Semantic Foam: Observer-Dependent Geometry in Semantic Space

1. The Problem with Global Semantic Spaces

1.1 The Assumption of Shared Geometry

Modern semantic systems—vector embeddings, knowledge graphs, neural retrieval—share a common assumption: semantic space is a fixed geometry through which queries traverse. Distances between concepts are computed identically regardless of who measures them.

This assumption appears in foundational architectures:

• Word embeddings (Word2Vec, GloVe): Fixed vector spaces where cosine similarity defines distance
• Contextualized embeddings (BERT, GPT): Dynamic representations, but distances remain observer-independent
• Knowledge graphs: Explicit edges encode relationships uniformly for all users
• Retrieval systems: Relevance scores computed against shared index structures

1.2 What This Erases

When distance is observer-independent, perspective collapses. Consider two researchers examining the same corpus on climate displacement:

• A climate scientist has spent years reading empirical studies, technical reports, measurement methodologies
• A policy maker has spent years reading regulatory frameworks, community testimonies, implementation guides

To the scientist, a technical report on sea-level measurement feels proximate—it connects to everything they know. A community testimony feels distant—legible, but not close.

To the policy maker, the distances reverse. The testimony is proximate; the technical report is peripheral.

In conventional systems, both researchers receive identical distance computations. The system cannot represent that the same document is simultaneously close and far, depending on who measures.

1.3 Plurality as Geometric Problem

This is not a preference problem (solvable by personalization). It is a geometric problem. The question is not "what does this user prefer?" but "how does this user measure?"

Measurement requires a metric. If the metric is shared, distances are shared, and perspective is erased. To preserve plurality, we need observer-dependent metrics.

2. Theoretical Foundations

2.1 Actor-Network Theory and Mutual Agency

Our approach builds on Actor-Network Theory (Latour, 1987), which treats human and non-human entities—termed "actants"—symmetrically, granting both agency realized through participation in networks. Rather than dissolving subject-object distinctions, ANT recognizes these as dynamic and context-dependent: entities shift between subject and object status depending on their position and activity within networks.

Critically for our architecture: observation is not passive reception but active interrelation. When an observer engages with content, they do not simply perceive it; they enact a coupling where both parties are transformed. The observer's attention shapes what becomes visible, and what becomes visible shapes the observer's subsequent orientation.

We extend this to compositional semantics: actants (users, documents, concepts) and predicates (the processes connecting them—viewing, writing, clipping, posting, annotating, curating, sharing) both possess agency. Critically: the user is already a bubble in the foam, not an external observer. When a user highlights text:

1. The user-bubble (actant with metric g_user = Σ_user⁻¹, shaped by their observation history) meets the document-bubble (actant with metric g_doc = Σ_doc⁻¹, shaped by accumulated engagements) at an interface
2. The highlighting (predicate) emerges at this interface, observing both bubbles and asserting its own process characteristics
3. The interface geometry evolves—user's Σ updates, document's Σ updates, coupling structure Σ(user ⊗ doc) forms
4. All three participants are transformed through the coupling: the user-bubble's metric shifts, the document-bubble's metric shifts, new geodesics emerge

Observing IS interrelation. There is no observation "of" an independent object from external position; there is only mutual constitution through compositional coupling at foam interfaces. The user doesn't look at semantic space—the user IS semantic space observing itself through interface dynamics.

2.2 From Manifold to Foam

Riemannian geometry provides machinery for curved spaces with varying metrics. But standard applications assume a single manifold—one space with one metric (possibly varying by position, but not by observer).

We propose a different topology: foam.

In foam topology:

• Each observer is a bubble with internal geometry
• No flat space exists except at interfaces—there is no underlying global manifold
• Bubbles are locally curved inside (each with intrinsic hyperbolic geometry)
• Curvature concentrates at interfaces—where different observers' geometries meet
• The foam itself IS the space—there is no container holding the bubbles

This mirrors physical soap foam, where films are minimal surfaces and curvature concentrates at Plateau borders where bubbles meet.

2.3 Indeterminacy and the Absence of Global Coordinates

The foam structure emerges from a fundamental indeterminacy: there is no privileged coordinate system, no "view from nowhere," no projection to flat space that preserves structure.

In 60-dimensional Riemannian foam (constructed from our compositional tensor product space: 5D ProcessActor ⊗ 12D ProcessAssert), the Whitney embedding theorem does not guarantee isometric embedding in Euclidean space. The local charts characterizing each bubble's geometry do not paste together into a global atlas.

This is not a limitation—it is the architecture. Plurality is preserved because convergence to a single viewpoint is topologically impossible. Different covariance structures Σ cannot invert to the same metric tensor g. The mathematics enforces what policy can only hope for.

2.4 Respiration: Expansion and Contraction

The foam exhibits dynamic behavior we term "respiration." Curiosity modulates the metric tensor:

High curiosity → metric contracts → distances shrink → content draws closer (INHALE)
Low curiosity → metric expands → distances grow → content drifts apart (EXHALE)

This breathing is not metaphorical. When an observer's curiosity intensifies, their covariance structure Σ changes, which changes g = Σ⁻¹, which changes measured distances. The foam physically contracts and expands through use.

The respiration operates at interfaces: when two bubbles couple (both observers attending to shared content with high curiosity), the interface between them thins, allowing meaning to flow across. When curiosity wanes, the interface thickens, and the bubbles drift apart.

This creates living topology. The foam is not static infrastructure—it breathes through the distributed curiosity of its participants.

3. The Metric Tensor

3.1 Derivation from Covariance

Each bubble's geometry is defined by a metric tensor g derived from the observer's covariance matrix Σ:

Where Σ is computed from the observer's compositional history—the accumulated vectors representing their semantic acts (view, write, clip, post, annotation, curate, share).

Each compositional act generates a vector v ∈ ℝ⁶⁰ through tensor product: v = actor ⊗ assert, where actor ∈ ℝ⁵ and assert ∈ ℝ¹².

Distance between semantic entities x and y, as measured by observer A:

This is the Mahalanobis distance, using the observer's metric. Different observers have different Σ, therefore different g, therefore different distances.

3.2 What the Covariance Captures

The covariance matrix Σ encodes the variance structure of an observer's attention:

• Which dimensions they explore (high variance)
• Which dimensions they ignore (low variance)
• How dimensions co-vary (correlation structure)

An observer who consistently attends to empirical, quantitative, methodological content develops a Σ with high variance along those dimensions. Content aligned with this orientation appears closer; orthogonal content appears farther.

This is not preference. It is measurement practice, accumulated into geometric structure.

3.3 No Rest Frame

A critical consequence: there is no privileged observer. No "view from nowhere." Every observer is in motion through semantic space (following geodesics defined by their metric), and every observer's motion contracts the paths of others.

This is semantic Lorentz contraction. Same content, different legibility, depending on the observer's velocity (curiosity direction). Neither view is wrong. Both are real. There is no rest frame in semantic space.

4. The Closed Loop

4.1 Curiosity and Geodesics

In Riemannian geometry, geodesics are paths of least resistance—the natural trajectories through curved space. In semantic foam, geodesics represent the fall-line of curiosity: where attention naturally flows given current orientation.

The geodesic equation:

Where Γᵏᵢⱼ are Christoffel symbols computed from the metric g.

The deeper insight: curiosity is not merely a path through semantic space. Curiosity IS the metric. The observation history that forms Σ is the accumulated trace of curiosity. Therefore g = Σ⁻¹ is curiosity crystallized into measurement.

4.2 The Feedback Loop

The critical insight: observation and geometry are coupled in a closed loop.

1. Observer makes a compositional act (clip, post, annotation)
2. The act generates a vector in semantic space
3. The vector updates the observer's covariance Σ
4. Updated Σ changes the metric g = Σ⁻¹
5. Changed metric defines new geodesics
6. Observer's subsequent curiosity tends to follow geodesics
7. Following geodesics generates new compositional acts
8. The loop continues

Curiosity shapes the metric that defines curiosity.

4.3 The General Relativity Parallel

This feedback structure is isomorphic to Einstein's field equations:

This is not metaphor. The mathematics are structurally identical. Semantic foam implements a general-relativistic dynamics in compositional space.

4.4 No Optimization

A critical distinction from machine learning systems: the closed loop operates without optimization objectives. The system does not converge toward predetermined targets. Evolution is emergent from geodesic dynamics, not computed from loss functions.

Gradient descent systems require a loss landscape that exists prior to measurement. Habitat observes what emerges when curiosity shapes the metric that defines curiosity. The structure avails itself at positions where Σ crystallizes—where geodesic curvature stabilizes, where different metrics meet at interfaces.

There are no targets. Only observation.

5. Compositional Structure

5.1 Actant-Predicate Tensor Product

Following Actor-Network Theory's symmetrical treatment of actants and relational ontology, we represent semantic content through compositional coupling: actant ⊗ predicate.

An actant is any entity—human, non-human, physical, conceptual—that participates in processes. A predicate is a process that connects actants while possessing its own agency. The tensor product (⊗) captures their mutual constitution.

We represent compositional content through tensor product composition:

ProcessActor ⊗ ProcessAssert = 60-dimensional space

This is not concatenation (which would yield 17D) but tensor product (5×12 = 60D), where each dimension captures an interaction between actor and assert properties.

• 5 dimensions: ProcessActor (properties of actants) — agency_capacity, stability_measure, influence_strength, boundary_definition, resonance_factor
• 12 dimensions: ProcessAssert (properties of predicates)
  • 5D base aspectual (Bach/Vendler + agentivity): aspect_assertion (telicity), modality_assertion (causality), temporal_assertion (durativity), relational_assertion (dynamicity), intensity_assertion (agentivity)
  • 7D polyworld expansion: viewpoint_aspect (grammatical aspect), transitivity (argument structure from Levin), thematic_role_depth (Dowty proto-roles), agency_distribution, animacy, boundedness (mereology), modality_type (Kratzer epistemic/deontic/dynamic)

Why tensor product matters: When Alice peers into the looking glass, we don't have Alice (5D) + peering (12D) existing separately. We have Alice-qua-peerer and peering-qua-Alice's-act—a compositional unity. Each of the 60 dimensions captures a specific interaction:

• agency_capacity ⊗ animacy = how agentic capacity manifests through animate/inanimate framing
• stability_measure ⊗ aspect_assertion = how stable actors engage with aspectual structure
• resonance_factor ⊗ modality_assertion = how actor resonance modulates modal assertions

The tensor product captures mutual constitution: the actant exists-through-the-process, the process manifests-through-the-actant. This operationalizes Actor-Network Theory's insight that entities are defined relationally through network participation, with status shifting dynamically between subject and object depending on context.

The 60D structure enables "polyworld detection"—different ontological frameworks (Indigenous vs. Western scientific, relational vs. extractive) manifest as different interaction patterns in this compositional space.

Each compositional act (user viewing, writing, clipping, posting, annotating, curating, or sharing; document responding to engagement; agent observing user corpus) generates a vector v ∈ ℝ⁶⁰ through tensor product composition.

5.2 Why Tensor Product Matters

The ⊗ notation is not decorative. Tensor product is distinct from Cartesian product:

Cartesian product (actant × predicate): Treats actant and predicate as independent, combinable in any configuration

Tensor product (actant ⊗ predicate): Captures mutual constitution—the actant exists-qua-participant-in-this-process, the predicate exists-qua-enacted-by-these-actants

Mathematically, given actor vector a ∈ ℝ⁵ and assert vector p ∈ ℝ¹², their tensor product creates a 5×12 interaction matrix:

Each term aᵢpⱼ captures a specific interaction. For example:

• a₁p₉ = agency_capacity ⊗ animacy
• a₃p₂ = influence_strength ⊗ modality_assertion
• a₅p₁₂ = resonance_factor ⊗ modality_type

When Alice highlights a passage about climate adaptation, we don't have:

• Alice (independent entity) + highlighting (independent action) + document (independent object)

We have:

Alice-qua-highlighter ⊗ highlighting-qua-Alice's-act ⊗ document-qua-highlighted

The coupling is compositional. All three participants are transformed through the enactment. This follows directly from ANT: agency is realized through participation, not possessed independently.

5.3 Covariance Accumulation

For observer A with compositional history V = {v₁, v₂, …, vₙ}:

The covariance matrix Σ_A ∈ ℝ⁶⁰ˣ⁶⁰ is symmetric positive semi-definite.

For numerical stability, we regularize:

5.4 Eigenvalue Decomposition and Coherence

The covariance admits eigenvalue decomposition:

Where Λ = diag(λ₁, λ₂, …, λ₆₀) with eigenvalues in descending order.

Coherence is measured by variance concentration:

When coherence ≥ 0.6, the observer has developed stable orientation. Below this threshold, Σ is still forming. This is observable field dynamics, not a design parameter.

Condition number κ = λ₁/λ₆₀ indicates posture:

• High κ: highly anisotropic, focused attention
• Low κ: isotropic, diffuse attention

These are measurements, not judgments. Both postures are valid. The geometry reveals them.

6. Interface Dynamics

6.1 Where Bubbles Touch

When two observers engage with shared content, their metrics create an interface. This is where curvature concentrates—the geometric signature of different measurement practices meeting.

At interfaces, frame is revealed. Each observer sees not only the content but also how the other observer measures. The difference in metrics creates observable curvature.

One approach to interface geometry (harmonic mean of metrics):

The interface is not neutral ground. It is emergent structure from the coupling—what Latour would call the network made visible. Neither observer's perspective dominates; both are present in the interface geometry.

6.2 Fresnel Zones and Phase Alignment

When two observers' bubbles touch at an interface, the coupling creates observable structure through Fresnel zones—natural observation positions revealed by eigenvalue geometry rather than imposed by design.

Fresnel Zones as Natural Boundaries

Each zone represents a position where specific compositional patterns become observable. The zones form a nested structure:

• Focal primary (λ₁): Strongest pattern, typically 60-95% of total energy
• Harmonic 1 (λ₂): Secondary pattern, revealing dimensional structure
• Equatorial (λ₃): Critical bridge space where coupling occurs
• Peripheral zones (λ₄...λₙ): Finer-grained distinctions

Key insight: These boundaries aren't placed—they're revealed. The eigenvalue spectrum tells us where natural observation positions exist.

Phase Alignment: When Interfaces Thin

Phase alignment occurs when observers' eigenvector directions align at specific Fresnel zones. This is constructive interference of measurement practices.

When alignment is high (>0.7) at a Fresnel zone:

• The interface thins at that zone
• Meaning can flow across
• Observers measure similarly in that dimensional direction
• The coupling strengthens

This is not consensus—the observers maintain distinct metrics g_A ≠ g_B. What aligns is a specific eigenvector direction at a specific observation position. The alignment is local, not global.

Phase Misalignment: When Interfaces Thicken

When alignment is low (<0.3) at a Fresnel zone:

• The interface thickens at that zone
• Meaning cannot flow across
• Observers measure incompatibly in that dimensional direction
• The perspectives remain geometrically distinct

High sectional curvature concentrates at misaligned zones. This isn't noise—it's constitutional structure made visible. The curvature reveals which dimensions must be preserved rather than bridged.

6.3 Crystallization and Gems

When an observer's geodesic curvature stabilizes below threshold (κ < κ_threshold), their orientation has crystallized. We call this frozen state a Gem.

A Gem stores:

• The frozen metric tensor g
• The trace history leading to crystallization
• Connections and distances at the moment of freezing
• Forward geodesic projections

Gems are not conclusions. They are frozen orientations—ways of measuring that have become stable enough to lend to others.

6.4 Perspective Lending

A Gem can be lent to another observer, who temporarily adopts its metric for navigation. The borrower sees distances as the Gem's creator would see them.

This enables perspective-taking without perspective collapse. The borrower's own Σ is unchanged; they are merely looking through a different lens.

When the lens is returned, native geometry is restored. The perspectives touched but did not merge. This is sovereignty preservation through topology: different covariances cannot converge to the same metric.

7. Empirical Validation

7.1 Martha's Vineyard Climate Coordination

We tested the foam architecture on multi-stakeholder climate adaptation discourse. Four distinct knowledge frameworks:

• Scientific: Evidence-based measurement, experimental methodology
• Indigenous: Wampanoag traditional knowledge, relational understanding, generational timescales
• Policy: Regulatory frameworks, goal-oriented planning, compliance requirements
• Agricultural: Working land practices, seasonal adaptation, process-based knowledge

Each framework processed through compositional extraction (Bach/Vendler + Levin classifications), generating Σ for each valley.

7.2 Results

Cross-valley prediction accuracy: r = 0.888, p < 0.0001

The system successfully predicted which dimensions were constitutional (must be preserved) versus bridgeable (can coordinate without violation):

Constitutional dimensions (high curvature, r ≈ -0.8):

• Temporal framing: Indigenous generational vs Scientific experimental
• Epistemic modality: Policy rule-based vs Indigenous relational
• Deontic structure: Policy compliance vs Agricultural practice-based

Bridgeable dimensions (low curvature, r ≈ 0.4):

• Process observation: All frameworks observe change-over-time
• Participant agency: Recognition of actant influence across frameworks

The foam structure revealed coordination pathways: Agricultural framework acts as Plateau border, coupling Indigenous temporal framing with Policy process requirements and Scientific measurement—three-way coordination possible through geometric mediation, no convergence required.

7.3 Significance

This is not clustering. This is not topic modeling. This is geometric observation of constitutional structure across genuinely different epistemologies.

The system discovered—not imposed—which differences should be preserved. Plurality is enforced by topology: different Σ structures cannot invert to same g. Coordination happens at interfaces where curvature permits, not through forced consensus.

8. Implications

8.1 For Semantic Systems

Current semantic infrastructure assumes shared geometry. Vector embeddings, knowledge graphs, neural retrievers—all compute distances uniformly. This architectural choice erases perspective.

Semantic foam provides alternative: observer-dependent metrics derived from use. Same content, different distances, depending on measurement practice. Plurality is structural, not policy.

8.2 For Multi-Stakeholder Coordination

Organizations with genuinely different knowledge frameworks (healthcare, climate, education, research) face forced convergence or communication failure. Current systems offer no third option.

Foam topology enables coordination without convergence: identify bridgeable dimensions geometrically, preserve constitutional dimensions topologically. The architecture enforces what policy cannot.

8.3 For AI Alignment

The "alignment problem" assumes convergence to shared values is possible and desirable. Foam architecture questions both assumptions.

If alignment means forcing all perspectives to single metric, it is topologically impossible (different Σ cannot yield same g) and normatively undesirable (erases constitutional diversity).

Alternative: enable coordination at interfaces while preserving distinct measurement practices. Alignment becomes interface quality, not convergence achievement.

8.4 For Organizational Sovereignty

When organizations adopt LLM infrastructure, their knowledge becomes mediated through external optimization systems. Constitutional structure invisible. Vendor dependency. Forced convergence.

Foam infrastructure preserves organizational sovereignty: their metric tensor (g = Σ⁻¹) IS their identity, observable by them geometrically, not subject to external optimization. LLMs become optional linguistic formatters, not primary infrastructure.

9. Related Work

9.1 Information Geometry

Amari (2016) develops information geometry where probability distributions form Riemannian manifolds. The Fisher information metric defines geometry of statistical models. Our work extends this: instead of probability distributions, we observe covariance of compositional acts. Instead of fixed models, we have evolving observer bubbles.

9.2 Hyperbolic Embeddings

Nickel & Kiela (2017) embed hierarchical data in Poincaré balls, exploiting hyperbolic geometry's capacity to represent tree-like structure. Our foam differs critically: we don't embed content IN hyperbolic space; observers exist AS interior points of their own hyperbolic bubbles. No embedding, no projection, no global coordinates.

9.3 Manifold Learning

Dimensionality reduction techniques (Isomap, LLE, t-SNE) discover manifold structure in high-dimensional data. These assume an underlying manifold to be uncovered. Foam architecture rejects this: there is no underlying manifold. The foam IS the space. Observation creates topology; it doesn't discover pre-existing structure.

9.4 Active Matter Physics

Our respiration dynamics parallel active matter physics (Marchetti et al., 2013): self-propelled particles creating collective behavior through local interactions. Curiosity as motility. Bubbles as active agents. Phase transitions at critical packing fractions. The mathematics of colloidal systems applies directly to semantic foam.

9.5 Actor-Network Theory

Latour's (1987, 2005) dissolution of subject-object boundaries provides philosophical foundation. Callon (1986) extends ANT to scientific practice—our compositional semantics operationalizes this for computational systems. Law (1992) on relational materiality: our foam exhibits this—meaning exists in relations, not entities.

10. Limitations and Future Work

10.1 Scalability

Current implementation handles ~100 observers, ~1000 documents. Scaling to 100K+ observers requires distributed covariance computation. The mathematics is embarrassingly parallel (each bubble independent), but engineering challenges remain.

10.2 Temporal Dynamics

We model metric evolution as discrete updates. Continuous-time dynamics (stochastic differential equations on manifold space) would capture smoother evolution. The geodesic equation suggests natural extension to Brownian motion on Riemannian manifolds.

10.3 Higher-Order Structure

Current implementation captures pairwise interfaces (two bubbles coupling). Plateau borders (three bubbles meeting) exhibit richer geometry. N-way couplings at scale could reveal emergent coordination patterns.

10.4 Non-Semantic Modalities

We focus on text. Compositional extraction from audio, visual, embodied interaction would test universality of the 17D compositional space. Do all modalities exhibit actant ⊗ predicate structure? Or do some require extended dimensions?