Apoth3osis
<_MODELS/SEMANTIC_CLOSURE

Semantic Closure Genesis Model

Watch a mathematical proof come alive. This visualization is driven by data exported directly from verified Lean 4 theorems—the numbers you see are certified by the proof assistant, not hand-tuned by artists.

Lean-driven scene

Narrative

Semantic Closure

Proof snapshot (offline)

Fixed-point convergence visualization. Drag to rotate.

phases

3

selectors

1 -> 2 -> 0

What You're Seeing

The three glowing nodes represent the components of Rosen's (M,R)-system: metabolism (produces components), repair (rebuilds machinery), and the environment(provides resources). The system must "close" into a self-sustaining loop.

BARS
The vertical bars show actual vector values from the Lean proof. Cyan = positive, red = negative. Watch them transform as the closure operator is applied.
PHASES
Preimage: Raw input vectors (unstable). Closure: ReLU transformation applied. Fixed: Threshold transformation reaches stable point.
GLOW
When closure is achieved, the center pulses emerald and particles burst outward—the mathematical moment when self-reference becomes stable.
SELECTOR
The indices [1, 2, 0] in the corner show which node maps to which—the "wiring" of the closure operator that the Lean proof certifies is consistent.

The Mathematical Core

The key theorem is closeSelector.idem: applying the closure operator twice gives the same result as applying it once. This proves self-repair doesn't lead to infinite regress.

>NUCLEUS_OPS

The Nucleus Transformations

The visualization animates two key operations from the Lean proofs. These aren't arbitrary—they're the mathematical "gates" that allow self-reference to stabilize into closure.

ReLU Nucleus

Like a neural network's rectified linear unit: negative values become zero, positive values pass through. The theorem reluNucleus_idempotent proves applying ReLU twice equals applying it once.

input [-0.8, 0.35, 0.9] -> [0, 0.35, 0.9]

Threshold Nucleus

Clamps values to a boundary: if below threshold, snap to threshold. The theorem thresholdNucleus_fixed_iff proves exactly when a point is stable.

threshold [-0.1, 0.5, 0.6]
output [-0.1, 0.5, 0.9]
>SIGNIFICANCE

Why This Matters

For Biology

This is a mathematical model of how life maintains itself. Every cell must continuously rebuild its own machinery while using that machinery—the closure operator shows this isn't paradoxical.

For AI

Self-improving systems face the same challenge: how do you modify the system that's doing the modifying? Closure operators provide a rigorous foundation for stable self-reference.

For Verification

This demonstrates a new paradigm: using proof assistants not just to verify code, but to generate visualizations guaranteed to be mathematically accurate.

>GALLERY
Lean-driven visualization

Lean-driven visualization

Captured from the proof-parametrized WebGL scene showing closure convergence.

Concept reference

Concept reference

The (M,R)-system triangle: metabolism, repair, and environment in closed loop.

>STATUS

This model runs entirely in your browser using WebGL. The proof data is loaded from JSON files exported by the Lean build process. For the full interactive experience with sound design, contact rgoodman@apoth3osis.io.