mod30-residue-lanes / lab report

Notebook 29 — Sparse Feature Emergence at the Reset Boundary

Sparse emergence analysis across the modulo-30 reset manifold:

23 → 29 → 01

View Repository Colab Notebook 29

Overview

Notebook 29 studies sparse feature emergence near the modulo-30 reset boundary.

Instead of treating residue lanes as isolated categories, the notebook models:

23 → 29 → 01

as a cyclic transition manifold where sparse activations, local pressure changes, and reset-boundary dynamics accumulate over rolling windows.

The resulting framework provides a lightweight emergence-monitoring pipeline using rolling residue statistics, sparse feature transforms, boundary-pressure metrics, and temporal event detection.

Core Features

Feature	Description
`23`, `29`, `01`	Rolling residue counts near reset boundary.
`gap_23_29`	Transition spacing between 23 and 29.
`gap_29_01`	Reset spacing across modulo rollover.
`reset_pressure`	Boundary imbalance metric across the reset edge.
`boundary_pressure`	Amplified local transition intensity.
`z_*` features	Normalized sparse-event indicators.

Reset Boundary Graph

Graph showing the reset boundary transition between residue lanes 23, 29, and 01.

Minimal reset-boundary geometry: 23 → 29 → 01

Sparse Feature Heatmap

Heatmap of sparse feature activations across rolling windows and reset-boundary derived features.

Sparse activations cluster across coupled reset-boundary features.

Reset Pressure

Rolling reset-pressure dynamics across the sparse boundary manifold.

Sparse Event Timeline

Timeline of sparse emergence events detected across rolling windows.

Sparse emergence events cluster into localized temporal regions.

Reset State Counts

Bar chart showing counts of balanced and lane-leading reset states.

Balanced states dominate while sparse transition states remain detectable.

Reset Boundary Lane Counts

Rolling residue counts for lanes 23, 29, and 01.

Rolling lane populations near the modulo-30 reset boundary.

Interpretation

Sparse emergence does not appear uniformly random.

Instead, reset-boundary transitions exhibit localized activation clustering, coherent pressure spikes, cross-feature coupling, and stable baseline regions interrupted by sparse transition bursts.

This suggests that lightweight residue-manifold monitoring may already contain useful emergence signals before large-scale latent modeling.

Relationship to Earlier Notebooks

Notebook 23 established graph Laplacians, spectral embeddings, lane-correlation structure, and manifold connectivity.

Notebook 29 extends this into sparse activation structure, transition-state monitoring, reset-boundary pressure tracking, and temporal emergence analysis.