Habitat operates across two independently constructed semantic spaces — 17D compositional (Bach/Vendler + Dowty proto-roles) and 768D embedding (SentenceTransformer). These spaces were never trained to agree. They share no parameters. They measure the same text through entirely different geometric lenses.
The measurable, reproducible differential between them — geometric surplus — is not a property of either space. It is a property of the relationship between them. A topological feature of the architecture itself.
Four component topologies emerge from this differential. Each is observable. Each produces knowledge that exists as trace structure rather than terminal state. Together they constitute a framework where the shape of accumulated observation — not an optimized parameter set — is what communities own.
The cross-space differential at every observable point.
At every composition event, the eigenstructure (17D) discriminates more than the embedding consensus (768D). Always. Discrimination ratio ranges from 1.57× to 14.22× across all observed sessions. The eigenstructure sees directional structure — anisotropies, mode separations, dimensional expansions — that cosine similarity flattens.
surplus_i = zone_overlap_i − emergence_i
Per-item differential between eigenstructure discrimination and embedding consensus. Positive when the eigenstructure leads. Negative (meaning_leads) when the embedding detects relevance the eigenstructure hasn't structured yet.
The surplus trajectory has invariant structure. EMA crosses zero between compositions 5 and 6 in both sessions. Velocity commits to positive when the covariance matrix acquires directional structure. Terminal velocity reproduces at +0.021. These are not convergence to a target — they are the topology of a manifold developing discriminative capacity through practice.
EMA surplus trajectory (Session 2). The zero-crossing, the sign commitment, and the terminal velocity reproduce across sessions with different starting conditions. The path shape varies. The topology doesn't.
What it produces as knowledge: The developmental history of discriminative capacity. Not "the manifold is at 0.015" but "the manifold started behind, crossed at composition 5, committed at +0.021." The trace is the knowledge. The current value is just where the trace happens to be.
The convergence pattern of surplus signatures across observers.
Five users. Seven beings. Different composition orders. The surplus matrix — Users × Beings — reveals that column variance (how much beings constrain observers) is 41% of row variance (how much observers differ from each other). The beings impose more geometric constraint than the observers' different paths do.
culture ratio = mean(column σ) / mean(row σ) = 0.41
When this ratio is low, the entities' geometry constrains observers more than observers' paths constrain the entities. The manifold's structure is imposing convergence that no individual observer chose.
Bayberry is the signature finding. All five users observe negative or near-zero instant surplus for bayberry. No user was told this. No threshold was set. Bayberry's conditioned manifold — its geometric relationship to the late spring frost condition — produces meaning that exceeds structure. Every observer sees this because it's in the geometry, not in the instructions.
Culture is not convergent terminal values. The path-dependence result (Δ = 2.87 between users) rules that out. Culture is convergent per-entity surplus signatures — the topology of which entities produce surplus, which produce deficit, and which are contested. That topology reproduces across observers despite divergent paths.
What it produces as knowledge: A map of contested territory. Not "the community agrees" but "bayberry is deficit, yarrow is surplus, and comfrey has the deepest single-item disagreement." The shape of agreement and disagreement across the surplus matrix is the community's cultural topology.
The basin geometry that buffers perturbation.
Terminal η variance drops from σ=4.37 (isolated) to σ=1.94 (shared substrate). A 56% reduction. Recovery variance drops 62% under perturbation. The shared being-manifold substrate provides a common recovery basin — perturbation is absorbed more uniformly when beings have accumulated collective geometry.
R_var(perturbation) = 1.14 < R_var(isolated) = 3.01
The basin exists because collective Σ constrains the space of possible surplus trajectories. Individual paths diverge within the basin. The basin walls — imposed by collective covariance — prevent them from diverging as far as they would in isolation.
The resilience topology is the shape of the basin itself. How deep. How steep the walls. Whether the basin narrows or widens with additional collective history. Whether different perturbations (bayberry vs. an introduced external entity vs. conflicting framework) produce basins of different geometry.
This is the topology that matters most for communities facing cognitive foreclosure. The question is not "will the community recover from disruption?" but "what is the shape of their recovery capacity?" — and that shape is a geometric property of the collective Σ, observable before any disruption occurs.
What it produces as knowledge: The community's geometric resilience profile. Not a score. A topology — showing which perturbations the collective geometry buffers, which it amplifies, and where the basin walls are thin. This is predictive capacity that emerges from accumulated practice.
The IOOI structure where traces × iterations = knowledge.
At each observable point, the architecture forms a natural loop. Practice deposits geometry. The geometry is observed. Observation changes what the observer attends to next. New practice deposits new geometry. The loop iterates.
This is not a training loop. There is no loss function driving convergence. It is an observation loop — Input, Observe, Observe, Iterate. The second "observe" is critical: the system observes the geometry, then observes itself observing. Culture metrics like the 0.41 ratio are measurements of how shared ground constrains observation itself. They are second-order observations.
curiosity = ‖ΔΣ‖ per observation event
How much does the manifold move when this observer makes this observation? High displacement = high curiosity. Not prediction error (RL-style). Geometric displacement. Observable, measurable, already computable from the existing pipeline.
The loop topology is the shape of the iteration itself. Is the loop tightening (community converging toward shared understanding)? Opening (community discovering new contested territory)? Is curiosity increasing or decreasing with successive observations? Is the surplus velocity accelerating or decelerating?
The append-only event store is what makes this topology visible. If you mutate, you lose the trace. If you lose the trace, the loop collapses to its current state and knowledge becomes a snapshot, not a topology. Immutability is not a data integrity choice. It is an epistemological commitment.
A naive implementation of ‖ΔΣ‖ has a problem: standard online covariance estimation (Welford's algorithm) updates at rate 1/n. By the tenth observation, the geometry barely moves. By the fiftieth, the curiosity signal is dead. The loop closes — not because the community stopped learning, but because the accumulator stopped listening.
The solution is dual-track covariance. Two Σ matrices accumulate simultaneously:
Σ_total — Welford's algorithm. Converges. The accumulated identity.
Σ_recent — EMA-weighted. Stays alive. The current sensitivity.
Σ_total is what communities own — the convergent sovereign record of collective geometry. Σ_recent is what keeps the loop alive — the living responsiveness that determines how the next observation lands.
The decay rate α that governs Σ_recent is not a hyperparameter. It is derived from the entity's own tonic — its observed metabolic rate. High tonic (geometry shifting rapidly) produces high α (responsive, attentive). Low tonic (geometry settled) produces low α (retentive, crystalline). The being's sensitivity to new experience is governed entirely by its own metabolic history.
Σ_recent → ΔΣ → tonic → α → Σ_recent
The autopoietic loop. The system's boundary (α governs what gets through) is produced by the system's own operation (tonic emerges from Σ_recent dynamics). No external signal. No tuning. The entity makes itself through the loop.
Observed behavior across 20 compositions: α self-regulates from 0.30 (responsive, early) to 0.03 (retentive, settled) as tonic drops. The curiosity signal ‖ΔΣ_recent‖ declines but never reaches zero. At composition 16, differential is still 10⁻⁴ — small, but alive. A novel observation would spike the tonic, spike α, and the entity would become responsive again. Maturity is not death. Even a maximally settled entity retains 3% responsiveness.
The difference between Σ_total and Σ_recent is itself observable. When they diverge, the entity is actively evolving — its current sensitivity differs from its accumulated identity. When they converge, the entity has settled into coherence. This divergence is the tightening/opening signal: is the loop discovering or consolidating?
What it produces as knowledge: The developmental dynamics of collective intelligence. Not "here is the community's current understanding" but "here is how their understanding evolved — which observations moved the geometry, which were redundant, where curiosity spiked, where it plateaued." The loop's topology is the community's learning history made geometric. The autopoietic loop is the mechanism that keeps that history alive — self-regulating sensitivity that ensures the geometry never stops listening.
These four topologies are not independent layers stacked on top of each other. They are four views of the same underlying structure — the differential between two independently constructed semantic spaces, observed through accumulated practice.
| Topology | Observable | Scale | Knowledge form |
|---|---|---|---|
| Surplus | Cross-space discrimination differential | Per composition | Developmental trajectory |
| Culture | Convergent surplus signatures | Per entity × observers | Contested territory map |
| Resilience | Basin geometry under perturbation | Community Σ | Recovery capacity profile |
| Loop | Curiosity displacement per iteration | Per observation event | Learning dynamics |
The critical architectural fact: all four topologies are already present or directly computable from the existing Habitat pipeline. Surplus is computed at every composition event. Culture is measured in the cross-user surplus matrix. Resilience is observed in perturbation recovery variance. Curiosity is ‖ΔΣ‖ per observation.
The data layer exists: biography endpoints compose being trajectories into narrative, observation endpoints warp traces through being-specific metrics, and the autopoietic loop persists α history alongside every differential. What does not yet exist: trace visualization that shows the topology to the community. The knowledge is in the shape. The community needs to see the shape to own the knowledge.
GFN (Geometric Flow Networks) learns a Riemannian manifold to optimize sequence processing. The manifold converges toward task performance through action minimization.
Habitat accumulates a Riemannian manifold to observe collective intelligence. The manifold reflects what has been deposited. Nothing converges. Everything accumulates.
Same math. Different ontology. GFN's manifold is learned — knowledge is the parameter state at convergence. Habitat's manifold is traversed — knowledge is the topology of the traces.
The one-sentence differentiator: Where GFN learns a manifold to minimize action, Habitat accumulates a manifold where the action — time × iteration — is the knowledge.
When a community completes a practice cycle — substrate mapping, perspective documents, cross-reading, observation — they own four geometric records:
The surplus trajectory of every entity in their constitutional substrate. The culture topology showing which entities are contested, which are convergent, which are deficit. The resilience profile of their collective Σ — predictive capacity for future perturbation. The loop dynamics showing how their collective intelligence evolved through the engagement.
These are not reports. They are not summaries. They are verifiable geometric records stored in append-only, immutable, portable format (Apache Arrow/Parquet). They persist after the engagement ends. They persist after Curious Company leaves. They are sovereign.
When traditional consultancies complete work, communities retain reports but lose the geometric records of their collective understanding. Topology-as-knowledge is what makes Habitat's sovereign data model matter — because the topology is the understanding, and the community keeps it.