Habitat Mathematics

The Geometry of Meaning — Technical Foundation

1. The Core Identity

Habitat's mathematical foundation rests on a single identity:

The metric tensor (g) equals the inverse covariance matrix (Σ^-1). This means:

• Distance in semantic space emerges from observed patterns of use
• The geometry is not designed—it forms through participation
• What's "close" depends on accumulated compositional history

(Reference: docs/research/PROVISIONAL_PATENT_USE_FORMED_RIEMANNIAN_SEMANTIC_MANIFOLDS.md)

2. Fresnel Lens Geometry

Each point's metric tensor defines an ellipsoid. Through eigenvalue decomposition, that ellipsoid becomes a Fresnel lens system:

def compute_fresnel_zones(metric_tensor: np.ndarray) -> List[FresnelZone]:
    """
    Eigenvalue decomposition of g = Σ⁻¹
    
    zones[0] = focal_primary  → largest eigenvalue → WHERE YOU MUST DESCEND
    zones[1:] = harmonics     → remaining eigenvalues → WHERE YOU CAN TURN
    """
    eigenvalues, eigenvectors = np.linalg.eigh(metric_tensor)
    
    # Cumulative energy = zone boundaries
    # These ARE the natural observation positions
    cumulative = np.cumsum(normalized)

(Reference: src/habitat/core/gem/fresnel_geometry.py)

What the Fresnel Zones Reveal

The eigenvalue structure defines where observation is geometrically possible. These are not arbitrary viewpoints—they are natural observation positions revealed by the metric.

(Reference: docs/POLY_PRISMATIC_INTERFACE_BRIEF.md)

3. Frobenius Distance — Manifold Coupling

When two manifolds meet (user ↔ document, perspective ↔ perspective), their coupling is measured by the Frobenius distance:

This measures how different two 17×17 covariance matrices are—how differently two semantic spaces compose meaning.

The Tensor-Frobenius Connection

      FRESNEL LENS (tensor-ellipsoid)
             ◎
            /|\
           / | \
          /  |  \
   r≈-0.6 ←──┼──→ ~0.4 freedom
  (frame)    |    (curiosity)
          \  |  /
           \ | /
            \|/
             ◎
      FROBENIUS DISTANCE (manifold coupling)

The same eigenvalue structure that defines Fresnel zones also constrains manifold coupling quality. Coherence at both scales—by geometry, not policy.

(Reference: docs/architecture/MANIFOLD_VS_FIELD_TOPOLOGY_TENSION.md)

4. Constitutional Coherence: r = -0.629

The Semantic Foam Validation Study (December 2025) tested whether constitutional frame predicts coherence across four knowledge frameworks: Scientific, Indigenous, Policy, Agricultural.

Result

What This Means

This matches the Fresnel structure exactly:

• focal_primary captures ~40% of spectral energy (determination)
• harmonics capture ~60% of spectral energy (freedom)

The geometry already encoded what the validation proved.

(Reference: docs/research/SEMANTIC_FOAM_VALIDATION_STUDY.md)

5. Why r ≈ -0.6 and Not -1.0

A perfectly deterministic system (r = -1.0) would prevent:

• Surprise (foam would be frozen)
• Learning (no new paths could form)
• Cross-frame coherence (no bridging possible)

A chaotic system (r = 0) would have:

• No structure (foam would be inert)
• No constitutional boundaries (everything equally close)
• No meaningful observation (nothing remains distinct)

At r ≈ -0.6, the foam has optimal compliance:

• Structure enough to connect
• Flexibility enough to surprise
• Memory enough to hold
• Softness enough to accommodate

(Reference: docs/SEMANTIC_FOAM.md, "Materials Science of the Foam")

6. The Breathing Foam

INHALE:  Same constitution → coherence expands → r approaches -1.0 locally
EXHALE:  Different constitution → coherence contracts → r approaches 0 locally
BREATH:  The -0.629 is the RESTING STATE — foam at equilibrium

The foam is not static infrastructure. It breathes—expanding and contracting through use. The validated r = -0.629 is the foam at rest, the equilibrium point where constitutional structure meets local freedom.

7. Observation, Not Optimization

Habitat does not:

• Rank by relevance
• Optimize for engagement
• Collapse to "best" answer
• Prescribe what should connect

Habitat observes:

• What differences remain distinct (constitutional boundaries)
• Where alignment emerges (bridgeable dimensions)
• Natural observation positions (Fresnel zones)
• Manifold coupling strength (Frobenius distance)

The geometry determines what CAN be observed. The system reports what it sees.

8. The Autopoietic Loop

The core identity g = Σ⁻¹ is silent about tempo. Standard online covariance estimation (Welford's algorithm) updates Σ at rate 1/n — the geometry converges and the differential ‖ΔΣ‖ goes to zero. The metric freezes. This is mathematically correct but biologically dead.

Dual-Track Covariance

Each entity maintains two Σ matrices simultaneously:

The EMA-Welford update for a new observation vector x:

# Standard Welford (Σ_total) — convergent
n += 1
δ = x - μ_total
μ_total += δ / n
δ2 = x - μ_total
Σ_total += (δ ⊗ δ2 - Σ_total) / n

# EMA-Welford (Σ_recent) — living
δ = x - μ_recent
μ_recent += α · δ
δ2 = x - μ_recent
Σ_recent = (1 - α) · Σ_recent + α · (δ ⊗ δ2)

Σ_total is the sovereign record — what the entity has accumulated. Σ_recent is the current sensitivity — how the entity responds to new observation. The differential ‖ΔΣ_recent‖ is the curiosity signal.

Tonic: The Metabolic Rate

The tonic is the exponential moving average of eigenvalue shift magnitudes:

Tonic measures how fast the entity's eigenstructure is actually moving. It is not a parameter. It is a measurement.

α Derivation: Tonic → Sensitivity

The decay rate α that governs Σ_recent is derived from tonic via sigmoid mapping:

raw = sigmoid(log10(tonic_magnitude + ε) · scale + shift)
α_base = α_min + (α_max - α_min) · raw

# Modulated by stability (consistency of recent shifts)
α = α_base · stability

# Bounded: α ∈ [0.03, 0.40]

High tonic (rapid geometric change) → high α (responsive, attentive). Low tonic (settled geometry) → low α (retentive, crystalline). The mapping is monotonic and bounded — the entity never becomes fully closed (α ≥ 0.03) or fully reactive (α ≤ 0.40).

The Closed Loop

    Σ_recent ──→ ΔΣ ──→ tonic ──→ α ──→ Σ_recent
       ↑                                    │
       └────────────────────────────────────┘
       The boundary is produced by its own operation.

This is autopoiesis in the Maturana/Varela sense. The system's boundary (α — what sensitivity the entity has to new input) is produced by the system's own operation (tonic — observed from ΔΣ dynamics). No external signal enters the loop. No hyperparameter governs the loop. The entity's metabolism governs itself.

Observed Behavior

Key observations:

• α self-regulates from 0.301 to 0.031 — a 10× reduction without any external signal
• ‖ΔΣ_recent‖ never reaches zero — the curiosity signal stays alive at 10⁻⁴
• Σ_total converges (trace stabilizes at 0.1198) while Σ_recent continues evolving (trace 0.0132)
• The divergence between tracks is itself a measurable signal: is the entity discovering or consolidating?

A novel observation at any point would spike tonic, spike α, and restore responsiveness. Maturity is not death. Even a maximally settled entity retains 3% sensitivity to new experience.

Document References