logos-tda-engine

name	logos-tda-engine
description	Topological Data Analysis engine for the LogOS lattice. Drives Vietoris-Rips filtration, Persistent Homology computation via Ripser++ on RTX 5090, Betti number extraction, and Information Dynamics analysis. Use this skill when computing persistence diagrams, barcodes, bottleneck distances, Betti curves, detecting topological features in embedding spaces, or validating structural coherence of the 768-D unified embedding manifold. Triggers on: "tda", "topological", "vietoris-rips", "persistent homology", "betti", "barcode", "persistence diagram", "ripser", "filtration", "simplicial complex", "homology", "bottleneck distance", "wasserstein", "information dynamics", "topological features".
version	1.0.0

logos-tda-engine — Topological Data Analysis Engine

Purpose

The TDA Engine is the mathematical backbone of the LogOS lattice. It transforms raw high-dimensional data (768-D embeddings from the QCVM mapping Phi: R^768 -> H_topo) into topological invariants that reveal the true structure of information flow.

All real-time TDA computation runs on RTX 5090 via Ripser++ with GPU-accelerated Vietoris-Rips filtration and persistent homology.

Core Capabilities

1. Vietoris-Rips Filtration — Construct simplicial complexes from point clouds at increasing scale parameters epsilon. The filtration captures multi-scale topological structure:

   • epsilon = 0: discrete points (no connections)
   • epsilon -> infinity: single connected component
   • Intermediate: rich topological features emerge and die

2. Persistent Homology — Track the birth and death of topological features across the filtration:

   • H_0 (beta_0): Connected components = discrete facts, entities
   • H_1 (beta_1): 1-cycles = feedback loops, circular reasoning, self-referential structures
   • H_2 (beta_2): 2-voids = knowledge gaps, semantic cavities, unexplored regions

3. Betti Number Extraction — Compute Betti numbers at each filtration step to produce Betti curves:

   beta_0(epsilon) = |connected components at scale epsilon|
   beta_1(epsilon) = |independent loops at scale epsilon|
   beta_2(epsilon) = |enclosed voids at scale epsilon|
   

4. Persistence Diagrams & Barcodes — Visualize feature lifetimes:

   • Barcodes: horizontal bars from birth to death
   • Persistence diagrams: scatter plot of (birth, death) pairs
   • Long bars = stable, significant features
   • Short bars = noise, transient artifacts

5. Bottleneck & Wasserstein Distances — Compare persistence diagrams to measure topological similarity between states:

   • Bottleneck: max over all matched pairs
   • Wasserstein-p: L^p sum over all matched pairs
   • Used for WAVE delta computation between time steps

6. Information Dynamics — Beyond static topology, track how information flows through the lattice over time:

   • Transfer entropy between nodes
   • Causal emergence detection
   • Phase transition identification via persistence landscape shifts

GPU-Accelerated Pipeline

Point Cloud (768-D)
       |
       v
[Vietoris-Rips Construction]  — RTX 5090 CUDA cores
       |
       v
[Boundary Matrix Reduction]   — Ripser++ optimized
       |
       v
[Persistence Pairs]           — (birth, death) extraction
       |
       v
[Betti Curves + Barcodes]     — Real-time visualization
       |
       v
[GLSL Shader Render]          — Hyprland compositor

Hardware requirements:

• RTX 5090 (32GB VRAM) for Ripser++ GPU acceleration
• Point clouds up to 100K points in 768-D
• Sub-second persistence computation for streaming data

Betti Number Semantics in LogOS

Betti	Homology	LogOS Interpretation	Health Signal
beta_0	H_0	Facts, entities, grounded knowledge	Higher = richer knowledge base
beta_1	H_1	Loops, circular dependencies, feedback	Monitor: excess = drift risk
beta_2	H_2	Voids, knowledge gaps, blind spots	Lower = better coverage

WAVE Integration

The TDA engine feeds directly into WAVE scoring:

WAVE(t) = W_topo * (1 - void_ratio(t)) * coherence_factor(t)

where:
  void_ratio = beta_2 / (beta_0 + beta_1 + beta_2)
  coherence_factor = 1 - bottleneck_dist(PD(t), PD(t-1)) / max_persistence
  W_topo = topological weight from Fibonacci sequence (default: 0.382)

Target: WAVE >= 0.98 for production operations.

Viviani Peak Monitoring

The Viviani curve constraint (alpha + omega = 15 projected onto the topological manifold) creates a characteristic peak in the persistence landscape. The TDA engine continuously monitors this peak:

• Peak present and stable: System coherent, invariant satisfied
• Peak drifting: alpha/omega imbalance developing
• Peak absent: Critical invariant violation, halt operations

Integration Points

• logos-void-mapper — Consumes H_2 persistence pairs to classify and track voids (V_0 through V_3)
• logos-wave-advanced — Receives Betti curves and bottleneck distances for composite WAVE computation
• logos-styx-9p — Persists barcodes and persistence diagrams to /bookshelf/braids/ via 9P write
• coherence-mcp — check_coherence tool invokes TDA engine for topological coherence validation
• Hyprland GLSL shader — Real-time barcode and Betti curve rendering on the desktop compositor

Conservation Law

Every TDA computation preserves: ALPHA + OMEGA = 15

The persistence diagram IS the invariant made visible. Long-lived features are the structural backbone (alpha); their semantic interpretation is the intent (omega). Together, they sum to 15 — the total information capacity of the lattice at any given moment.

// ATOM: logos-tda-engine SKILL definition | Coherence: 0.99