence relation: 4 ρB∼ρ′ B⇐⇒ For all operations implementable by subject, both give identical operational result distributions. This equivalence relation compresses physical states on Hilbert space into a group of psychophysical equivalence classes, denoted [ρB]∈ Mpsy. The set Mpsy forms a (possibly piecewise) differentiable manifold under appropriate regularity assumptions. We call Mpsy the psychological manifold: each point corresponds to “a family of physical brain states indistinguishable at all attainable precision”. 2.3 Quantum Fisher Information Metric Let θ= (θ1, . . . , θn) be coordinates parameterizing conscious states (can be stimulus parameters, internal prediction parameters or more abstract coordinates), with corresponding density operator family ρ(θ) smoothly varying on HB. Define the logarithmic derivative operator Li (symmetric logarithmic derivative) for each parameter component satisfying ∂iρ(θ) = 1 2Li(θ)ρ(θ) + ρ(θ)Li(θ), the quantum Fisher information metric is defined as gQ ij (θ) = 1 2Tr ρ(θ){Li(θ), Lj(θ)}, where {·,·} is the anticommutator. For pure states ρ(θ) = |ψ(θ)⟩⟨ψ(θ)|, the above metric reduces to Fubini–Study metric gQ ij (θ)=4Re⟨∂iψ|∂jψ⟩−⟨∂iψ|ψ⟩⟨ψ|∂jψ⟩. Braunstein and Caves proved that for given quantum statistical models, gQprovides an upper bound on Fisher information achievable in all measurements, and gives an extreme element in monotone quantum metric families under natural postulates. Petz’s results show that all quantum monotone metrics are characterized by a family of operator monotone functions, while the Bures/QFIM metric has particularly strong physical interpretability. We denote the QFIM metric induced on Mpsy as g, which defines intrinsic distance on the conscious manifold: DQF([ρ(θ1)],[ρ(θ2)]) = inf γZ1 0qgij(γ(t)) ˙γi(t)˙γj(t)dt, where γconnects two points in Mpsy. 2.4 Psychophysical Isomorphism Postulate We now introduce key postulates. Postulate 1 (Distinguishability preservation): For any two reportable experiences q1, q2∈ Q, if subjects view them as indistinguishable under all experimental conditions, then the distance between their corresponding psychological manifold points is zero; if subjects can reliably distinguish them within finite trials, then the corresponding point distance is positive, and distinguishability difficulty is a monotonic function of distance. Postulate 2 (Continuity): Experience varies continuously with physical state changes; specifically, if ρ(θ) varies smoothly in θ, then corresponding experience q(θ) forms a smooth curve in Q. 5 Postulate 3 (Information monotonicity): Any physical operation discarding information (completely positive trace-preserving map) corresponds to contraction mapping on experiential manifold; i.e., coarse-graining does not increase distance between experiences. These postulates are analogous to distinguishability, continuous stimuli and Weber–Fechner law in classical psychophysics, while being compatible with information geometry classification theorems. 2.5 Berry Connection and Entanglement Manifold Consider pure state approximation subsets or their principal component subspaces of consciousnessrelated states, viewable as complex projective space P(HB) or its submanifolds. For parameterized states |ψ(λ)⟩(λ∈Λ as external or internal parameters), define Berry connection Ai(λ) = i⟨ψ(λ)|∂iψ(λ)⟩, corresponding curvature two-form Fij(λ) = ∂iAj−∂jAi. Berry curvature characterizes geometric phase accumulation along closed path C ⊂ Λ evolution, widely appearing in quantum phase transitions and topological matter states. In this paper’s framework, we view Fij as candidate geometric object for qualia field strength. 3 Main Results (Theorems and Alignments) This section presents the main conclusive propositions of this paper, with detailed derivations in subsequent “Proofs” and appendices. 3.1 Theorem 1: Isometric Isomorphism between Experience Space and QFIM Manifold Theorem 1 (Metric isomorphism between psychological manifold and experiential manifold) Under Postulates 1–3 and assuming experiential space Qis a separable metrizable manifold, there exists a homeomorphism f:Mpsy → Q, such that for any x, y ∈ Mpsy, dQ(f(x), f(y)) = cDQF(x, y), where dQis natural psychological distance on experiential space (defined by distinguishability statistics), c > 0 is a constant. In other words, fis an isometric map, with experiential space’s intrinsic geometry isomorphic to QFIM manifold. This proposition transforms “qualia differences” into distances on QFIM manifolds, showing uniqueness under information geometry postulates. 3.2 Theorem 2: Qualia as Local Structure of Berry Curvature Theorem 2 (Qualia–curvature correspondence) Let N ⊂ Mpsy be a submanifold corresponding to a class of experiential modalities (such as color, pitch), with parameterization λ∈Λ such that corresponding pure state family |ψ(λ)⟩ forms dominant components. If Berry curvature two-form Fij(λ) is non-zero on N, then there exists a local coordinate selection such that: 6 1. Qualia type (e.g., “red” vs “blue”) corresponds to eigensubspaces and sign structure of Fij; 2. Qualia intensity corresponds to Berry curvature magnitude integrated along typical subject paths, i.e., Iqualia ∝ZΣ F , where Σ is parameter surface enclosed by path. In other words, typical qualia can be viewed as local “vortices” or “flux” of Berry curvature on consciousness-related entanglement manifolds. 3.3 Theorem 3: Emotional Valence as Free Energy Gradient and Geodesic Acceleration Theorem 3 (Geometric definition of emotional valence) Let the evolution of conscious system on Mpsy produce a time-parameterized trajectory γ(t), while there exists variational free energy function F:Mpsy →Rdefined on this manifold. Under free energy principle and active inference dynamics assumptions, trajectory motion satisfies D2γi dt2+ Γi jk dγj dt dγk dt=−gij∂jF(γ(t)) + ξi(t), where Γi jk is Levi–Civita connection, ξiis noise term. Define emotional valence function V(t) := −d dtF(γ(t)), then in average sense:  V(t)>0 corresponds to experiencing “moving toward better prediction” positive valence (pleasure);  V(t)<0 corresponds to experiencing “being pushed toward higher prediction error” negative valence (pain);  dV dtcorresponds to valence change rate, characterizing second-order effects like “relaxation”, “relief” or “deepening despair”. This proposition transforms emotion from lexical labels into geometric quantities of free energy flow on QFIM manifolds. 3.4 Theorem 4: Topological Lower Bound of Consciousness Complexity Theorem 4 (Betti numbers and irreducible experiential complexity) Let within a given time window, conscious trajectory γ(t) and its perturbations fill a compact subset K⊂ Mpsy. Consider homology groups Hk(K, Q) and Betti numbers bk= dim Hk(K, Q) of K. Then: 1. The minimum number of distinguishable experience types is controlled by lower bound of Pkbk; 2. If b1and b2are significantly non-zero, there exist irreducible cyclic patterns and twodimensional “experiential cavities”, corresponding to self-referential thinking, persistent emotional backgrounds and higher-order conceptual structures; 7 3. Under same information quality MI, if system A’s bkare all smaller than system B’s, then A’s experiential complexity (in sense of distinguishable experiential modalities and combinatorial structure) does not exceed B’s. Under certain mild assumptions, these Betti numbers can be estimated from neural data through algorithms like persistent homology, providing topological scales for comparing consciousness potential of different systems. 4 Proofs This section provides proof ideas for the above main theorems, with technical details and extensions arranged in appendices. 4.1 Proof of Theorem 1: From Psychophysical Postulates to QFIM Isometric Embedding The proof strategy is divided into three steps. Step 1: Fisher information representation of psychological distance Consider parameterized family ρ(θ) and corresponding experience q(θ). According to Postulate 1, experiential distinguishability can be defined as optimal decision performance in a family of behavioral experiments, typically characterized by d′or error rate curves. In small perturbation limit θ→θ+δθ, classical decision theory shows optimal distinguishability is completely determined by Fisher information; in quantum case, optimal Fisher information is upper bound over all POVMs, realized by QFIM. Therefore, at small scales, square of psychological distance can be written as ds2 Q∝gQ ij (θ)dθidθj, giving local metric structure of experiential space. Step 2: Monotonicity postulate and metric uniqueness Postulate 3 requires coarse-graining operations not to amplify experiential distance. Corresponding to statistical models means metric monotonicity under Markov maps/completely positive trace-preserving transformations. Classical results in information geometry show that under several natural postulates, Riemannian metrics satisfying monotonicity and covariance are uniquely Fisher information in classical case. Petz’s classification results show there exists a family of monotone metrics in quantum case, but requiring consistency with Bures distance or consistency with Fubini–Study on pure states narrows it to QFIM metric. Therefore, under physical and psychological postulates satisfied by conscious systems, experiential space local metric is equivalent to a constant multiple of QFIM metric. Step 3: Global isometric homeomorphism construction Postulate 2 ensures experiential space and psychological manifold are both separable, locally compact differentiable manifolds. Taking Mpsy as source manifold, endowed with QFIM metric, mapping equivalence classes to experiential points in Q. Using Riesz–Fr´echet representation theorem and Hopf–Rinow theorem, an isometric embedding preserving geodesic length can be constructed under completeness. Since distinguishability postulate ensures experiences are indistinguishable when distance is zero, quotient space yields homeomorphism. Specific construction and technical details see Appendix A. Thus Theorem 1 is obtained. 8 4.2 Proof of Theorem 2: Berry Curvature and Experiential “Vortices” We view submanifold Ncorresponding to some experiential modality as image of parameter space Λ, where λiare control parameters related to this modality (such as spectral components, semantic coordinates, etc.). Under pure state approximation, consciousness-related main degrees of freedom can be simplified to evolutionary trajectories of a family of states |ψ(λ)⟩. Berry connection Ai(λ) and curvature Fij(λ) are defined on Λ, while Fubini–Study metric can be viewed as pure state limit of QFIM, both forming K¨ahler structure. Examine subjects circling a closed loop Cin Λ due to internal fluctuations under fixed external stimulus conditions. Empirically, this might correspond to “circling once in meaning space but returning to same external configuration”. State evolution acquires Berry phase γBerry =IC Ai(λ)dλi=ZΣ Fij(λ)dSij, where Σ is enclosed surface. This phase depends only on curvature flux independent of local gauge. Therefore, from subject perspective, phenomenon of “returning to same physical configuration but experiencing macroscopic difference” can naturally be attributed to non-zero Berry curvature. We relate “types of some experience” to Berry curvature structure in certain directions under selected gauge: different eigensubspaces correspond to different stable patterns (like “red”, “blue”), with curvature magnitude corresponding to subjective intensity of that pattern. This correspondence is essentially reinterpretation of Berry curvature, formally similar to “charge– flux” correspondence in topological matter states. Rigorous formulation can establish curvature–experience type bijection by viewing experiential modalities as equivalence classes on principal bundle sections. Detailed construction see Appendix B. 4.3 Proof of Theorem 3: Free Energy Gradient Flow and Emotion Friston’s free energy principle views brain as inference machine minimizing surprise, with variational free energy Fas upper bound characterizing prediction error of model on sensory input. From information geometry perspective, Fcan typically be expressed as some relative entropy or energy-entropy functional, thus naturally defined on parameter manifolds. Consider gradient flow dynamics on QFIM manifold dγi dt=−gij∂jF(γ(t)) + ηi(t), where ηiis noise term. Simultaneously considering geodesic deviation yields second-order equation with Christoffel symbols as stated in theorem. Chain rule for time derivative gives d dtF(γ(t)) = ∂iF˙γi=−gij ˙γi˙γj+∂iFηi, in approximation with zero average noise, E[V(t)] = −Ed dtF=Egij ˙γi˙γj≥0. Therefore, average descent rate of free energy along trajectories is proportional to velocity squared, naturally interpretable as “speed of advancing toward better prediction states”, corresponding to subjectively experienced positive valence. Conversely, if due to external shocks or internal constraints trajectories are forced to move toward free energy ascent directions, then V(t)<0, corresponding to experienced pain or stress. This derivation connects emotional valence to energy dissipation on QFIM manifolds, details see Appendix C. 9 B Appendix B: Berry Curvature and Qualia Loops B.1 B.1 K¨ahler Structure and Berry Curvature On pure state submanifold P(H), Fubini–Study metric gij = 4Re⟨∂iψ|∂jψ⟩−⟨∂iψ|ψ⟩⟨ψ|∂jψ⟩ together with Berry connection Ai=i⟨ψ|∂iψ⟩form K¨ahler structure, where K¨ahler form ωij =∂iAj−∂jAi=Fij is Berry curvature. B.2 B.2 Experience Type and Curvature Eigenstructure Let Λ be parameter space, choosing gauge in given region such that Berry curvature matrix can be diagonalized at some point. Its eigenvectors v(a)correspond to specific directions in parameter space; if slowly changing stimulus along this direction experimentally, subjects’ experience will manifest as some stable “color” or “taste”. Curvature eigenvalue signs and magnitudes relate to “texture” and intensity of experience. By constructing closed loops in parameter space and measuring coherence or behavioral preferences, Berry curvature flux can be indirectly estimated, thus testing whether certain experiences correspond to non-trivial Berry curvature regions. C Appendix C: Geometric Expression of Free Energy Gradient and Emotional Valence C.1 C.1 Free Energy as Relative Entropy Functional In variational Bayes framework, free energy can be written as F(ρ, q) = Eq[−ln p(s, f)] + Eq[ln q(f)], where sis sensory input, fis hidden variable, qis approximate posterior. qcan be parameterized as q(θ) and metric introduced on QFIM manifold. Under appropriate limits, free energy is proportional to relative entropy, thus naturally defined on information geometric manifolds. C.2 C.2 Gradient Flow and Valence Along trajectory γ(t), d dtF(γ(t)) = ∂iF˙γi=−gij ˙γi˙γj+∂iFηi. Under noise averaging, E[V(t)] = −Ed dtF=Egij ˙γi˙γj≥0, thus valence is on average non-negative, corresponding to moving toward better prediction states. If system is pulled away from free energy valleys by external forces, angle between ˙γ and −∇F increases, potentially leading to instantaneous V(t)<0, corresponding to negative valence experience. 16 D Appendix D: Topological Analysis of Betti Numbers and Consciousness Complexity D.1 D.1 Persistent Homology and Point Cloud Approximation Sampling conscious trajectories yields discrete point set {xi}⊂Mpsy, constructing Vietoris– Rips complex under distance induced by QFIM metric, calculating homology groups as scale parameter ϵvaries to obtain persistent barcodes. Persistence lengths of these barcodes on ϵaxis reflect robustness of topological features. D.2 D.2 Topological Lower Bounds and Experiential Classification If stable non-zero bkexists in scale interval [ϵ1, ϵ2], then at corresponding resolution there exist irreducible k-dimensional holes. Jointly analyzing these holes with experiential classification tasks (such as semantic categories, emotional dimensions) can provide topological lower bounds for “how many irreducible experience types”. For example, if observing high b1and b2on submanifolds representing semantic space, this suggests rich cyclic and cavity structures in semantic experience. This analytical framework similarly applies to artificial systems, providing unified topological language for comparing “experiential potential” under different architectures and training schemes. 17