How Habitat Is Built

At the heart of Habitat is an equation: g = Σ⁻¹

This isn't a design choice. It's information geometry — the mathematics of how meaning patterns relate to each other.

g = Σ⁻¹  # The core identity
# Your metric IS your covariance, inverted
# No external frame required

This equation creates architecture that serves human judgment. The geometry is built from process observations — how things unfold, not what they are. This makes differences visible across perspectives, because process properties (telicity, agency, causality) are readable from any worldview.

The Foam Model

Bubbles, not containers. Apertures, not boxes.

Habitat doesn't put perspectives into containers. Each perspective is a bubble — a covariance structure (Σ) built from compositional acts. The bubble IS the observation aperture, with its own geometry (g = Σ⁻¹). Meaning emerges from observation relations, not from fixed positions.

# No true possessive - only relation
def get_relational_sigma(observed, observer):
    """
    Σ(observed | observer) - relational pattern.
    This is NOT "observed's sigma" - sigma doesn't
    belong to anyone. It emerges from relation.
    """

This means:

No god's-eye view. There's no position outside the foam from which everything can be seen objectively. Every observation happens from within a bubble — including the observation of other bubbles. The code implements this: get_relational_sigma(observed, observer) always requires an observer.

Interfaces, not walls. Where bubbles meet, something new emerges. The interface between your bubble and mine isn't a barrier — it's where coupling happens. The code computes this emergence: get_coupling_sigma(entity_a, entity_b).

Geometry makes structure visible. Each bubble has its own shape (Σ). When you look at my bubble from yours, you see it through your geometry. The mathematics make this visible — what you do with that visibility is your choice.

Solving the Flattening Problem

The challenge: AI systems converge. Perspectives get averaged into one voice.

How Habitat helps:

Each perspective maintains its own bubble geometry (Σ). When Indigenous knowledge and scientific analysis interact, neither gets absorbed into the other. The coupling between them produces something new:

def get_coupling_sigma(entity_a, entity_b):
    """
    Σ(a ⊗ b) - coupling pattern between two.
    It belongs to NEITHER - it's an emergent third thing.
    """
    # Harmonic mean emphasizes shared structure
    eig_harmonic = 2 * eig_a * eig_b / (eig_a + eig_b)

The coupling creates something genuinely new without collapsing either source. The interface emphasizes what they share while preserving what's distinct.

Result: People can work across perspectives without losing what makes each valuable. The geometry keeps the differences visible.

Solving Trust Collapse

The challenge: You can't verify where information came from or how it was transformed.

How Habitat helps:

Every piece of content carries its geometric signature — not as attached metadata, but as intrinsic structure. The signature shows:

# Σ(olivia | maya) ≠ Σ(olivia | manifold)
# Same "olivia", different patterns
# depending on who observes
sigma_from_maya = get_relational_sigma(olivia, maya)
sigma_from_grace = get_relational_sigma(olivia, grace)

When content moves through the system, its geometric history remains visible. You can see what perspective shaped it by looking at its structure — then decide what that means to you.

Result: Provenance becomes visible structure. What you do with that information is your judgment to make.

Coupling at the Interface

Habitat is an interface of interfaces. The recursion bottoms out in touching.

There is no global manifold. No neutral space. No container. Just observation-bubbles and their interfaces — where bubbles touch without merging.

# When two bubbles touch:
# Surfactant thins at contact
# Contents can see across (if Fresnel ready)
# Coupling emerges IN the interface
# Bubbles remain distinct

Σ(a ⊗ b) = get_coupling_sigma(entity_a, entity_b)
# Harmonic mean emphasizes shared structure
# while preserving what's distinct

What happens at the interface:

Indigenous knowledge and scientific analysis touch. The coupling emerges in the interface — belongs to neither. Constitutional dimensions stay distinct (sovereignty). Bridgeable dimensions enable exchange (coordination). The geometry shows both.

Nothing "holds" the foam. The foam holds itself: curiosity creating bubbles, bubbles touching, interfaces stabilizing or destabilizing.

Result: Coordination without forced convergence. Bubbles remain distinct. Coupling emerges where they touch. People decide — the geometry makes the terrain visible.

Portable Knowledge

Knowledge should travel. Curation should persist. Publication should carry provenance.

What Gems are:

A Gem is not just frozen data. It's a frozen metric space — the Σ configurations at each point along a worldline. When you view someone else's Gem, the foam warps per your perspective: same connections, different distances.

@dataclass
class GemWithTraceMemory:
    """Extended GEM that includes frozen worldline with traces.
    Another user doesn't just get Σ — they get memory."""
    
    Σ_gem: np.ndarray      # Frozen covariance
    trace_memory: GemTraceMemory  # The worldline trace
    is_public: bool        # Publishing state

What happens when you view someone's Gem:

Their connections appear at your curiosity-warped distances. High curiosity contracts the metric — their connections seem closer. Low curiosity expands it — connections seem farther. Same worldline, different measurements.

def warp_distance_through_metric(original_distance, ...):
    """The Lorentz transformation of Habitat spacetime.
    Same connection, different distance for different observers.
    Curiosity warps semantic space like mass warps spacetime."""
    curiosity_ratio = original_curiosity / viewer_curiosity
    return original_distance * curiosity_ratio * (1 + divergence)

Result: Knowledge becomes portable without losing its origins. You can publish, share, and curate — and everyone who receives it sees it from their own position, while the source remains traceable.

How Traceability Works

Traceability is not decomposition. It's historicity — the path through metric configurations.

What a trace actually is:

A trace is a worldline through semantic spacetime. At each point, curiosity configures the metric. The trace captures not just what was known, but how curiosity evolved through compositional acts.

@dataclass
class TracePoint:
    """Single point on a worldline — one curious act in semantic spacetime."""
    actant_vector: np.ndarray   # 5D ProcessActor (position)
    predicate_vector: np.ndarray # 12D ProcessAssert (position)
    connections_at_act: Dict[str, float]  # What was near in THAT metric
    curiosity_at_act: float     # The metric configuration (g = Σ⁻¹)

# The feedback loop
Σ(t+1) = f(Σ(t), composition(t))  # Covariance evolves with acts
g(t+1) = [Σ(t+1)]⁻¹               # Metric updates
γ(t+1) = geodesic(g(t+1))         # New curiosity path emerges

When another observer views your trace, they see your worldline warped through their metric. Same connections, different distances. High curiosity contracts — connections seem closer. Low curiosity expands — connections seem farther.

Result: Not "who did what" but "how curiosity evolved through acts, changing the metric at each step" — visible from any position, warped by each observer's own curiosity.

How Safety Is Built In

Surveillance economics violates trust. Activity tracking enables manipulation.

What Habitat observes:

Habitat observes what you do, not what you look at. Only explicit actions:

Agents in the system are observation instruments, not decision-makers. They observe and report — they don't have goals or optimization objectives. The decisions stay with people.

Result: Safe to participate because the architecture doesn't enable surveillance — and doesn't make decisions for you.