mod30-residue-lanes / lab report

Notebook 17 — Reflected Manifold Decomposition

Boundary pressure becomes spectral structure through reflected manifold modes: 

13 | 17 → reflected modes → low-rank manifold structure
View Repository Colab Notebook 17

Overview

Notebook 17 turns boundary pressure into spectral structure.

Notebook 13 measured pressure around the central split between lane 13 and lane 17. Notebook 17 anchors the reflected side of that split and decomposes rolling residue-lane trajectories into shared modes, reflected modes, and local deviations. 

boundary pressure → covariance structure → spectral modes

The result is a low-rank view of the residue manifold: most of the rolling dynamics can be understood through a small number of coordinated eigenmodes.

Core Features

Feature Description
rolling_prime_lane_matrix Window-by-lane matrix of rolling prime-count trajectories.
lane_correlation_matrix Pairwise correlations between residue-lane trajectories.
spectral_modes Eigenvectors of the covariance matrix, interpreted as coordinated lane modes.
explained_variance Variance captured by each spectral mode.
mode_scores Temporal activation of each mode across rolling windows.
low_rank_reconstruction Reconstruction error after retaining only the leading modes.

Rolling Prime Lane Matrix

Heatmap of rolling prime-count values across the eight admissible modulo-30 residue lanes.
Rolling lane trajectories form the matrix that spectral decomposition analyzes.

Lane Correlation Matrix

Correlation matrix showing relationships between rolling residue-lane trajectories.
Correlations remain broadly positive but split into structured lane neighborhoods.

Spectral Explained Variance

Plot of explained variance and cumulative explained variance across spectral modes.
A small number of modes captures most of the rolling manifold structure.

Eigenmode 1 — Shared Background Mode

Bar chart showing lane loadings for eigenmode 1.
Mode 1 behaves like a shared global background mode across the residue manifold.

Eigenmode 2 — Reflected Lane Partition

Bar chart showing lane loadings for eigenmode 2.
Mode 2 separates lane groups into opposite directions of coordinated variation.

Eigenmode 3 — Local Deviation Structure

Bar chart showing lane loadings for eigenmode 3.
Mode 3 exposes another local lane-family split inside the rolling manifold.

Mode Scores

Line chart showing temporal scores for the first spectral modes.
Mode scores track when each spectral structure becomes active across rolling windows.

Low-Rank Reconstruction Error

Line plot showing reconstruction error as more spectral modes are added.
Low-rank reconstruction tests whether a few modes recover the manifold's observable structure.

Reflected Pair Counts: 17 vs 13

Line chart comparing rolling prime counts for reflected lanes 17 and 13.
Lane 17 and lane 13 remain coupled while retaining local differences.

Reflection Gap 17 − 13

Line chart showing the rolling reflection gap between lane 17 and lane 13.
The reflection gap oscillates around zero, reinforcing reflected-pair behavior.

Interpretation

Notebook 17 shows that rolling residue-lane dynamics are not just noisy fluctuations.

They decompose into:

This is the first report where the mod30 residue manifold becomes explicitly spectral.

Relationship to Neighboring Notebooks

Notebook 13 measured boundary pressure around the 13 | 17 split.

Notebook 17 turns that reflected boundary into covariance structure and spectral modes.

Notebook 19 then studies how those spectral structures evolve through time using rolling eigenspace stability, spectral entropy, and phase dynamics.