HodgeMesh Technical Architecture

Component Layout

graph TD
    subgraph UserSpace [Simulation & User Interface]
        Dashboard[Obs Dashboard HTML/JS]
        Sim[Simulation Script]
    end

    subgraph LedgerSpace [State & Network Layer]
        NodeA[HodgeNode A]
        NodeB[HodgeNode B]
        Server[SSE Telemetry Server]
    end

    subgraph GeometricSpace [Geometric Foundations]
        SSC[SimplicialStateComplex]
        Operators[DEC Operators: d0, d1, delta]
        Reconciler[Geodesic Reconciler]
        Solver[Conjugate Gradient Solver]
    end

    Sim --> NodeA
    Sim --> NodeB
    NodeA <-->|Sync Merge| NodeB
    NodeA -->|Post Telemetry| Server
    Server -->|SSE event-stream| Dashboard

    NodeA & NodeB --> SSC
    NodeA --> Reconciler
    Reconciler --> Operators
    Reconciler --> Solver
                

1. SimplicialStateComplex (complex.py)

Tracks causal event histories as a directed simplicial complex:

• 0-simplices (Vertices \(V\)): Events. Each vertex tracks its causal parents.
• 1-simplices (Edges \(E\)): Directed transitions. If vertex \(u\) is a parent of \(v\), the edge is \((u, v)\).
• 2-simplices (Triangles \(T\)): Formed automatically when a vertex has multiple parents that are themselves causally connected. This preserves the simplicial topology and represents concurrent branches.

2. DEC Operators (operators.py)

Computes the linear operators on \(k\)-cochains:

• Exterior Derivative \(d_0\) (Gradient): Maps 0-cochains (vertex potentials) to 1-cochains (edge flows).
• Exterior Derivative \(d_1\) (Curl): Maps 1-cochains (edge flows) to 2-cochains (triangle circulations).
• Codifferential \(\delta_1\) (Adjoint): Maps 1-cochains to 0-cochains. Implemented as the adjoint \(d_0^T\).
• Laplacian Operators:
  • \(L_0 = d_0^T d_0\) (Vertex Laplacian)
  • \(L_2 = d_1 d_1^T\) (Triangle Laplacian)

3. Iterative CG Solver (solver.py)

A pure-standard-library Conjugate Gradient solver optimized for sparse symmetric positive semi-definite (SPSD) systems. It solves \(A x = b\) for Laplacians where \(A\) has a non-trivial null space, running to a convergence tolerance of \(10^{-6}\) squared residual norm.

4. Conflict Reconciler (reconciler.py)

Performs the Hodge Decomposition:

1. Computes the curl discrepancy vector \(b_2 = d_1 \Delta S\).
2. Solves the coexact system \(L_2 \beta = b_2\).
3. Computes the coexact conflict flow: \(S_{\text{coexact}} = d_1^T \beta\).
4. Reconciles the flow: \(\Delta S_{\text{reconciled}} = \Delta S - S_{\text{coexact}}\).
5. Integrates \(\Delta S_{\text{reconciled}}\) into vertex state potentials using a topological line integral starting from genesis.