mod30-residue-lanes / lab report

Notebook 23 — Graph Embeddings of Residue-Lane Manifolds

Rolling lane trajectories become graph geometry:

rolling trajectories → lane similarity graph → Laplacian embedding

View Repository Colab Notebook 23

Overview

Notebook 23 turns temporal lane relationships into graph geometry.

Notebook 19 tracked temporal spectral phases across rolling residue manifolds. Notebook 23 converts those lane relationships into graph nodes, weighted edges, Laplacian embeddings, graph modes, and graph signal trajectories.

19 → temporal spectral dynamics
23 → graph manifold embeddings
29 → sparse reset-boundary emergence

The result is a learned residue-lane network: eight admissible lanes become graph nodes, and their weighted edges emerge from rolling prime-lane trajectory similarity.

Core Features

Feature	Description
`lane_similarity_graph`	Weighted graph where residue lanes are nodes and trajectory similarity defines edges.
`graph_adjacency_matrix`	Matrix of weighted relationships between residue lanes.
`laplacian_embedding`	Two-dimensional graph embedding from Laplacian eigenvectors.
`node_centrality`	Weighted degree of each residue lane inside the learned graph.
`graph_modes`	Laplacian eigenmodes that describe graph-level lane structure.
`graph_signal_scores`	Temporal projections of rolling lane vectors onto graph modes.

Lane Similarity Graph

Graph visualization showing weighted similarity relationships between mod30 residue lanes.

Residue lanes become graph nodes, with weighted edges learned from rolling prime-lane trajectories.

Laplacian Embedding

The graph Laplacian turns lane relationships into a learned geometric embedding.

Graph Adjacency Matrix

Adjacency matrix of weighted graph relationships between residue lanes.

Raw edge weights show how strongly each lane connects to every other lane.

Lane Correlation Matrix

Correlation matrix showing pairwise relationships between rolling residue-lane trajectories.

Correlation structure supplies one component of the weighted lane graph.

Weighted Degree Centrality

Weighted degree measures how centrally each lane participates in the graph manifold.

Sorted Edge Weights

Bar chart of graph edge weights sorted by strength.

Strong edges reveal the dominant lane couplings inside the learned residue graph.

Graph Mode 1 Loadings

Graph modes expose structured directions of variation across the lane network.

Graph Mode 2 Loadings

Mode 2 separates another graph-level lane relationship pattern.

Graph Mode 3 Loadings

Mode 3 adds local lane-family structure beyond the first graph splits.

Graph Signal Mode Scores

Rolling lane vectors become graph signals evolving across Laplacian modes.

Interpretation

Notebook 23 shows that residue lanes can be treated as graph nodes whose weighted edges emerge from rolling prime-lane trajectories.

The graph is not imposed as a fixed wheel. It is learned from:

trajectory similarity,
cosine alignment,
covariance structure,
and graph Laplacian geometry.

This report makes the residue manifold explicitly topological: lane relationships become a graph that can support centrality, community structure, Laplacian modes, and graph-signal dynamics.

Relationship to Neighboring Notebooks

Notebook 19 studied temporal spectral dynamics and phase continuity.

Notebook 23 converts those lane relationships into graph geometry.

Notebook 29 then uses this manifold framing to study sparse reset-boundary emergence along the transition path 23 → 29 → 01.