mod30-residue-lanes / lab report

Notebook 19 — Temporal Spectral Dynamics of Residue Manifolds

Spectral modes become temporal phase dynamics: 

spectral modes → rolling eigenspaces → temporal phases
View Repository Colab Notebook 19

Overview

Notebook 19 turns spectral modes into temporal phase dynamics.

Notebook 17 decomposed rolling residue-lane trajectories into spectral modes. Notebook 19 asks whether those spectral structures remain stable, rotate, fragment, or reorganize over time. 

17 → spectral decomposition
19 → temporal eigenspace dynamics
23 → graph manifold embeddings

The result is a phase-system view of the mod30 residue manifold: stable global modes persist while secondary modes rotate through bounded temporal transitions.

Core Features

Feature Description
rolling_prime_manifold Rolling prime-count matrix across the eight admissible residue lanes.
rolling_explained_variance Time-varying variance explained by the leading spectral modes.
spectral_entropy Concentration or fragmentation of variance across modes.
eigenspace_cosine_drift Cosine similarity between adjacent rolling eigenspaces.
mode_rotation_angles Angular rotation of spectral modes across rolling windows.
transition_heatmap Similarity matrix showing temporal continuity between rolling manifold states.

Rolling Prime Manifold

Heatmap of rolling prime-count values across the eight admissible modulo-30 residue lanes.
Rolling prime counts form the temporal manifold analyzed by eigenspace dynamics.

Rolling Explained Variance

Line chart showing the explained variance of leading spectral modes over rolling windows.
The leading modes remain coherent while their relative strengths fluctuate over time.

Spectral Entropy

Line chart showing spectral entropy across rolling windows.
Spectral entropy tracks when the manifold becomes more concentrated or more distributed.

Eigenspace Cosine Drift

Line chart showing eigenspace cosine similarity over rolling windows.
High similarity indicates persistent manifold structure rather than random collapse.

Mode Rotation Angles

Line chart showing spectral mode rotation angles across rolling windows.
Rotation spikes mark temporary reorientation of secondary manifold structure.

Transition Heatmap

Heatmap showing similarity between rolling manifold states across time.
Temporal similarity remains high, showing continuous evolution across manifold states.

Phase Cluster Embedding

Two-dimensional embedding of rolling manifold states colored by temporal order.
The phase embedding reveals a curved temporal path rather than disconnected clusters.

Interpretation

Notebook 19 shows that spectral structure persists through time.

The manifold does not collapse into randomness. It:

This report turns the residue manifold into a temporal phase system, bridging spectral decomposition and graph-manifold structure.

Relationship to Neighboring Notebooks

Notebook 17 identified spectral modes and low-rank residue-manifold structure.

Notebook 19 tracks how those modes evolve through rolling time.

Notebook 23 then converts lane relationships into graph embeddings, Laplacian modes, and graph signal structure.