feat(deep_causality_topology): Added Lattice Gauge Field

deep_causality_physics/src/theories/electroweak/electroweak_impl.rs

@@ -9,8 +9,7 @@ use crate::{
 use deep_causality_metric::{LorentzianMetric, WestCoastMetric};
 use deep_causality_num::RealField;
 use deep_causality_tensor::{CausalTensor, TensorData};
-use deep_causality_topology::GaugeFieldWitness;
-use deep_causality_topology::{BaseTopology, GaugeField, Manifold, U1};
+use deep_causality_topology::{BaseTopology, GaugeField, GaugeFieldWitness, Manifold, U1};
 
 impl<S> ElectroweakOps<S> for ElectroweakField<S>
 where

deep_causality_physics/tests/theories/general_relativity/gr_ops_impl_tests.rs

@@ -11,7 +11,6 @@ use deep_causality_physics::theories::general_relativity::{GrOps, schwarzschild_
 use deep_causality_physics::{NEWTONIAN_CONSTANT_OF_GRAVITATION, SPEED_OF_LIGHT};
 use deep_causality_tensor::CausalTensor;
 use deep_causality_topology::{GaugeField, Manifold, Simplex, SimplicialComplexBuilder};
-
 // ============================================================================
 // GR Gauge Field Integration Tests
 // ============================================================================

deep_causality_topology/BUILD.bazel

@@ -15,6 +15,7 @@ rust_library(
         "//deep_causality_metric",
         "//deep_causality_multivector",
         "//deep_causality_num",
+        "//deep_causality_rand",
         "//deep_causality_sparse",
         "//deep_causality_tensor",
     ],

deep_causality_topology/Cargo.toml

@@ -18,6 +18,10 @@ path = "src/lib.rs"
 [features]
 # Default means CPU execution
 default = []
+# Enables random number generator from the host OS for secure random numbers.
+# Enabling os-random adds an indirect depdency on getrandom and libc.
+# Disabled by default to use native Rust prng instead.
+os-random = ["deep_causality_rand/os-random"]
 # MLX acceleration for Apple Silicon (M Series).
 # Forwards to deep_causality_tensor/mlx and deep_causality_multivector/mlx
 # for MLX / GPU-accelerated operations.
@@ -35,6 +39,10 @@ path = "../deep_causality_num"
 version = "0.1.12"
 default-features = false
 
+[dependencies.deep_causality_rand]
+path = "../deep_causality_rand"
+version = "0.1"
+
 [dependencies.deep_causality_sparse]
 path = "../deep_causality_sparse"
 version = "0.1.4"
@@ -64,20 +72,29 @@ name = "basic_graph"
 path = "examples/basic_graph.rs"
 
 [[example]]
-name = "manifold_analysis"
-path = "examples/manifold_analysis.rs"
+name = "chain_algebra"
+path = "examples/chain_algebra.rs"
+
+[[example]]
+name = "complex_operators"
+path = "examples/complex_operators.rs"
+
+[[example]]
+name = "differential_field"
+path = "examples/differential_field.rs"
 
 [[example]]
 name = "hkt_graph_convolution"
 path = "examples/hkt_graph_convolution.rs"
 
 [[example]]
-name = "chain_algebra"
-path = "examples/chain_algebra.rs"
+name = "lattice_gauge_simulation"
+path = "examples/lattice_gauge_simulation.rs"
 
 [[example]]
-name = "differential_field"
-path = "examples/differential_field.rs"
+name = "manifold_analysis"
+path = "examples/manifold_analysis.rs"
+
 
 [[bench]]
 name = "bench_causal_topology"

deep_causality_topology/README.md

@@ -2,39 +2,60 @@
 
 **Topological Data Analysis (TDA) and Causal Geometry for Rust**
 
-`deep_causality_topology` is a core crate of the `deep_causality` project, providing rigorous topological data structures and algorithms for causal modeling, geometric deep learning, and complex systems analysis.
+`deep_causality_topology` is a core crate of the `deep_causality` project, providing rigorous topological data
+structures and algorithms for causal modeling, geometric deep learning, and complex systems analysis.
 
-It bridges the gap between discrete data (graphs, point clouds) and continuous geometric structures (manifolds, simplicial complexes), enabling advanced reasoning about the "shape" and connectivity of causal systems.
+It bridges the gap between discrete data (graphs, point clouds) and continuous geometric structures (manifolds,
+simplicial complexes), enabling advanced reasoning about the "shape" and connectivity of causal systems.
 
 ## Features
 
-*   **Comprehensive Topological Types**:
-    *   **Graph**: Efficient sparse-matrix based graphs for causal networks.
-    *   **Hypergraph**: Modeling higher-order relationships (hyperedges) between multiple nodes.
-    *   **SimplicialComplex**: Generalizing graphs to higher dimensions (triangles, tetrahedra) to capture voids and holes.
-    *   **Manifold**: Validated geometric structures for differential geometry operations.
-    *   **PointCloud**: Raw multi-dimensional data with Vietoris-Rips triangulation capabilities.
-*   **Topological Algorithms**:
-    *   **Vietoris-Rips Triangulation**: Convert point clouds into simplicial complexes at a given scale.
-    *   **Euler Characteristic**: Compute topological invariants ($\chi$) to classify shapes (e.g., healthy vs. pathological tissue).
-    *   **Boundary/Coboundary Operators**: Sparse matrix operators for algebraic topology computations.
-*   **Algebraic Topology & Differential Geometry**:
-    *   **Chain Algebra**: Perform algebraic operations on chains (formal sums of simplices) and verify fundamental topological theorems like `∂∂=0`.
-    *   **Differential Operators**: Compute the exterior derivative (`d`), Hodge star (`⋆`), codifferential (`δ`), and Hodge-Laplacian (`Δ`) on discrete differential forms.
-    *   **Hodge Theory**: Detect topological features like holes and voids by finding harmonic forms (solutions to `Δω = 0`).
-*   **Higher-Kinded Types (HKT)**:
-    *   Implements `Functor`, `BoundedComonad` (Extract/Extend), and `BoundedAdjunction` (Unit/Counit) via `deep_causality_haft`.
-    *   Enables functional geometric patterns like "neighborhood extraction" (Comonad) and "geometric realization" (Adjunction).
+* **Comprehensive Topological Types**:
+    * **Graph**: Efficient sparse-matrix based graphs for causal networks.
+    * **Hypergraph**: Modeling higher-order relationships (hyperedges) between multiple nodes.
+    * **SimplicialComplex**: Generalizing graphs to higher dimensions (triangles, tetrahedra) to capture voids and
+      holes.
+    * **Manifold**: Validated geometric structures for differential geometry operations.
+    * **PointCloud**: Raw multi-dimensional data with Vietoris-Rips triangulation capabilities.
+* **Topological Algorithms**:
+    * **Vietoris-Rips Triangulation**: Convert point clouds into simplicial complexes at a given scale.
+    * **Euler Characteristic**: Compute topological invariants ($\chi$) to classify shapes (e.g., healthy vs.
+      pathological tissue).
+    * **Boundary/Coboundary Operators**: Sparse matrix operators for algebraic topology computations.
+* **Algebraic Topology & Differential Geometry**:
+    * **Chain Algebra**: Perform algebraic operations on chains (formal sums of simplices) and verify fundamental
+      topological theorems like `∂∂=0`.
+    * **Differential Operators**: Compute the exterior derivative (`d`), Hodge star (`⋆`), codifferential (`δ`), and
+      Hodge-Laplacian (`Δ`) on discrete differential forms.
+    * **Hodge Theory**: Detect topological features like holes and voids by finding harmonic forms (solutions to
+      `Δω = 0`).
+* **Higher-Kinded Types (HKT)**:
+    * Implements `Functor`, `BoundedComonad` (Extract/Extend), and `BoundedAdjunction` (Unit/Counit) via
+      `deep_causality_haft`.
+    * Enables functional geometric patterns like "neighborhood extraction" (Comonad) and "geometric realization" (
+      Adjunction).
 
 ## Core Concepts
 
-| Type | Description | Mathematical Structure |
-| :--- | :--- | :--- |
-| **PointCloud** | Set of points in $\mathbb{R}^n$ with metadata. | Discrete Metric Space |
-| **Graph** | Nodes and binary edges. | 1-Complex |
-| **Hypergraph** | Nodes and hyperedges (subsets of nodes). | Hypergraph |
-| **SimplicialComplex** | Collection of simplices (points, lines, triangles...) closed under sub-simplices. | Simplicial Complex $K$ |
-| **Manifold** | A topological space that is locally Euclidean. | Manifold $M$ |
+| Type                  | Description                                         | Mathematical Structure      |
+|:----------------------|:----------------------------------------------------|:----------------------------|
+| **PointCloud**        | Set of points in $\mathbb{R}^n$ with metadata.      | Discrete Metric Space       |
+| **Graph**             | Nodes and binary edges.                             | 1-Complex                   |
+| **Hypergraph**        | Nodes and hyperedges (subsets of nodes).            | Hypergraph                  |
+| **SimplicialComplex** | Collection of simplices closed under sub-simplices. | Simplicial Complex $K$      |
+| **Manifold**          | A topological space that is locally Euclidean.      | Manifold $M$                |
+| **Lattice**           | Regular discrete grid with periodic boundaries.     | $\mathbb{Z}^D$ Lattice      |
+| **CellComplex**       | Generalized simplicial complex (CW-complex).        | CW-Complex                  |
+| **Chain**             | Formal sum of simplices for algebraic topology.     | Chain Group $C_n$           |
+| **Skeleton**          | k-skeleton of a complex (simplices up to dim k).    | Skeleton $K^{(k)}$          |
+| **Simplex**           | Basic building block (vertex, edge, triangle...).   | $n$-Simplex                 |
+| **Topology**          | Abstract topological space with graded data.        | Graded Vector Space         |
+| **DifferentialForm**  | Discrete differential k-forms on a complex.         | $\Omega^k(M)$               |
+| **CurvatureTensor**   | Riemann/Ricci curvature tensor.                     | $R^{\mu}_{\nu\rho\sigma}$   |
+| **ReggeGeometry**     | Discrete gravity via deficit angles.                | Regge Calculus              |
+| **GaugeField**        | Gauge field on a manifold (connections).            | Principal Bundle Connection |
+| **LatticeGaugeField** | Wilson-formulation lattice gauge theory.            | $U_\mu(n) \in G$            |
+| **LinkVariable**      | Group element on a lattice edge.                    | $\text{SU}(N)$ Element      |
 
 ## Usage
 
@@ -57,9 +78,9 @@ fn main() -> Result<(), Box<dyn std::error::Error>> {
     // Adjacency: 0->1, 1->2
     let adj = CsrMatrix::from_triplets(3, 3, &[(0, 1, 1), (1, 2, 1)])?;
     let data = CausalTensor::new(vec![1.0, 2.0, 3.0], vec![3])?;
-    
+
     let graph = Graph::new(adj, data, 0)?;
-    
+
     println!("Neighbors of 1: {:?}", graph.get_neighbors(1)?);
     Ok(())
 }
@@ -88,15 +109,15 @@ Convert raw sensor data into structured geometry to detect anomalies (e.g., tumo
 use deep_causality_topology::PointCloud;
 
 // 1. Ingest Raw MRI Data
-let pc = PointCloud::new(points_tensor, metadata, 0)?;
+let pc = PointCloud::new(points_tensor, metadata, 0) ?;
 
 // 2. Triangulate (Vietoris-Rips)
-let complex = pc.triangulate(0.6)?;
+let complex = pc.triangulate(0.6) ?;
 
 // 3. Diagnose based on Topology
 let chi = complex.euler_characteristic();
 if chi <= 0 {
-    println!("Pathological: Detected Void/Necrosis");
+println ! ("Pathological: Detected Void/Necrosis");
 }
 ```
 
@@ -113,35 +134,37 @@ use deep_causality_topology::{Manifold, PointCloud};
 
 // Time-step the heat equation: ∂u/∂t = -Δu
 for _ in 0..num_steps {
-    let laplacian = manifold.laplacian(0);
-    // ... update data using: new_data = current_data - dt * laplacian
+let laplacian = manifold.laplacian(0);
+// ... update data using: new_data = current_data - dt * laplacian
 }
 ```
 
 ## Higher-Kinded Types (HKT)
 
 This crate leverages `deep_causality_haft` to provide functional geometric abstractions.
 
-*   **Functor**: Map functions over the data stored in the topology (e.g., transform node weights).
-*   **BoundedComonad (Extract/Extend)**:
-    *   *Extract*: Get the value at the current "cursor" (focus).
-    *   *Extend*: Apply a local computation (convolution) over the neighborhood of every point to produce a new topology. This is the foundation of **Graph Neural Networks (GNNs)** and **Cellular Automata**.
-*   **BoundedAdjunction (Unit/Counit)**:
-    *   *Unit*: Embed discrete data into a topological structure.
-    *   *Counit*: Project/Integrate topological data back into a flat representation.
+* **Functor**: Map functions over the data stored in the topology (e.g., transform node weights).
+* **BoundedComonad (Extract/Extend)**:
+    * *Extract*: Get the value at the current "cursor" (focus).
+    * *Extend*: Apply a local computation (convolution) over the neighborhood of every point to produce a new topology.
+      This is the foundation of **Graph Neural Networks (GNNs)** and **Cellular Automata**.
+* **BoundedAdjunction (Unit/Counit)**:
+    * *Unit*: Embed discrete data into a topological structure.
+    * *Counit*: Project/Integrate topological data back into a flat representation.
 
 ## Examples
 
-| File Name | Description | Engineering Value |
-| :--- | :--- | :--- |
-| `basic_graph.rs` | Graph construction & traversal | Foundation for large-scale causal network modeling. |
-| `manifold_analysis.rs` | 1-Manifold validation & Euler Char. | Ensuring geometric validity for differential operators. |
-| `point_cloud_triangulation.rs` | MRI Tissue Segmentation | Bridging raw sensor data with topological reasoning for diagnosis. |
-| `chain_algebra.rs` | Chain complex algebra & `∂∂=0` | Foundational verification for homological algebra. |
-| `differential_field.rs` | Solving the Heat Equation on a manifold | Simulating physical diffusion processes on complex shapes. |
-| `hodge_theory.rs` | Finding harmonic forms to detect holes | Advanced topological feature detection using the Hodge-Laplacian. |
+| File Name                      | Description                             | Engineering Value                                                  |
+|:-------------------------------|:----------------------------------------|:-------------------------------------------------------------------|
+| `basic_graph.rs`               | Graph construction & traversal          | Foundation for large-scale causal network modeling.                |
+| `manifold_analysis.rs`         | 1-Manifold validation & Euler Char.     | Ensuring geometric validity for differential operators.            |
+| `point_cloud_triangulation.rs` | MRI Tissue Segmentation                 | Bridging raw sensor data with topological reasoning for diagnosis. |
+| `chain_algebra.rs`             | Chain complex algebra & `∂∂=0`          | Foundational verification for homological algebra.                 |
+| `differential_field.rs`        | Solving the Heat Equation on a manifold | Simulating physical diffusion processes on complex shapes.         |
+| `hodge_theory.rs`              | Finding harmonic forms to detect holes  | Advanced topological feature detection using the Hodge-Laplacian.  |
 
 To run examples:
+
 ```bash
 cargo run -p deep_causality_topology --example point_cloud_triangulation
 ```
@@ -152,10 +175,11 @@ This project is licensed under the [MIT license](LICENSE).
 
 ## Security
 
-For details about security, please read the [security policy](https://github.com/deepcausality-rs/deep_causality/blob/main/SECURITY.md).
+For details about security, please read
+the [security policy](https://github.com/deepcausality-rs/deep_causality/blob/main/SECURITY.md).
 
 ## Author
 
-*   [Marvin Hansen](https://github.com/marvin-hansen).
-*   Github GPG key ID: 369D5A0B210D39BC
-*   GPG Fingerprint: 4B18 F7B2 04B9 7A72 967E 663E 369D 5A0B 210D 39BC
+* [Marvin Hansen](https://github.com/marvin-hansen).
+* Github GPG key ID: 369D5A0B210D39BC
+* GPG Fingerprint: 4B18 F7B2 04B9 7A72 967E 663E 369D 5A0B 210D 39BC