The Commons Cooperative Company
December 2025
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:
When distance is observer-independent, perspective collapses. Consider two researchers examining the same corpus on climate displacement:
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.
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.
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.
MANIFOLD FOAM ──────── ──── Single continuous space Multiple bubbles One metric (possibly varying) Each bubble has own metric All observers share geometry Observers maintain distinct geometries Curvature distributed Curvature concentrated at interfaces
In foam topology:
This mirrors physical soap foam, where films are minimal surfaces (flat), and curvature concentrates at Plateau borders where bubbles meet.
Each bubble's geometry is defined by a metric tensor g derived from the observer's covariance matrix Σ:
g = Σ⁻¹
Where Σ is computed from the observer's compositional history—the accumulated vectors representing their semantic acts (reading, clipping, annotating, writing).
Distance between semantic entities x and y, as measured by observer A:
d_A(x,y) = √[(x-y)ᵀ g_A (x-y)]
This is the Mahalanobis distance, using the observer's metric. Different observers have different Σ, therefore different g, therefore different distances.
The covariance matrix Σ encodes the variance structure of an observer's attention:
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.
In Riemannian geometry, geodesics are paths of least resistance—the natural trajectories through curved space. A ball rolling on a curved surface follows geodesics.
In semantic foam, geodesics represent the fall-line of curiosity: where attention naturally flows given current orientation. The geodesic equation:
d²xᵏ/dt² + Γᵏᵢⱼ (dxⁱ/dt)(dxʲ/dt) = 0
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. The metric doesn't describe where curiosity goes—the metric IS curiosity, frozen into geometric structure.
The critical insight: observation and geometry are coupled in a closed loop.
Observation → Composition → Σ → g → Geodesic → Observation
↑ │
└──────────────────────────────────────────────┘
Curiosity shapes the metric that defines curiosity.
This feedback structure is isomorphic to Einstein's field equations:
| General Relativity | Semantic Foam |
|---|---|
| Matter tells space how to curve | Observation tells Σ how to evolve |
| Space tells matter how to move | g tells curiosity where to flow |
| Stress-energy tensor Tμν | Compositional acts |
| Metric tensor gμν | Metric tensor g = Σ⁻¹ |
| Geodesic = free fall | Geodesic = curiosity fall-line |
This is not metaphor. The mathematics are structurally identical. Semantic foam implements a general-relativistic dynamics in compositional space.
A consequence: there is no privileged observer. No "view from nowhere." Every observer is in motion through semantic space, and every observer's motion contracts the paths of others.
Content aligned with your curiosity direction appears close and legible. Content orthogonal to your direction appears present but illegible—contracted by your motion through the space.
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.
When an observer's geodesic curvature stabilizes below threshold, their orientation has crystallized. We call this frozen state a Gem.
A Gem stores:
Gems are not conclusions. They are frozen orientations—ways of measuring that have become stable.
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.
The foam architecture preserves sovereignty at every scale:
This is the mathematical foundation for pluralistic infrastructure: systems that enable coordination without requiring consensus on how to measure.
We represent semantic content in a d-dimensional compositional space. In our implementation, d = 17:
Each compositional act (clip, post, annotation) is extracted to a vector v ∈ ℝ¹⁷.
For observer A with compositional history V = {v₁, v₂, ..., vₙ}:
μ_A = (1/n) Σᵢ vᵢ (mean / stance) Σ_A = (1/(n-1)) Σᵢ (vᵢ - μ_A)(vᵢ - μ_A)ᵀ (covariance)
The covariance matrix Σ_A ∈ ℝ¹⁷ˣ¹⁷ is symmetric positive semi-definite.
For numerical stability, we regularize:
Σ_A ← Σ_A + εI where ε ≈ 10⁻⁶
The metric tensor is the inverse of the covariance:
g_A = Σ_A⁻¹
This is standard information geometry. The metric defines the local inner product:
⟨u, v⟩_A = uᵀ g_A v
Distance between semantic entities x and y, as measured by observer A:
d_A(x,y) = √[(x-y)ᵀ g_A (x-y)]
This is the Mahalanobis distance. For different observer B with different g_B:
d_B(x,y) = √[(x-y)ᵀ g_B (x-y)] d_A(x,y) ≠ d_B(x,y) in general
The same content, different distances.
The covariance admits eigenvalue decomposition:
Σ_A = V Λ Vᵀ
Where Λ = diag(λ₁, λ₂, ..., λ_d) with eigenvalues in descending order.
Coherence is measured by variance concentration:
coherence = (λ₁ + λ₂ + λ₃) / Σᵢ λᵢ
When coherence ≥ 0.6, the observer has developed stable orientation. Below this threshold, Σ is still forming.
Condition number κ = λ₁/λ_d indicates posture:
Geodesics in Riemannian space satisfy:
d²xᵏ/dt² + Γᵏᵢⱼ (dxⁱ/dt)(dxʲ/dt) = 0
Christoffel symbols of the second kind:
Γᵏᵢⱼ = (1/2) gᵏˡ (∂gₗᵢ/∂xʲ + ∂gₗⱼ/∂xⁱ - ∂gᵢⱼ/∂xˡ)
In our implementation, each observer has a single global Σ (not position-dependent within their bubble). This means the metric is constant within an observer's space, Christoffel symbols vanish, and geodesics are straight lines in that observer's coordinates.
Curvature emerges only at interfaces between different observers' metrics, or across temporal evolution of a single observer's metric.
The feedback loop operates as a discrete dynamical system:
Σ(t+1) = f(Σ(t), v(t)) g(t+1) = [Σ(t+1)]⁻¹ γ(t+1) = geodesic(g(t+1), x(t), ẋ(t))
Where f is the covariance update function (incremental or batch).
The system does not optimize toward any target. Evolution is emergent from the coupling between observation and geometry.
When two observers A and B engage with shared content, their metrics create an interface. One approach to interface geometry:
g_AB = (g_A⁻¹ + g_B⁻¹)⁻¹ (harmonic mean of metrics)
Or weighted by engagement:
g_AB = (w_A g_A⁻¹ + w_B g_B⁻¹)⁻¹
The interface is where coupling occurs—where frame is revealed, where perspectives can touch without merging.
Semantic foam provides a mathematical framework for observer-dependent semantics. By deriving metric tensors from observation history (g = Σ⁻¹), we create a space where the same content has different distances for different observers. The closed loop between observation and geometry—where curiosity shapes the metric that defines curiosity—enables dynamic, self-organizing semantic structures.
This architecture preserves plurality not through policy, but through mathematics. There is no rest frame in semantic space. Every observer measures from their own motion. Neither is wrong. Both are real.
Bach, E. (1986). The algebra of events. Linguistics and Philosophy, 9(1), 5-16.
Dowty, D. (1991). Thematic proto-roles and argument selection. Language, 67(3), 547-619.
Kratzer, A. (1991). Modality. In von Stechow & Wunderlich (Eds.), Semantik: Ein internationales Handbuch.
Levin, B. (1993). English Verb Classes and Alternations: A Preliminary Investigation. University of Chicago Press.
Vendler, Z. (1957). Verbs and times. The Philosophical Review, 66(2), 143-160.
Amari, S. (2016). Information Geometry and Its Applications. Springer.
Correspondence: patrick@thecommons.ai